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ABSTRACT The theory of General Relativity has been in existence for 90 years and has stood up to all tests it has been subjected to in that time. The PPN parameter γ is a measure of the accuracy of theories of gravity and assumes diﬀerent values in diﬀerent theories. By measuring the Shapiro time delay of light it is possible to constrain γ and thereby constrain gravitational theories. This Shapiro time delay can be measured in our solar system but it is only in the vicinity of extremely compact objects such as pulsars and black holes that it can be tested under the immense gravitational ﬁelds that can only be found there. A pulsar in a binary orbit about another compact object is the ideal system in which to test this eﬀect. In this work we have gone from Kepler’s laws of simple planetary motion to deriving the equations that explain binary orbits to incorporating General Relativity into these equations in order to obtain the equations for relativistic particle orbits. We then evolved this theory even further so as to be able to explain relativistic light ray orbits and then used this knowledge to model the Shapiro delay in a binary system. With a working model it became possible to simulate the Shapiro delay in a wide range of possible systems and then to use these simulations to say something about what type of system should be focussed on in future so as to measure the Shapiro delay and thereby constrain more tightly the parameter γ. CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. The PPN Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 The 10 PPN parameters and their origin . . . . . . . . . . . . . . 9 2.3 The Nordtvedt eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Variation of Newton’s constant . . . . . . . . . . . . . . . . . . . 14 2.4.1 Spin tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4.2 Orbital decay tests . . . . . . . . . . . . . . . . . . . . . . 15 2.4.3 Changes in the Chandrasekhar mass . . . . . . . . . . . . 15 3. Orbital mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 Orbital geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Orbital parameters . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Keplerian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Binary pulsar evolution . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Evolutionary processes . . . . . . . . . . . . . . . . . . . . 26 4. Topics from the Theory of Curved Space . . . . . . . . . . . . . . . . . 28 4.1 Curvilinear coordinates . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 The metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5. Relativistic particle orbits in a Schwarzschild Geometry . . . . . . . . . 34 5.1 Physical and Mathematical concepts . . . . . . . . . . . . . . . . 34 5.1.1 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . 35 5.2 Derivation of relativistic particle orbits . . . . . . . . . . . . . . . 36 6. Shapiro delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.2 Mathematical derivation for the Shapiro time delay of light . . . . 41 6.2.1 Deﬂection angle . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2.2 The Shapiro delay of light . . . . . . . . . . . . . . . . . . 46 7. Strong-ﬁeld Shapiro delay . . . . . . . . . . . . . . . . . . . . . . . . . 48 Contents 1 8. Results of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.1 Procedure followed . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.2 Graphs and Results . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8.3 Predicted Future Results . . . . . . . . . . . . . . . . . . . . . . . 67 8.4 3-Dimensional Parameter spaces . . . . . . . . . . . . . . . . . . . 74 9. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Appendix 88 A. The connection or covariant derivative . . . . . . . . . . . . . . . . . 89 A.1 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 B. MatLab code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 1. INTRODUCTION The process of stellar evolution is a long and complicated one with numerous factors inﬂuencing the ﬁnal state of a star. Generally the death of stars with a birth-mass below about 8 Solar masses (M ) is a relatively gentle aﬀair. They will expand to many times their original diameter and go through a red giant phase before the core, with a left-over mass below the Chandrasekhar mass of 1.4M , collapses to form a White Dwarf that is sustained against further collapse by electron degeneracy pressure. This is the pressure created through the Pauli exclusion principle preventing 2 electrons from occupying the same energy state and the highest energy level that is ﬁlled is referred to as the Fermi energy. Those stars of birth-mass greater than 8M , come to a far more spectacular end. These stars will explode as supernovae, releasing in an instant up to 1044 J of energy and having a luminosity of billions of times that of the sun. Then, as before, the core re-collapses and in the event of the core still having a mass greater than 2.5M , although this limiting mass is still a matter of debate, there is no degeneracy pressure that can retard the massive gravitational collapse and it collapses to form a black hole. It is those stars whose cores have a mass of between 1.4M and 2.5M that, upon collapse, form an extremely interesting class of astrophysical object known as a neutron star. In 1932 the English physicist, Sir James Chadwick, discovered the neutron, for which he won the Nobel Prize in 1935, and unwit- tingly started a cascade of discoveries and theories in astrophysics that have lasted to the present day. Two years after this discovery Walter Baade and Fritz Zwicky1 surmised that a supernova explosion in a star that has insuﬃcient mass for complete gravitational collapse into a black hole, could be sustained by neu- tron degeneracy pressure against further collapse. This is analogous to electron degeneracy pressure that sustains white dwarf stars against collapse. Under or- dinary circumstances the nuclear reactions of neutron and neutrino formation, p+ + e− n + ν, tend toward the production of proton-electron pairs but under the massive self-gravitation of a forming neutron star, the forward reaction dom- inates and neutron formation is the primary reaction. This is because, as central density increases, the electron Fermi energy always increases up to the point where inverse beta decay takes place and forces the electrons into the nuclei. The 1 Baade, W.,Zwicky, F., “Cosmic Rays from Super-novae”, Proc. Nat. Acad. Sci., 20, 254, (1934) 3 reverse reaction barely takes place because all of the electron states into which the electrons could move are already occupied, preventing decay of the neutrons in the nucleus into protons and electrons. As a result these objects would consist primarily of neutron matter and have a very small radius and enormous densities. The likelihood of ﬁnding such a star, however, was deemed virtually nonexistent due to the high probability that such a star would be extremely small and emit very little visible light. The luminosity function is given by L ∝ R2 T 4 and the predicted radius of such a star is approximately 10km. When this is compared to a 1 solar mass star’s radius of approximately 700000km, the diﬀerence when these values are squared is of the order of 109 . In mitigation of this the neutron star’s temperature is also about 100 times greater than that of the sun and, as the luminosity function contains a T 4 term, the loss in luminosity due to de- creased radius would largely be compensated for by the gain due to increased temperature. The fundamental diﬀerence is that, as a blackbody, the neutron star’s radiation will peak short of the optical band and as such be scarcely visible in the optical wavelengths. Theoretical debate on the properties of neutron stars continued unabated as numerous great minds tried to elucidate their properties from ﬁrst principles. In 19672 , Antony Hewish and his research student, Jocelyn Bell, were un- dertaking an interplanetary scintillation study using a large antenna that was sensitive to relatively long, 3.7 m, wavelengths. Within a month Miss Bell had detected a large periodic ﬂuctuation that could not be scintillation. After further observations it was realised that the signal had a 1.337 s period and thus could not be a random eﬀect. Initially it was feared/hoped to be a signal from an extra-terrestrial source and was given the designation LGM, standing for Little Green Men, but this thought was quickly laid to rest when the signal showed no variation in period whatsoever. The link between this pulsating object and neutron stars was not immediately made, however, and even once the association had been made, it was not readily accepted. Although both Franco Pacini3 and Thomas Gold4 published papers linking rapidly rotating neutron stars to observ- able radio pulses, competing theories of rotating or pulsating white dwarf stars remained. It took a year for the matter to largely be put to rest with the dis- covery of the 89 ms Vela pulsar5 . This settled one side of the argument as it was realised that a white dwarf star rotating at that speed would quickly be ripped apart by the centrifugal forces created. Only a star with the small diameter of the predicted neutron star could sustain such periodicities. With the discovery 2 Hewish, A., Bell, J., et al., “Observations of a Rapidly Pulsating Radio Source”, Na- ture, 217, 709, (1968) 3 Pacini, F., “Energy Emission from a Neutron Star”, Nature, 216,567 , (1967) 4 Gold, T.,“Rotating Neutron Stars as the Origin of the Pulsating Source of Radio”, Na- ture, 218,731 , (1968) 5 Large, M.I., Vaughan, A.E., Mills, B.Y.,“A Pulsar Supernova Association”, Nature, 220, 340-341, (1968) 4 that the periodicity was gradually decreasing, the last doubt was laid to rest as only a rotating object would slow down as it lost energy while a vibrating one would not. At last this class of object could be categorised and became known to the world as pulsars with general designation of PSR, or pulsating source of radio. The extremely large mass of neutron stars concentrated in such a small volume means that the gravitational ﬁelds generated by these objects are just slightly less than that of the super-dense black holes that general relativists have for so long sought and studied. The obvious, and very large, advantage of neutron stars is that they are visible in radio and occasionally in optical frequencies and changes in behaviour can therefore be observed. A neutron star in a binary system is a phenomenal testing ground for gravitational theories as the observed electromagnetic beam generated just above the surface of the star will often pass through the gravitational ﬁeld generated by a binary companion. Should this companion be another neutron star, or even better a black hole, the beam would pass through extremely strong gravitational ﬁelds and thus strong ﬁeld tests of gravity could be performed that 30 years ago would have been mere speculation. When the ﬁrst double neutron star system was discovered by Russell Hulse and Joseph Taylor (1975)6 , such tests became a reality and have continued unabated with ever larger and more accurate data sets. The discovery of a double pulsar system by Lyne, Burgay, Kramer et al. (2004)7 , was yet another step closer to an understanding of matter under these extreme conditions. The ability to observe the gravitational eﬀects of two pulsars as their beams pass through one another’s gravity is paramount to deepening our understanding of the manner in which space-time is curved. It is clear that pulsars are observed through reception of the signal on earth and the subsequent analysis of the signal and any changes contained therein. In practice, however, the process is not quite so simple. Generally the pulse of radiation received on earth is very small and is swallowed up in the noise of the electronics and even the noise of the cosmic microwave background (CMB). The method used to extract the information of the pulse is that of folding, or integrating, the signal until the peak emerges from the noise. In a pulsar with a known frequency, a long observation time of the pulsar is cut into segments of one pulsar period’s length and all of these segments are added together so that the pulse emerges from the noise. This is possible because the noise is stochastic and does not add to any signiﬁcant degree while the pulse intensity adds up over each folding. In pulsar searches, the signal is folded over using a series of frequencies until the peak emerges for a given frequency. The correct period can then be found by reﬁning this process to obtain the maximum signal to noise ratio for 6 Hulse, R., Taylor, J.,“Discovery of a Pulsar in a Binary System”, Ap. J., 195, L51-L53, (1975) 7 Lyne, A.G., Burgay, M., Kramer, M., et al.,“”, Science, 303, 1153, (2004) 5 the signal. A complicating eﬀect in this process is known as the Roemer delay. This eﬀect occurs in signals originating from pulsars that are in binary orbits. As the pulsar itself moves through space, traversing periodically away from and then towards the earth, the signal originating from the pulsar has to travel a varying distance to our earth-based detectors. The result of this is a signal that varies with a very similar periodicity to that of the Shapiro delay we are trying to measure. Fortunately the Shapiro delay in the strong ﬁeld regime has been measured so all is not lost. In a system that is correctly oriented so as to make the Shapiro delay a maximum, the delay can be measured and this has been done in the highly relativistic binary system PSR J0737-30398 . So even though the eﬀect is diﬃcult to detect, discovery of more relativistic systems and a better understanding of the Shapiro delay should assist in measuring this eﬀect more easily. This eﬀect also occurs due to the earth’s orbit about the sun but fortunately the orbital periods of the systems in which we are interested in this thesis are so short that the motion of the earth during that time can be neglected. The calculation of the magnitude of this eﬀect is a fairly straightforward process that requires only some simple orbital mechanics. This calculation will be dealt with in a later chapter. In pulsar timing one also encounters the problem of dispersion. This is the phenomenon where the higher frequency signals travel faster through the inter- stellar medium and therefore arrive on earth earlier than lower frequency bands resulting in a ’smearing out’ of the pulsar signal. For this reason de-dispersion is necessary to obtain the true information from a signal by separating the in- coming signal into frequency bands and compensating for the dispersive eﬀect in each band. There are a myriad of other eﬀects that pulsar astronomers are aware of and take into account when timing pulsars but these are not the topic of this thesis and as such will not be discussed in detail. Suﬃce it to say that through the application of various tools it is possible to measure pulse arrival times to an accuracy of up to 10−15 s and therefore even minuscule variations in the arrival times can be observed and analysed as caused by possible physical eﬀects occurring in and around the neutron star. These eﬀects can have their source in the interior of the star, its crust, the surrounding magnetosphere, the gravitational well generated by these strongly self-gravitating bodies or a combination of these and only through continued observation and study can an improved understanding of these dynamic objects be reached. Gravity is one of those aspects for which a better understanding is desired. The theory of General Relativity (GR) formulated by Einstein in 19169 is still a frontrunner in the race to understand gravity but further testing of GR and its 8 Burgay, M., D’Amico, N., Possenti, A., et al.,“The Highly Relativistic Binary Pulsar PSR J0737-3039: Discovery and Implications”, ASP Conf. series, 328, 53, (2005) 9 a Einstein, A., “Die Grundlage der Algemeinen Relativit¨tstheorie”, Ann. der Physik, 49, 769-822, (1916) 6 competitors is essential in laying the matter to rest. The Parameterised Post- Newtonian formalism was thus developed in the 1970’s10,11,12,13,14,15 as a frame- work in which to compare theories of gravity through their eﬀects on a set of 10 parameters. Although only a few of these parameters are applicable to pul- sars, many tests were quickly formulated to allow for the testing of gravity in the strong-ﬁeld regime about pulsars. One of these tests is the Shapiro time delay16 of light; an eﬀect manifested through a retardation of an electromagnetic wave as it moves through the warping of spacetime caused by mass and energy. Within the PPN framework, the Shapiro delay is a function of the PPN parameter γ and as such, measurements of the eﬀect should allow for a tighter constraint to be placed upon that parameter. Through simulations of the timing eﬀects occurring in binary systems, such as Roemer delay and Shapiro delay, and the categorising of which systems prove promising in detecting these eﬀects, it becomes possible to ﬂag the type of system that should be used for the measurement of the Shapiro delay and the resultant constraint on γ. To what extent has γ been constrained and what hope is there for further constraint? The PPN parameter γ is constrained via two eﬀects that are inti- mately linked as we shall see later. These eﬀects are the Shapiro time delay and the deﬂection of light. The deﬂection of light is an eﬀect that was predicted by Einstein in 1916 and ﬁrst tested by Eddington17 , unfortunately to low (around 30%) accuracy. Very Long Baseline Interferometry (VLBI) measurements made in 1995 of the sources 3C273 and 3C279 as they approached superior conjunction with our sun led to a constraint of 1 (1 + γ) = 0.9996 ± 0.001718 while an anal- 2 10 Nordtvedt, K. Jr.,“Post Newtonian Metric for a General Class of Scalar-Tensor Gravita- tional Theories and Observational Consequences”,Ap. J., 161, 1059, (1970) 11 Thorne, K. S., Will, C.M., “Theoretical Framework for Testing Relativistic Gravity I: Foundations”,Ap. J., 163, 595, (1971) 12 Will, C.M.,“Theoretical Framework for Testing Relativistic Gravity II: Parametrized Post- Newtonian Hydrodynamics and the Nordvedt Eﬀect”,Ap. J., 163, 611, (1971) 13 Will, C.M.,“Theoretical Framework for Testing Relativistic Gravity III: Conservation Laws, Lorentz Invariance and Values of the PPN parameters”,Ap. J., 169, 125, (1971) 14 Will, C.M., Nordtvedt, K. Jr.,“Conservation Laws and Preferred Frames in Relativistic Gravity I: Preferred-Frame Theories and an Extended PPN Formalism”,Ap. J., 177, 757, (1972) 15 Will, C.M., Nordtvedt, K. Jr.,“Conservation Laws and Preferred Frames in Relativistic Gravity II: Experimental Evidence to Rule Out Preferred-Frame Theories of Gravity”,Ap. J., 177, 775, (1972) 16 Shapiro, I.I., “The Fourth Test of General Relativity”,Phys. Rev. Lett., 13, 798, (1964) 17 Eddington, A.S., “The Total Eclipse of 1919 May 29 and the Inﬂuence of Gravity on Light”, The Observatory, 42, 119-122, (1919). 18 Lebach, D. E., Corey, B. E., Shapiro, I. I., et al., Measurement of the solar gravitational de- ﬂection of radio waves using very-long-baseline interferometry, Phys. Rev. Lett., 75, 14391442, (1995). 7 ysis made in 2004 of almost 2 million VLBI measurements made of 541 sources using 87 VLBI sites improved this constraint to 0.99992 ± 0.0002319 , the tightest constraint on this parameter using this technique. The Shapiro time delay of light is the second means of constraining γ. First tested in 197720 with the Viking mission to Mars, it has been most accurately constrained with the recent Cassini mission to Saturn21 where Doppler tracking constrained γ − 1 to (2.1 ± 2.3) × 10−5 meaning that γ must be within 0.0012% of unity, an even tighter constraint than light deﬂection tests can oﬀer. Within the weak ﬁeld regime it is hoped that NASA’s Laser Astrometric Test Of Relativity (LATOR) mission will constrain γ to better than 1 part in 108 but as launch is expected no sooner than 2009-2010, much work can still be done in the interim. The testing of gravity in the strong ﬁeld regime is still regarded as a fairly ﬂedgling science in that systems suitable for this testing are scarce; only 8 dou- ble neutron star binaries are known22 and it is these systems that are required for such investigations. As the population of compact binaries increases, and existence/non-existence of a pulsar-black hole binary is determined, we can only improve our knowledge of the universe and the gravity that holds it together. 19 Shapiro, S. S., Davis, J. L., Lebach, D. E., and Gregory, J. S., Measurement of the solar gravitational deﬂection of radio waves using geodetic very-long-baseline interferometry data, 1979-1999, Phys. Rev. Lett., 92, 121101, (2004). 20 Shapiro, I.I., Reasenberg, R.D., et al., “The Viking Relativity Experiment”,J. Geophys. Res., 82, 4329-4334, (1977). 21 Bertotti, B., Iess, L., and Tortora, P., A test of general relativity using radio links with the Cassini spacecraft, Nature, 425, 374376, (2003). 22 Stairs, I.H., “Pulsars in Binary Systems: Probing Binary Stellar Evolution and General Relativity”, Science, 304, 547, (2004). 2. THE PPN FORMALISM 2.1 Background The testing of General Relativity through the timing of radio pulsars falls into two broad categories: setting limits on the magnitudes of parameters that describe vi- olation of equivalence principles, and verifying that the measured post-Keplerian timing parameters of a given binary system match the predictions of strong ﬁeld GR better than competing theories. This thesis will primarily focus on the latter but it would be negligent not to mention the former and give some explanation of what is meant in the following few pages. The Equivalence Principle basically states that there exists no experiment by which one could tell the diﬀerence between a uniform gravitational ﬁeld and a uniform acceleration in the absence of a gravitational ﬁeld and that these two frames may thus be regarded as equivalent. Over time, it was realised that three diﬀerent forms of the Equivalence Principle needed to be distinguished. These are the Weak Equivalence Principle (WEP), the Einstein Equivalence Principle (EEP) and the Strong Equivalence Principle (SEP) and diﬀer in the constraints they impose on the system. The WEP states that objects of varying composi- tions and masses will experience the same acceleration in an external gravitational ﬁeld. The EEP goes a little further and adds Lorentz invariance (non-existence of preferred reference frames) and positional invariance (non-existence of preferred locations) for non-gravitational experiments. This means that experiments will have the same outcomes in inertial and freely falling reference frames. Lastly the SEP adds Lorentz invariance and positional invariance for gravitational experi- ments, thereby including experiments on objects with strong self-gravitation. For some years now, the WEP and EEP have been experimentally veriﬁed and there is little doubt with regards to their accuracy in explaining observed phenomena. The Strong Equivalence Principle, however, still needs to stand up to experimental scrutiny. In order to more easily determine whether a violation of the SEP had occurred, a formalism was developed to facilitate comparisons with other gravitational theories in the strong gravitational limit. This parameterised post-Newtonian (PPN) formalism, mentioned in the previous chapter, uses ten parameters (γP P N , β, ξ, α1 , α2 , α3 , ζ1 , ζ2 , ζ3 , ζ4 ,) to compare competing theories with one another. Pulsar timing allows for strict limits to be placed on α1 , α3 and ζ2 as well as placing limits on other SEP violations that constrain combinations of PPN parameters. Possible violations are the Nordtvedt eﬀect, the existence of 2.2 The 10 PPN parameters and their origin 9 dipolar gravitational radiation and possible time variation of Newton’s constant. The parameters and SEP violations will now be discussed. 2.2 The 10 PPN parameters and their origin The PPN metric is given by1 , g00 = −1 + 2U + 2βU 2 − 2ξΦW + (2 + 2γ + α3 + ζ1 − 2ξ2 )Φ1 + 2(3γ − 2β + 1 + ζ2 + ξ)Φ2 + 2(1 + ζ3 )Φ3 − 2(3γ + 3ζ4 − 2ξ)Φ4 − (ζ1 − 2ξ)A − (α1 − α2 − α3 )ω 2 U − α2 ω i ω j Uij + (2α3 − α1 )ω i Vi + O( 3 ), (2.1) 1 1 g0i = − (3 + 4γ + α1 − α2 + ζ1 − 2ξ2 )Vi − (1 + α2 − ζ1 + 2ξ)Wi 2 2 1 i j 5/2 − (α1 − 2α2 )w U − α2 w Uij + O( ), (2.2) 2 gij = (1 + 2γU + O( 2 ))δij (2.3) (2.4) where the various potential terms are given by ρ U= d3 x , (2.5) |x − x | ρ(x − x )i (x − x )j 3 Uij = d x, (2.6) |x − x |3 ρ ρ (x − x ) (x − x ) (x − x ) 3 3 ΦW = 3 · − d xd x , (2.7) |x − x | |x − x | |x − x | ρ [v · (x − x )]2 3 A= d x, (2.8) |x − x |3 ρv2 3 Φ1 = d x, (2.9) |x − x | ρU Φ2 = d3 x , (2.10) |x − x | ρΠ Φ3 = d3 x , (2.11) |x − x | ρ Φ4 = d3 x , (2.12) |x − x | 1 Will, C.M., “The Confrontation Between General Relativity and Experiment”,Liv. Rev. Rel., submitted (astro-ph:gr-qc/0510072),(2005) 2.2 The 10 PPN parameters and their origin 10 ρ vi 3 Vi = d x, (2.13) |x − x | ρ [v · (x − x )](x − x )i 3 WI = d x, (2.14) |x − x |3 It is within these equations that the 10 PPN parameters appear and below is a brief description of each. The ﬁrst parameter, γP P N , measures the space curvature produced by a unit rest mass. In GR it takes the value 1 and is measured through observing light deﬂection due to the presence of mass. In theories that predict some violation of the SEP, known as semi-conservative theories, as well as some fully conservative theories, that don’t violate the SEP, this parameter takes on a value other than 1 and so is a prime candidate for testing GR. The parameter β measures any non-linearity in the superposition law for gravity and has been accurately measured through measurement of the precession of Mercury’s perihelion. This solar system test was one of the ﬁrst big successes of general relativity and over the years has not been found to contradict GR in any way. The parameter, usually measured as |β − 1| has been constrained to 3 × 10−3 , in agreement with the prediction of GR that β = 1. ξ is the parameter that measures any violation of positional invariance or the existence of preferred-location eﬀects. In GR the value is zero as GR does not support the supposition that the outcome of local gravitational experiments depends on the location of the laboratory with respect to the source of gravitation. The basic result of such an eﬀect is the inconstancy of the gravitational constant as locally measured. The result of this, in terms of the parameter ξ, is the existence of anomalous torques that cause a random alignment of the sun’s spin axis with the ecliptic. ξ has been constrained to a few parts in 107 . The parameters α1 , α2 and α3 all measure various aspects of preferred-frame eﬀects. As with ξ, GR predicts that there is no eﬀect of the velocity of the laboratory with respect to the mean rest frame of the universe on the outcome of local experiments and so these parameters should all be zero. An orbital evolution of a binary system due to a relative velocity between it and the “universal” background reference frame given by the CMB, will result in a non-zero α1 , however. Similarly a random alignment of the sun’s spin axis with the ecliptic, as with a non-zero ξ, will result in a non-zero α2 . In the event of a self-acceleration of a rotating body orthogonal to both its spin and its absolute velocity, the term α3 would be non zero and violate both local Lorentz invariance and conservation of momentum. The last four PPN parameters (ζ1 , ζ2 , ζ3 and ζ4 ) all indicate a non- conservation of momentum. ζ2 along with α3 predicts an acceleration of the center of mass of a binary system according to πm1 m2 (m1 − m2 ) acm = (α3 + ζ2 ) enp . (2.15) Pb [(m1 + m2 )a(1 + e2 )]3/2 2.2 The 10 PPN parameters and their origin 11 Fig. 2.1: A brief description of the 10 PPN parameters with their values in some of the competing theories (Will, 2005). It should be noted that there is no discrepancy in the values of the last 5 parameters. This was not always the case and certain theories predicted non-zero values for these parameters but over time it was realised that these theories were ﬂawed and only the surviving theories are shown here. Fig. 2.2: An overview of a few of the PPN parameters with their values in some of the more plausible competing theories (Will, 2005). Again it should be noted that there is no discrepancy in the values of the last 5 parameters. 2.3 The Nordtvedt eﬀect 12 In this equation np is a unit vector from the center of mass to the periastron of body 1 with a mass m1 . m2 is the mass of the companion object, e the eccentricity of the system, a the semi-major axis length and Pb is the binary orbital period. ˙ This acceleration leads to a Doppler type contribution to the pulsar’s P , which is the ﬁrst time derivative of its spin period, in that, in the case of higher eccentricity orbits2 , the projection of acm along the line of sight to the earth will change with time producing a second derivative to the pulse period given by 2 ¨ P 2π X(1 − X) eω cos ω ˙ P = (α3 + ζ2 )m2 sin i 2 (1 − e2 )3/2 (2.16) 2 Pb (1 + X) ˙ where X = m1 /m2 , ω is the periastron angle, ω its rate of change and i is the orbital inclination. The PPN parameters are summarised in Figures (2.1) and (2.2). 2.3 The Nordtvedt eﬀect The SEP states that diﬀering masses will experience the same acceleration in an external gravitational ﬁeld even when taking their self-gravitation into account. If the SEP were violated, it would imply that the earth and the moon, containing diﬀerent fractional contributions from self-gravitation, would fall diﬀerently in the sun’s gravitational ﬁeld. This would result in a “polarisation” of the orbit in the direction of the sun. This was ﬁrst pointed out by Nordvedt3 when he suggested testing the eﬀect using laser ranging to the moon. The Nordtvedt eﬀect does not constrain any one parameter but rather a combination of seven of them given by 10 2 2 1 η = 4β − γ − 3 − ξ − α1 + α2 − ζ1 − ζ2 . (2.17) 3 3 3 2 1 In General Relativity η = 0 while in tensor-scalar theory η = 2+ω + 4Λ. In the case of η = 0 the earth will fall towards the sun with a slightly diﬀerent acceleration to that of the moon resulting in a perturbation of the earth-moon orbit causing a polarisation of the orbit. This results in a perturbation of the earth moon distance given by: δr = 13.1η cos (ωm − ωs )t [m], (2.18) where ωm , ωs are the angular frequency of the moon and sun about the earth, respectively. Lunar laser-ranging allows for measurement of the earth-moon dis- tance to an accuracy of about 1 cm. This in turn allows for a limit to be placed of η < 0.001. 2 Will, C.M., “Is momentum conserved? A test in the binary system PSR 1913+16”,Astrophys. J., 393. L59-L61. (1992) 3 Nordtvedt, K., “Testing Relativity with Laser Ranging to the Moon”, Phys. Rev., 170. 1186-1187. (1968) 2.3 The Nordtvedt eﬀect 13 In the strong-ﬁeld formalism, however, η is not used directly but rather as part of a larger function ∆i , that contains higher order η terms and which relates the gravitational and inertial masses for various objects, i, through mgrav = 1 + ∆i (2.19) minertial i 2 E grav E grav = 1+η +η + .... (2.20) mc2 i mc2 i where η’ is the ﬁrst order correction to η. This equation perhaps highlights the diﬀerence between strong-ﬁeld and weak-ﬁeld gravity better than any other. In the case of the earth, the quantity E grav /mc2 takes on a value of the order of 10−10 and even for the entire solar system the quantity takes on a value no bigger than 10−5 . For neutron stars and black-holes, however, this quantity takes on values of 0.2 and 0.5, respectively, and therefore the higher order terms in equation (2.20) cannot be ignored. It is this factor that distinguishes weak from strong ﬁeld gravity. In the event of a SEP violation, the two components of a binary system will fall diﬀerently in the gravitational ﬁeld of the galaxy, denoted g, and the equations of motion will contain an extra acceleration ∆net g, where ∆net = ∆pulsar −∆companion in the case of a binary pulsar system. This ∆net g term will aﬀect the evolution of the system, in particular the evolution of the eccentricity. This implies that there will be competition between the general relativistic evolution of eccentricity through advance of periastron and this new forcing of the eccentricity towards alignment with the projection of g, g⊥ , onto the orbital plane, and this term will be constant. The time-dependent eccentricity vector can thus be written as e(t) = eF + eR (t). (2.21) Here eR (t) is the relativistic evolution of the eccentricity vector due to advance ˙ of periastron (ω) and eF is the forced component. The equation for this forced component for low eccentricity is given by4 1 ∆net g⊥ c2 |eF | = , (2.22) 2 F GM (2π/Pb )2 where M is the total mass of the system, G is Newton’s constant, Pb is the orbital period and F = 1 in the case of GR. There are a few considerations to be taken into account when selecting binary pulsar systems for useful testing of this ˙ eﬀect. The rate of advance of periastron (ωR ) must be greater than the rate of galactic rotation in order for g to be constant. Furthermore, the pulsar must be ˙ suﬃciently old that the ω-induced rotation of the eccentricity e has completed 4 Wex, N., “New limits on the violation of the Strong Equivalence Principle in strong ﬁeld regimes”. Astron. Astrophys., 317, 976-980. (1997). 2.4 Variation of Newton’s constant 14 Fig. 2.3: Illustration of the polarisation of a low eccentricity orbit due to the presence of a forcing vector g. The diagram shows the forced eccentricity eF and the eccentricity evolving under general relativistic advance of periastron eR through an angle θ enough turns that the component eR (t) can be assumed to be randomly oriented in equation (2.21). One can only assume this randomness has been achieved in pulsars with a characteristic age of τc ˙ 2π/ω. For this reason young pulsars can be ruled out as valid systems in which to perform this test. 2.4 Variation of Newton’s constant As mentioned in the discussion of the PPN parameters themselves, some theo- ries that allow violations of the SEP also allow for a time varying gravitational constant, G. Although this variation is expected to occur on a Hubble timescale, ˙ G/G ∼ H◦ ∼ 0.7 × 10−10 yr−1 , it should be testable with the aid of neutron stars, white dwarfs and binary systems. 2.4.1 Spin tests It is fairly obvious that any variation in the gravitational constant would lead to a variation in the binding strength between atoms in a neutron star and thereby aﬀect its moment of inertia. This would lead to a variation in the spin period on the same timescale as the variation in G. This change has been written as5 P˙ ∂ ln I G˙ = (2.23) P ˙ G ∂ ln G N G, 5 Goldman, I., “Upper Limit on G Variability Derived from the Spin-Down of PSR 0655+64”,Mon. Not. R. Astron. Soc., 244,184-187, (1990). 2.4 Variation of Newton’s constant 15 for I the moment of inertia at a constant number of baryons N . The neutron star equation of state will give the baryon density for the star and, since there is no agreement as to what the correct equation of state is, there is still much debate ˙ over what the baryon density and therefore the ratio G/G is. The value has, however, been calculated for the pulsar PSR B0655 + 64 by Goldman (1990) and ˙ found to be |G/G| ≤ (2.2 − 5.5) × 10−11 yr−1 . This indicates that only pulsars with fairly large characteristic ages, of the order of tens of Gyr, can be used to perform this test as in younger pulsars the eﬀect would simply not have had enough time to become manifest. 2.4.2 Orbital decay tests Since the orbital period of a binary system is dependent on G, it is obvious that a variation therein would lead to an orbital variation with time. This problem was ﬁrst considered in 1988 by Damour, Gibbons and Taylor6 who found that ˙ Pb ˙ G = −2 . (2.24) Pb ˙ G G It was realised, however7 , that a changing G would also aﬀect the internal struc- ture of the neutron star and thus the total mass and angular momentum. Taking these factors into consideration it was shown that the change should be given by ˙ Pb m 1 c1 + m 2 c2 3 m1 c2 + m2 c1 ˙ G =− 2− − (2.25) Pb ˙ G m1 + m2 2 m1 + m2 G where ci is the compactness of body “i”. Compactness is deﬁned by Nordtvedt (1990) to be G δmi ci = , (2.26) mi δG where δmi is the variation in mass of body “i” and δG is the variation in grav- itational constant. Equation (2.25) shows that with improved timing and mea- ˙ surement of Pb , and with tighter constraints on the component masses, tighter constraints can be placed on a variation of G. 2.4.3 Changes in the Chandrasekhar mass During the formation of white dwarf stars, complete core collapse is prevented through electron degeneracy pressure. The mass at which this electron degener- 6 Damour, T., Gibbons, G.W., Taylor, J.H., “Limits on the Variability of G Using Binary Pulsar Data”, Phys. Rev. Lett., 61, 1151-1154, (1988) 7 ˙ Nordtvedt, K., “G/G and a Cosmological Acceleration of Gravitationally Compact Bod- ies”,Phys. Rev. Lett., 65, 953-956, (1990). 2.4 Variation of Newton’s constant 16 acy pressure can no longer oppose gravitational collapse is known as the Chan- drasekhar mass, MCh , and is expressed as 3/2 ¯ hc 1 MCh ∼ . (2.27) G m2 n ¯ Here h is Planck’s constant and mn is the neutron mass. Clearly a change in the value of G would aﬀect the Chandrasekhar mass and so by comparing the masses of very old to very young pulsars, it should be possible to identify any change in mass over time taking into account the obvious mass-loss that older pulsars would have experienced over that time. Such an analysis was done by Thorsett8 ˙ who found a limit in the change of G to be G/G = (−0.6 ± 4.2)−12 yr−1 at a 95% conﬁdence level, the strongest limit found thus-far. In summary, it has been shown that the PPN formalism has been used for many years to distinguish between competing theories of gravity, and in some cases even to lead physicists to discard a theory for not meeting experimental criteria. Although constraining a parameter perfectly is never a scientiﬁcally possible outcome, the placing of tighter limits through both strong and weak ﬁeld tests of gravity allows for the elimination of certain aspects of gravitational theories, bringing us closer to an understanding of this most fundamental of forces. Pulsars stand in the forefront of this testing process by allowing for the direct testing of three of the parameters and three further combinations thereof. Only through improved and more extensive timing and analysis of these most dynamic of objects, will the scientiﬁc community have suﬃcient data available to them to put gravitational theories under the microscope and ﬁnally put the theory ﬁrst elucidated by Sir Isaac Newton in his Philosophæ Naturalis Principia Mathematica in 1686 to rest. 8 Thorsett, S.E., “The Gravitational Constant, the Chandrasekhar Limit, and Neutron Star Masses”,Phys. Rev. Lett., 77, 1432-1435, (1996) 3. ORBITAL MECHANICS In a study of the eﬀects of General Relativity on signals emitted by a pulsar in a binary system, it is important to understand the mechanics of binary orbits. In order to do this eﬃciently, it is necessary to develop the theory from simple elliptical orbits to that of slightly more complex binaries. In the next sections we will evolve this theory, along with a fairly detailed discussion of how the binary orbits under discussion actually formed. 3.1 Orbital geometry The analysis of the geometry describing orbits leads to the understanding of the parameters that are used in calculating the many eﬀects occurring in such systems. It has long been known that all orbits can be described via the family of conic sections known as ellipses. The analysis of ellipses illustrates relationships between parameters leading to equations for position and velocity as functions of time. An analysis of Keplerian orbits now follows; 3.1.1 Orbital parameters There are seven parameters needed to describe a satellite in orbit. These seven parameters, known as the Keplerian parameters, deﬁne the ellipse of the orbit, the position of the ellipse about the parent body and the position of the orbiting body within the ellipse. In classical Keplerian mechanics, the ellipse has constant shape and orientation and for this analysis we will use this approximation, generalising the concepts in later sections. The Keplerian parameters are: the Epoch, the orbital inclination, the right ascension of the ascending node, the argument of perigee, the eccentricity, the mean motion and lastly the mean anomaly. 1. The Epoch is simply a number that speciﬁes the time at which all other parameters were measured. It is usually measured in Mean Julian Days and is often corrected to the solar system barycenter in order to have a standard time measurement on the earth’s globe. 2. The orbital inclination is the angle between the plane of the orbit and the plane of the sky, that is the perpendicular plane to the line of sight from earth. An inclination angle of 0◦ is a face-on orbit while an inclination angle of 90◦ represents an edge-on orbit. 3.1 Orbital geometry 18 Fig. 3.1: Basic vectors deﬁning the parameters for a Keplerian orbit 3. The right ascension (RA) of the ascending node is the celestial equivalent of terrestrial longitude. The RA is measured as an increasing angle in the east-west direction, along the equator from the zero point at the vernal equinox of the component of interest’s orbit. The ascending node is the point at which the orbiting body crosses through the ecliptic of the binary system moving from south to north where north is deﬁned as the direction to the perigee. 4. The argument of perigee is the angle that determines the rotation of the ellipse within its orbital plane. It is the angle between the line of nodes and the semi-major axis measured at the center of the ellipse. 5. Eccentricity is a measure of how distorted from a circle the orbit is. An eccentricity of 0 is a circular orbit while an eccentricity of close to 1 is a long and narrow orbit. Eccentricities of 1 are physically impossible as that would imply a linear oscillatory motion from perigee to apogee. 6. The mean motion, n, is a concept used to indicate the approximate size of the orbit. It is the inverse of the orbital period and so is measured in number of revolutions per unit time. It is also related to the semi-major axis a via GM n= (3.1) a3 Where G is the gravitational constant and M the mass of the parent body. 3.1 Orbital geometry 19 7. Lastly the mean anomaly is used to describe the position of the orbiting body at any instant in time. It is an angle that traces out the 360◦ of an orbit in equal time increments and is measured from the semi-major axis at the center of the ellipse. For circular orbits (eccentricity = 0) the line to which the angle is measured would point directly to the orbiting body. For eccentric orbits, however, the velocity of the orbiting body is a function of position within the orbit and so the line to which the mean anomaly is measured is not commensurate with the position of the body. Consider ﬁgure (3.1). For a system with primary focus f and origin o, the mean anomaly M is deﬁned such that the area of triangle boz is always equal to the area of triangle af z. In this scenario a is the point on the auxiliary circle created by extending the perpendicular to the semi-major axis through the position of the orbiting body, p, and b is the point on the auxiliary circle creating the line which deﬁnes the angle that is the mean anomaly. Thus only at perigee and apogee does the line to which the mean anomaly is measured, point towards the orbiting body. At this point it is also perhaps useful to indicate the other two anomalies often referred to when dealing with orbits. The eccentric anomaly, E, is measured from the semi-major axis to the line ao, is a measure of the eccentricity of the orbit and is related to the mean anomaly via M = E − e sin E. Secondly the true anomaly, T , is measured at the focus of the ellipse and is the angle between the semi-major axis and the line from the orbiting body to the prime focus of the ellipse pf . Continuing with our geometrical analysis: For an ellipse with semi-major axis a, eccentricity e and true anomaly θ such as the one illustrated in ﬁgure (3.2) it is always true that Fig. 3.2: Diagram illustrating ellipse parameters 3.1 Orbital geometry 20 r + r = constant (3.2) but when θ = θ = 0 as at perigee we can write the distances r and r as r = 2ae + (a − ae) (3.3) and r = a − ae, (3.4) and so r + r = 2a. (3.5) But, from geometrical considerations r cos θ = r cos θ − 2ae (3.6) and r sin θ = r sin θ . (3.7) Squaring these equations we have r 2 cos2 θ = r2 cos2 θ + 4aer cos θ + 4a2 e2 (3.8) and r 2 sin2 θ = r2 sin2 θ (3.9) which then gives the relationship r 2 = r2 cos2 θ + 4aer cos θ + 4a2 e2 + r2 sin2 θ. (3.10) Using equation (3.5) then yields (2a − r)2 = r2 + 4aer cos θ + 4a2 e2 , (3.11) such that r(1 + e cos θ) = a(1 − e2 ), (3.12) or in ﬁnal form a(1 − e2 ) r= ; (3.13) 1 + e cos (θ + θ◦ ) where the θ◦ takes into account any possible rotation of the ellipse about the origin. This equation then relates the position of the orbiting body to the fun- damental parameters explained above. Furthermore, this equation is used for the determination of the Roemer delay in a binary orbit as discussed in Chapter 1. With some knowledge of the orientation of the orbit in space and the above equation, the variation of distance through which the pulsar signal propagates can be determined. Using r and the inclination angle of the orbit, i, one ﬁnds that the distance to the pulsar as it moves in it’s orbit is given by R = [(r cos(π/2 − i) sin(θ))2 + (r cos(π/2 − i) cos(θ))2 ]1/2 . (3.14) 3.2 Keplerian mechanics 21 3.2 Keplerian mechanics In a system with a mass m orbiting a mass M there are no preferred directions or points other than the vector r going from mass M to mass m and possibly the vector v which is the rate of change of r with time. Therefore it is irrelevant how the basis vectors are chosen and as such we may as well choose them at M ˆ ˆ with r pointing outward and θ in the direction of increasing θ, as shown in ﬁgure (3.3). Fig. 3.3: Basis vectors for an orbiting system Newton II states ¯ a F = m¯ (3.15) and ¯ GM m ˆ F = − 2 R, (3.16) r ¯ r and since r = rˆ we have r d¯ dr r dˆ ¯ v= = ˆ r+r (3.17) dt dt dt ˙r ˆ ˙θ; = rˆ + rθ (3.18) so v d¯ dr˙ dˆ dr ˙ ˆ r ˙ dθ ˆ ˆ a= ¯ = r + r + θθ + r θ + rθ ˆ ˙ ˙ dθ (3.19) dt dt dt dt dt dt = rr + rθ ¨ˆ ˙ ˆ ˙ ˙ˆ ¨ˆ ˙ ˙r ˙θ + rθθ + rθθ + rθ(θˆ) (3.20) ˙ r ¨ ˙ ˆ = (¨ − rθ2 )ˆ + (rθ + 2rθ)θ. r (3.21) And so it is seen that ¯ ¨ˆ ¨ˆ a = rr + rθθ + extra terms. (3.22) It is these extra terms that indicate how the basis is changing as the object moves through the space and, in combination, are often referred to as a centrifugal force despite being only an ”imaginary” force. One therefore ﬁnds GM m ˙ r ¨ − ˙˙ ˆ r = m(¨ − rθ2 )ˆ + m(rθ + 2rθ)θ, ˆ r (3.23) r2 3.2 Keplerian mechanics 22 thus GM ˙ − = r − rθ2 ¨ (3.24) r2 and ¨ ˙˙ 0 = rθ + 2rθ. (3.25) The second of these ordinary diﬀerential equations can be multiplied out by r to show ˙˙ ¨ 0 = 2rrθ + r2 θ (3.26) d 2 ˙ d ˙ = (r ) θ + r2 θ (3.27) dt dt d 2˙ = (r θ) (3.28) dt and so ˙ r2 θ = k, a constant (3.29) which implies k 2 GM 0=r− ¨ + 2 . (3.30) r3 r Solving these diﬀerential equations to ﬁnd r = r(t) and θ = θ(t) is done by letting r = u−n . In this case d −n r= ˙ u = −nu−n−1 u; ˙ (3.31) dt but du dθ ˙ du ˙ u= =θ , (3.32) dθ dt dθ and ˙ k θ = 2 = ku2n , (3.33) r therefore du u = ku2n ˙ . (3.34) dθ Equation (3.31) can then be written as du r = −nu−n−1 ku2n ˙ (3.35) dθ du = −nun−1 k , (3.36) dθ such that the second derivative is d du dθ r = ¨ −nkun−1 (3.37) dθ dθ dt 2 du d2 u = −nk(n − 1)un−2 − nkun−1 2 ku2n . (3.38) dθ dθ 3.2 Keplerian mechanics 23 ¨ But equation (3.30) already gave us a relationship for r that can be rewritten as r = k 2 u3n − Gmu2n ¨ (3.39) from which we ﬁnally get 2 du d2 u k 2 u3n − Gmu2n = −nk 2 (n − 1)u3n−2 − nk 2 u3n−1 . (3.40) dθ dθ2 Choosing n = 0, 1 the du term is eliminated thus simplifying our equation but dθ the n = 0 solution is trivial, therefore we choose n to be 1. In this case d2 u 0 = −k 2 u2 − k 2 u3 + GM u2 , or (3.41) dθ2 d2 u GM 0= 2 +u− 2 ; (3.42) dθ k GM but k is a constant and this allows us to write d2 GM GM 0 = 2 u− 2 +u− (3.43) dθ k k2 2 d = (s) + s. (3.44) dθ2 An equation of this form has the solution s = A cos (θ + θ0 ) (3.45) as it is analogous to the analysis of the harmonic oscillator. We thus have GM 1 u = A cos (θ + θ0 ) + 2 = (3.46) k r and therefore 1 r = GM (3.47) k2 + A cos (θ + θ0 ) 2 k 1 = Ak2 . (3.48) GM 1+ GM cos (θ + θ0 ) This is analogous to equation (3.13) and so we must have that Ak 2 k2 e= and a(1 − e2 ) = . (3.49) GM GM At perigee/apogee the separation of the two bodies will be a min/max, ˆ rmin /rmax but more importantly the velocity will be entirely in the θ direction, ¯ ˙ˆ v = rθθ. (3.50) 3.2 Keplerian mechanics 24 This implies ¯ 1 1 ˙ Ek = m¯2 = m(rθ)2 . v (3.51) 2 2 But from equation (3.29) we can deduce that ˙ (rm θ)2 k2 = 2, (3.52) 2 2rm the kinetic energy per unit mass. The total energy per unit mass will then be the combination of kinetic and potential energies k2 GM = 2 − , (3.53) 2rm rm and therefore 2 GM k rm + rm − =0 (3.54) 2 which obviously leads to −1 MG M 2 G2 2 rm = 2 ± + 2 . (3.55) k k4 k Here we have used equation (3.53). Upon comparison with equation (3.13) and (3.49) and remembering that at perigee/apogee cos (θ + θ0 ) = ±1, we see that M 2 G2 2 A= + 2. (3.56) k4 k Now by combining equations (3.13) and (3.49) one gets k2 rm = , (3.57) M G(1 + e) such that M 2 G2 (1 + e)2 M 2 G2 (1 + e) = − (3.58) 2k 2 k2 2 2 M G (1 + e)[1 + e − 2] = (3.59) 2k 2 2 2 2 M G (e − 1) = (3.60) 2k 2 MG = − , (3.61) 2a 2 k since a(1 − e2 ) = 2M . From this it is then easily seen that the total energy per unit mass is given by MG v2 M G − = − , (3.62) 2a 2 r 3.2 Keplerian mechanics 25 and upon rewriting we obtain the orbital speed for a mass m orbiting a mass M 2 1 v2 = M G − . (3.63) r a 3.3 Binary pulsar evolution 26 3.3 Binary pulsar evolution It is fairly obvious that a binary pulsar is a system of two, mutually orbiting stellar bodies, one of which is a detectable pulsar. In the case of strong ﬁeld General Relativistic tests, however, what is meant by a binary pulsar must be stated more precisely. It means a binary system wherein there exists a pulsar with a compact binary companion such as a white dwarf star, another neutron star or in the most extreme case, a black hole. The reason that an extrasolar planet or main-sequence star are excluded is that the gravitational ﬁeld generated by such bodies is not suﬃcient to aﬀect the signal from the pulsar suﬃciently to allow tests of General Relativity. Stellar evolutionary models have indicated that only one in a hundred stars will evolve to become a neutron star or black hole and furthermore have shown that around ﬁfty percent of stars occur in binary systems. This implies that only one out of every two hundred stellar systems will form a compact binary of some sort. Of course the chances that one member of the binary system is a pulsar beaming towards earth is again many times smaller and therefore the short list of a few hundred detected compact binary systems is understandable. The list of only 8 systems where both components are neutron stars is thus also understandable in light of the rarity of such systems. 3.3.1 Evolutionary processes There are believed to be four evolutionary scenarios1 that can give rise to a com- pact binary system, with the development of a neutron star-black hole binary being ignored due to the non-detection of such a system at the time of writing of this thesis. The ﬁrst scenario results in the formation of a long-period binary system with a millisecond pulsar and a low-mass white dwarf companion. One starts with a main-sequence (> 8M ) star and a low mass (∼ 1M ) companion. The primary star undergoes standard stellar evolution and explodes as a super- nova with the core collapsing to form a neutron star. The lower mass companion undergoes its usual, much slower evolution and, during its expansion phase, over- ﬂows its Roche-lobe allowing accretion of the matter onto the companion neutron star. This is a very slow steady process and results in the system being visible in the x-ray regime due to production of hard and soft x-rays when the incident ac- creted matter is pulled onto the pulsar along the magnetic ﬁeld lines and strikes the pulsar magnetosphere. The ﬁnal result is a low-mass white dwarf in slow orbit about a recycled millisecond pulsar. In the case of a (> 8M ) main-sequence star with an intermediate-mass (∼ 5M ) companion, one again has neutron star formation through a super- nova explosion of the primary star. However, due to the much larger size of the companion, one does not have an accretion driven evolution but rather common- 1 Stairs, I.H., “Pulsars in Binary Systems: Probing Binary Stellar Evolution and General Relativity”, Science, 304, 547, (2004). 3.3 Binary pulsar evolution 27 envelope evolution where the neutron star spirals into and eventually expels the envelope of the companion. This results in a mildly-recycled pulsar in a close orbit with a massive (∼ M ) white dwarf. The third scenario is where the evolutionary process begins to get very in- teresting. In the event that neither star is massive (> 8M ), but rather there exists one star at (∼ 7M ) while the other is slightly smaller at (∼ 5M ), mass transfer will occur from the primary star to its lighter companion due to the primary’s faster evolution. Through this process the primary loses enough mass to evolve normally into a white dwarf star. The companion is now much larger than it initially was and so with its standard evolution it expands to envelop the white dwarf primary until the white dwarf spirals in and expels the envelope. The result of this is a white dwarf - helium star binary that, upon supernova explosion of the helium star, forms a young pulsar in orbit about a relatively massive white dwarf companion. The last possibility for this sort of binary formation is certainly the rarest. For two main-sequence stars, both of which have mass (> 8M ), the primary explodes as a supernova to form a neutron star. There then comes a period of mass transfer from the companion to the neutron star in the form of a companion wind. Again common-envelope evolution occurs with the neutron star expelling the envelope of the secondary to leave a neutron star - helium star binary system. At this stage Roche-lobe overﬂow can occur from the helium star to the neutron star, resulting in a high-mass x-ray binary, but the eventual supernova explosion of the helium star to form a second neutron star is what makes this such an interesting system. The ﬁrst such system, PSR 1913+16, was detected by Russell Hulse and Joseph Taylor (1975) and their study of this system resulted in their being awarded the 1993 Nobel prize in physics. The ﬁrst double pulsar system, PSR J0737-3039, in which both neutron stars are beaming towards earth was only discovered in 2004 by Lyne, Burgay, Kramer et al. (2004). A neutron star is formed during the ﬁnal stages of stellar evolution when a star has converted almost all the hydrogen in its core to heavier elements. When electron degeneracy pressure can no longer support the the star, the inner layers collapse under the inﬂuence of gravity while the outer layers are blasted oﬀ into space in a supernova explosion. Under this free-fall acceleration, the matter in the core of the star collapses into itself reducing its volume by up to 109 times. At these volumes with the large mass still present in the core of the star, the density exceeds nuclear density and electrons are forced into the nuclei to form neutrons. This then allows for neutron degeneracy pressure to sustain the star against further collapse. In this process a star is formed with a density in excess of 1013 g/cm3 and magnetic ﬁelds of the order of 1010 Gauss or more. As a result, any free charged particles near the surface of the star would be seized by the magnetic ﬁeld and accelerated along the ﬁeld lines. By the process of bremsstrahlung, these particles would radiate electromagnetic radiation and this energy could be detected over many parsecs of space. 4. TOPICS FROM THE THEORY OF CURVED SPACE In any attempt to test a theory, one must have a ﬁrm enough grasp of the subject matter so as to understand any behaviour that deviates from prediction. An overview of some of the basic principles of General Relativity is mandatory and will help in the full understanding of the concepts that will be introduced later. The understanding of General Relativity requires the ability to switch coor- dinate systems at will, knowing that events can be described within an inﬁnity of diﬀerent coordinate systems. 4.1 Curvilinear coordinates Consider ﬁgure (4.1). Construct a rectilinear, or cartesian, 2-dimensional coor- dinate system according to X i = X i (x) with basis vectors Ei . Pick any point within that space such that it has coordinates x1 and x2 . The x1 grid line go- o o ing through that point will then be given, in terms of the rectilinear coordinate system, by X 1 = X 1 (x1 , x2 ) 0 (4.1) and X 2 = X 2 (x1 , x2 ). 0 (4.2) Just as the components of the x2 grid line will be X 1 = X 1 (x1 , x2 ) 0 (4.3) and X 2 = X 2 (x1 , x2 ). 0 (4.4) In this case the components of the basis e1 in the (X 1 , X 2 ) coordinate system will be given by ∂X 1 1 2 (x , x ) (4.5) ∂x1 0 0 and ∂X 2 1 2 (x , x ); (4.6) ∂x1 0 0 and therefore the basis vectors are ∂X i e1 = Ei (4.7) ∂x1 4.1 Curvilinear coordinates 29 Fig. 4.1: Curvilinear coordinates embedded in a cartesian coordinate system and ∂X i e2 = Ei ; (4.8) ∂x2 or in index notation ∂X i eα = Ei . (4.9) ∂xα So within the coordinate system X i = X i (x) we have a new basis given by equation (4.9). Both sets of bases must be expressible in terms of one another, else they would not be basis vectors, and therefore the relation ∂xα Ei = eα (4.10) ∂X i must hold. Taking the derivative of eα will determine how this new basis changes as we move through the curvilinear coordinate system. This is done as follows: ∂eα ∂2X i ∂X i ∂Ei = (x)Ei + (4.11) ∂xβ ∂xβ ∂xα ∂xα ∂xβ 2 i ∂ X = (x)Ei (4.12) ∂xβ ∂xα ∂Ei where the second term is zero since ∂xβ = 0 because the Euclidian basis does not change. Thus ∂X i eα (x + δx) = (x + δx)Ei (4.13) ∂xα ∂X i ∂2X i = (x) + β α (x)δxβ Ei (4.14) ∂xα ∂x ∂x 4.2 The metric 30 ∂2X i = eα + β ∂xα (x)δxβ Ei (4.15) ∂x ∂2X i ∂xσ = eα + β α (x)δxβ eσ . (4.16) ∂x ∂x ∂X i We now deﬁne ∂ 2 X i ∂xσ ≡ Γσ ; αβ (4.17) ∂xβ ∂xα ∂X i and as such equation (4.16) becomes eα (x + δx) − eα (x) = Γσ δxβ eσ . αβ (4.18) If we then divide through by δxβ and take the limit as it goes to zero, we get ∂eα = Γσ eσ . αβ (4.19) ∂xβ This equation tells us the way in which the bases, in the context of ﬂat spaces, change while one moves through a curvilinear coordinate system. By determining the Γ’s for a given coordinate system, the basis can be found for every point within that coordinate system. This is an essential piece of information needed for any further calculations within a given spacetime. 4.2 The metric How does one measure distance on a 2-dimensional surface embedded within a 3-dimensional Euclidian space? Well, consider ﬁgure (4.2). In this system Fig. 4.2: Diagram highlighting measurement on a 2-dimensional plane embedded in 3-dimensions. the question becomes, how do we measure the distance A→ B? Ordinarily on 4.3 Geodesics 31 a ﬂat space, such as can be approximated over an inﬁnitesimal region of the curved space, we would sum over all the inﬁnitesimal lengths δs by using the pythagorean relationship δs2 = δu2 + δv 2 but on curved surfaces this relationship does not hold. The relationship δs2 = δx2 + δy 2 + δz 2 does, however. But since x = x(u, v) we can write ∂x ∂x δx = δu + δv, (4.20) ∂u ∂v and squaring this yields 2 2 2 ∂x 2 ∂x ∂x ∂x δx = δu + δv 2 + 2 δuδv. (4.21) ∂u ∂v ∂u ∂v The line element δs2 can then be written as 2 2 2 2 2 2 ∂x ∂y 2 2 ∂z 2∂x 2 ∂y δs = δu + δu + δu + δv + δv 2 + ∂u ∂u ∂u ∂v ∂v 2 ∂z ∂x ∂x ∂y ∂y ∂z ∂z δv 2 + 2 + + δuδv; (4.22) ∂v ∂u ∂v ∂u ∂v ∂u ∂v which is written more concisely in index notation as ∂xi ∂xj α β δs2 = δij δu δu (4.23) ∂uα ∂uβ = gαβ δuα δuβ . (4.24) ∂x ∂x i j Here gαβ = δij ∂uα ∂uβ and the δij term is the Euclidian metric in a Cartesian system of coordinates. It is this gαβ term that is deﬁned as the metric for a particular curved space. 4.3 Geodesics A straight line may be described as the shortest distance between two points. Light paths are also often used to describe straight lines because they naturally follow the straightest route between two points. It is this property of light that is of fundamental importance to this thesis. Mathematically, how is this indicated? Consider 2 points connected by a curve C with the property (by Hamilton’s variational principle) that S= ˙ L(x, x, λ)dλ (4.25) C where the curve is parameterised by xi = xi (λ). In this case the curve C for which S is an extremum is the solution to the Euler-Lagrange equations d ∂L ∂L 0= − (4.26) dλ ∂ xi ˙ ∂xi 4.3 Geodesics 32 Now consider curves of shortest length in the surface xi = xi (u1 , u2 ). 2 S= ds, (4.27) 1 but from equation (4.24) we know that ds2 = gαβ δuα δuβ and so 2 S= gαβ δuα δuβ . (4.28) 1 But the u’s are also parameterised by λ and as such duα duα = (λ)dλ, (4.29) dλ and hence we can write ∂uα ∂uβ gαβ δuα δuβ = gαβ (u(λ)) (λ)dλ (λ)dλ. (4.30) ∂λ ∂λ Finally we can rewrite equation (4.28) as 2 S= gαβ uα uβ dλ. ˙ ˙ (4.31) 1 ∂L Now in order to apply the Euler-Lagrange solution it is necessary to ﬁnd ∂ uα ˙ . In order to avoid confusion of indices this is written as ∂L ∂ α = (gρσ uρ uσ )1/2 ˙ ˙ (4.32) ∂u ˙ ∂ uα ˙ 1 ∂gρσ ρ σ ∂ uρ ˙ ∂ uσ ˙ = (gρσ uρ uσ )−1/2 ˙ ˙ u u + gρσ α uσ + gρσ uρ α ˙ ˙ ˙ ˙ (4.33) 2 ∂ uα ˙ ∂u˙ ∂u˙ 1 = (gρσ uρ uσ )−1/2 (0 + gρσ δα uσ + gρσ uρ δα ) ˙ ˙ ρ ˙ ˙ σ (4.34) 2 1 = (gρσ uρ uσ )−1/2 (2gασ uσ ); ˙ ˙ ˙ (4.35) 2 where we have used that facts that the metric gρσ is not a function of uα and ˙ ρ that gρσ δα = gασ . Furthermore we have ∂ 1 ∂gρσ ρ σ (g uρ uσ )1/2 = (gρσ uρ uσ )−1/2 α ρσ ˙ ˙ ˙ ˙ ˙ ˙ u u +0+0 . (4.36) ∂u 2 ∂uα The Euler-Lagrange equation then becomes d 1 ∂gρσ 0= (gρσ uρ uσ )−1/2 gασ uσ − (gρσ uρ uσ )−1/2 α uρ uσ ˙ ˙ ˙ ˙ ˙ ˙ ˙ (4.37) dλ 2 ∂u 4.3 Geodesics 33 Writing out the ﬁrst term of this equation we get d d ∂gασ ∂uβ σ (gρσ uρ uσ )−1/2 gασ uσ ˙ ˙ ˙ = [gρσ uρ uσ ]−1/2 gασ uσ + (gρσ uρ uσ )−1/2 ˙ ˙ ˙ ˙ ˙ ˙ u dλ dλ ∂uβ ∂λ +(gρσ uρ uσ )−1/2 gασ uσ ˙ ˙ ¨ (4.38) ∂gασ = 0 + (gρσ uρ uσ )−1/2 β uβ uσ ˙ ˙ ˙ ˙ ∂u +(gρσ uρ uσ )−1/2 gασ uσ ˙ ˙ ¨ (4.39) This is because (gρσ uρ uσ )1/2 = L and since L = L(x, x) is not a function ˙ ˙ ˙ d d −1 of λ, dλ L = 0 and therefore dλ L = 0. This implies that L is constant along the Euler-Lagrange curve and therefore equation (4.37) ﬁnally becomes (using (4.39)): ∂gασ β σ 0 = 0 + (gρσ uρ uσ )−1/2 ˙ ˙ u u + (gρσ uρ uσ )−1/2 gασ uσ ˙ ˙ ˙ ˙ ¨ ∂uβ 1 ∂gµν − (gρσ uρ uσ )−1/2 α uµ uν ˙ ˙ ˙ ˙ (4.40) 2 ∂u ∂gασ β σ 1 ∂gρσ ρ σ = (gρσ uρ uσ )−1/2 ˙ ˙ β u u + gασ uσ − ˙ ˙ ¨ ˙ ˙ u u . (4.41) ∂u 2 ∂uα Now with a little mathematical wizardry we are able to change some summation indices, without inﬂuencing the content of the equation in the slightest, and use the symmetry condition of the metric to show ∂gασ β σ ∂gαβ σ β 1 ∂gαβ σ β 1 ∂gασ β σ ˙ ˙ u u = ˙ ˙ u u = u u + ˙ ˙ ˙ ˙ u u (4.42) ∂uβ ∂uσ 2 ∂uσ 2 ∂uβ and with this identity equation (4.41) becomes 1 ∂gρσ ρ σ 1 ∂gαρ σ ρ 1 ∂gασ ρ σ 0 = gασ uσ − ¨ ˙ ˙ u u + u u + ˙ ˙ ˙ ˙ u u (4.43) 2 ∂uα 2 ∂uσ 2 ∂uρ = gασ uσ + Γαρσ uρ uσ ¨ ˙ ˙ (4.44) (see equation (4.17) above). But we can use the properties of metrics and the µ identity g µν gνλ = δλ and write 0 = g βα gασ uσ + g βα Γαρσ uρ uσ ¨ ˙ ˙ (4.45) β β ¨ = u + Γσρ (4.46) These concepts are suﬃcient to understand the details of the derivations done in later sections of this work. Other general relativistic concepts can be found in the appendices. 5. RELATIVISTIC PARTICLE ORBITS IN A SCHWARZSCHILD GEOMETRY 5.1 Physical and Mathematical concepts The motion of particles under the inﬂuence of a gravitational ﬁeld is an impor- tant aspect in the understanding of how light behaves under the same conditions. Taking the relativistic corrections into account is essential to developing an ac- curate picture of space-time curvature. For a star represented as an isolated, non-rotating source of gravity the Einstein ﬁeld equations can be solved to de- scribe the geometry of the spacetime surrounding the star. The solution to this problem gives rise to the Schwarzschild metric1 whose line element is given by −1 2GM 2GM ds2 = − 1 − 2r (cdt)2 + 1 − 2 dr2 + r2 (dθ2 + sin2 (θ)dφ2 ) (5.1) c cr When this equation is compared with the static, weak-ﬁeld metric 2Φ(xi ) 2Φ(xi ) ds2 = − 1 + (cdt)2 + 1 − (dx2 + dy 2 + dz 2 ) (5.2) c2 c2 one sees that they are the same when one realises that Cartesian coordinates (x, y, z) have been converted to spherical (r, θ, φ) and the Newtonian potential Φ is given as − GM . So M must be the total mass of the source of curvature. r This means that M is comprised of any source of mass as well as any source of energy such as electromagnetic ﬁelds, nuclear interaction energy and even the energy in the spacetime curvature itself. In this regime the spacetime curvature for spherical symmetry is dependent only on the total mass M and not on how it is distributed inside the source. Two fundamental properties of the Schwarzschild solution are time indepen- dence and spherical symmetry. These properties give rise to, amongst others, two vectors, known as Killing vectors, that highlight the symmetries in a system. Named after the German physicist Wilhelm Killing, Killing vectors are vectors that allow for the isolation of certain symmetries within a solution of Einstein ﬁeld equations and these two Killing vectors will be used extensively in later sections. 1 ¨ Schwarzschild, A., “Uber das Gravitationsfeld eines Massanpunktes nach der Einsteinschen Theorie”, Proc. Royal Prus. Ac. Sci., 1, 189-196, (1916). 5.1 Physical and Mathematical concepts 35 5.1.1 Gravitational Redshift Consider a stationary observer emitting a light signal with frequency ω at time t◦ from a ﬁxed Schwarzschild coordinate radius R. The photon will lose some energy in climbing out of the gravitational well of the central mass. Since the energy of a photon is related to its frequency, as measured by an observer through E = hω; another stationary observer situated at r ¯ R (or ≈ ∞), will measure a lower frequency for the emitted photon. This is the gravitational redshift. The magnitude of this eﬀect is most easily shown using the time independence property of the Schwarzschild geometry and the conserved quantity arising as a result of this. When considering the two symmetries of spatial, (φ), independence and time, (t), independence, the Killing vectors are ξ and η (not to be confused with the η introduced in the Nordtvedt eﬀect) and the conserved quantity due to the spatial symmetry is given by ξ · u = const (5.3) for a four-velocity u. It is therefore obvious that ξ · p must also be conserved for the four-momentum p because p = mu. The energy of a photon as measured by an observer is given by E = −p · uobs , (5.4) and thus hω = −p · uobs . ¯ (5.5) But the observer is stationary and so the spatial components of the four-velocity are zero and only the time component contributes anything to the frequency. Normalisation of the four velocity yields uobs(r) · uobs (r) = gαβ uα (r)uβ (r) = −1, obs obs (5.6) and since all but the time component are zero, we have gtt [ut (r)]2 = −1. obs (5.7) The coeﬃcient of the tt component of the metric, from equation (5.1), is −1/2 2M 1− r and thus in general uα (r) = [(1 − 2M/r)−1/2 , 0, 0, 0] = (1 − 2M/r)−1/2 ξ α , obs (5.8) while for a stationary observer at a radius r this becomes uobs (r) = (1 − 2M/r)−1/2 ξ. (5.9) If we combine this with equation (5.5) we can see that the frequency measured by the stationary observer at R, the Schwarzschild radius, is given by −1/2 2M hωR = 1 − ¯ (−ξ · p)R . (5.10) R 5.2 Derivation of relativistic particle orbits 36 Clearly for the observer at inﬁnite radius hω∞ = (−ξ · p)∞ , ¯ (5.11) but it has already been noted that ξ · u is conserved and so must be the same at inﬁnity as at R. Therefore it is clear that the frequency redshift due to gravity must be given by 2M −1/2 ω∞ = ωR 1 − . (5.12) R 5.2 Derivation of relativistic particle orbits In this analysis it has been seen that the two Killing vectors for spatial and time independence are ξ and η, whose components are (1, 0, 0, 0) and (0, 0, 0, 1) in the coordinate basis associated with the Schwarzschild solution. Consider the 4-velocity of the particle, u. Because of the conservative nature of Killing vectors, the quantities formed by taking the products of the the 4-velocity and the two Killing vectors are also conserved. They only return the dt and dz components respectively. This is because the metric is both t and φ independent. These quantities are given the names 2M dt e = −ξ · u = 1 − (5.13) r dτ and dφ l = η · u = r2 sin2 θ (5.14) dτ and represent the conservation of energy per unit mass and the conservation of angular momentum per unit rest mass, respectively. This is seen from the fact that for ﬂat space the energy of the particle is given by dt E = mut = m (5.15) dτ and for suﬃciently large r, e approximately reduces to this. Similarly l becomes the angular momentum per unit mass when dealing with suﬃciently small veloc- ities. For simplicity it is convenient to restrict one’s attention to particles moving in a meridional ”plane” about the gravitating body. Consider the 4-velocity of a particle at an instant in time. If the coordinates are oriented in such a way that dφ/dτ = 0 at the instant that φ = 0, then l is zero and by the conservation of angular momentum and equation (5.3), dφ/dτ must remain zero. As a result the particle must remain within the meridional plane of φ = 0. Changing our viewpoint slightly we could simply remain within the plane of θ = π/2 in which case uθ = 0. 5.2 Derivation of relativistic particle orbits 37 The 4-velocity is normalised such that u · u = gαβ uα uβ (5.16) dxα dxβ = gαβ = −1, (5.17) dτ dτ and when the Schwarzschild metric is substituted in, one gets −1 2M 2M − 1− (ut )2 + 1 − (ur )2 + r2 (uφ )2 = −1. (5.18) r r Here it must be noted that geometrised units have been used such that G = c = 1. Now, in general, uα = dα/dτ ; using that knowledge and equations (5.14) and (5.15) to eliminate dt/dτ and dφ/dτ one gets −1 −1 2 2M 2M dr l2 − 1− e2 + 1 − + = −1. (5.19) r r dτ r2 Solving for e2 − 1 and dividing by two: 2 e2 − 1 1 dr 1 2M l2 = + 1− 1+ −1 , (5.20) 2 2 dτ 2 r r2 which illustrates the similarity to the Newtonian energy integral if one identiﬁes the term on the left as , the total energy per unit mass, and the second term on the right as the eﬀective potential Vef f (r). Thus orbits in the Schwarzschild geometry can also be described using an equivalent of the Newtonian eﬀective potential, with the diﬀerence lying in the existence of a third, relativistic term. This is seen by multiplying out the right-hand side term of equation (5.20) to get 2M l2 2M l2 2M l2 1− 1+ 2 −1 =− + 2− 3 (5.21) r r r r r This third, cubic term prevents runaway to inﬁnity for very small r as happens in the Newtonian case but rather brings it back down to a ﬁnite value. This is clearly shown in ﬁgure (5.1). What we thus have can be written as: 2 1 dr ε= + Vef f (r) (5.22) 2 dτ What does the eﬀective potential actually mean? In chapter 3 we saw that the energy per unit mass for a body in orbit about another was given by equation (3.62). That equation was the classical form that indicated a potential going to inﬁnity for small r. By implication no particle can approach a gravitating source beyond a certain point in the Keplerian theory due to the runaway of the 5.2 Derivation of relativistic particle orbits 38 Fig. 5.1: Illustration of the Eﬀective Potential for a particle a distance r from a source mass M , comparing the Newtonian and Relativistic cases. potential to inﬁnity. The Relativistic correction applied in this section introduces a cubic term that serves to bring the potential back down for small values of r, see ﬁgure (5.1). As a result it becomes possible to push a particle over that increased potential until it falls into the well that captures the particle. By analysing this potential for various energies we see in ﬁgure (5.1) that there exist four possible types of orbit, depending on energy, labeled a, b, c and d. 1. In (a) we have a particle approaching a gravitating source with suﬃcient energy to overcome the potential ’hump’ of the relativistic scenario but an impact parameter low enough so as to get caught. Such a particle will be pulled into the gravitating source and be absorbed in what is known as a plunge orbit. 2. In scenario (b) we ﬁnd two positions of equilibrium where a particle will orbit the parent body in a perfectly circular orbit. The upper position, however, is very unstable and a small perturbation in energy will send it either crashing into the parent body or ﬂying oﬀ to inﬁnity and so only in the case where the incident particle’s energy lies at the minimum will a stable circular orbit be found. 3. Case (c) is known as a scattering orbit as the incident particle will approach from inﬁnity, encounter the potential ’hump’, which it cannot overcome, and thus slide back out to inﬁnity. The trajectory of the incident particle is merely bent about the parent body to continue on an altered, but otherwise 5.2 Derivation of relativistic particle orbits 39 ordinary path. In this case the there exists an angle δ through which the trajectory has been bent and it is this angle that is fundamental to the Shapiro delay that will be discussed later. This is the type of orbit many comets follow as they are only temporarily captured by our sun on their journey through space. 4. The last scenario is one where the particle rolls back and forth along between the points making up the line (d) in an orbit described as precessing. In such an orbit the particle’s eccentric orbit gradually moves about the parent body tracing out a large circle over time. This orbit is perhaps the strangest, but the existence of which came as the ﬁrst proof of Einstein’s theory of Relativity when it explained the precession of Mercury’s perihelion. The precessing orbit is one where the positions of the perihelion (point of closest approach) and aphelion (point of farthest distance) gradually rotate about the parent body and trace out a circle with time. The precession of perihelia is another interesting test of gravitational theories and much eﬀort has gone into understanding and predicting this eﬀect. In following chapters we will see that for the case of light rays, not all these orbits exist and that the scattering orbit is the one of most interest in the testing of GR using signals from radio pulsars. 6. SHAPIRO DELAY 6.1 Background The Shapiro delay (Shapiro, 1964) is a general relativistic eﬀect occurring every- where that an electromagnetic wave passes through the gravitational ﬁeld gener- ated by mass or energy density. The eﬀect does, however, require a suﬃciently large mass to become detectable using current instruments. An electromagnetic wave will always follow a path that in the geometrical approximation is the null geodesic about a gravitating body. Keplerian geometry obviously does not take into account the bending of spacetime and predicts a linear path of propagation for the electromagnetic wave. There is thus a diﬀerence between the predicted paths taken by the beam, for Keplerian and Einsteinian geometry respectively, and therefore a corresponding diﬀerence in path length. Since, in the absence of interfering media, the wave propagates at velocity c, there will be a diﬀerence in propagation time for the beam. This diﬀerence in arrival time computed for Keplerian and Einsteinian geometries is known as the Shapiro delay. This eﬀect was accurately measured for the ﬁrst time in the weak ﬁeld regime with the Viking mission to Mars in 19761 . Both of the landers and both of the orbiters carried transponders that transmitted signals to earth at or close to grazing incidence with the sun. The landers transmitted signals at the 10 cm wavelength while the orbiters transmitted at both the 10 cm wavelength and at 3 cm. This was to compensate for the dispersive eﬀect of the solar corona on the propagating signals. The experiment was run when the Earth and Mars were moving in opposing directions relative to one another i.e. on opposite sides of the sun (See ﬁgure (6.1)). In this way the eﬀect could be observed to grow and then fade as the signal approached and then receded from grazing incidence with the sun. That experiment measured the delay to be a maximum of approximately 250 µsec, a remarkable achievement given the fairly primitive equipment used and the complexity of the problem. Using this result it was possible to constrain the PPN parameter, γ, to 0.2% within Einstein’s theory. Further solar system experiments2 constrained this parameter even further and a point was reached 1 Shapiro, I.I., Reasenberg, R.D., et. al., “The Viking Relativity Experiment”,J. Geophys. Res., 82, 4329-4334, (1977). 2 Lebach, D. E., Corey, B. E., Shapiro, I. I., et al., “Measurement of the solar gravitational de- ﬂection of radio waves using very-long-baseline interferometry”, Phys. Rev. Lett., 75, 14391442, (1995) 6.2 Mathematical derivation for the Shapiro time delay of light 41 Fig. 6.1: Method for measuring the Shapiro time delay of light as performed during the Viking mission to Mars in 1976/7. (Shapiro, et al., 1977) where only strong ﬁeld tests could possibly allow for further constraints on the theory. The discovery of the ﬁrst binary pulsar system in 1974 (Hulse & Taylor, 1975) promised to be the system where the time delay could be measured in the strong ﬁeld3,4 but unfortunately the orbital elements weren’t favourable to accurate measuring of the delay and more than thirty years would pass before the challenge could be taken up again. In 2004, the ﬁrst double pulsar binary system was discovered (Lyne, et al., 2004) and allowed for the best strong ﬁeld testing of Relativity to date. The orbital elements are such that measurement of the Shapiro time delay of light is measurable in the strong ﬁeld regime and has so far yielded excellent results. A detailed discussion of this eﬀect follows below. 6.2 Mathematical derivation for the Shapiro time delay of light This derivation largely parallels that which was done to calculate the expected delay in the signal during the Viking mission. A beam is emitted from Earth at a certain time, reﬂected oﬀ a reﬂector the far side of the sun (in the case of the Viking mission, at Mars) such that the beam passes as close to the sun as possible. 3 van Straten, W., et al., “A test of General Relativity from the Three-Dimensional Orbital Geometry of a Binary Pulsar”, Nature, 412, 158-160, (2001) 4 Camilo, F., Foster, R.S., Wolszczan, A., “High Precision Timing of J1713+0747: Shapiro Delay”, Ap. J., 437, L39-L42, (1994) 6.2 Mathematical derivation for the Shapiro time delay of light 42 The round-trip travel time is then measured here on earth using an atomic clock. In the case of pulsars we do not have the luxury of having reﬂectors in place but fortunately pulsars are such stable clocks that, if our theories on the composition of the pulsar are correct, the emission time of every pulse is extremely accurately known and so the predicted arrival time equally well known. Any deviation from this arrival time can therefore be attributed to some physical eﬀect such as the Shapiro delay and the highly predictable arrival times fulﬁl the role of the reﬂector. Consequently, a similar derivation is suﬃcient to determine the Shapiro delay for the Viking experiment and for binary pulsars. In the case of orbits of light rays, a very similar treatment is followed to that done for particles in the previous chapter. The fundamental diﬀerence is that light cannot be treated as a beam of particles with rest mass and so the concepts of energy and momentum per unit mass have no meaning. This is not a problem if one describes the world line of the light rays in terms of the family of aﬃne parameters that parameterise the path. Additionally, it is useful to deal with the ratio between e and l as opposed to these parameters independently, in order to eliminate the mass dependence of these parameters. As before, because the Schwarzschild metric is independent in both t and φ, one gets 2M dt e ≡ −ξ · u = 1 − (6.1) r dλ and dφ l ≡ η · u = r2 sin2 θ , (6.2) dλ where λ is the aﬃne parameter that parameterises the path. The tangent vectors that are described by this parameter must thus also be null because they are the derivatives of points lying along the light path which is described by the aﬃne parameter of the curve that traces out the geodesics and therefore dxα dxβ u · u = gαβ = 0, (6.3) dλ dλ as opposed to the particle case where this equation equated to -1. Upon sub- stituting in the Schwarzschild metric and focussing on the equatorial plane of θ = π/2, one gets 2 −1 2 2 2M dt 2M dr 2 dφ − 1− + 1− +r = 0. (6.4) r dλ r dλ dλ This reduces to −1 −1 2 2M 2M dr l2 − 1− e2 + 1 − + = 0, (6.5) r r dλ r2 6.2 Mathematical derivation for the Shapiro time delay of light 43 when equations (6.1) and (6.2) are used to eliminate the diﬀerentials in favour of the e and l terms. Multiplying by (1 − 2M/r)/l2 and rewriting yields 2 e2 1 dr 1 2M 2 = 2 + 2 1− . (6.6) l l dλ r r This again has the form of a Newtonian energy integral where the second term on the right is the eﬀective potential and the ’energy’ is given by (l2 /e2 )−1 which, for reasons that will become obvious, is designated 1/b2 . What is this b2 term? If the light ray is suﬃciently far away from the source of gravitation then one can approximate the surrounding space-time to be ﬂat and the Schwarzschild polar coordinates can be replaced by ordinary Cartesian coordinates. For a light ray traveling parallel to the x-axis a distance d from it; l r2 dφ/dλ dφ b≡ ≈ = r2 . (6.7) e dt/dλ dt Geometrically it is obvious that for very large r, φ ≈ d/r and dr/dt ≈ −1. Therefore dφ dφ dr d = = 2, (6.8) dt dr dt r which shows that b = d. Therefore b is just the impact parameter of the incident light ray approaching a source of gravitation from inﬁnity. This impact parameter b is in geometrised units and determines the various types of orbit a light ray can take about a gravitating body. Unlike for particle orbits, a light ray can only follow circular, scattering or plunge orbits although it must be mentioned that the circular orbit is only possible about a black hole; see below. For neutron stars, that leaves the scattering and plunge orbits that depend upon the impact parameter, b. As a result, equation (6.6) can be written as: 2 1 1 dr 2 = 2 + Vef f (r), (6.9) b l dλ for 1 2M Vef f (r) = 1− . (6.10) r2 r This eﬀective potential clearly contains a cubic as well as a quadratic term but not a linear one. Runaway to positive inﬁnite potential still does not happen although the shape of the eﬀective potential graph is slightly diﬀerent to that of the particle case as is seen in ﬁgure (6.2). The light ray orbits possible due to the Schwarzschild geometry parallel those available to particles in the Schwarzschild geometry, with the diﬀerence that precessing orbits are not available to light rays. This means that circular, scattering and plunge orbits are all possible for light rays although the circular orbit is energetically unstable as it only occurs for the maximum eﬀective potential and so will quickly decay into a scattering or plunge orbit with any perturbation in energy. 6.2 Mathematical derivation for the Shapiro time delay of light 44 Fig. 6.2: Eﬀective potential for a light ray approaching a gravitating source of mass M at a distance r 6.2.1 Deﬂection angle It is clear that all material bodies bend light to some degree and this bending results only in scattering and plunge orbits. For the case of the Shapiro delay, the plunge orbit can be ignored as an absorbed photon cannot be detected by an earthbound observer and so we will only consider the case of the scattering orbit. When dealing with a scattering orbit, the light ray will be deﬂected through an angle δφdef as shown in ﬁgure (6.3). In order to determine the deﬂection of the light one needs to know how the angle φ changes as a function of distance from the gravitational source, or dφ/dr. Once this diﬀerential equation is obtained it can be easily solved to ﬁnd the desired relationship between r and the deﬂection angle φ. To do this, equations (6.2) and (6.6) are solved for dφ/dλ and dr/dλ, respectively, and one divided into the other. The result is: −1/2 dφ l2 = ±l(r2 sin2 θ)−1 2 − l2 Vef f (r) (6.11) dr b Simplifying yields −1/2 dφ 1 1 = ± 2 2 − Vef f (r) . (6.12) dr r b where the sign gives the direction of the orbit. Deﬁning the total angle swept out by the beam as it approaches from inﬁnity, is bent about the source and continues onwards as ∆φ with the turning point being the bisector of the angle. Thus ∆φ is the same as twice the angle swept out from the turning point, or point 6.2 Mathematical derivation for the Shapiro time delay of light 45 of closest approach r1 , to inﬁnity. This can is seen in ﬁgure (6.3). Substituting Fig. 6.3: Deﬂection angle for an incident electromagnetic wave upon a source of gravi- tation in the deﬁnition of Vef f (r) this is then: ∞ −1/2 dr 1 1 2M ∆φ = 2 − 2 1− (6.13) r1 r 2 b2 r r The point r1 is where 1/b2 = Vef f (r1 ), so by introducing a new variable r ≡ b/w, the previous expression becomes: −1/2 b dw w2 1 ∞ w2 2M ∆φ = 2 − 2 2 2 − 2 1− w ; (6.14) b/w1 w b b b b that easily reduces to: w1 −1/2 2M ∆φ = 2 dw 1 − w2 1 − w . (6.15) o b In this equation w1 is the point where 1/b2 = Vef f (r1 ) as before. It should be clear that the value 2M/b determines the amount through which the beam is bent. In the case of our sun, b ≥ R (= 6.96 × 105 km) and M = M (= 1.47 km) which means that 2M/b ≈ 10−6 , a very small number. As a result we can expand equation (6.15) in powers of 2M/b. We ﬁrst re-write (6.15) as: −1/2 −1 −1/2 w1 2M 2M ∆φ = 2 dw 1 − w 1− w − w2 . (6.16) o b b 6.2 Mathematical derivation for the Shapiro time delay of light 46 Expand both inverse powers in terms of 2M/b to yield −1/2 w1 M 3 M2 2 2M M2 ∆φ = 2 dw 1 + w + w + ... 1+ w + 4 2 w2 + ... − w2 . o b 2 b2 b b (6.17) To ﬁrst order this is M w1 1+ b w ∆φ = 2 dw 1/2 . (6.18) o 2M 1+ b w − w2 This integral can be now be looked up in integral tables to show that the ﬁnal solution is simply 4M ∆φ ≈ π + , (6.19) b for small M/b. The relationship between ∆φ and δφdef is clearly shown in ﬁgure (6.3) and is simply δφdef = ∆φ − π. (6.20) Finally the relativistic deﬂection of light can be written out; Remembering that we used geometrised units for the derivation and replacing the correct factors for G and c, this is 4GM δφdef = (for small GM/bc2 ). (6.21) bc2 6.2.2 The Shapiro delay of light The derivation of the equation giving the amount by which a source will delay a light beam is now shown. This derivation proceeds in a similar manner to that done for the deﬂection angle. We require the diﬀerential equation dt/dr in terms of the parameters e and l and this is obtained by solving equation (6.1) for dt/dλ, equation (6.6) for dr/dλ and dividing the second into the ﬁrst to obtain −1 −1/2 dt 1 2M 1 =± 1− − Vef f , (6.22) dr b r b2 where the ± indicates increasing or decreasing radius. In this derivation we are in- terested in the variation of the total travel time for the beam in the Schwarzschild coordinates centered on the sun. The total travel time for the beam is equal to the travel time taken from the source of the signal at rp to the point of closest approach indicated by r1 , plus the time taken from that point of closest approach to the detector on earth at rR . In other words we measure time both forwards and backwards from the point of closest approach to the source of curvature deﬁned as r1 . This is written as: (∆t)total = t(rp , r1 ) + t(r1 , rR ) (6.23) 6.2 Mathematical derivation for the Shapiro time delay of light 47 where in general r −1 −1/2 1 2M 1 t(r, r1 ) = dr 1− − Vef f (r) . (6.24) r1 b r b2 The expansion of this is then r 2M 4M 2 1 3 t(r, r1 ) = dr 1 + + 2 + ... 2 1 + b2 Vef f (r) + b4 Vef f (r) + ... r1 r r 2 8 (6.25) r 2 2M 1 M b Vef f (r) = dr 1 + + b2 Vef f (r) + + ... . (6.26) r1 r 2 r Also we know that 1 1 2M 2 = Vef f (r1 ) = 2 1 − (6.27) b r1 r1 therefore −1/2 2M b = r1 1− (6.28) r1 and, to ﬁrst order in M, M + ...). b = r1 (1 + (6.29) r1 To ﬁrst order we can use this to eliminate b from the integral and integrate it to get 2 2 r+ r 2 − r1 r − r1 1/2 t(r, r1 ) = r2 − r1 + 2M ln +M . (6.30) r1 r + r1 In this equation we ﬁnd that the ﬁrst term is the Newtonian contribution to the propagation time that would be present even in the absence of the gravitating source. The other terms are the relativistic corrections due to the curvature of the spacetime in the vicinity of the source. The total travel time for the electro- magnetic signal is then given by substituting equation (6.30) into equation (6.23) for the various radial parameters of interest. These parameters are rR which is the distance from the source of curvature to the detector i.e. the earth, rp is the distance from the source of curvature to the source of the electromagnetic ray i.e. the pulsar; and r1 is the distance of closest approach between the electromagnetic ray and the gravitating body as shown in ﬁgure (6.3). Using these parameters the excess time of ﬂight of the signal, which is the Shapiro delay, in the weak ﬁeld is written as 2GM 4rR rp ∆tShapiro ≈ 3 ln 2 +1 , (6.31) c r1 upon un-geometrising. 7. STRONG-FIELD SHAPIRO DELAY The necessary extension to the previous chapter is the elucidation of the Shapiro delay in the strong ﬁeld regime. Such a regime is found in the vicinity of a double neutron star system consisting of a pulsar and a neutron star, a rarer pulsar-pulsar binary or possibly even the theorised pulsar-black hole binary. One of the ﬁrst analyses of this type of compact binary system was done by Blandford and Teukolsky (1976)1 who corrected earlier work by Wheeler2 and although reﬁnements have been made since,3,4 the original analysis has been used here. In order to simplify the problem somewhat, the relativistic eﬀects are regarded as small perturbations of the classical orbit and it is assumed that this orbit is a slowly precessing Keplerian ellipse. The reason for this approximation is that the pulsars are so small compared to the orbit that they may be regarded as point particles in a rotating frame centred on the centre of mass of the binary system. When dealing with rotating frames, the Kerr metric is to be used but due to the approximation made, a diﬀerent version is used to solve the problem. This version is1 : ds2 = −[1+2Φ+O(v 4 )]dt2 +O(v 3 )dxi dt+[1−2Φ+O(v 4 )](dx2 +dy 2 +dz 2 ). (7.1) where O(v 4 ) represents all the higher order terms. To ﬁrst approximation, the Newtonian potential Φ can be regarded as the sum of two Newtonian potentials centred on each of the neutron stars in the binary, hence −M1 M2 Φ(r, t) = Φ1 + Φ2 = − , (7.2) |r − r1 (t)| |r − r2 (t)| using geometrised units and with M1 and M2 representing the masses of the pulsar and it’s companion, respectively. One can then construct the equatorial plane of a polar coordinate system such that it coincides with the orbital plane 1 Blandford, R., Teukolsky, S.A., “Arrival-time analysis for a pulsar in a binary system”,Ap. J., 205, 580-591, (1976). 2 Wheeler, J.C., “Timing eﬀects in Pulsed Binary Systems”, Ap. J. (Letters), 196, L67, (1975). 3 Haugan, M., “Post-Newtonian Arrival-Time Analysis for a Pulsar in a Binary System”, Ap. J., 296, 1-12, (1985). 4 Backer, D., Hellings, R., “Pulsar Timing and General Relativity”, Ann. Rev. Astron. Astrophys., 24, 537-575, (1986). 49 of the binary system and in doing so the (r, θ, φ) coordinates of the pulsar and companion are simply, r1 = (r1 , π/2, φ), r2 = (r2 , π/2, φ + π). (7.3) In these equations, φ is the true anomaly as described in Chapter 3 and r1 and r2 relate the positions of each object within their own ellipses and given by, M2 M1 a(1 − e2 ) r1 = r, r2 = r, r= . (7.4) M1 + M 2 M1 + M2 1 + e cos φ The time that would be measured by an ideal clock on the pulsar, the proper time Tp , is determined in terms of the coordinate time, t, and is is found from the Kerr metric deﬁned above. The relationship is, 1 (dT )2 = −ds2 = dt2 1 + 2Φ + O(v 4 ) − (dx2 + dy 2 + dz 2 ) dt2 2 = dt2 [1 + 2Φ + O(v 4 ) − v1 ]. (7.5) This can be re-written as, dTp 2 = [1 + 2Φ + O(v 4 ) − v1 ]1/2 (7.6) dt which upon performing the Taylor expansion and incorporating higher order terms into O(v 4 ) yields dTp 1 2 = 1 + Φ(r1 ) − v1 + O(v 4 ). (7.7) dt 2 From (7.2) it may appear that −M1 M2 Φ(r1 ) = Φ1 (r1 ) + Φ2 (r1 ) = − . (7.8) |r1 (t) − r1 (t)| |r1 (t) − r2 (t)| This would imply that Φ1 (r1 ) is inﬁnite. However, Φ should be evaluated at the surface of the pulsar i.e. at r = r1 + ε. It is thus a constant gravitational redshift term. Also, − 1 v1 can be regarded as the transverse Doppler shift and is deﬁned 2 2 by 2 2 M2 2 1 v1 = − , (7.9) M1 + M2 r a as was seen in chapter 3(cf. the end of section 3.2). Upon dropping the constant terms in equation (7.7) one obtains 2 dTp M2 M2 1 = 1− − (7.10) dt r M1 + M2 r M2 (M1 + 2M2 ) = 1− (7.11) r M1 + M2 50 The reason we can drop the constant terms becomes apparent after looking at the timing equation for time of emission of the N -th pulse from a pulsar with a rotational frequency ν, 1 2 1 3 N = N0 + νTp + νTp + ν Tp . ˙ ¨ (7.12) 2 6 We want to ﬁnd Tp and so after integration, any overall multiplicative constants aﬀecting Tp emerging from the integration can be absorbed into the ν of the timing equation while any additive constants that come out can be absorbed into N0 without aﬀecting the ﬁnal form of the result. From the mechanics introduced in chapter 3, we know that r = a(1 − e cos E), (7.13) for E the eccentric anomaly. Equation (7.11) can thus be written as dTp M2 M1 + 2M2 1 =1− (7.14) dt a M1 + M2 (1 − e cos E) since the constant term can be dropped. It has been stated in chapter 3 that the mean anomaly is related to the eccentric anomaly via: M = E − e sin E (7.15) But we know that for a binary with orbital period Pb , 2π M= (t − t0 ) (7.16) Pb and since (2π/Pb )t0 can be dropped because it is a constant that relates to the time of periastron passage, we have Pb Pb t= E− e sin E. (7.17) 2π 2π Therefore: Pb Pb dt = dE − e cos EdE; (7.18) 2π 2π such that M2 M1 + 2M2 1 dTp = dt 1 − , (7.19) a M1 + M2 (1 − e cos E) which can ﬁnally be written as M2 M1 + 2M2 Pb dTp = dt 1 − dE . (7.20) a M1 + M2 2π 51 Equation (7.14) can therefore be integrated to yield M2 M1 + 2M2 Pb E Tp = t − (7.21) a M1 + M2 2π after one has transformed the integration variable from dt to dE as was shown above. The propagating signal has to cross the large distances of space and in doing so will be aﬀected by the interstellar medium’s electron content. It will therefore travel at a group velocity of less than the speed of light, which in the geometrised units in which we are working is unity. Deﬁne from the line element (7.1) ˆ dt = (1 + Φ)dt, |dˆ| = (1 − Φ)|dx|, x (7.22) in order to accurately measure coordinate time at a constant point and to accu- rately measure distances at a constant coordinate time. This lower velocity can then be described by x dˆ =1− . (7.23) ˆ dt And so dx = 1 − + 2Φ. (7.24) dt We now integrate along a straight line joining re to r1 by approximating the world line of the photon as a straight line. This is because light rays follow null geodesics and in the absence of a large gravitating source to warp the spacetime between the pulsar and the earth, this path will be approximately straight. We further integrate between the time of emission and the time of arrival to obtain: re (tarr ) tarr − tem = (1 + − 2Φ)|dx|. (7.25) r1 (tem ) In this equation there are three terms to be integrated. The constant is a trivial matter. The will result in a term that contains the frequency of the signal as well as the dispersion constant, D, that is a function of the electron content, ne , along the path to be integrated and is thus itself an integral given by: e2 re D= ne |dx|. (7.26) 2πm r1 Electron content models are notoriously diﬃcult to obtain due to the variations constantly occurring in the interstellar medium but the most common model currently used is that of Cordes and Lazio 5 . 5 Cordes, J.M., “NE2001: A New Model for the Galactic Electron Density and it’s Fluctua- tions” in Milky Way Surveys: The Structure and Evolution of our Galaxy, Proceedings of ASP Conference 317, Eds, Clemens, D., Shah, R., Brainerd, T., p211, (2004) 52 It is however, the integral of Φ that is responsible for the relativistic time delay across the orbit and it is this value that varies over the orbit while the others depend only on the beam path taken. If one focuses only on this integral and remembers that Φ1 can be ignored because it only adds a constant term, we have only the contribution from Φ2 and it becomes possible to write the integral as, re tarr dt −2 Φ|dx| ≈ 2M2 . (7.27) r1 tem |x(t) − r2 (tem )| Since we are integrating along a straight path along the line joining re and r1 , we know that the tangent vector to the curve has constant direction and so we can write [x(t) − r1 (tem )][tarr − tem ] = [re (tarr ) − r1 (tem )][t − tem ]. (7.28) Therefore t − tem x(t) = r1 (tem ) + [re (tarr ) − r1 (tem )], (7.29) tarr − tem which is substituted into the integral to obtain tarr dt tarr dt 2M2 = 2M2 t−tem tem |x(t) − r2 (tem )| tem |r1 (tem ) + [r (t ) tarr −tem e arr − r1 (tem )] − r2 (tem )| (7.30) The solution to this integral is then tarr − tem |re − r1 ||r + re − r1 | + |re − r1 |2 + r · (re − r1 ) 2M2 ln . (7.31) |re − r1 | |re − r1 |r + r · (re − r1 ) Looking at equation (7.29), to ﬁrst approximation tarr − tem = |re − r1 | and the expression outside the ln is simpliﬁed. Furthermore re r1 and so the ﬁnal solution for the relativistic time delay in the strong ﬁeld regime is, 2re ∆trel = 2M2 ln (7.32) r+r·n where n = re and represents the unit vector pointing to the solar system barycen- re ter. The distance between the binary system and the earth is practically constant and so the numerator in the above expression is eﬀectively constant and can be ignored. The dot-product term is expressible as −r sin (ω + φ) sin i using the Keplerian mechanics of chapter 3 and since r = 1 + e cos φ, we have 1 + e cos φ ∆trel = 2M2 ln . (7.33) 1 − sin i sin (ω + φ) It must be noted that, from the start, this analysis was done assuming the cor- rectness of General Relativity. Fortunately the analysis varies very little within 53 the PPN framework, only introducing the parameter γ that replaces the number 2 in equation (7.33)with the expression (1 + γ). In GR γ takes on the value 1 while in certain other theories it is other than unity. The ﬁnal equation in the PPN formalism for the relativistic time delay of a pulsar signal in a compact binary system is thus GM2 1 + e cos φ ∆trel = (1 + γ) ln , (7.34) c2 1 − sin i sin (ω + φ) once we un-geometrise. 8. RESULTS OF SIMULATIONS 8.1 Procedure followed The results displayed in the following sections were done with a program writ- ten in the MATLAB programming language, for which the code can be found in appendix B. A program was written to accommodate those orbital parameters that are relatively easy to determine through simple observation of the pulsar in its orbit. These parameters, such as the orbital period, eccentricity and semi- major axis length, were then used to determine the relativistic eﬀect of interest, the Shapiro time delay. Essential parameters in this calculation were the in- clination of the orbit and the mass of the companion body. These, although extremely diﬃcult to determine experimentally, were also included and allowed to vary through a wide range of values so as to see the eﬀects on the Shapiro time delay. As mentioned previously, it has proven experimentally very diﬃcult to measure the Shapiro delay due to the fact that it varies quasi-sinusoidally with orbital phase. This in itself is not problematic but the eﬀect virtually disappears into the Roemer time-of-ﬂight delay caused by the ﬁnite size of the orbit and ﬁnite speed of light. Because the pulsar is in orbit, its signal has to propagate periodically through larger and then shorter distances to earth. This results in a quasi-sinusoidal variation in the arrival time of the signal as it is received on earth. Furthermore the magnitude of this signal is much greater than that of the Shapiro delay and through this combination of similar periodicity and diﬀer- ing amplitude, the Shapiro delay signal is easily buried within the Roemer delay signal and lost. For this reason the Roemer delay was also modeled so that the contrast in signal could be observed. Ordinarily the Roemer delay refers to the annual sinusoidal variation in arrival time due to the orbit of the earth about the sun but in the case of binary pulsars, the timescale of variation of the signal of the pulsar in its orbit is much smaller than that of the solar system Roemer delay. For this reason the Roemer delay referred to in this thesis represents the time of ﬂight delay of only the binary system signal. In order to understand all the possible eﬀects a large number of simulations were run in which the various orbital parameters were varied so as to note any trends in the signal that might be observed using current and future equipment, and identify promising systems types. 8.2 Graphs and Results 55 8.2 Graphs and Results The ﬁrst set of 6 results shows the comparison of the Roemer and Shapiro delays in a simulated, yet typical, compact binary system where both components have a mass of 1.4M . The semi-major axis of the pulsar’s orbit is set at a typical value of 1 000 000 km. Each ﬁgure of the 6 ﬁgures (8.1-8.6) below contains 6 graphs. Successive graphs within a ﬁgure have higher inclination angles and show both the Roemer and Shapiro time delays on the same set of axes. The Shapiro delay has been scaled up 105 times to make it visible and to show the similarities in phase of the two curves. Successive ﬁgures within the set (8.1-8.6) have a higher eccentricity to show how the variation of this parameter aﬀects the curves by inducing a phase shift between the two competing curves. 8.2 Graphs and Results 56 Fig. 8.1: The eccentricity of the system in this set is 0. Please note that the two delays are not on the same scale. The Shapiro delay is 105 times smaller than the Roemer delay. The top left graph is at an inclination angle of 15◦ with the top right being at an inclination of 30◦ . The middle left graph is then at 45◦ , the middle right at 60◦ , the bottom left at 75◦ and ﬁnally the bottom right is at an inclination angle of 90◦ . 8.2 Graphs and Results 57 Fig. 8.2: The eccentricity of the system in this set is 0.1. Please note that the two delays are not on the same scale. The Shapiro delay is 105 times smaller than the Roemer delay. The top left graph is at an inclination angle of 15◦ with the top right being at an inclination of 30◦ . The middle left graph is then at 45◦ , the middle right at 60◦ , the bottom left at 75◦ and ﬁnally the bottom right is at an inclination angle of 90◦ . 8.2 Graphs and Results 58 Fig. 8.3: The eccentricity of the system in this set is 0.2. Please note that the two delays are not on the same scale. The Shapiro delay is 105 times smaller than the Roemer delay. The top left graph is at an inclination angle of 15◦ with the top right being at an inclination of 30◦ . The middle left graph is then at 45◦ , the middle right at 60◦ , the bottom left at 75◦ and ﬁnally the bottom right is at an inclination angle of 90◦ . 8.2 Graphs and Results 59 Fig. 8.4: The eccentricity of the system in this set is 0.4. Please note that the two delays are not on the same scale. The Shapiro delay is 105 times smaller than the Roemer delay. The top left graph is at an inclination angle of 15◦ with the top right being at an inclination of 30◦ . The middle left graph is then at 45◦ , the middle right at 60◦ , the bottom left at 75◦ and ﬁnally the bottom right is at an inclination angle of 90◦ . 8.2 Graphs and Results 60 Fig. 8.5: The eccentricity of the system in this set is 0.6. Please note that the two delays are not on the same scale. The Shapiro delay is 105 times smaller than the Roemer delay. The top left graph is at an inclination angle of 15◦ with the top right being at an inclination of 30◦ . The middle left graph is then at 45◦ , the middle right at 60◦ , the bottom left at 75◦ and ﬁnally the bottom right is at an inclination angle of 90◦ . 8.2 Graphs and Results 61 Fig. 8.6: The eccentricity of the system in this set is 0.8. Please note that the two delays are not on the same scale. The Shapiro delay is 105 times smaller than the Roemer delay. The top left graph is at an inclination angle of 15◦ with the top right being at an inclination of 30◦ . The middle left graph is then at 45◦ , the middle right at 60◦ , the bottom left at 75◦ and ﬁnally the bottom right is at an inclination angle of 90◦ . 8.2 Graphs and Results 62 The ﬁgures (8.1 - 8.6) can be summarised best by combining the Shapiro delays for varying eccentricities in a single plot. In ﬁgures (8.7) and (8.8) we see how the Shapiro delay varies in a system with inclination 90◦ and one with inclination 45◦ . It has been noted that the Shapiro delay is most easily measured in systems with high inclination angles because of the sharp peak in the delay curve and in this instance it can be seen that increasing eccentricity has little eﬀect. Only in the lower inclination systems does an increase in eccentricity lead to a marked phase shift that should make it separable from the Roemer delay. In ﬁgures (8.7) and (8.8) a simulated system was used with a 10M companion in order to maximise the eﬀect of the Shapiro delay and a semi-major axis length of 3.3 ls in order to minimise the Roemer delay for ease of identiﬁcation. Fig. 8.7: Variation in Shapiro delay with an increase in the eccentricity of a system at an inclination angle of 90◦ and keeping all other orbital parameters constant. 8.2 Graphs and Results 63 Fig. 8.8: Variation in Shapiro delay with an increase in the eccentricity of a system at an inclination angle of 45◦ and keeping all other orbital parameters constant. 8.2 Graphs and Results 64 These ﬁgures have illustrated the fact that the Roemer and Shapiro delays both vary sinusoidally with phase and as a result it is extremely problematic to distinguish the Shapiro delay from the Roemer delay owing to its much smaller (10−5 s) size. It is in systems with a high inclination angle that it becomes more favourable to measure the Shapiro delay due to the sharp peak that forms when nearing an eclipsing orbit. It is also apparent that a high eccentricity causes a distinct phase shift between the Roemer and Shapiro delays, making the Shapiro delay distinguishable under certain circumstances. It is exactly these circumstances that are important to understand so that future measurements of the delay can be made in a wider range of binary system. It can be seen from ﬁgure (8.10) that a decreasing semi-major axis length, i.e. an increasingly compact binary system, leads to a decrease in the magnitude of the Roemer delay. Fortunately a decrease in a does not aﬀect the Shapiro delay and thus extremely compact binary systems are ideal for separating these two eﬀects. Furthermore an increasing companion mass leads to an increasing Shapiro delay, ﬁgure (8.9), without aﬀecting the Roemer delay directly. Of course an increasing companion mass would alter the orbital dynamics and thereby aﬀect the Roemer delay; but in the absence of any evolution of the orbital parameters due to the increase in companion mass, say through the process of accretion, an increasingly heavy companion will greatly assist in determination of the Shapiro delay. Fig. 8.9: Figure illustrating the increase in the Shapiro delay as the companion mass, mc increases. Note that mc is given in units of solar masses. 8.2 Graphs and Results 65 Fig. 8.10: Figure illustrating the increase in the Roemer delay as the semi-major axis length, a, increases. Note that a is given in units of light-seconds. Fig. 8.11: Figure illustrating the variation in the Roemer delay as the eccentricity in- creases in a system with an inclination of 45◦ . This eﬀect is shown for a binary system with a companion mass of 1.4M . 8.2 Graphs and Results 66 Fig. 8.12: Figure illustrating the variation in the Shapiro delay as the position of peri- astron (θ◦ ) is rotated about the center of mass from 90◦ to the right of the line of sight to 90◦ to the left of the line of sight. The system used was one with an eccentricity of 0.9, a companion mass of 1.4M and an inclination of the orbit of 45◦ . Fig. 8.13: Figure illustrating the variation in the Roemer delay as the position of pe- riastron θ◦ is rotated about the center of mass. The system used was one with an eccentricity of 0.9, a companion mass of 1.4M and an inclination of the orbit of 45◦ . 8.3 Predicted Future Results 67 8.3 Predicted Future Results The following set of graphs parallels the ﬁrst 6 shown in this chapter, in com- paring Shapiro and Roemer delays in a binary system with all parameters except inclination and eccentricity ﬁxed. Each set of the 6 ﬁgures is plotted at a constant eccentricity with the inclination varying from 15o to 90o as one moves through the graphs going from left to right and top to bottom. Each consecutive ﬁgure increases the eccentricity. The major diﬀerence in these graphs is a companion mass, mc , equal to 10M . This means our companion is a low mass black hole and the system is the very exotic and as yet unfound pulsar-black hole binary. Furthermore the simulation was done using a compact, but not impossible semi- major axis length of 2 000 000 km so as to minimise the Roemer delay of the system. The reason for including this set of data is that a pulsar-black hole binary is widely regarded as the holy grail of relativistic astrophysics and some indication of the expected results would assist in the identiﬁcation of the Shapiro delay in such systems. 8.3 Predicted Future Results 68 Fig. 8.14: The following set of 6 results shows the expected Roemer and Shapiro delays for a hypothetical system containing a black-hole as the companion object. Successive ﬁgures have increasing eccentricity while sequential graphs within each ﬁgure have increasing inclination angles. The eccentricity of the system in this set is 0. Please note that the two delays are not on the same scale. The Shapiro delay is 104 times smaller than the Roemer delay. 8.3 Predicted Future Results 69 Fig. 8.15: The eccentricity of the system in this set is 0.2. Please note that the two delays are not on the same scale. The Shapiro delay is 104 times smaller than the Roemer delay. 8.3 Predicted Future Results 70 Fig. 8.16: The eccentricity of the system in this set is 0.4. Please note that the two delays are not on the same scale. The Shapiro delay is 104 times smaller than the Roemer delay. 8.3 Predicted Future Results 71 Fig. 8.17: The eccentricity of the system in this set is 0.6. Please note that the two delays are not on the same scale. The Shapiro delay is 104 times smaller than the Roemer delay. 8.3 Predicted Future Results 72 Fig. 8.18: The eccentricity of the system in this set is 0.8. Please note that the two delays are not on the same scale. The Shapiro delay is 104 times smaller than the Roemer delay. 8.3 Predicted Future Results 73 Fig. 8.19: The eccentricity of the system in this set is 0.9. Please note that the two delays are not on the same scale. The Shapiro delay is 104 times smaller than the Roemer delay. 8.4 3-Dimensional Parameter spaces 74 8.4 3-Dimensional Parameter spaces In the following series of surface plots, the various parameters of the Shapiro and Roemer delays have been plotted against one another to generate surfaces of maximal delay. In this manner it is possible to see in 3-dimensions the regions where the Shapiro delay is maximal and contrast it with where the Roemer delay is minimal. 8.4 3-Dimensional Parameter spaces 75 Fig. 8.20: This 2-dimensional plot shows the combined eﬀect of eccentricity and incli- nation on the Shapiro delay for a standard compact binary system. What is meant by a standard binary system is one in which the companion is a 1.4M neutron star, the semi-major axis length is 6.6 ls or 2 000 000 km, the periastron is positioned at 45◦ to the right of the line of sight, the eccentricity is 0.3 and the inclination angle is 45◦ . In all further simulations these values are used except for where the two parameters of interest are varied over the full range of possible values such as the eccentricity and inclination angle in this ﬁgure. In this plot it is seen that increasing inclination is the dominant parameter in the Shapiro delay but it must be noted that there does exist an increase due to increasing eccentricity, particularly for lower inclination angles as evidenced by the slope of the projected contour lines. 8.4 3-Dimensional Parameter spaces 76 Fig. 8.21: This 2-dimensional plot shows the combined eﬀect of eccentricity and incli- nation on the Roemer delay for a standard compact binary system. 8.4 3-Dimensional Parameter spaces 77 Fig. 8.22: This 2-dimensional plot shows the combined eﬀect of eccentricity and the position of the periastron on the Shapiro delay for a standard compact binary system. There is a clear symmetry about 1.6 rad (or 90◦ ) as at this angle the periastron is directly behind the companion thus maximising the eﬀect. 8.4 3-Dimensional Parameter spaces 78 Fig. 8.23: This 2-dimensional plot shows the combined eﬀect of eccentricity and the position of the periastron on the Roemer delay for a standard compact bi- nary system. The symmetry noticed in the previous graph is apparent here again but in this case the gradient of the increase in delay is much greater and implies that at periastron positions of 0◦ and 180◦ there is a maximal diﬀerence between Shapiro and Roemer delay. 8.4 3-Dimensional Parameter spaces 79 Fig. 8.24: This 2-dimensional plot shows the combined eﬀect of companion mass and semi-major axis length on both the Shapiro and Roemer delay for a stan- dard compact binary system. This ﬁgure illustrates the earlier statement that Shapiro delay is not dependent on orbit size and Roemer delay is not dependent on companion mass. 8.4 3-Dimensional Parameter spaces 80 Fig. 8.25: This 2-dimensional plot shows the combined eﬀect of inclination and the position of the periastron on the Shapiro delay for a standard compact binary system. There is a slight increasing eﬀect due to the position of the periastron but this is clearly buried in the increase due to inclination angle 8.4 3-Dimensional Parameter spaces 81 Fig. 8.26: This 2-dimensional plot shows the combined eﬀect of inclination and the position of the periastron on the Roemer delay for a standard compact binary system. In this ﬁgure there is also a bulge in the Roemer delay at a periastron position of around 1.6 rad and a decrease at positions around 0 and 3.2 rad. This would indicate a diﬃculty in separating Shapiro and Roemer delays purely through systems with the periastron of the pulsar in the line of sight of the center of mass of the system. 9. CONCLUDING REMARKS General Relativity is still, after 90 years of tests, the front-runner in the race to understand gravity. If it has stood up so well to experiment, why continue to go to such pains to develop new methods to test it? There are two reasons for doing this: ﬁrstly gravity is one of the fundamental forces of nature and if we do not understand the fundamental forces we cannot hope to understand the stranger aspects of the universe. Secondly we are fairly conﬁdent that General Relativity as it stands is not suﬃcient to allow us to unify the four fundamental forces of nature into one Grand Uniﬁed Theory. Failures to quantise gravity have led us to believe that to some degree it must be adjusted before unifying it with the other forces. And thus we continue to test its framework and predictions in the hope of ﬁnding, or not-ﬁnding, some discrepancy. General Relativity predicts a Shapiro time delay in an electromagnetic wave propagating through the gravitational ﬁeld of any object. The magnitude of this delay is dependent on the region of the gravitational ﬁeld that the electromag- netic wave passes through. General Relativity takes the view that gravity is not an exchange force but simply a geometry where all objects bend, or warp, the spacetime fabric that makes up the universe. The greater the warp that the elec- tromagnetic signal travels through, the greater the delay it experiences. It should be immediately obvious that the heavier the body responsible for the bending of spacetime or the closer that the signal passes to that body, the greater the delay experienced by the signal. We have seen how this eﬀect has been thoroughly tested in solar systems tests from the Viking mission to Mars in the 70’s to the Cassini mission to Saturn a few years ago. The PPN parameter testing the valid- ity of various theories with respect to the Shapiro delay has been constrained to within 0.012% of unity. Although this does not eliminate all competing theories as can be seen from ﬁgure (2.2), it does indicate that in the Brans-Dicke the- ory, for example, the parameter ωBD must be greater than 40000 to meet those experimental results. These facts highlight the success of General Relativity in the weak-ﬁelds of the solar system but more needs to be done in the strong-ﬁeld regime found only in the near vicinity of pulsars and black holes. It is for this reason that this thesis involves the simulation of the Shapiro delay in compact binary systems where one component is a pulsar and the other a neutron star or black hole. We have seen that the Shapiro delay is occulted by the Roemer time of ﬂight delay of the signal as it traverses the ﬁnite size of the orbit. The greater the size of 83 the orbit, measured through the semi-major axis and described by the parameter a, the greater the size of the Roemer delay and the more diﬃcult it becomes to separate the very small Shapiro delay from it. Therefore it is essential for future tests that we search for and time binary pulsar systems that are as compact as possible with as heavy a companion as can be found. These two properties, if found in the same system, will maximise the Shapiro delay while minimising the Roemer delay and accordingly increase our chances of constraining γ. Simulations done here have shown, and it has been discussed in theory, that high inclination systems are the most promising in which to measure the Shapiro delay. High inclination means the pulsar will move in an almost eclipsing orbit forcing its beam to pass at close to grazing incidence to the companion and thereby maximise the delay. Furthermore it can be argued that only in these high inclination systems does the beam pass suﬃciently close to the companion so as to pass through the strong ﬁeld we wish to test. This may be true but there are many factors to consider. For a binary system with a semi-major axis of 1 000 000 km at an inclination angle of 89◦ , the impact parameter is approximately 20 000 km, 1000 times the diameter of the companion, assuming a neutron star companion. At an inclination of 80◦ , which is still regarded as high inclination, the impact parameter has increased by an order of magnitude and so drawing a line beyond which the beam no longer experiences strong ﬁeld is diﬃcult. Additionally there is the position of periastron to take into account. The maximum delay occurs when the pulsar is behind the companion within the line of sight. Should the semi-major axis line be at right angles to the line of sight, the pulsar will be in that maximal position i.e. behind the companion, at a distance much closer than the semi-major axis length that is used to estimate the impact parameter. So should periastron lie in the line of sight, the impact parameter is increased while if it lies at 90◦ to the line of sight, it will be minimised. Furthermore the companion could be a black hole in which case the region of strong ﬁeld gravity is greatly enlarged. Clearly many considerations need to be taken into account when determining what is meant by strong ﬁeld. For the remainder of this discussion we will assume that the entire pulsar binary system falls within the category of strong ﬁeld. Varying the eccentricity of the orbital system indicates that this could be a very useful parameter to consider when testing for the Shapiro delay. In the absence of a suﬃciently high inclination, the Shapiro delay cannot be separated from the Roemer delay as the two signals are in phase. As the eccentricity increases, however, there is an increase in the phase shift between the two signals with the result that subtracting out the Roemer delay should not eliminate the Shapiro delay along with it. Additionally there is the advantage that, as the inclination of the orbit decreases, so too does the Roemer delay. Unfortunately there are some problems with this line of investigation. The very nature of binary pulsar searches makes ﬁnding high eccentricity orbits extremely diﬃcult. The problem arises with the searching of binary systems in the 5-dimensional 84 parameter space of parameters aﬀecting binary orbits. Because of the diﬃculty in this, one method of searching is to assume low eccentricity and longitude, or position, of periastron resulting in a 3-dimensional parameter space that is easier to search. A simpler method frequently used is known as an acceleration search which assumes a constant acceleration over each orbital period. Obviously this approximation will deviate further from reality as eccentricity increases and so this indicates that most binary pulsar searches are only looking for close to circular orbits. New methods of pulsar searching could assist in ﬁnding more high eccentricity systems that could be used for the testing of our earlier supposition. Although this thesis has not attempted to constrain γ to any degree, it has shown how the measurement of arrival times of pulses emitted by a pulsar in a binary system can be used to constrain γ through determining the Shapiro delay. We have seen how the parameter appears in the governing equations and how it could be determined using pulsar timing data. 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(1992). • Will, C.M., “The Confrontation Between General Relativity and Experi- ment”,Liv. Rev. Rel., submitted (astro-ph:gr-qc/0510072),(2005). APPENDIX A. THE CONNECTION OR COVARIANT DERIVATIVE This is a mathematical function fundamental to GR with some very interesting and special properties. The quantity X Y is a function that, on a diﬀerentiable manifold M , takes in two vector ﬁeld inputs and outputs a third vector ﬁeld. Practically what it does is measure the rate of change of the Y-vector ﬁeld in the direction of the X-vector ﬁeld. Mathematically this is written as: : χ(M ) × χ(M ) → χ(M ) : (X, Y ) −→ X Y As previously mentioned, has some very important properties that have deep consequences on the subject matter to be dealt with. In order to highlight these properties we shall look at the function in terms of its subscript and main arguments. It turns out that is linear in its subscript argument but not in the second argument and as such is not a tensor. To see this one must ﬁrst note that X Y can be re-written as δ i XY =Yj Y Ei = (LX )Y i Ei (A.1) δxj where LY is the diﬀerential operator acting on the vector ﬁeld Y such that Ly = δ Y i δyi and Ei is the basis of the space. Using this we can now show that is linear in its subscript argument. So X+X Y = (LX+X Y i )Ei = (LX Y i + LX Y i )Ei = (LX Y i )Ei + (LX Y i )Ei = XY + X Y; (A.2) and f XY = (Lf X Y i )Ei = (fLX Y i )Ei = f (LX Y i )Ei = f X Y. (A.3) In the second argument, however, it is not linear A.1 Torsion 90 X (Y + Y ) = (LX [Y i + Y i )Ei = (LX Y i + LX Y i )Ei = (LX Y i )Ei + (LX Y i )Ei = XY + XY ; (A.4) and X (f Y) = (LX [fY i ])Ei = ([LX f ]Y i + f [LX Yi ])Ei = [LX f ](Y i Ei ) + f ([LX Y i ]Ei ) = (LX Y) + f X Y. (A.5) This shows that, although the ﬁrst criterion for linearity is met, the second is clearly a Leibniz multiplication law and not linear. For to be a tensor it must be linear in all its arguments. Rather, is known as a connection. A.1 Torsion Is there a diﬀerence between XY and Y X? i XY = X (Y Ei ) i i = X Y (Ei ) + Y ( X Ei ) = (LX Y i )Ei (A.6) since X Ei = 0 because the basis does not change. Similarly we ﬁnd that YX = (LY X i )Ei , (A.7) and therefore the diﬀerence is XY − YX = (LX Y i + LY X i )Ei . (A.8) This is simply the commutator bracket [X, Y]. This property of the connection is known as symmetry. A connection is said to be symmetric if, on a diﬀerentiable manifold, XY − Y X = [X, Y] (A.9) and thus, in a Euclidean space XY − YX − [X, Y] = 0 (A.10) A.2 Curvature 91 The next step is to generalise this statement so that it applies to any case and not just to an Euclidean space. The general format of this equation introduces us to the concept of Torsion and the Torsion tensor T . In general XY − YX − [X, Y] = T (X, Y) (A.11) In order to understand the signiﬁcance of this it is necessary to under- stand the signiﬁcance and geometrical interpretation of the commutator bracket [X, Y]. Consider the ﬁeld lines X and Y parameterised by σ and λ respectively as shown in ﬁgure (3.1). We then pick a point (xo , yo ) and follow the X ﬁeld line that passes through that point by a set parameter value σ to the point P1 . Next one moves along the Y ﬁeld line passing through that point by a set parameter value λ to the point P2 . If we then move the same parameter ’distance’, λ along the Y ﬁeld line passing through the point (xo , yo ) to a point P3 and then the set parameter value σ along the X ﬁeld line that passes through the point P3 to a point P4 , the vector from point P2 to point P4 is the commutator [X, Y]. Fig. A.1: Geometric interpretation of the commutator bracket [X, Y] A.2 Curvature In many cases one would be expected to use the connection twice in order to determine the second covariant derivative of a vector ﬁeld. The order of this diﬀerentiation is very important to the outcome of the problem; by analysing the diﬀerence between the orders in which diﬀerentiation can be applied, valuable insight can be obtained. In mathematical notation: A.2 Curvature 92 If X, Y and Z are vector ﬁelds on a manifold M, then both X Z and Y Z will be vector ﬁelds on M. We are interested in the second derivatives and so consider ﬁrst X Y Z. We know from our analysis of torsion that i YZ = Y (Z Ei ) = ( Y Z i )Ei = (LY Z i )Ei ; (A.12) Therefore: i X YZ = X ((LY Z )Ei ) = ( X (LY Z i ))Ei = (LX (LY Z i ))Ei . (A.13) Similarly: Y XZ = (LY (LX Z i ))Ei . (A.14) If we construct the diﬀerence between these two terms we have X YZ − Y XZ = (LX LY − LY LX )Z i )Ei = (L[X,Y] Z i )Ei = ( [X,Y] Z i )Ei i = [X,Y] (Z Ei ) = [X,Y] Z. (A.15) Therefore it can be seen that, for a Euclidean space: X YZ − Y XZ − [X,Y] Z =0 (A.16) A space in which this last statement holds is said to be ﬂat or to have zero curvature. If we now, as before, generalise this statement to all spaces, we ﬁnd that the above statement is usually not equal to zero but is equal to a rank 3 tensor that takes 3 vector ﬁeld inputs and gives one vector ﬁeld output. This is because there is nothing in the deﬁnition of to guarantee that this tensor must be zero. This tensor is known as the Riemannian curvature tensor, R and is deﬁned by ˙ R(X, Y)Z = X Y Z − Y X Z − [X,Y] Z (A.17) The geometrical interpretation of the [X,Y] Z illustrates its meaning far more clearly. First we consider a point (xo , yo ) with ﬁeld lines X and Y going through that point and parameterised by λ and σ respectively. At the point (xo , yo ) we have ¯ ¯ tangent vectors (xo , y o ) to the ﬁeld lines. We then parallel transport these tangent vectors along the ﬁeld lines to the points (xo +Xλ, yo ) and (xo , yo +Y σ) to obtain A.2 Curvature 93 Fig. A.2: Geometric interpretation of [X,Y] Z ¯ ¯ vectors (x1 , y 1 ). At these new points there are yet again tangent vectors to the ¯ ¯ ﬁeld lines and these we shall call (x2 , y 2 ). In this scheme [X,Y] Z ¯ ¯ ¯ ¯ = (x2 , x1 ) − (y 2 , y 1 ) (A.18) which equals zero in the case of a coordinate system because the commutator [X, Y] equals zero in a coordinate system. B. MATLAB CODE