1 Problems with the Schwarzschild Metric

Transcription

1 Poblems with the Schwazschild Metic The spacetime outside of a non-otating sta (o planet o whateve) of total mass m is descibed by the metic tenso ds 2 = dt 2 + d d 2 This metic solves the vacuum Einstein equations and, accoding to Bikho s Theoem, is the only spheically symmetic metic that does. It obviously has a poblem at =. One metic coe cient blows up while anothe goes to zeo. Conside a clock that is holding its position at a constant value of the adial coodinate. If it does this fo an inteval t of coodinate time, then the time elapsed on the clock will be = t Fo nea vey little pope time will elapse on the clock even though a geat deal of coodinate time elapses. This esult tells us that the t = constant hypesufaces ae failing to advance in time nea =. Duing the ealy days of geneal elativity, the poblem with the Schwazschild solution was egaded mostly as a cuiosity of little consequence because, fo nomal astonomical objects such as the sun and the eath, the citical value of the adius is extemely small. Fo an object with the mass of the eath, the citical adius is about a centimete. Fo the sun, it is a kilomete. The metic inside an odinay sta is not given by the Schwazchild vacuum solution and is quite egula eveywhee. Fo the extenal vacuum spacetime to extend to the citical adius, the sun would have to be compessed to a adius of a kilomete. Until 939, (Oppenheime and Volko, Oppenheime and Snyde) no astophysicist seiously believed that a mass equal to that of the sun could be compessed into an object only a kilomete in adius. Not too many believed it afte that eithe. Fom a mathematical point of view, the coodinate-independent natue of spacetime geomety was not well undestood duing the ealy days, so it was some time befoe someone consideed the possibility that the = singulaity was simply the esult of bad coodinates. One indication that this might be the case is that none of the scala cuvatue invaiants such as R; R R and invaiant atios of lightlike components of cuvatues become in nite at =. 2 The Kuskal Extension 2. Othogonal Suface Metic Once it is ealized that the = singulaity is eally just a poblem with the coodinates, it is faily easy to x. The angle coodinates ; ' ae xed by the

2 spheical symmety goup, so nothing could be wong with them. That leaves the coodinates ; t and the metic 2 ds 2 = fdt 2 + f d 2 ; f = on the timelike two-sufaces othogonal to the goup obits. We need to nd new coodinates in which this metic looks moe egula. An obvious geometical featue of this two dimensional metic is the light cone. On any such suface, it is always possible to nd coodinates U; V such that cuves at constant U ae lightlike and so ae cuves at constant V. Fo example, in two dimensional Minkowski spacetime, one can take U = t+; V = t and obtain the metic in the fom 2 ds 2 = dudv. In geneal, so long as the metic takes the fom 2 ds 2 = dudv then an inteval with eithe U = 0 o V = 0 will be lightlike. Thus, we seek coodinates U; V such that fdt 2 + f d 2 = dudv fo some function. Once we nd such coodinates, we can constuct space and time coodinates u = U V; v = U + V in which the light cones look exactly like the ones in Minkowski spacetime. If thee is any coodinate system in which this metic tenso is egula, then this has to be the one. 2.2 Conditions on Advanced and Retaded Time Coodinates Repesent patial deivatives with espect to t; by subscipts so that du = U d + U t dt; dv = V d + V t dt and theefoe fdt 2 + f d 2 = (U d + U t dt) (V d + V t dt) 2

3 o U t V t = f; U V = f U V t + U t V = 0 Divide all of these equations by U V and obtain the esults U t U V t V = f 2 ; V t V + U t U = 0; which can be solved fo the two atios in the fom U t U = f; V t V = f: Thus, if the coodinates exist, then they must satisfy these two conditions. Convesely, the agument can be evesed to show that solving these two conditions is enough to put the spacetime metic into the desied fom whee the confomal facto can be found fom the equation U t V t = f. In moe explicit fom the conditions to be @ 2.3 Solving the Conditions: Totoise Coodinate To see what to do next, ecall what these conditions would look like fo null coodinates in @ ; which would yield the esult that U depends only on t + and V depends only on t. To get the conditions into this fom, we need a new adial coodinate @ Fom the chain ule fo patial deivatives, so we have o which can be e-witten @ = + 3

4 and integated diectly: = + ln j j This coodinate has been called the totoise coodinate afte Zeno s paadox and the way that the coodinate ceeps up on the value = ; which is achieved only as!. Fo lage values of, the new adius coodinate becomes quite simila to : In tems of the totoise coodinate, we have the conditions which @ ; U = U (t + ) ; V = V (t ) : We still have two functions, U (x) ; V (x) of a single vaiable to choose. One potential di culty needs to be anticipated at this point. The totoise coodinate is not continuous acoss the bounday at =. When we get the nal fom of the metic, we need to be sue that it is egula thee. 2.4 Regulaizing the Metic Now all we have to do is nd the confomal facto and see if thee is any way to choose the functions U; V so that the singulaity at = goes away. Use pimes to denote deivatives of functions of a single vaiable and wite the condition that is to be solved fo U t V t = f U U (t + ) = U 0 (t + ) U 0 V 0 = = : Fo this to wok, the confomal facto must not vanish at = o we will still have a singulaity thee. Thus, the functions U; V must be chosen so that they poduce a facto of. This can be aanged because and U = U (t + ) = U (t + + ln j j) V = V (t ) = V (t ln j j) depend explicitly on. To pull out the ove-all facto of that we need, choose the functions to be exponentials U (x) = Ae + x 4m ; V (x) = Be x 4m : 4

5 The constants A and B will be chosen late. The condition to be solved fo the confomal facto is then AB 4m e t+ 4m 4m e t 4m = o AB 6m 2 e = o AB 6m 2 e + lnj j = AB 6m 2 e +lnj j = o AB j j = 6m2 e : Now we see why the constants A and B have been caied along. They can each be eithe + o but thei de nitions must switch as the = bounday is cossed so that which yields the nal esult AB j j = ; = 6m2 e and puts the obit-othogonal 2-metic into the fom 2 ds 2 = 6m2 e dudv; which no longe has any tace of iegulaity at = : It emains singula at = 0, howeve. 2.5 Undestanding Kuskal s Coodinates The lightlike Kuskal coodinates U and V ae elated to the usual Schwazschild coodinates by U = Ae 4m (t+) ; V = Be 4m (t ) so that UV = ABe = ABe (+ lnj j) = ABe 4m j j o and UV = e 4m ( ) V U = B A e t 5

6 All of these elationships have been deived fo the egion of spacetime whee the Schwazchild coodinates make sense. Within this egion we have U > 0; V < 0 with the = singulaity happening at U = V = 0. Fo the de nitions of U and V to wok in this egion, we need A = ; B =. Since > in this egion, we con m that the elation AB j j = woks thee. Now switch to the space and time coodinates de ned by U = v + u; V = v u fo which the spacetime metic takes the fom 2 ds 2 = 6m 2 e dv 2 + du 2 : In these coodinates the sufaces of constant ae given by v 2 u 2 = e 4m ( ) ; which implicitly de nes (u; v). Similaly, the constant t sufaces ae given by v u v + u = A B e t The egion, within which we have deived the new coodinate system, coesponds to a wedge of the full u; v plane: : v > u; v > u o equivalently u > jvj : The bounday of this egion at u = jvj coesponds to =. Within this egion, sufaces of constant > coespond to hypebolas. Sufaces of constant time coespond to staight lines though the oigin with t = at 6

7 u = v and t = + at u = +v. The natue of the coodinate poblem at = can now be seen vey clealy. The Schwazchild time coodinate t fails to advance at that point. 2.6 Extending the Solution Now conside the othe quadants of the Kuskal plane. In the uppe quadant, U > 0; V > 0 so that the constants A and B can each be chosen to be +. The sign of is negative, so we con m that AB j j = in this egion as equied. Similaly, the equied signs wok out in each of the othe quadants. Since nothing goes wong with the spacetime metic at the = boundaies of the coodinates that we ae using, we can simply use the whole solution, fo all values of u; v. The only place whee something still goes wong is at = 0 whee the timelike 2-suface metic is still egula, but the goup obits have zeo aea. A map of the full solution in Kuskal coodinates is called the Kuskal diagam. The egion between the two hypebolas is the maximal extension of the Schwazschild metic. The Schwazschild metic itself coesponds to the ight-hand quadant of the map. To use this diagam, note that light ays move along 45 lines so that the allowed wold-lines of paticles always move upwad at less than 45 to the vetical. 7

8 The diagonal lines in the pictue coespond to =. Light signals fom the ight-hand quadant of the pictue can each the outside univese. Howeve, if an object s wold-line caies it acoss the = suface, then any light signals that it might emit will hit the singulaity and neve get out. It is impotant to emembe that the Kuskal solution is the maximal extension of the vacuum solution. Most physical situations will use only pat of it as an exteio solution, which is matched to an inteio solution with matte souces. Fo a nomal stable object such as the Eath o the Sun, only a pat of the ight-hand quadant is used and the = suface neve appeas. A collapsing sta uses moe of the solution because its suface passes though = and uncoves a bit of the uppe quadant. Even in that exteme case, howeve, the bottom quadant and the left-hand quadant do not appea. Thee is no known situation that equies the use of the bottom quadant of the Kuskal extension. Such a situation would be extemely awkwad because signals could tavel fom the initial singulaity to the outside univese. Because no initial conditions can be de ned at the singulaity, anything at all could come out of it and the entie univese would become completely unpedictable. The nal singulaity in the Kuskal extension does not pose such a poblem because it is cloaked by the = event hoizon. A singulaity that is not cloaked by a hoizon in this way is called a naked singulaity. The Cosmic Censoship Conjectue assets that natue always manages to avoid ceating naked singulaities. 8

9 3 Black Holes 3. A little astophysics Stas spend most of thei lives fusing hydogen to make helium. In smalle stas such as ou Sun, the fusion eaction takes place only at the cente of the sta. When the hydogen fuel uns out at the cente, the nuclea e buns outwad and causes the suface of the sta to swell up by pehaps a facto of a hunded. The sta then becomes a ed giant. Fo ou sun, the hydogen fuel will eventually un out and the ed-giant stage will be followed by a cooling white dwaf sta, held up by the fact that electons obey an exclusion pinciple that causes them to esist being foced into the same states. Usually the collapse to a white dwaf is accompanied by the elease of a last bust of enegy that blows away pat of the sta s atmosphee as an expanding planetay nebula such as the Helix nebula below. The newbon white dwaf sta can be seen at the cente of the expanding cloud. Hundeds of newbon white dwafs ae seen in the sky. Because the expanding cloud dissipates in just a few thousand yeas, thee ae many moe olde white dwaf stas that have lost thei nebulae. Stas much moe massive than ou sun go though a moe elaboate seies of stages but still end up in a nal collapse. Often, they have too much mass to be stable white dwafs and collapse until they ae just a few kilometes in diamete. The electons no longe suppot the sta because they no longe exist. All of the electons have been absobed into neutons though invese beta decay and it is the exclusion pinciple fo neutons that holds up the sta. The gavitational potential enegy eleased when the sta collapses to such a small size is compaable to all of the nuclea enegy eleased by the sta duing its entie lifetime. The sudden elease of such a lage amount of enegy causes much of the sta to be blown away in a supenova explosion. Hee is an image 9

10 of a supenova that ocued in 987: Eventually, all that is left of the sta is an expanding cloud of gas with a apidly otating neuton sta at its coe. In some cases the neuton sta dags its magnetic eld though the neaby nebula, acceleating paticles and geneating enough enegy to light up the whole nebula. The Cab nebula is an example of such a system. Thee ae some tuly massive stas with 50 to 00 times the mass of ou sun. Such supe-massive stas ae vey unstable, as the case of eta cainae shown 0

11 hee demonstates: when stas of this sot nally un out of ewoks, the mass left behind can be too much fo even a neuton sta and the suface aea of the sta dops below 4 () 2. The suface of the sta is then inside the = limit of the Kuskal metic and all timelike cuves lead towad smalle values of and thus smalle values fo the suface aea. The sta is then doomed to nal collapse, no matte what physical pocesses go on inside it. 3.2 Ae black holes eally out thee? Bie y: Yes. An isolated black hole would be impossible to nd since it is only a few miles acoss and, of couse, it is black because it absobs all light that hits it. Howeve, matte that falls into a black hole eleases enomous amounts of enegy. Fo example, matte falling into a sola-mass size hole would fall into a gavitational eld identical to that of the sun, a distance of a half million miles close to the cente of the eld than the suface of the sun, with the gavitational eld inceasing the whole way. Figue out the amount of wok a given mass m could do if it is loweed gadually into a black hole at the end of a sting. The answe tuns out to be mc 2. A black hole has the potential to convet mass into enegy with 00% e ciency. It is geneally believed that the most violent pocesses in the univese ae poweed by matte acceting onto black holes. When a black hole is consuming matte, the matte usually stats out with some angula momentum so, instead of going staight into the hole, it foms

12 an accetion disk aound the hole. In such a disk, the faste inne ing usually couples to the slowe oute ing (often though embedded magnetic elds) and tansfes the angula momentum outwad until it is lost in a adial plume of ejected matte. The gavitational eld of the black hole is not Newtonian and, fo matte that comes too close, thee ae no longe any stable obits. As the inne ing of matte is slowed, it dives into the hole. The enegy eleased by that matte s descent into the hole s gavitational eld is eleased as X-ays. Meanwhile, the twisting magnetic eld geneated by the otating accetion disk causes some matte to escape along the otation axis of the system and be focused into jets. Such a jet can be seen in some detail in the pictues of the galaxy M87 below: The black hole esponsible fo jets such as the one seen in M87 ae thought to have masses that ae millions of times the mass of ou sun. It has long been believed that a giant black hole of this sot exists at the centes of many galaxies, including ou own. 4 Womholes 4. The Einstein-Rosen Bidge Suppose that we somehow have a spacetime that is the full Kuskal extension of the Schwatzchild vacuum solution. Conside the space geomety at the instant of time symmety, de ned in Kuskal coodinates by v = 0. Specializing the 2

13 implicit de nition of the Schwatzchild luminosity adius coodinate to this case gives u 2 = e 4m ( ) : Fo the dimensionless coodinates we have the elation k = u p ; s = k 2 = e s 2 (s ) ; which, when plotted, shows that the luminosity adius goes though a minimum value of = at k = 0 o u = 0. / = The luminosity adius goes to in nity, coesponding to being fa fom the souce of the gavitational eld when u inceases to in nity. It does exactly the same thing when u deceases to minus in nity. Thus, we have two outside egions. Recall that the luminosity coodinate is de ned in tems of the aea of the spheical goup obits. Thus, the two outside egions ae connected by a sphee of minimum aea, called the womhole thoat. To make a pictue of this stuctue, take the dimensionless luminosity adius s to be a adius given by s = p x 2 + y 2 and plot the esulting cicles as a function of the educed Kuskal coodinate, k. The esulting suface is de ned by k = e p x 2 +y p 2 2 x2 + y 2 : Tun this into a paameteized plot. Fo this to wok, you need to st de ne a function K by p e 2 ( ) > K () = 0 and then fom the coodinate vecto [ cos ; sin ; K ()] 3

14 Select ComputejPlot 3Djectangula with the cuso anywhee in this vecto. Click on the esulting pictue and then on the box in the lowe ight cone in ode to adjust the paamete anges set to > 0; and 0 < 2. The esulting plot is Womhole Thoat If you ae viewing this text with Scienti c Notebook o Scienti c Wokplace, you can use you mouse to otate the pictue and view it fom di eent angles (double click and dag). 4.2 Womholes?? An Einstein-Rosen Bidge can be thought of as connecting two di eent univeses, each of which looks like at space-time fa fom the bidge. Imagine cutting the bidge and looking at each of these two univeses. To make a pictue, de ne the function q () = 4 > 0:5 6 0:5 4

15 and do a 3D plot of the suface twice [ cos ; sin ; q ()] Because these two univeses ae so close to at spacetime at geat distances, we only have to bend them a little to t them togethe and have them both be the same univese with two widely sepaated holes like this: cos ; sin ; q q ( cos 3) sin 2 + q q ( cos + 3) sin 2 Now we have two moe loose edges to take cae of, namely the two thoats. They tted togethe befoe, so they will suely still match up now. Identify the two thoats so that anything going down into one comes out the othe. The esult is an agument that Einstein s eld equations should pemit a spacetime with a kind of built-in subway that lets you go fom one pat of the univese to anothe without tavesing the odinay space and time between. 5

16 4.3 Intestella supehighways? Bie y: No. Nea the womhole thoat, the spacetime geomety should be almost exactly the same as fo an Einstein-Rosen Bidge and will be descibed by the Kuskal extended vacuum solution. Fo two ships passing though the womhole, we would daw a diagam like this: The ship that makes it though without hitting the singulaity and getting cushed to in nite density is moving on a faste-than light tajectoy. The othe ship has obviously su eed a failue of its ftl dive system and is in big touble. A faste-than-light dive is even moe poblematic than womholes, so these things might not be all that useful. A wose poblem can also be seen in the diagam: The initial singulaity is uncoveed (a naked singulaity) and can send completely unpedictable things into the egion of the thoat. If the initial singulaity is coveed by the suface of a collapsing sta, then that sta will block the thoat. Moe complex sots of womholes ae possible. Fo example, the gavitational eld of a otating, electically chaged sta can be extended in much the same way as the eld of an unchaged, non-otating sta. In that case, the singulaities may be avoidable but naked singulaities ae still pesent and thee is no known way to fom these things. 4.4 Then what good ae they? The fact that Einstein s theoy admits solutions like the womhole spacetimes indicates that the undelying topology of spacetime might not be simple. It also aises the question of how that topology is detemined. In Einstein s theoy one poceeds by assuming an undelying manifold with a given topology. The womhole solutions and many othe examples suggest that Einstein s eld equations can be solved fo any undelying manifold. The question then is What detemines the undelying manifold? That question has become fa moe seious now that an eve-expanding vaiety of undelying manifold stuctues is being tied in paticle and quantumgavity theoies, fom sting theoy (2 dimensions) though M-theoy (with an dimensional low enegy limit). Some theoies have exta dimensions that ae 6

17 olled up into closed manifolds of seveal sots while othe theoies have exta dimensions that ae not limited at all. The pictue on this couse s home page depicts stings that end on 4-dimensional manifolds called banes. How does one nd the undelying topological stuctue? The only answe that we have so fa is vey unsatisfactoy: One guesses an undelying stuctue and builds a theoy on it. 7