Advanced general relativity by Eric Poisson

Author:Eric Poisson
Language: eng
Format: epub
Tags: General Relativity,

ds 2 = -dT 2 + (dr + ^2M/rdT) 2 + r 2 dti 2 .

ds 2 = t]ab dz A dz B = -(dz 0 ) 2 + (dz 1 ) 2 + (dz 2 ) 2 + (dz 3 ) 2 + (dz 4 ) 2 ;

z = a sinh(f/a)

z 1 = a cosh(t/a) cosx,

z 2 = a cosh(t/a) sin \ cos 6,

3.13 Problems

89

z 3 = a cosh(i/a) sin x sin 6 cos <f>, z 4 = a cosh(i/a) sin x sin 9 sin <j>, where a is a constant.

a) Compute the unit normal n A and the tangent vectors e A = dz A /dx a to

the hypersurface.

b) Compute the induced metric g a p. What is the physical significance of this

four-dimensional metric? Does it satisfy the Einstein field equations?

c) Compute the extrinsic curvature K a p. Use the Gauss-Codazzi equations

to prove that the induced Riemann tensor can be expressed as

Ra/3-yd = {da-jdpS ~ 9ad9P"/) ■

This implies that the four-dimensional hypersurface is a spacetime of constant Ricci curvature.

3. In this problem we consider a spherically symmetric space at a moment of time symmetry. We write the three-metric as

ds 2 =d£ 2 +r 2 (£) dtf,

where £ is proper distance from the centre.

a) Show that in these coordinates, the mass function introduced in Sec. 3.6.5

is given by

m(r) = r -[l-(dr/dl) 2 ].

b) Solve the constraint equations for a uniform energy density p on the hy-

persurface. Make sure to enforce the asymptotic condition r(£ — > 0) — > £, so that the three-metric is regular at the centre.

c) Prove that r(£) can be no larger than r max = >/3/87r/9.

d) Prove that 2m(r max ) = r max , and that m(r max ) is the maximum value of

the mass function.

e) What happens when £ —»7rr max ?

4. Prove the statement made toward the end of Sec. 3.7.5, that [K a t] = 0 is a sufficient condition for the regularity of the full Riemann tensor at the hypersurface S.

5. Prove that the surface stress-energy tensor of a thin shell satisfies the conservation equation

s a | 6 = -4i a L

where j a = T a pe"n' 3 . Interpret this equation physically. (Consider the case where the shell is timelike.)

6. The metric

ds 2 = -dt 2 +d£ 2 +r 2 (£)dn 2 ,

where r(£) = £ when 0 < £ < £ 0 and r(£) = 2£ 0 - £ when £ 0 < £ < 2£ 0 , describes a spacetime with closed spatial sections. (What is the volume of a hypersurface t = constant?) The spacetime is flat in both f~ (£ < £ 0 ) and y + (£ > £o), but it contains a surface layer at £ = £ 0 .

a) Calculate the surface stress-energy tensor of the thin shell. Express this in terms of a velocity field u a , a density a, and a surface pressure p.

Hyp ersurfaces

b) Consider a congruence of outgoing null geodesies in this spacetime, with

its tangent vector k a = —d a (t — I).

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