General Relativity: An Introduction for Physicists · Problem 12.11

Further spherically symmetric geometries — Hobson, Efstathiou & Lasenby

Physics · cosmic-string-einstein-equations

Problem

Consider a static infinitely long cylindrical configuration of matter that is invariant to translations and Lorentz boosts along the axis of symmetry (a cosmic string). Adopting 'cylindrical polar' coordinates (ct,r,ϕ,z)(ct, r, \phi, z), show that a self-consistent solution to the Einstein field equations may be obtained if the stress-energy tensor for the matter is of the form [Tμν]=diag(ρc2,0,0,−ρc2),[T^{\mu\nu}] = \mathrm{diag}(\rho c^2, 0, 0, -\rho c^2), such that there is a negative pressure (or tension) along the string, and the line element is of the form ds2=c2 dt2−dr2−B(r) dϕ2−dz2,ds^2 = c^2\,dt^2 - dr^2 - B(r)\,d\phi^2 - dz^2, where B(r)B(r) satisfies B′′2B−(B′)24B2=−κρc2.\frac{B''}{2B} - \frac{(B')^2}{4B^2} = -\kappa\rho c^2. Show further that b(r)=B(r)b(r) = \sqrt{B(r)} satisfies b′′=−κc2ρbb'' = -\kappa c^2 \rho b. Hint: You may find your answers to Exercises 8.9, 9.28 and 9.29 useful.

Related theory

Browse the problem library