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General Relativity
Introduction
 GR is Einstein’s theory of gravitation that
builds on the geometric concept of spacetime 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
forces on the person to ma (=0 since a=0)
•Fs - mg = 0 so Fs = mg
•Fs gives the weight of the person.
Let upward
forces be
positive,
thus gravity
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 value of a
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”
Dichotomy in the concept of "mass"
“mass”  a measure of an object’s resistance to changes in
movement (F=ma)  inertial mass
“mass”  a measure of an objects response to gravitational
attraction (F=GMm/r2)  gravitational mass.
Dichotomy resolved 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 equivalence means that 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 - relates
length interval to coordinate system (matrix)
Christoffel symbols – cross terms
Gμν = 8πTμν = Rμν – 1/2gμνR
how space is curved
location and motion of matter
(energy-momentum tensor)
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
In our Solar System, the effect
is greatest for Mercury - closest
to Sun and high eccentricity
•Mercury’s perihelion position
advances by 5600 arc seconds
per century.
•All but 43 arc seconds can be
accounted for by Newtonian
effects and the perturbations of
other planets.
•Einstein was able to explain
those 43 arc seconds via GR.
a) Curved space-time for Mercury’s
orbit around the Sun. Since
Mercury’s orbit is elliptical, its
distance from the Sun changes. It
therefore passes through regions of
different curvature. b) This causes
the orbit to precess (amount of shift
exaggerated in this figure!)
Bending of Light
Einstein said that the warping of spacetime 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  is the angle in radians and b is
the 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
Measurements 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 as the Sun would move in
front of a cluster of distant stars.
Bent light path also causes 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
A large galaxy or cluster can act as
a gravitational lens
light emitted from objects behind
the lens display distortion and
spherical aberration.
Measuring the degree of lensing
can be used to calculate the mass
of the intervening body.
One of the techniques to detect
the presence of dark matter.
Abell 2218
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
in a specific way).
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/ (energy of a photon)
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
Full GR calculation agrees with approximation in the limit 2GM/(rc2)<<1
If we let r2 go to infinity and use an approximation for small shifts
2/1 = 1 + GM/r1c2
The wavelength shift due to gravitational redshifting is then
/ = GM/rc2
What gravitational redshift would be measured for spectral lines
originating in the atmosphere of the Sun (in terms of /)?
M = 2 x 1033 g, r = 7 x 1010 cm and G = 6.67 x 10-8 dyn cm2/g2
How about a 1 solar mass compact stellar remnant (white dwarf)
with r = 7x108 cm?
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 effect).
Gravitational time dilation has been measured using clocks
on airplanes, rockets. When the clocks return, they ran
slightly faster (ahead) compared to 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
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html#c5
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 stellar remnants (e.g. neutron stars).
LIGO – 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.
LISA – NASA’s version in space!
Black Holes
 Stellar remnants of the highest mass
stars (see Chapter 18)
 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 into 1/4 its current radius?
Vesc = 620 x 2 = 1240 km/s
What if we squeezed the Sun to ~10 km radius (Neutron star
size)?
Vesc = 163,000 km/s ( ~half the speed of light!)
Compressing the Sun further, we would eventually have the
escape speed equal to the speed of light.
No objects could then escape, including photons
 Black Hole
The critical radius at which the escape speed equals the speed
of light is called the Schwarzschild Radius.
The sphere around a Black Hole at the Schwarzschild Radius is
called the “event horizon,” because no event inside that sphere
can ever be known outside of it.
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 occurs!
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.
Schwarzschild radii for objects with different masses:
1 earth mass: 1 cm
1 solar mass: 3 km  Rs = 3km (M/Msun)
106 solar masses: 3 x 106 km – supermassive BH
109 solar masses: 3 x 109 km – supermassive BH
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
to the Black Hole…
(We’ll remain behind at a safe
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).
When she hovers at a distance very close to the event horizon, the
radiation from her flashlight gets gravitationally redshifted even more….
to the infrared and finally radio….
/ = GM/(rc2)
Another effect is seen in the
paths of the photons – photons
only follow “straight” 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
For animations and descriptions of the event horizon and photon sphere see
http://apod.nasa.gov/htmltest/rjn_bht.html
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!
T = To/(1-2GM/Rc2)1/2
If the astronaut was your twin sister, after her trip to the
Black Hole
…you would be older than her!
(Your clock really would be 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!
The observer outside the blackhole can
not tell anything about what is going on
inside the blackhole.
Non-rotating BH
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).
Rotating BH
Stationary limit – objects within this limit
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)