Introduction to General Relativity. Note 6

Introduction to General Relativity. Note « 6 »

N.B.

There will be NO teaching on week 42 (13-17/10). There WILL be teaching on week 43 (20-24/10).

On Tuesday, 21/10 we have to change to another auditorium: 520-616.

The Hilbert's action for the gravitational field The most successful action for the gravitational field is the Hilbert's action

S_g = -¹/_2κ ∫ R√(-g)dΩ ,

where κ is a constant (Einstein's constant). Its variation δS_g under a variation δg^ab can be (easily :) calculated

δS_g = - ¹/_2κ ∫(R_ab-¹/₂g_abR)δg^ab√(-g)dΩ.

The equations of the gravitational field (Einstein's equations) From the least action principle δS_g+δS_m = 0, where the variation of the matter action δS_m is

δS_m = ¹/₂ ∫T_abδg^ab√(-g)dΩ,

we find the gravitational field equations

R_ab-¹/₂g_abR = κ T_ab.

The Newton's gravity law as a limit of Einstein's equations for slow and weak gravitation fields In this limit we assume, that all velocities are small, all fields are weak. Thus, only the T₀₀=m component of the energy-momentum tensor is non-vanishing. Therefore we shall only consider the ₀₀ component of the Einstein's equation

R₀₀ = κ (T₀₀ - ¹/₂ g₀₀ T).

Now, in this limit

R₀₀ = Δφ, where g₀₀ = 1 + 2φ

and we thus get the Poisson's equation

Δφ = ¹/₂ κ μ,

which is equivalent to the Newtonian theory if we put

κ = ^8πG/_c⁴,

where G=6.67×10^-11m³kg^-1s^-2 is the Newton's gravitational constant.

Exercises

The lagrangian density for the electromagnetic field is
L = -¹/_16πF_abF^ab .
Calculate the corresponding energy-momentum tensor using
¹/₂ √(-g) T_ab = ^∂√(-g)L/_∂g^ab
Answer: T_ab = ¹/_4π (-F_acF_b^c + ¹/₄ F_cdF^cdg_ab)