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Gravitational waves

In a weak gravitational field the space-time is almost flat and the metric tensor gab is equal to the flat metric ηab plus a small correction hab, gab = ηab+hab. The Riemann tensor to the lowest order in hab is
Rabcd = 1/2(had,bc + hbc,ad - hac,bd - hbd,ac).
If we choose coordinates such that
(hab - 1/2 ab),b=0,
the Ricci tensor is simply
Rab = -1/2hab,c,c
and the vacuum Einstein's equations turn into the ordinary wave equations
( 2/∂t2 - Δ )hab = 0.
The intensity of gravitational radiation by a system of slowly moving bodies is determined by its quadrupole moment Dαβ
- dE/dt = G/45c5 (D'''αβ)2

Schwarzschild metric

Schwarzschild metric is a static spherically symmetric solution of the vacuum Einstein's equations,
ds2 = (1 - R/r) dt2 - (1 - R/r)-1 dr2 - r2(dθ2 + sin2θ dφ2),
where R=2Gmc-2 is the gravitational (Schwarzschild) radius of the source of the field, and m is its mass.

Exercises

  1. For the Schwarzschild metric
    1. calculate the length of a circle of radius r with a center at the origin.
    2. Calculate the distance between two points, r1 and r2, on the same radius ("radius" is a (straight) line from the center outwards). Compare with r2 - r1.
    3. Write down the equations of motion of a massive test body.
    4. Write down the equation of the trajectory of a light ray.
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