Introduction to General Relativity. Note 4

Introduction to General Relativity. Note « 4 »

Geodesic from the variational principle (d'Inverno,7.6; t'Hooft,page 29) Geodesic is the straightest possible curve in a curved space. Along this curve the integral

S=∫_A^B ds,

between two points 1 and 2 in space-time has an extremum:

δ S = 0,

Maxwell equations in the gravitation field (d'Inverno,12.5; t'Hooft,10) The first pair does not change its form

∂_aF_bc+∂_bF_ca+∂_cF_ab = 0.

In the second pair we use the covariant derivative ^D/_dx^k

^{DF^ab}/_dx^b = ^4π/_c j^a .

Equation of motion of a charged particle in the presence of both gravitational and electro-magnetic field

m^{Du^a}/_ds = e F^a_bu^b .

The Riemann curvature tensor. (d'Inverno 6.5,6.12; t'Hooft 5.) If we make a parallel transform of a vector A along an infinitesimal closed contour the vector will change if the space is curved. The change ΔA_a of the a-th component will be proportional to the vector itself and the area ΔS^bc of the surface enclosed by the contour

ΔA_a = ¹/₂ R^d_abcA_dΔS^bc .

The factor R^d_abc here is called the Riemann tensor. As we have easily calculated

R^d_abc = ∂_bΓ^d_ac-∂_cΓ^d_ab +Γ^d_ebΓ^e_ac+Γ^d_ecΓ^e_ab .

The Riemann tensor defines also the commutator of covariant derivatives

(D_aD_b-D_bD_a)A_c = R^d_abcA_d .

Exercises

Show that the first pair of Maxwell equations do not change if one substitutes the ordinary derivatives ∂_i with covariant derivatives D_i.
(non-obligatory) Try explain the twin paradox from the point of view of the accelerating observer.