From Gravitational Collapse to Black Holes

Transcription

1 Fom Gavitational Collapse to Black Holes T. Nguyen PHY 391 Independent Study Tem Pape Pof. S.G. Rajeev Univesity of Rocheste Decembe 0, Intoduction The pupose of this independent study is to familiaize ouselves with the fundamental concepts of geneal elativity and, in paticula, black-hole physics. In this pape, I examine the spacetime geomety of a sta collapsing into a black-hole. Towads the end of its lifetime, a sta has bunt up almost all of its fuel. As the themonuclea fusion pocess begins to slow down and elease less enegy, the sta begins to contact due to the foce of gavity. Fo stas of about the mass of the Sun, the gavitational collapse is usually halted by degeneacy pessue by the Pauli exclusion pinciple two femions cannot be at the same quantum state). Howeve, if the mass of the emnant exceeds 3 4 sola masses the initial mass of the sta exceeds 5 sola masses), in ode fo degeneacy pessue to suppot against gavitational collapse, the electons have to move faste than the speed of light. As a esult, the sta collapses into what is known as a black hole. The gavitational influence of a black hole is so stong that even light. cannot escape it. To undestand the spacetime geomety of a black hole, it is cucial that we undestand the geomety of a collapsing sta. This pape is based on my lectue fo the Kapitza Society. In Section, I will intoduce the Schwazschild geomety and use it to descibe the geomety aound a spheically symmetical black hole. In Section 3, I will examine a spheically symmetical collapse of a sta. I will also geneate the spacetime diagam using Python to descibe the Schwazschild geomety. The Schwazschild Black Hole.1 The Schwazschild Geomety In 1916, Kal Schwazschild solved Einstein s field equations and deived the spacetime geomety outside of a spheically symmetical body. The Schwazschild geomety could be used to descibe the geomety outside of a collapsing sta if we make the appoximation that 1

2 the collapse is spheically symmetic. Note that this is hadly the case fo any collapsing body because the otation of the body aound its axis will distot the shape. Howeve, if the otation angula velocity is small enough, the body is spheically symmetical to the fist ode appoximation. The line element in Schwazschild geomety outside of a body of mass M is given by: ds = 1 M ) dt + 1 M ) 1 d + dθ + sin θdφ ) Hee t is the time measued by a stationay obseve at infinity, theta and phi ae the pola and azimuthal angle, and is the adius measued as the cicumfeence, divided by π, of a sphee centeed aound the body). To simplify ou equation, we use the natual unit whee the gavitational constant G and the speed of light c ae equal to 1 G = c = 1). The coesponding metic is: g αβ = 1 M/) M/) sin θ ) The Schwazschild geomety beaks down at = 0 and = M, the latte called the Schwazschild adius. These ae known as singulaities. The singulaity at = M is a coodinate singulaity it can be emoved by changing the coodinate system), while the singulaity at = 0 is a physical singulaity it cannot be emoved by changing the coodinate system and indicates some special popeties of spacetime).. Eddington-Finkelstein Coodinates When a sta collapses, its adius appoaches the Schwazschild adius. In such cases, the Schwazschild coodinate system faces many difficulties in descibing the spacetime geomety due to the singulaity at = M and the change in the signs of g tt and g in Eq.. Howeve, because = M is a coodinate singula, we may be able to ovecome it by defining a diffeent paameteization fo the Schwazschild geomety. We define a paamete v such that: t = v M log M 1 3) Expessing dt in tems of dv and d, the line element in Eq. 1 becomes: dt = dv + 1 M ) d 1 M ) 1 dvd 4) ds = 1 M ) dv + dvd + dθ + sin θdφ ) 5) This is known as the Eddington-Finkelstein coodinates. The Eddington-Finkelstein coodinates ae especially useful fo studying gavitational collapse because the singulaity at

3 = M vanishes i.e. it connects smoothly the spacetime egions inside and outside this adius). On the contay, the singula at = 0 does not vanish, indicating that thee is some special popeties of spacetime at this adius. Fo the study of gavitational collapse, we only conside the case whee > 0. The coesponding metic of the Eddington-Finkelstein coodinate v,, θ, φ) is: g αβ = 1 M/ sin θ 6) Unlike the Schwazschild metic, the non-diagonal tems g v and g v do not vanish. On the othe hand, like the Schwazschild metic, at lage, the metic is appoximately flat and t v fo the logaithm tem in Eq. 3 vanishes..3 Light cones in Eddington-Finkelstein coodinates In geneal elativity, to undestand a spacetime geomety, we have to examine the behavio of a light ay in said geomety. In this section, we solve fo the equation of motion of a light ay descibed by the Eddington-Finkelstein coodinates. Conside a adial light ay. By definition, the wold line of the light ay will have dθ = dφ = 0 adial) and d = 0 null inteval). The line element in Eq. 5 gives the diffeential equation; ds = 1 M ) dv + dvd = 0 7) The fist and most obvious solution is dv = 0, o the light tavels along the cuve: v = const o t + + M log M 1 = const 8) Fo < M and > M, deceases as t inceases. Fo an obseve as infinity, this is thus an equation of an ingoing adial light ay. The second solution can be obtained by solving the educed equation: 1 M dv = v log ) dv + d = 0 9) 1 M ) 1 d 10) M 1 = const o t M log M 1 = const 11) When > M, the solution coesponds to an outgoing light ay, because inceases as t inceases. On the othe hand, when < M, the solution coesponds to an ingoing light ay, because deceases as t inceases. As a esult, we see that light behaves diffeently inside and outside of the adius = M. When = M, the dv tem in Eq. 9 vanishes. We obtain the thid solution to the diffeential equation d = 0, o = M. 3

4 By solving fo the equation of motion of a adial light ay, we find the egions inside and outside of the adius = M have diffeent physical popeties. To futhe examine these diffeences, we examine the spacetime diagam. We plot t = v vesus. As biefly mentioned above, t = v is the time measued by an obseve infinitely fa fom the body. In Fig. 1, each dotted blue line coesponds to a possible solution of v = const, while each blue line coesponds to a possible solution of Eq. 11. The black vetical line coesponds to the solution = M, and the bold vetical line at = 0 coesponds to the physical singulaity. At each point in the spacetime diagam, a futue light cone is defined by a Figue 1: The spacetime diagam of the Schwazschild geomety in the Eddington-Finkelstein coodinates. As the adius deceases, the futue light cone gets inceasingly tipped towads the singulaity at = 0. At M, all timelike geodesics will head towads the singulaity. At = M, a null geodesic allows fo light to obit the black hole in a cicula motion. solution of v = const and a solution of Eq. 11. In Fig. 1, thee light cones ae shown at the intesections. As the adius deceases, the light cone is inceasingly tipped towads the singulaity at = 0. At the bounday = M, one adial light ay points towads the singulaity, while anothe obits the black hole in a cicula obit. At M, all futue light cones will point towads the singulaity..4 The Event Hoizon Because at each point in spacetime, no object with mass may move outside of the futue light cone as they would be moving faste than the speed of light), any object falling pass 4

5 = M has to move faste than the speed of light to avoid being sucked into the singulaity! This implies no infomation inside = M can be obtained, and thus egions inside and outside this bounday ae physically disconnected! The bounday = M is called the event hoizon of the black hole because it divides spacetime into two egions with diffeent physical popeties. It is geneated by light ays that neithe escape to infinity o fall into the singulaity. Outside the event hoizon, light ays may escape into infinity. Inside the event hoizon, light ays can no longe escape and will always fall into the singulaity. Hence, physicists name these objects black holes. It is almost) impossible to detect a black hole via light-based telescope 1 ; its location can only be infeed fom its massive gavitational influence on neaby stas o the emission gavitational waves. Conside in the case of = const d = 0), the line element becomes: ds = 1 M ) dv + dθ + sin θdφ ) = { + 1 M 1 M dv + dθ + sin θdφ ) > 0, fo < M dv + dθ + sin θdφ ) > 0, fo > M When < M, the tem g vv in Eq. 6 is positive, so the line element is spacelike. In contast, when > M, the line element is timelike. This is also evident fom Fig. 1. The vetical line = const is always inside the local light cone at > M. Howeve, at < M, the local light cone is tipped such that the vetical line is always lying outside of the cone. At the event hoizon, the vetical line contibutes to one of the two null cuves. Outside = M, = const is a space in spacetime, while inside = M, = const is a time in spacetime. Geometically, once we coss the event hoizon, space and time tades place with each othe! At any time t dt = 0), the suface of the event hoizon of a Schwazschild black hole is descibed by dσ = dθ + sin θdφ ) = M) dθ + sin θdφ 4) The aea of the event hoizon thus given by 13) A suface = 16πM 15) This aea is time-independent and can only be changed by alteing the mass of the black hole. Because nothing can escape the black hole, the mass, and by extension, the aea of the event hoizon, cannot classically decease 1. 3 Gavitational Collapse 3.1 The Collapsing Dust Sphee When a sta collapses due to its own gavity, the mateials inside the suface of the sta get compessed and heated up. As the density and tempeatue inceases, adiation pessue, 1 A black hole can still be detected by Hawking adiation. Theoetically pedicted by Stephen Hawking in 1974, it is the adiation due to the captue of a spontaneous paticle-antipaticle pai just outside of the event hoizon. Because a paticle o antipaticle escapes via Hawking adiation, it is esponsible fo the loss in mass and enegy of a black hole. 5

6 which is popotional to the fouth powe of tempeatue, goes up damatically and pushes the mateials outwad, counteacting gavitational collapse. Futhemoe, in some cases, the incease in density, tempeatue, and pessue may also e-ignite nuclea fusion and polong the life of the sta. Fo the pupose of ou study, we will ignoe these effects and conside the gavitational collapse of a sphee of non-inteacting, pessueless dust. Because the Schwazschild geomety descibes only spacetime outside a spheically symmetic body, we conside only what happens outside of the dust sphee. The equation of motion fo a adially plunge obit is: τ) = ) M) 3 τ τ) 3 16) whee τ is the pope time, o the time measued by an obseve iding along with the suface of the dust sphee, and τ is an integation constant at = 0. Fo a distant obseve, the adius of the sphee is instead given by the equation: t = t + M [ ) 3 + log 3 M M [ + 1 M ] log M ] 1 Substitute in Eq. 3, we aive at the equation of motion fo the suface of a spheically symmetical collapsing dust sphee in the coodinate v,, θ, φ): v = t + M v) M = [ 3 M ) 3 M M ) 3 + log 3 M M ) [ 1 + log M [ + 1 M ) ]] ) 18) ] + const 19) Once the suface of a sta collapses to a adius smalle than the adius of the event hoizon = M, the adius of the sta can only decease. The mateial on the suface must move along a geodesic inside the futue local light cone, as indicated by special elativity, and all light cones ae point towads the singulaity. Hence, the sta will inevitably collapse into an infinitely dense singulaity at = Gavitational Redshift Conside a distant obseve at R and an obseve iding along with the suface of a collapsing sta. The falling obseve communicates with the distant obseve by sending signals at a fequency of ω. Befoe the falling obseve cosses the event hoizon, the signal can still each the distant obseve. The signal is emitted at a time inteval τ = π/ω. Let the signal emitted at v E, E ) be eceived by the distant obseve at the pope time t R. The time inteval between successive signals is t R t R ) is a function of t R. The equation of motion of the signal is descibed by Eq. 11: v log M 1 = const 6

7 Because this holds tue fo all v, ), the constant on the ight-hand sided at v R, R ) is equal to the constant at v E, E ). Fo E M, the logaithm tem dominates and thus the left-handed side becomes: v E E log E M 1 E log M 1 0) On the othe hand, fo R M, the logaithm tem vanishes. Futhemoe, t R v R R. The left-handed side becomes: v R R log R M 1 vr R t R R 1) We find the elation between R, E, and t R to be log E M 1 tr R ) [ )] R t R E = M 1 + exp 3) Take the deivative with espect to t R, and assuming that the inteval t R 1: E d E = 1 ) t R dt R exp R t R 4) If the falling obseve has a fou-velocity of u α = u v, u, 0, 0) the negative sign is due to the adius deceasing), then we may expess the space inteval between successive signals E as the time inteval between successive signals τ: E = u τ. We may expess t R in tems of τ and t R : E = u τ = 1 ) t R t R exp R t R 5) ) t R = u tr R τ exp 6) ω R = ω ) u exp R t R 7) ω R ω exp t ) R 8) We have expessed the eceived fequency as a function of the emitted fequency and the pope time of the distant obseve. The constant of popotionality is exp R /)/ u. As long as the adial component of the fou-velocity does not change, Eq. 8 will hold. Because the exponential tem is deceasing as t R inceases, the distant obseve will see the signal moe edshifted ove time. Because the photon enegy is popotional to its fequency, the obseve will see the signal less luminous ove time. Once the falling obseve cosses the event hoizon = M, the signals, along with any paticle with non-zeo mass, can no longe each the distant obseve but instead fall towads the singulaity = 0. As indicated above, thee can be no infomation exchanged between the two obseves. 7

8 To examine the complete pictue, we plot the spacetime diagam t vesus in Fig.. The suface of the sta is descibed in Eq. 19. The shaded egion is inside the suface of the sta and thus cannot be descibed by the Schwazschild geomety. The dotted lines ae the wold lines of the signal fom the falling obseve. Fom Fig., as the falling obseve is heading towads the event hoizon, the time inteval between successive signals as obseved by the distant obseve gets lage and lage. Once the falling obseve cosses the event hoizon, no signal can escape as all null geodesics point towads the singulaity. Figue : The suface of a collapsing sta in the Eddington-Finkelstein coodinates. As the falling obseve gets close to the event hoizon, the distant obseve will see time inteval between successive signals longe and the signals moe edshifted. Once the falling obseve falls inside the event hoizon, no signal can escape and the communication between the two obseves is inteupted. 4 Conclusion In this pape, we used the Eddington-Finkelstein coodinates to descibe the Schwazschild geomety of a spheically symmetical collapsing sta. We deived the null geodesics of a adial light ay and found that inside and outside the event hoizon, light ay behaves qualitatively diffeently. Outside the hoizon, light ays may escape into infinity o fall towads the hoizon. Howeve, inside the hoizon, all light ays cannot escape and will instead convege into a singulaity. This singulaity coesponds to an infinitely dense egion of spacetime, whee the laws of physics as we know beak down. We also deived the equation 8

9 of motion of the suface of a collapsing dust sphee and examined the case of the two obseves. One obseve is infinitely fa fom the collapsing sta, while the othe ides along on the suface of the sta. We found as the suface of the sta collapses, the signals emitted by the falling obseve will get exponentially edshifted as seen by the distant obseve. All of ou deivations ae based on the assumption that the collapse is spheically symmetical. This is not necessaily the case because any otation of the body aound an axis will distot the shape. Howeve, this is ou fist step to undestanding the geomety of a nonspheical collapse and eventually the geomety of a black hole. The following popeties of a spheical collapse will hold in the case of a nonspheical collapse: the fomation of a singulaity, the fomation of an event hoizon, and the incease in the aea of the hoizon. Refeences [1] J.B. Hatle. Gavity: An Intoduction to Einstein s Geneal Relativity. Addison-Wesley,