General Relativity, the Schwarzschild Solution, and the Precession ...

Thomas Rudelius
September 27,2011

Some familiar examples of line elements:

These metrics have signature (3,0) and (2,0),
respectively.
In relativity, our “metric” has signature (3,1)

A particle moves in space-time along a path
that extremizes the proper time
T
For special relativity,
x

Typically, we set G = c = 1.

For sufficiently large r/sufficiently small M,
Setting
We get

So,
Expanding to first order in the velocity

In classical mechanics, particle moves to
extremize action,
Setting
We get same expression, up to negligible
constants

We could write out all the Christoffel symbols
and calculate that way.
Instead, we exploit symmetries (Killing
vectors)
Planar motion, set

If the line element is independent of a
coordinate (t and φ here), we get a conserved
quantity from taking the inner product

We have the identity,
Plugging in e and l,
We get a 1-D problem

Rearranging terms, we find
where

Compare this expression to that for
Newtonian mechanics,
Identical except for an O(r -3 ) correction,

Solving for extrema of the potential gives
Can show

http://physics.ucr.edu/~wudka/Physic
s7/Notes_www/node98.html

The precession of the perihelion of Mercury is
measured to be 5600 arc seconds per
century.
Newton’s equations, accounting for
gravitational interaction with other planets,
Mercury’s rotation, and the fact that Earth is
not an inertial reference frame, predicts 5557
arc seconds per century– off by 43.
GR accurately accounts for discrepancy.

Recall,
Yielding,

Solving to lowest order in c 2 gives,

Recall that our coordinates break down at
r=2M.
Can eliminate this by introducing new
coordinates:

No more singularity at r = 2M
Light rays travel at 45º angles in U-V diagram

Otto

Questions

Hartle, James B. Gravity: An Introduction to
Einstein’s General Relativity. San Francisco:
Addison Wesley, 2003.
McAllister, Liam. Physics 4445 Lecture Series,
2010.
“Precession of the perihelion of Mercury,”
http://physics.ucr.edu/~wudka/Physics7/Not
es_www/node98.html