The odd thing about gravity . . .
•The force of gravity is proportional to the mass of the object it acts on
GMmrˆ
F 
r2
M
•The acceleration is the same, no
matter the mass you drop
F  ma
GMrˆ
a 2
r
•Suppose you are in an elevator
•Now, we cut the cable
•You fall
•Other things fall at the same rate
•To you, it looks like you are weightless
•This is why astronauts are weightless - they
are always falling
•They are not far from the Earth
F
r
m
The equivalence principle
•Are there any other force formulas that are proportional to mass?
•Suppose you are on a merry-go-round 2
mv rˆ
•Centrifugal force
F
r
•Like gravity, everything “accelerates” equally
•You can make it go away by letting go
Can we create “gravitational” effects through acceleration?
•Put the person out in space in the elevator
•Attach to an accelerating rocket
•The effects are indistinguishable from
gravity
The effects of gravity are indistinguishable from
the effects of being in an accelerated reference frame
The metric again
•We will start by reexamining the metric – the distance formula
•Or in this case, the proper time formula
c 2  2  c 2 t 2  x 2  y 2  z 2
•This formula works if you move in a straight line
•What if you don’t move in a straight line?
•You can divide any longer path into several short segments
•Then, find the distance formula for each one
c d  c  dt    dx    dy    dz 
2
2
2
2
2
2
2
c   c  dt    dx    dy    dz 
2
2
2
 dx   dy   dz 
2
c   dt c         
•We won’t be using this formula
 dt   dt   dt 
2
2
2
2
2
•Interestingly, you can show that the straight line has the longest proper time path –
called a geodesic
An object with no forces acting on it will always follow a geodesic,
which is the longest proper time path between two points in spacetime
Changing Coordinates
•It is important to be able to change coordinates to a different system x  r sin  cos 
c d  c  dt    dx    dy    dz 
2
2
2
2
2
2
2
y  r sin  sin 
z  r cos 
•Let’s change to spherical coordinates: (x,y,z,t)  (r,,,t)
dy  d  r sin  sin  
  dr  sin  sin   r  d sin   sin   r sin   d sin  
 sin  sin   dr   r cos sin   d   r sin  cos   d 
•Similarly,
dx   dr  sin  cos   r  d sin   cos   r sin   d cos  
 sin  cos   dr   r cos  cos   d   r sin  sin   d 
dz   dr  cos   r  d cos    cos   dr   r sin   d 
c d 
2
2
 c  dt    dr   r
2
2
2
2
 d 
2
 r sin   d 
2
2
2
Moving in curved coordinates
•The geodesic principle works in any coordinate system
An object with no forces acting on it will always follow a geodesic,
which is the longest proper time path between two points in spacetime
•It is possible, starting from the metric, to find equations that describe geodesic motion
c d  c  dt    dr   r
2
2
2
2
x    t , r , ,  




1
2
g



2
g

2
 d 
2
  g  
 r sin   d 
2
2
1
 g g g 
   a   
x
x 
 x
g 
 c2

0


0

0
a



d 2 x
dx
dx





 
2
d
 
 d  d
2
0
0
1 0
2
0 r
0

0

0





2
2
r sin  
0
0
0
Moving in curved coordinates (2)
•The effects of moving in curved coordinates looks like acceleration
d r
 d 
 d 
2
 r
  r sin  

2
d
 d 
 d 
2
2
2
y
•But it really isn’t
•It is moving as straight as it can in curved
coordinates
•You can always eliminate this apparent acceleration,
simply by returning to “flat” coordinates
•It is possible, starting from the metric alone, to
prove that spacetime is really just flat
x
Curved Space
•It is possible to live in a spacetime that is inherently curved
•It’s not just the coordinates, it’s spacetime itself that is curved
•Consider the surface of the Earth
•Think of it as a 2D object
Imagine two explorers setting out from the North Pole in
“straight lines” (geodesics)
•At first they are traveling away from each other
•When they reach the equator, they will be traveling
“parallel” to each other; their distance is no longer
increasing
•They then start traveling towards each other
•They meet at the south pole
•The curvature is real
•It is space itself that is curved, not the coordinates only
ds  a  d   a sin   d 
2
2
2
2
2
2
Curvature
•How can we tell, looking only at the distance formula (the metric), if the curvature is
real or a consequence of our coordinate choice?
•The Riemann Tensor tells you if it is curved
R










            
x
x


•If the Riemann tensor is zero, space is not curved; if it is non-zero, it is curved
•Changing coordinates doesn’t make it go away
Some other measures of curvature:
•The Ricci tensor: R 
R




R

R
g
•The Ricci scalar:
 

•The Einstein tensor:

G  R  12 g R
The Stress-Energy Tensor
V
What causes gravity? Energy and momentum
•The presence of matter, or mass density, is the cause of gravity
E
•Mass density is proportional to energy density
u
•If energy makes a difference, why not momentum as well?
V
•Momentum density also contributes to gravity
P
g
The flow of energy and momentum also causes gravity
V
•Another way of looking at momentum density is the transfer of energy
P
•It is like power flowing through an area P  A  g
F
•You can also transmit momentum across a boundary
•This is what forces do
F   A
 u gx g y gz 
•Pressure is a good example


•A more general example is called the stress tensor
g



x
xx
xy
xz 

•Gravitational effects in general relativity are
T 
 g y  yx  yy  yz 
caused by the energy density u, the momentum


density vector g, and the stress tensor
g 


A

z
zx
zy
zz

Einstein’s Equations
The central idea of Einstein:
•Gravity looks just like acceleration, except
•When you have a source of gravity, parallel doesn’t remain parallel
•This tells you gravity has to do
m
a
with curvature
M
m
•The stress-energy tensor Tµ must be related to the curvature
•Einstein found a relationship that worked, now called Einstein’s equations
G
8 G
 4 T
c
•It may look simple, but it isn’t
•This is really 16 equations (since  and  each take on 4 values)
•The expression on the left contains hundreds of terms
•It is highly non-linear
Geodesics in curved spacetime
•Why do particles curve under the influence of gravity?
•Because spacetime itself is curved!
An object with no non-gravitational forces acting on it will always follow a
geodesic, which is the longest proper time path between two points in spacetime
G
8 G
 4 T
c
a



d 2 x
dx
dx





 
2
d
d

 

 d

0

Matter tells space how to curve, and
space tells matter how to move
The Schwarzschild Solution
•Assume you have a spherically symmetric source of gravity
•Doesn’t matter how it’s distributed
•You have to be outside it
•The solution you get for the Metric is called the Schwarzschild Solution
M
dr 

2
2
2
 2GM 
2
2
2
c d  c 1  2   dt  
 r  d   r sin   d 
2GM
cr 

1 2
cr
2
2
2
2
•If you let M = 0, you get the same solution as before (flat spacetime)
•The factors of 1-2GM/c2r are the only change
•The change in the radial term (dr2) tells you that the relationship between radius and
circumference isn’t the obvious one
•C  2r !
•The dt2 term tells you that time runs more slowly when you are near a mass!
Time slows down in proximity to a mass!
Gravitational time effects
•Suppose you are standing still (dr = d = d = 0) near a mass
•Time runs more slowly
2GM
2


c d  c 1  2   dt 
cr 

2
2GM
d  dt 1  2
cr
2
2
2GM
•This can be measured directly by comparing atomic   t 1  2
cr
clocks in airplanes vs. atomic clocks on the ground
M
f0, 0
f, 
•Next generation atomic clocks: measure which floor you are on
2GM
•GPS wouldn’t even work if they didn’t take GR into account f  f 1 
0
2
c
r
This has an important effect when studying spectral lines from
high mass/small radius stars
0

•The frequency f we observe away from the star’s gravitational
2GM
field is increased compared to f0
1 2
•Since wavelength is inversely proportional to frequency, the
cr
wavelength observed is longer, “red shifted”
Gravitational Forces
•Why do objects accelerate in a gravitational field?
An object with no non-gravitational forces acting on it will
always follow a geodesic, which is the longest proper time path
between two points in spacetime
•I want to toss a ball to myself – release it and catch it one
seconds later, at the same place
•What path should I take
•Goal: maximize the proper time the ball experiences
•Strategy #1 – hope it levitates in place
•Bad – it is stuck near the Earth, where time runs slowly
•Strategy #2 – toss it very high in the air
•Bad – high speed causes time to slow down too much
•Strategy #3 – toss it moderately fast into the air
•Good – a compromise – it gains effect of going away from
Earth without too much speed
time fast
time
medium
time
slow
Gravitational Deflection of Light
•Gravity affects all objects, because it is a curvature of spacetime
•The effect is small, for the Sun, and almost
impossible to measure except from space or during
eclipses
•For objects like galactic clusters, the effect is
large and can be used to measure the mass of the
cluster
Black Holes
dr 

2
2
2
 2GM 
2
2
2
c d  c 1  2   dt  
 r  d   r sin   d 
2GM
cr 

1 2
cr
2GM
RS  2
•There is a radius where the equation looks like it goes crazy
c
•Called the Schwarzschild radius
2
2
2
2
•This radius is inside the physical radius of the Earth, Sun, and all planets in the solar
system
•About 3 km for the Sun
•The formulas are no longer valid here
•If you managed to squeeze the Sun to this radius, something remarkable would happen
•The Sun would become a black hole
Precession of the Perihelion
Mercury
•Newtonian gravity and General Relativity have almost the same
predictions
•Some slight discrepancies
•Newtonian gravity says planets orbit the Sun in ellipses
•General relativity says the direction of the ellipse slowly changes
with time
•This effect is called precession of the perihelion
•Depends on the speed
•Discovered, but not explained, before Einstein
•For Mercury, it works out to 43 arc-seconds per century
•This effect has since been seen in other systems
•Spacecraft
•Orbiting pairs of neutron stars
Sun
Gravity Probe B
•Four superaccurate gyroscopes that orbited the Earth from 2004-2005
•Accurate tests of special and general relativity
gyroscope
Two effects it is looking for:
•As the object orbits, its motion and
Earth’s gravity combine to make the direction
of the axis tilt in the direction of motion
•This motion is called the “geodetic effect”
•April 2007: effect matches expectation at the 1% level
•September 2009: effect matches expectation at the 0.1% level
•The Earth is simultaneously rotating on ITS axis
•It drags spacetime along with it.
•This effect is called “frame dragging”
•September 2009: effect matches expectation at the 15% level
Earth
Gravity Probe B
Gravitational Waves
•Distortions of spacetime are produced by masses
•Just like electromagnetic fields are produced by charges
•If you have accelerating masses, you can make gravity waves
•Just like accelerating charges make electromagnetic waves
•These distortions affect everything that feels gravitational forces
•i.e. everything
•These effects are truly tiny
•Motions ~ 0.01 fm
•Several gravity wave detectors are
looking for these effects
•No positive results yet
Gravitational Wave Detection
•The direct way to detect gravity waves is via interferometry
•Make light interfere after going along two LONG paths
•Extremely sensitive method
•LIGO – Laser Interferometer
Gravity Wave Observatory
•Livingston, Louisiana
•Hanford Washington
•Indirect detection –
•If two objects are circling each other, they must
lose energy
•They must move closer
•Frequency of orbit
increases
•Detected in Pulsar pair
PSR 1913+16
Hanford, Washington