VIEWS: 7 PAGES: 29 POSTED ON: 11/17/2011 Public Domain
General Relativity Chapter 8 Introduction GR is Einstein’s theory of gravitation that builds on the geometric concept of space- time introduced in SR. Is there a more fundamental explanation of gravity than Newton’s law? GR makes specific predictions of deviations from Newtonian gravity. Curved space-time Gravitational fields alter the rules of geometry in space-time producing “curved” space For example the geometry of a simple triangle on the surface of sphere is different than on a flat plane (Euclidean) On small regions of a sphere, the geometry is close to Euclidean How does gravity curve space-time? •With no gravity, a ball thrown upward continues upward and the worldline is a straight line. •With gravity, the ball’s worldline is curved. No gravity gravity t t x x •It follows this path because the spacetime surface on which it must stay is curved. •To fully represent the trajectory, need all 4 space-time dimensions curving into a 5th dimension(!) •Hard to visualize, but still possible to measure Principle of Equivalence A uniform gravitational field in some direction is indistinguishable from a uniform acceleration in the opposite direction Keep in mind that an accelerating frame introduces pseudo-forces in the direction opposite to the true acceleration of the frame (e.g. inside a car when brakes are applied) Elevator experiment •First, elevator is supported and not moving, but gravity is present. Equate Let upward forces be forces on the person to ma (=0 since a=0) positive, •Fs - mg = 0 so Fs = mg thus gravity •Fs gives the weight of the person. is -g •Second, no gravity, but an upward acceleration a. The only force on the person is Fs and so •Fs = ma or Fs = mg if “a” value is the same as “g” •Person in elevator cannot tell the difference between gravitational field and accelerating frame •Third, there is gravity and the elevator is also in free-fall •Fs - mg = -mg or Fs = 0 See also http://www.pbs.org/wgbh/nova/einstein/relativity/ •“Weightless” Einstein was bothered by what he saw as a dichotomy in the concept of "mass." On one hand, by Newton's second law (F=ma), "mass" is treated as a measure of an object’s resistance to changes in movement. This is called inertial mass. On the other hand, by Newton's Law of Universal Gravitation, an object's mass measures its response to gravitational attraction. This is called gravitational mass. As we will see, Einstein resolved this dichotomy by putting gravity and acceleration on an equal footing. The principle of equivalence is really a statement that inertial and gravitational masses are the same for any object. This also explains why all objects have the same acceleration in a gravitational field (e.g. a feather and bowling ball fall with the same acceleration in the absence of air friction). The GR equations relate the curvature of spacetime with the energy and momentum within the spacetime (Matter tells spacetime how to curve, and curved space tells matter how to move). Where and vary from 0 to 3, thus this equation really represents 16 equations Ricci curvature tensor - R Metric coefficients - g Christoffel symbols Gμν = 8πTμν = Rμν – 1/2gμνR how space is curved location and motion of matter Tests of General Relativity Orbiting bodies - GR predicts slightly different paths than Newtonian gravitation Most obvious in elliptical orbits where distance to central body is changing and orbiting object is passing through regions of different space-time curvature The effect - orbit does not close and each perihelion has moved slightly from the previous position Effect is greatest for Mercury - closest to Sun and high eccentricity of orbit (why?) •Mercury’s perihelion position advances by 5600 arcsec per century. •All but 43 arcsec can be accounted for by Newtonian effects and the perturbations of other planets. •Einstein was able to explain the 43 arcsec exactly via GR calculations. Bending of Light Einstein said that the warping of space- time alters the path of light as it passes near the source of a strong gravitational field (i.e. photons follow geodesics). When viewing light from a star, the position of the star will appear different if passing near a massive object (like the Sun). = 4GM/bc2 Where angle is in radians and b is distance from light beam to object of mass M If b is radius of Sun (7x1010cm), is 8.5x10-6 rad or 1.74 arcseconds Meaurements must be made during a solar eclipse, when light from Sun is blocked and stars near the Sun’s edge can be seen. Sir Arthur Eddington headed the attempt to verify Einstein’s prediction during an eclipse in 1919 and did so with only a 10% error. Since then, the same experiment has been done with radio sources (better positional accuracy) with much lower error and higher accuracy. Similarly, the bent path of light also means a delay in the time for a signal to pass the Sun. This effect has been measured by bouncing radio waves off Mercury and Venus as they pass behind the Sun, and observing signals from solar system space craft. GR effects have been confirmed to an accuracy of 0.1% using these measurements. Gravitational Lensing Any large galaxy or galaxy cluster can act as a gravitational lens; the light emitted from objects behind the lens will display angular distortion and spherical aberration. Measuring the degree of lensing can be used to calculate the mass of the intervening body (galaxy clusters usually). Good way to detect dark matter.. Abell 1689 Light waves passing through areas of different mass density in the gravitational lens are refracted to different degrees. Produces double galaxy images and Einstein Rings (if observer, lens, and source are aligned correctly. Gravitational Redshift A photon’s wavelength is effected by a gravitational field Gravitational potential energy -GMm/r To determine PE for a photon we assign an effective mass based on E=mc2 m=E/c2 And since E=hc/ m=h/(c) Conservation of energy for a photon moving from r1 to r2 hc/1 - GMh/(r1c1) = hc/2 - GMh/(r2c2) This gives 2/1 = [1-GM/(r2c2)]/[1-GM/(r1c2)] Not a rigorous treatment, but a dimensional analysis approximation which agrees with full GR calculation 2/1 = ([1-2GM/(r2c2)]/[1-2GM/(r1c2)])1/2 Actual result (which agrees with approximation in the limit 2GM/(rc2)<<1) If we let r2 go to infinity and use the approximation for small shifts 2/1 = 1 + GM/r1c2 And the wavelength shift is / = GM/rc2 What gravitational redshift would be measured for spectral lines originating in the atmosphere of the Sun (in terms of /)? M = 2 x 1033g, r = 7 x 1010cm and G = 6.67 x 10-8 dyn cm2/g2 How about a 1 solar mass white dwarf with r = 7x108cm? Best cases of measured line shifts due to GR are the white dwarfs Sirius B (3x10-4) and 40 Eridani (6x10-5) Gravitational Time Dilation t2/t1 = ([1-2GM/(r2c2)]/[1-2GM/(r1c2)])1/2 All clocks run slower in a strong gravitational field than they do in a weaker field. The clock at r1 will run slower than that at r2 (r2 is the position further from the source of gravity and thus experiencing a weaker gravitational pull). Effect has been measured with clocks on airplanes, rockets (further from Earth). When the clocks returned, they were slightly fast wrt those on the ground. Let r2 go to infinity to get T = To/(1-2GM/Rc2)1/2 where T is time interval far from mass source On Earth’s surface T = To/(1-2gR/c2)1/2 Time dilation is about 1 part in 109 Gravitational Radiation Just as accelerated charged particles give off EM radiation, GR predicts that certain systems should emit gravitational radiation. Massive objects distort spacetime and a moving mass will produce “ripples” in spacetime which should be observable (e.g. two orbiting or colliding neutron stars). LIGO - Will try to detect the ripples in space-time using laser interferometry to measure the time it takes light to travel between suspended mirrors. The space-time ripples cause the distance measured by a light beam to change as the gravitational wave passes by. Also see LISA – NASA’s version in space! Black Holes Endpoints of high mass (25-30 Msun) stars The most compact objects in the Universe and therefore represent the most extreme gravitational fields Perfect place to investigate the effects of GR The escape speed for an object with mass M and size R is 2GM Vesc = R For the Sun, Radius = 700,000 km Vesc = 620 km/s What if we squeezed the Sun down to 1/4 its current radius? Vesc = 620 x 2 = 1240 km/s What if we squeezed the Sun down to 10 km radius (Neutron star size)? Vesc = 163,000 km/s ( ~half the speed of light!) Eventually (squeezing even more) the escape speed would exceed the speed of light. Nothing could get out, including light! That’s a Black Hole. The critical radius at which the escape speed equals the speed of light is called the Schwarzschild Radius. The sphere around the Black Hole at the Schwarzschild Radius is called the “event horizon,” because no event inside that sphere can ever be seen, heard or known by anyone outside. Schwarzschild worked out the curvature of space-time around a point mass to arrive at the radius where a singularity occurs (some quantity becomes infinite) Rs = 2GM/c2 If an object is completely contained within its Rs, a singularity with occur!! Recall 2/1 = ([1-2GM/(r2c2)]/[1-2GM/(r1c2)])1/2 If we set r1 to the Schwarzschild radius, 2 becomes infinite for any r2. No light can escape from within Rs. Scwarzschild radii for different masses: 1 earth mass: 1 cm 1 solar mass: 3 km 106 solar masses: 3 x 106 km 109 solar masses: 3 x 109 km The average density inside a 1 M blackhole is 1017 g/cm3 Greater than the density of an atomic nucleus! But, density decreases for more massive BHs = (1 x 1017 g/cm3)(M/M)-2 Density for 108 M is ~few g/cm3, not much denser than water. Tidal effects are significant near the Rs - gravitational force falls off very quickly with small changes in distance. Since g(r) = GM/r2 Differentiation yields dg(r)/dr = -2GM/r3 ….so tidal forces are most significant at small r What is the difference between the acceleration of gravity at the feet and head of an astronaut just outside a 1 solar mass blackhole? Strange goings on near a Black Hole. As you get close to a Black Hole, the previous exercise shows that you would get stretched, then torn apart… …because the gravitational pull at your feet is 2x1012 cm/s2 greater than at your head (about 2 billion times gravity on Earth!) Let’s imagine an indestructible astronaut, and give her a clock and a flashlight for her journey (We’ll remain behind at a safe to the Black Hole… distance.) Strange goings on near a Black Hole. As our astronaut friend approaches the Black Hole, we notice that her flashlight appears redder and redder (to us). From a distance very close to the event horizon, the radiation from her flashlight gets gravitationally redshifted even more…. to the infrared and finally radio…. Another effect is that the photons only follow straight-line paths when directed straight up (away) from the blackhole. All other light beams will bend. Only light aimed into the exit cone will escape. At r = 1.5Rs, photons aimed horizontally will orbit the blackhole - photon sphere Strange goings on near a Black Hole. Now note that light is like a clock… …electromagnetic oscillations at a given frequency. We see our friend’s oscillations slowing down (due to gravitational redshift). Thus her clock slows down due to gravitational time dilation. At the event horizon, her clock would appear (to us) to stop. We would never see her cross the event horizon….! Strange goings on near a Black Hole. What does our astronaut friend see? Her flashlight looks the same to her. Her clock seems to run at the same speed. Looking back at us, she sees… Our flashlight gets bluer. Our clock seems to speed up! If the astronaut was your twin sister, after her trip to the Black Hole …you would be older than her! (You would have aged more! Your clock really was running faster than hers!) Strange goings on near a Black Hole. What if our friend continued on through the Event Horizon? She would pass through and not perceive the event horizon in any way. With a supermassive blackhole, even the tidal forces might be survived. There is no physical boundary there but … she could never come back! What would our friend find inside the event horizon? The astronaut would be pulled to the center and crushed down to a point - the singularity What actually happens is not known, because: 1) Current theories are not up to the task. 2) We can never do the experiment! Non-rotating BH The observer outside the blackhole can not tell anything about what is going on inside the blackhole. The only properties that can be deduced are its mass, electric charge, and angular momentum “blackholes have no hair” In a rotating blackhole, ang mtm is non-zero. The structure differs from non-rotating BH (Kerr found the solutions to Einsteins equations for a rotating BH in 1963). Stationary limit – objects within this limit Rotating BH will be dragged around BH due to rotation. Objects moving at light speed here would appear “stationary” from outside the limit since the reference frame is moving at light speed. Touches EH at poles and stretches to non-rotating EH value. Ergosphere - Energy can be extracted from BH via particles in this region moving on specific trajectories – Penrose process (1969)