# General Relativity by 4T1pg658

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```									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
•“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
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
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
during a solar eclipse, when light
from Sun is blocked and stars
near the Sun’s edge can be
seen.

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/(r1c1) = hc/2 - GMh/(r2c2)
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
Just as accelerated charged
particles give off EM radiation, GR
predicts that certain systems

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,
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.

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
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
Another effect is that the photons
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)

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