What is the gravitational redshift of light emitted from the surface of a neutron star as observed by a faraway observer? A neutron star has typically a mass of 1.2 solar masses and a radius of about 20km. 10.13. The Reissner-Nordstr(cid:246)m repulsion Consider a radially infalling neutral particle in the Reissner-Nordstr(cid:246)m spaceQ time with M > . Show that when the particle comes inside the radius j j r = Q2=M it will feel a repulsion away from r = 0 (i.e. that d2r=d 2 < 0 for the proper time of the particle). Is this inside or outside the outer horizon r+? Show further that the particle can never reach the singularity at r = 0. 10.14. Light-like geodesics in the Reissner-Nordstr(cid:246)m spacetime We will in this problem consider radial photon paths in the Reissner-Nordstr(cid:246)m Q2, and spacetime. The horizons of this spacetime are at r we will assume that M > = M M 2 Q . j j

274) where dr r + Q2 2M r = : 1 Z r2 Show that u is a constant of motion for outgoing photons, while v is a constant of motion for ingoing photons. Show further that r = r + r2 + r+ r ln 1 r 2M j r+j + r2 r+ r ln 1 r 2M j r j :(10.275) (c) Draw the light-cones in the tr-plane for the three regions r < r r < r+ and r+ < r. , r < 255 256 The Schwarzschild Solution and Black Holes 10.15. Birkhoffs theorem We will in this problem consider a spherically symmetric metric describing the spacetime external to some region. We will (cid:2)rst assume that the metric is time dependent, but will show that, under some assumptions, that this cannot be possible. A spherically symmetric metric outside a source can always be put onto the canonical form

276) Assume also that the spacetime is asymptotically (cid:3)at; i.e. lim r !1 (r; t) = lim !1 r (r; t) = 0: (a) Outside some r0 we have T = 0. Denote the derivative r with a prime t with a dot. Show that Einsteins (cid:2)eld equations in vacuum (for and r > r0) can be written as e e 2 2 (cid:181) (cid:181) 20 r 1 r2 1 r2 20 r + 1 r2 = 0 1 r2 = 0 _ r 2 2e = 0 (10.277) (10.278) (10.279) e 2 (cid:181) 00 + 0 2 + 0 e 2 0 r + _2 00

_ _ = 0: (10.280) (b) Show that for r > r0 we have (r; t) = (r). Show also that 0 = 0, (r) for r > r0. Explain that the metric has and by integrating (r; t) = to have the form of metric (10.4). This is what is called Birkhoffs theorem: If a spacetime contains a region which is spherically symmetric, asymptotically (cid:3)at, and empty (T = 0) for r > r0, then the metric in this region is time independent and hence independent of the dynamical properties of its source. 10.16. Gravitational mass (a) Use the line-element (10.4) and show that the surface gravity of a Schwarzschild black hole can be written = e 0: (10.281) (b) Show, using Einsteins (cid:2)eld equations, that 4r2e+ T 0 0 T 1 1 T 2 2 T 3 3 = r2e 0 0 : (10.282) Hence, deduce that the surface gravity can be written = 4 r2 r Z0 T 0 0 T 1 1 T 2 2 T 3 3 e+r2dr: (10.283) Problems (c) De(cid:2)ne the gravitational mass MG inside a radius r of a spherical mass distribution by =