Introduction to General Relativity. Note 7

Introduction to General Relativity. Note « 7 »

Gravitational waves

In a weak gravitational field the space-time is almost flat and the metric tensor g_ab is equal to the flat metric η_ab plus a small correction h_ab, g_ab = η_ab+h_ab. The Riemann tensor to the lowest order in h_ab is

R_abcd = ¹/₂(h_ad,bc + h_bc,ad - h_ac,bd - h_bd,ac).

If we choose coordinates such that

(h^a_b - ¹/₂hδ^a_b)_,b=0,

the Ricci tensor is simply

R_ab = -¹/₂h_ab^,c_,c

and the vacuum Einstein's equations turn into the ordinary wave equations

( ^∂²/_∂t² - Δ )h_ab = 0.

The intensity of gravitational radiation by a system of slowly moving bodies is determined by its quadrupole moment D_αβ

- ^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

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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