1. The odd thing about gravity . . .
General Relativity
The odd thing about gravity . . .
The force of gravity is proportional to the mass of the object it acts on M F m
The acceleration is the same, no matter the mass you drop 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

2. The equivalence principle
Are there any other force formulas that are proportional to mass?
Suppose you are on a merry-go-round
Centrifugal force
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

3. The metric again
We will start by reexamining the metric – the distance formula
Or in this case, the proper time formula
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
We won’t be using this formula
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

4. Changing Coordinates
It is important to be able to change coordinates to a different system
Let’s change to spherical coordinates: (x,y,z,t)  (r,,,t)
Similarly,

5. 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

6. Moving in curved coordinates (2)
The effects of moving in curved coordinates looks like acceleration 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

7. 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

8. 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
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:
The Ricci scalar:
The Einstein tensor:

9. The Stress-Energy TensorV
What causes gravity? Energy and momentum
The presence of matter, or mass density, is the cause of gravity
Mass density is proportional to energy density
If energy makes a difference, why not momentum as well?
Momentum density also contributes to gravity
The flow of energy and momentum also causes gravity
Another way of looking at momentum density is the transfer of energy
It is like power flowing through an area
You can also transmit momentum across a boundary
This is what forces do
Pressure is a good example
A more general example is called the stress tensor P A F
Gravitational effects in general relativity are caused by the energy density u, the momentum density vector g, and the stress tensor

10. 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 with curvature M a m a m
The stress-energy tensor Tµ must be related to the curvature
Einstein found a relationship that worked, now called Einstein’s equations
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

11. Matter tells space how to curve, and space tells matter how to move
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
Matter tells space how to curve, and space tells matter how to move

12. Time slows down in proximity to a mass!
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
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!

13. Gravitational time effects
Suppose you are standing still (dr = d = d = 0) near a mass
Time runs more slowly
f0, 0
This can be measured directly by comparing atomic clocks in airplanes vs. atomic clocks on the ground
Next generation atomic clocks: measure which floor you are on
GPS wouldn’t even work if they didn’t take GR into account
f, 
This has an important effect when studying spectral lines from high mass/small radius stars
The frequency f we observe away from the star’s gravitational field is increased compared to f0
Since wavelength is inversely proportional to frequency, the wavelength observed is longer, “red shifted”

14. Gravitational Forces time fast
Why do objects accelerate in a gravitational field?
time medium
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
time slow
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

15. 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

16. Black Holes
There is a radius where the equation looks like it goes crazy
Called the Schwarzschild radius
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

17. Precession of the Perihelion
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
Mercury
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

18. Gravity Probe B
Four superaccurate gyroscopes that orbited the Earth from
Accurate tests of special and general relativity
Earth
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

19. Gravity Probe B

20. 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

21. 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
Hanford, Washington