Such objects are called black holes, and there is extremely strong circumstantial evidence that they exist.

Chapte 11 Spheical black holes One of the most spectacula consequences of geneal elativity is the pediction that gavitational fields can become so stong that they can effectively tap even light. Space becomes so cuved that thee ae no paths fo light to follow fom an inteio to exteio egion. Such objects ae called black holes, and thee is extemely stong cicumstantial evidence that they exist. In this chapte we apply the Einstein theoy of gavity to the idea of black holes using the Schwazschild solution. In the next chapte we shall take a fist step in consideing how gavitational physics is alteed if the pinciples of quantum mechanics come into play Hawking black holes), In the chapte afte that we shall conside how the Schwazschild solution is modified if a black hole is assumed to possess angula momentum Ke black holes.) 369

370 CHAPTER 11. SPHERICAL BLACK HOLES 11.1 Schwazschild Black Holes Thee is an event hoizon in the Schwazschild spacetime at S = 2M, which implies a black hole inside the event hoizon escape velocity exceeds c). Place analysis on fime gound by consideing a spacecaft appoaching the event hoizon in fee fall engines off). Fo simplicity, we assume the tajectoy to be adial. Conside fom two points of view 1. Fom a vey distant point at constant distance fom the black hole pofessos sipping matinis). 2. Fom a point inside the spacecaft students). Use the Schwazschild solution metic) fo analysis.

11.1. SCHWARZSCHILD BLACK HOLES 371 11.1.1 Appoaching the Event Hoizon: Outside View We conside only adial motion. Setting dθ = dϕ = 0 in the line element ds 2 = dτ 2 = 1 2M = 1 S ) dt 2 + 1 2M ) dt 2 + 1 S ) 1 d 2 ) 1 d 2. As the spacecaft appoaches the event hoizon its velocity as viewed fom the outside in a fixed fame is v=d/dt. Light signals fom spacecaft tavel on the light cone ds 2 = 0) and thus fom the line element v= d dt = 1 S ). As viewed fom a distance outside S, the spacecaft appeas to slow as it appoaches S and eventually stops as S. Thus, the distant obseve would neve see the spacecaft coss S : its image would emain fozen at = S fo all etenity.

372 CHAPTER 11. SPHERICAL BLACK HOLES But let s examine what this means a little moe caefully. Rewite d dt = 1 ) S dt = d 1 S /. As S time between successive wave cests fo the light wave coming fom the spacecaft tends to infinity and theefoe λ ν 0 E 0. The extenal obseve not only sees the spacecaft slow apidly as it appoaches S, but the spacecaft image is obseved to stongly edshift at the same time. This behavio is just that of the coodinate time aleady seen fo a test paticle in adial fee fall: /M Pope time τ Schwazschild coodinate time t S = 2M -Time/M Theefoe, moe popely, the extenal obsevation is that the spacecaft appoaching S apidly slows and edshifts until the image fades fom view befoe the spacecaft eaches S.

11.1. SCHWARZSCHILD BLACK HOLES 373 11.1.2 Appoaching the Event Hoizon: Spacecaft View Things ae vey diffeent as viewed by the doomed) students fom the inteio of the spacecaft. The occupants will use thei own clocks measuing pope time) to gauge the passage of time. Stating fom a adial position 0 outside the event hoizon, the spacecaft will each the oigin in a pope time τ = 2 3 3/2 0 2M) 1/2. The spacecaft occupants will geneally notice no spacetime singulaity at the hoizon. Any tidal foces at the hoizon may be vey lage but will emain finite Riemann cuvatue is finite at S ). The spacecaft cosses S and eaches the eal) singulaity at = 0 in a finite amount of time, whee it would encounte infinite tidal foces Riemann cuvatue has components that become infinite at =0). The tip fom S to the singulaity is vey fast Execise): 1. Of ode 10 4 seconds fo stella-mass black holes. 2. Of ode 10 minutes fo a billion sola mass black hole.

374 CHAPTER 11. SPHERICAL BLACK HOLES 11.1.3 Lightcone Desciption of a Tip to a Black Hole It is highly instuctive to conside a lightcone desciption of a tip into a Schwazschild black hole. Assuming adial light ays dθ = dϕ = 0 }{{} adial the line element educes to ds 2 = 1 2M ) dt 2 + ds 2 = 0 }{{} light ays 1 2M ) 1 d 2 = 0. Thus the equation fo the lightcone at some local coodinate in the Schwazschild metic can be ead diectly fom the metic dt d =± 1 2M ) 1. whee The plus sign coesponds to outgoing photons inceasing with time fo > 2M) The minus sign to ingoing photons deceasing with time fo >2M) Fo lage dt d =± 1 2M ) 1. becomes equal to ±1, as fo flat spacetime. Howeve as S the fowad lightcone opening angle tends to zeo as dt/d.

11.1. SCHWARZSCHILD BLACK HOLES 375 s =2M Time 0 1 2 3 /2M Figue 11.1: Lightcone stuctue of Schwazschild spacetime. Integating dt d =± 1 2M ) 1. gives 2M ln /2M 1 + constant Ingoing) t = + 2M ln /2M 1 + constant Outgoing) The null geodesics defined by this expession ae plotted in Fig. 11.1. The tangents at the intesections of the dashed and solid lines define local lightcones coesponding to dt/d, which ae sketched at some epesentative spacetime points.

376 CHAPTER 11. SPHERICAL BLACK HOLES Singulaity woldline Obseve woldline Exteio univese D E C Singulaity Inteio of black hole Event hoizon 0 S B Spacecaft woldline A obs ta t = t0 Figue 11.2: Light cone desciption of a tip into a black hole. The woldline of a spacecaft is illustated in Fig. 11.2, stating well exteio to the black hole. The gavitational field thee is weak and the light cone has the usual symmetic appeaance. As illustated by the dotted line fom A, a light signal emitted fom the spacecaft can intesect the woldline of an obseve emaining at constant distance obs at a finite time t A > t 0. As the spacecaft falls towad the black hole on the woldline indicated the fowad light cone begins to naow since dt d =± 1 2M ) 1.

11.1. SCHWARZSCHILD BLACK HOLES 377 Singulaity woldline Obseve woldline Exteio univese D E C Singulaity Inteio of black hole Event hoizon 0 S B Spacecaft woldline A obs ta t = t0 Now, at B a light signal can intesect the extenal obseve woldline only at a distant point in the futue aow on light cone B). As the spacecaft appoaches S, the opening angle of the fowad light cone tends to zeo and a signal emitted fom the spacecaft tends towad infinite time to each the extenal obseve s woldline at obs aow on light cone C). The extenal obseve sees infinite edshift.

378 CHAPTER 11. SPHERICAL BLACK HOLES g11 + g00 = 2 M gµν /M g11 Figue 11.3: Spacelike and timelike egions fo g 00 and g 11 in the Schwazschild metic. Now conside light cones inteio to the event hoizon. Fom the stuctue of the adial and time pats of the Schwazschild metic illustated in Fig. 11.3, we obseve that d and dt evese thei chaacte at the hoizon = 2M). 1. This is because the metic coefficients g 00 and g 11 switch signs at that point. 2. Outside the event hoizon the t diection, / t, is timelike g 00 < 0) and the diection, /, is spacelike g 11 > 0). 3. Inside the event hoizon, / t is spacelike g 00 > 0) and / is timelike g 11 < 0). Thus inside the event hoizon the lightcones get otated by π/2 elative to outside space time coodinates).

11.1. SCHWARZSCHILD BLACK HOLES 379 Singulaity woldline Obseve woldline Exteio univese D E C Singulaity Inteio of black hole Event hoizon 0 S B Spacecaft woldline A obs ta t = t0 The woldline of the spacecaft descends inside S because the coodinate time deceases it is now behaving like ) and the decease in epesents the passage of time, but the pope time is continuously inceasing in this egion. Outside the hoizon is spacelike and application of enough ocket powe can evese the infall and make incease. Inside the hoizon is timelike and no application of ocket powe can evese the diection of time. Thus, the adial coodinate of the spacecaft must decease once inside the hoizon, fo the same eason that time flows into the futue in nomal expeience whateve that eason is!).

380 CHAPTER 11. SPHERICAL BLACK HOLES Singulaity woldline Obseve woldline Exteio univese D E C Singulaity Inteio of black hole Event hoizon 0 S B Spacecaft woldline A obs ta t = t0 Inside the hoizon thee ae no paths in the fowad light cone of the spacecaft that can each the extenal obseve at 0 the ight vetical axis) see the light cones labeled D and E. All timelike and null paths inside the hoizon ae bounded by the hoizon and must encounte the singulaity at = 0. This illustates succinctly the eal eason that nothing can escape the inteio of a black hole. Dynamics building a bette ocket) ae ielevant: once inside S the geomety of spacetime pemits no fowad light cones that intesect exteio egions, and no fowad light cones that can avoid the oigin.

11.1. SCHWARZSCHILD BLACK HOLES 381 Thus, thee is no escape fom the classical Schwazschild black hole once inside the event hoizon because 1. Thee ae liteally no paths in spacetime that go fom the inteio to the exteio. 2. All timelike o null paths within the hoizon lead to the singulaity at =0. But notice the adjective classical... Moe late.

382 CHAPTER 11. SPHERICAL BLACK HOLES 11.1.4 Eddington Finkelstein Coodinates The peceding discussion is illuminating but the intepetation of the esults is complicated by the behavio nea the coodinate singulaity at = 2M. In this section and the next we discuss two altenative coodinate systems that emove the coodinate singulaity at the hoizon. Although these coodinate systems have advantages fo intepeting the inteio behavio of the Schwazschild geomety, the standad coodinates emain useful fo descibing the exteio behavio because of thei simple asymptotic behavio. In the Eddington Finkelstein coodinate system a new vaiable v is intoduced though t = }{{} v 2M ln, 2M 1 new whee, t, and M have thei usual meanings in the Schwazschild metic, and θ and ϕ ae assumed to be unchanged. Fo eithe > 2M o < 2M, insetion into the standad Schwazschild line element gives Execise) ds 2 = 1 2M ) dv 2 + 2dvd+ 2 dθ 2 + 2 sin 2 θdϕ 2.

11.1. SCHWARZSCHILD BLACK HOLES 383 The Schwazschild metic expessed in these new coodinates ds 2 = 1 2M ) dv 2 + 2dvd+ 2 dθ 2 + 2 sin 2 θdϕ 2. is manifestly non-singula at = 2M The singulaity at =0 emains. Thus the singulaity at the Schwazschild adius is a coodinate singulaity that can be emoved by a new choice of coodinates. Let us conside behavio of adial light ays expessed in these coodinates. Set dθ = dϕ = 0 adial motion) Set ds 2 = 0 light ays). Then the Eddington Finkelstein line element gives 1 2M ) dv 2 + 2dvd=0.

384 CHAPTER 11. SPHERICAL BLACK HOLES v - Hoizon v - 2 + 2M ln /2M - 1 ) = constant Ingoing Outgoing v - Singulaity 4 dvdt = 0 3 2 dv = 0 Ingoing) Singulaity = 0) 1 a) b) 0 = 2M 0 = 2M Figue 11.4: a) Eddington Finkelstein coodinates fo the Schwazschild black hole with on the hoizontal axis and v on the vetical axis. Only two coodinates ae plotted, so each point coesponds to a 2-sphee of angula coodinates. b) Light cones in Eddington Finkelstein coodinates. This equation has two geneal solutions and one special solution [see Fig. 11.4a)]: 1 2M ) dv 2 + 2dvd=0. Geneal Solution 1: dv = 0, so v = constant. Ingoing light ays on tajectoies of constant v dashed lines Fig. 11.4a)).

11.1. SCHWARZSCHILD BLACK HOLES 385 v - Hoizon v - 2 + 2M ln /2M - 1 ) = constant Ingoing Outgoing v - Singulaity 4 dvdt = 0 3 2 dv = 0 Ingoing) Singulaity = 0) 1 a) b) 0 = 2M 0 = 2M Geneal Solution 2: If dv 0, then divide by dv 2 to give 1 2M ) dv 2 + 2dvd=0 dv d = 2 1 2M ) 1, which yields upon integation v 2 + 2M ln ) 2M 1 = constant. This solution changes behavio at = 2M: 1. Outgoing fo > 2M. 2. Ingoing fo < 2M deceases as v inceases). The long-dashed cuves in Fig. 11.4a) illustate both ingoing and outgoing woldlines coesponding to this solution.

386 CHAPTER 11. SPHERICAL BLACK HOLES v - Hoizon v - 2 + 2M ln /2M - 1 ) = constant Ingoing Outgoing v - Singulaity 4 dvdt = 0 3 2 dv = 0 Ingoing) Singulaity = 0) 1 a) b) 0 = 2M 0 = 2M Special Solution: In the special case that = 2M, the diffeential equation educes to 1 2M ) dv 2 + 2dvd=0 dvd=0, which coesponds to light tapped at the Schwazschild adius. The vetical solid line at =2M epesents this solution.

11.1. SCHWARZSCHILD BLACK HOLES 387 v - Hoizon v - 2 + 2M ln /2M - 1 ) = constant Ingoing Outgoing v - Singulaity 4 dvdt = 0 3 2 dv = 0 Ingoing) Singulaity = 0) 1 a) b) 0 = 2M 0 = 2M Fo evey spacetime point in Fig. 11.4a) thee ae two solutions. Fo the points labeled 1 and 2 these coespond to an ingoing and outgoing solution. Fo point 3 one solution is ingoing and one coesponds to light tapped at = S. Fo point 4 both solutions ae ingoing.

388 CHAPTER 11. SPHERICAL BLACK HOLES v - Hoizon v - 2 + 2M ln /2M - 1 ) = constant Ingoing Outgoing v - Singulaity 4 dvdt = 0 3 2 dv = 0 Ingoing) Singulaity = 0) 1 a) b) 0 = 2M 0 = 2M The two solutions passing though a point detemine the light cone stuctue at that point ight side of figue). The light cones at vaious points ae bounded by the two solutions, so they tilt inwad as deceases. The adial light ay that defines the left side of the light cone is ingoing geneal solution 1). If 2M, the adial light ay defining the ight side of the light cone coesponds to geneal solution 2. 1. These popagate outwad if > 2M. 2. Fo < 2M they popagate inwad. Fo < 2M the light cone is tilted sufficiently that no light ay can escape the singulaity at = 0. At = 2M, one light ay moves inwad; one is tapped at = 2M.

11.1. SCHWARZSCHILD BLACK HOLES 389 The hoizon may be viewed as a null suface geneated by the adial light ays that can neithe escape to infinity no fall in to the singulaity.

390 CHAPTER 11. SPHERICAL BLACK HOLES 11.1.5 Kuskal Szekees Coodinates Thee is anothe set of coodinates exhibiting no singulaity at = 2M: Kuskal Szekees coodinates. Intoduce vaiables v,u,θ,ϕ), whee θ and ϕ have thei usual meaning and the new vaiables u and v ae defined though ) 1/2 u= 2M 1 e /4M t ) cosh >2M) 4M = 1 ) 1/2 t ) e /4M sinh <2M) 2M 4M ) 1/2 t ) v= 2M 1 e /4M sinh >2M) 4M = 1 ) 1/2 e /4M t ) cosh <2M) 2M 4M The coesponding line element is ds 2 = 32M3 e /2M dv 2 + du 2 )+ 2 dθ 2 + 2 sin 2 θdϕ 2, whee = u,v) is defined though ) 2M 1 e /2M = u 2 v 2. This metic is manifestly non-singula at = 2M, but singula at =0.

11.1. SCHWARZSCHILD BLACK HOLES 391 Futue singulaity = 0) II v Hoizon = 2M, t = = 2M, t = Constant t = 0 v Futue singulaity = 0) Hoizon = 2M, t = Constant II III Past singulaity = 0) IV a) I u III Hoizon = 2M, t = Past singulaity IV = 0) b) I = 2M, t = t = 0 u Figue 11.5: a) Schwazschild spacetime in Kuskal Szekees coodinates. Only the two coodinates u and v ae displayed, so each point is eally a 2-sphee coesponding to the vaiables θ and ϕ. Spacetime singulaities ae indicated by jagged cuves. The hatched egions above and below the = 0 singulaities ae not a pat of the spacetime. Cuves of constant ae hypebolas and the dashed staight lines ae lines of constant t. b) Woldline of a paticle falling into a Schwazschild black hole in Kuskal Szekees coodinates. Kuskal diagam: lines of constant and t plotted on a u and v gid. Figue 11.5 illustates.

392 CHAPTER 11. SPHERICAL BLACK HOLES = 1.75 M t = 0 v Singulaity = 0) Hoizon = 2M, t = = 2.75 M t = 0 u Timelike woldline Fom the fom of 2M 1 ) e /2M = u 2 v 2. lines of constant ae hypebolae of constant u 2 v 2. Fom the definitions of u and v t ) v=utanh 4M u = tanht/4m) > 2M) < 2M). Thus, lines of constant t ae staight lines with slope tanht/4m) fo > 2M 1/ tanht/4m) fo < 2M.

11.1. SCHWARZSCHILD BLACK HOLES 393 = 1.75 M t = 0 v Singulaity = 0) Hoizon = 2M, t = = 2.75 M t = 0 u Timelike woldline Fo adial light ays in Kuskal Szekees coodinates dθ = dϕ = ds 2 = 0), and the line element ds 2 = 32M3 e /2M dv 2 + du 2 )+ 2 dθ 2 + 2 sin 2 θdϕ 2 yields dv = ±du: 45 degee lightcones in the uv paametes, like flat space. Ove the full ange of Kuskal Szekees coodinatesv,u,θ,ϕ), the metic component g 00 = g vv emains negative and g 11 = g uu, g 22 = g θθ, and g 33 = g ϕϕ emain positive. Theefoe, the v diection is always timelike and the u diection is always spacelike, in contast to the nomal Schwazschild coodinates whee and t switch thei chaacte at the hoizon.

394 CHAPTER 11. SPHERICAL BLACK HOLES 50 a) Schwazschild coodinates b) Kuskal-Szekees coodinates 2 45 1 t/m v 0 40-1 -2 35 0 2 4 0 1 2 3 /M u Figue 11.6: A tip to the cente of a black hole in standad Schwazschild coodinates and in Kuskal Szekees coodinates. The identification of =2M as an event hoizon is paticulaly clea in Kuskal Szekees coodinates Fig. 11.6). The light cones make 45-degee angles with the vetical and the hoizon also makes a 45-degee angle with the vetical. Thus, fo any point within the hoizon, its fowad woldline must contain the =0singulaity and cannot contain the = 2M hoizon.

11.1. SCHWARZSCHILD BLACK HOLES 395 v v - Singulaity = 0) Hoizon = 2M Extenal obseve fixed ) Singulaity = 0 Hoizon = 2 M Extenal obseve fixed ) Suface of collapsing sta u Suface of collapsing sta a) Kuskal-Szekees coodinates b) Eddington-Finkelstein coodinates Figue 11.7: Collapse to a Schwazschild black hole. Fig. 11.7 illustates a spheical mass distibution collapsing to a black hole in Kuskal Szekees and Eddington Finkelstein coodinates. A distant obseve emains at fixed and obseves light signals sent peiodically fom the suface of the collapsing sta. Light pulses, popagating on the dashed lines, aive at longe and longe intevals as measued by the outside obseve. At the hoizon, light signals take an infinite length of time to each the extenal obseve. Once the suface is inside the hoizon, no signals can each the outside obseve and the entie sta collapses to the singulaity. Note: the Schwazschild solution is valid only outside the sta. Inside GR applies but the solution is not Schwazschild.

396 CHAPTER 11. SPHERICAL BLACK HOLES 11.2 Black Hole Theoems and Conjectues In this section we summaize in a non-igoous way) a set of theoems and conjectues concening black holes. Some we have aleady used in vaious contexts. Singulaity theoems: Loosely, any gavitational collapse that poceeds fa enough esults in a spacetime singulaity. Cosmic censoship conjectue: All spacetime singulaities ae hidden by event hoizons no naked singulaities). Classical) aea incease theoem: In all classical pocesses involving hoizons, the aea of the hoizons can neve decease. Second law of black hole themodynamics: Whee quantum mechanics is impotant the classical aea incease theoem is eplaced by 1. The entopy of a black hole is popotional to the suface aea of its hoizon. 2. The total entopy of the Univese can neve decease an any pocess.

11.2. BLACK HOLE THEOREMS AND CONJECTURES 397 The no-hai theoem/conjectue: If gavitational collapse to a black hole is nealy spheical, All non-spheical pats of the mass distibution quadupole moments,... ) except angula momentum ae adiated away as gavitational waves. Hoizons eventually become stationay. A stationay black hole is chaacteized by thee numbes: * the mass M, * the angula momentum J, and * the chage Q. M, J, and Q ae all detemined by fields outside the hoizon, not by integals ove the inteio. The most geneal solution chaacteized by M, J, and Q is temed a Ke Newman black hole. Howeve, It is likely that the astophysical pocesses that could fom a black hole would neutalize any excess chage. Thus astophysical black holes ae Ke black holes the Schwazschild solution being a special case of the Ke solution fo vanishing angula momentum). The No Hai Theoem : black holes destoy all details the hai) about the matte that fomed them, leaving behind only global mass, angula momentum, and possibly chage as obsevable extenal chaacteistics.

398 CHAPTER 11. SPHERICAL BLACK HOLES Bikhoff s theoem: The Schwazschild solution is the only spheically symmetic solution of the vacuum Einstein equations. The assumptions that we made of * no time dependence and * spheical symmety in deiving the Schwazschild solution ae in fact not independent. The static assumption is, in fact, a consequence of the spheical symmety assumption. These theoems and conjectues place the mathematics of black holes on easonably fim gound. To place the physics of black holes on fim gounds, these ideas must be tested by obsevation, which we shall take up in following chaptes.