SCHWARZSCHILD SOLUTION IN GENERAL RELATIVITY. Marko Vojinović

Transcription

1 SCHWARZSCHILD SOLUTION IN GENERAL RELATIVITY Mako Vojinović Mach 200

2 Contents The meaning of the metic tenso 2 2 Einstein equations and Schwazschild solution 5 3 Physical intepetation and consequences 9 3. Gavitational time dilatation Gavitational edshift Newton s law of gavity Embedding diagam Black holes 4 4. Time stops at event hoizon Finite pope time to fall though the hoizon Black holes ae black Black holes ae holes

3 Chapte The meaning of the metic tenso We begin with the definition of distance in Euclidean 2-dimensional space. Given two points A and B in the plane R 2, we can intoduce a Catesian coodinate system and descibe the two points with coodinates x A, y A ) and x B, y B ) espectively. Then we define the distance between these two points as: Note: s = x A x B ) 2 + y A y B ) 2. One can pove that the distance is invaiant with espect of the choice of the coodinate system, povided that the new coodinate system is also othogonal and has the same unit scale. Below we shall genealize this statement to all coodinate systems.) One can natually genealize this definition to the case of D-dimensional Euclidean space, R D, as: s 2 = D x µ A xµ B )2, µ= and even futhe to so-called pseudo-euclidean spaces, like the Minkowski spacetime M 4 : s 2 = t A t B ) 2 + x A x B ) 2 + y A y B ) 2 + z A z B ) 2. If one daws a pictue of the two points in a coodinate system, one can immediately see that the key ingedient in the above definition is the theoem of Pythagoa one can daw a ight tiangle, whee the hypotenuse is connecting points A and B, while the two catheti ae paallel to the two coodinate axes. If points A and B lie on a cuve C, one may wish to calculate the length of the cuve between the two points. In ode to do so, one usually employs the diffeential definition of distance between the infinitesimally neighboing points x µ and x µ + dx µ, and integates it along the cuve: ds 2 = D dx µ ) 2.) µ= s = Fo the Minkowski case we use the spacelike metic convention, and we denote time as the zeo-coodinate, ie. C ds. t = x 0, x = x, y = x 2, z = x 3. 2

4 The diffeential expession fo the distance is called a line element, and is fundamental to calculating lengths of all geometic objects. Suppose now that we want to intoduce new coodinates, via a change of vaiables. In geneal, new coodinates ae called cuvilinea coodinates, and ae defined as x µ = x µ x ν ), µ, ν {,...,D}. The only condition imposed on the functions x µ x ν ) is that they need to be invetible, meaning that we can go back to the oiginal coodinates via x ν = x ν x µ ). The necessay and sufficient condition fo this to be tue is that the Jacobian of the tansfomation must be nonzeo 2 : ] [ x µ I = det x ν 0. A typical and most common example of cuvilinea coodinates in thee dimensions is the choice of spheical coodinates,, θ, ϕ, intoduced as: x = sin θ cosϕ, y = sinθ sin ϕ, z = cosθ..2) If one diffeentiates these expessions and substitutes them into the line element, one obtains the following nontivial expession: ds 2 = d dθ sin 2 θdϕ 2. We see that this is vey peculia, since the definition of distance appeas to depend in a nontivial way) on the choice of the point whee it is to be calculated. In geneal cuvilinea coodinates one obtains a completely geneal expession fo the line element: ds 2 = D D g µν x)dx µ dx ν µ= ν= = g µν x)dx µ dx ν by Einstein summation convention) = g x)dx ) 2 + g 2 x)dx dx g DD x)dx D ) 2. The matix with elements g µν x) is called the metic tenso. Note: The metic tenso can always be assumed symmetic, because the diffeentials dx µ and dx ν commute. If one stats fom the coodinates x µ with metic g µν x) and tansfoms to coodinates x µ with metic g µ ν x ), one can pove that the absolute value of the Jacobian of the tansfomation is: ] [ x µ g I = det x ν = g, whee g = detg µν and g = detg µ ν. 2 Except maybe fo some paticula choices of coodinates which ae called singula. 3

5 As a consequence of the condition I 0 and the above equation, we see that the metic is always egula, detg µν x) 0. This in tun means that the invese matix always exists, and is denoted g µν x). The statement that these two tensos ae invese to each othe can be conveniently witten as g µν g νλ = δ λ µ { when λ = µ, 0 when λ µ. The δ µ ν is called the Konecke symbol.... Since we stated fom the line element.), thee always exists a coodinate tansfomation fom any cuvilinea coodinates to the Catesian, such that in Catesian coodinates the metic has the fom 3 : [g µν x)] =. At this point we make a huge genealization and dop the final popety: Define the distance via the line element ds 2 = g µν x)dx µ dx ν whee g µν x) is a completely abitay function of coodinates x µ, satisfying only the condition detg µν 0. This definition of distance is the genealization of the theoem of Pythagoa to cuved spaces. Intuitively, one can undestand this as follows. Dopping the above popety means that, stating fom some cuvilinea coodinates, it might not be possible to tansfom to Catesian coodinates. This means that all possible coodinates might be cuvilinea, which in tun means that thee might be no staight lines in such a space. Thus, such a space is called cuved. 3 In Minkowski spacetime it has the fom 2 [g µνx)] = , and is denoted η µν. 4

6 Chapte 2 Einstein equations and Schwazschild solution The Einstein equations ae usually witten in the following fom : Note: G µν R µν 2 Rg µν = 8π T µν. The quantity G µν is called the Einstein tenso, while T µν is called stess-enegy tenso. The Ricci tenso R µν and scala cuvatue R ae defined as: R µν = λ Γ λ µν ν Γ λ λµ + Γ λ λσγ σ µν Γ λ σνγ σ λµ, R = g µν R µν. 2.) The Ricci tenso is symmetic. The Chistoffel symbol Γ λ µν is defined as: It is symmetic with espect to two lowe indices. Γ λ µν = 2 gλσ µ g νσ + ν g µσ σ g µν ). 2.2) All in all, we see that on the left-hand side of Einstein equations we have G µν which is a function of the metic, its fist deivatives and its second deivatives. On the ight-hand side we have T µν which epesents matte. So essentially, Einstein equations epesent a set of 0 second-ode patial diffeential equations fo the 0 components of the metic tenso, with stess-enegy as a souce tem. Thus, We wok in a natual system of units whee speed of light and Newton s gavitational constant ae defined to be equal to one, c = m s =, m3 γ = kg s 2 =. This means that we have edefined the second and kilogam in tems of a mete, as: s m, kg m. If we wish to etun fom the natual system of units to the intenational SI metic system, we need to detemine what ae the SI units of a given physical quantity, and then multiply it with, suitably expessed in those units using the above fomulas. The coespondence is unique. Also, as well as in the SI system, in the natual system of units and one can employ the technique of dimensional analysis to check one s calculations the dimensions in metes) of the left-hand side of any equation must always match the ight-hand side. 5

7 Einstein equations detemine the geomety of spacetime by poviding the definition of distance theoem of Pythagoa), based on the matte content in that spacetime. In addition, motion of matte is detemined by this geomety. The fact that the motion of matte is detemined by popeties of geomety is called the equivalence pinciple, and is built in the Einstein equations. While this is not obvious, we shall demonstate it late in seveal examples. Simply put, matte tells geomety how to cuve, while geomety tells matte how to move. In this way, geomety ceases to be just an aena whee physics happens, but athe becomes an active paticipant in physical pocesses. This paticipation of dynamics of geomety in physical pocesses is called gavitational inteaction. Now we will demonstate all this in the simplest nontivial case the static spheically symmetic solution of Einstein equations, called Schwazschild geomety. Begin fom flat Minkowski spacetime with the line element ds 2 = dt 2 + dx 2 + dy 2 + dz 2, intoduce spheical coodinates via the change of vaiables.2), and obtain: ds 2 = dt 2 + d dθ sin 2 θdϕ 2. This is still a flat-spacetime line element, just expessed in cuvilinea coodinates. Genealize this lineelement in such a way to allow fo cuvatue, while peseving the equiements of geomety being static and spheically symmetic. Static means that the metic should not depend on time, while spheically symmetic means that it should not depend on angles θ and ϕ, othe than though aleady pesent tems. One can then show that it is enough to conside the following genealization fo the line element: ds 2 = e 2F) dt 2 + e 2H) d dθ sin 2 θdϕ 2, whee F) and H) ae two functions to be detemined by Einstein equations. Note that we have chosen to wite them in the exponent only because of late computational convenience. So constuct fist the left-hand side of Einstein equations. Read the metic and invese metic tensos fom the line element: [g µν ] = e 2F e 2H 2 2 sin 2 θ, [gµν ] = e 2F e 2H Use this to calculate the Chistoffel symbols using 2.2). The nonzeo symbols ae: 2 2 sin 2 θ Γ t t = F, Γ tt = F e 2F 2H, Γ = H, Γ θθ = e 2H, Γ ϕϕ = sin 2 θe 2H, Γ θ θ =, Γ θ ϕϕ = sinθ cosθ, Γ ϕ ϕ =, Γ ϕ θϕ = cotθ. Next use this to constuct the Ricci tenso, using 2.). The nonzeo components ae: R tt = e 2F 2H F + F ) 2 F H + 2 ) F, R = F + F ) 2 F H 2 ) H,. R θθ = e 2H + F H ), R ϕϕ = R θθ sin 2 θ. Next contact the Ricci tenso with the metic to obtain the Ricci scala: R = 2e [F 2H + F + 2 ) F H ) + e 2H )] 2. 6

8 Finally, put all this togethe to fom the Einstein tenso: G tt = 2 e2f 2H 2H e 2H), G = 2 + 2F e 2H), G θθ = 2 e [F 2H + F + ) ] F H ), G ϕϕ = G θθ sin 2 θ. Note that the G tt component of the Einstein tenso can be ewitten in the fom G tt = 2 e2f d d [ e 2H )], which will be convenient fo integation late on. Next we concentate on the ight-hand side of the Einstein equation. We ae inteested in modeling the simplest possible stess-enegy tenso, namely one that epesents a static ball of adius R and density ρ) with the cente in = 0. Remembeing the geneal fomula fo the stess-enegy tenso of a fluid element with density ρ, pessue p, and 4-velocity u µ, T µν = ρ + p)u µ u ν + pg µν, we wish to descibe the static fluid u = u θ = u ϕ = 0). So the stess-enegy obtains the fom T tt = ρu t u t + p u t u t + g tt ), T = pg, T θθ = pg θθ, T ϕϕ = pg ϕϕ, while othe components vanish. Next, the 4-velocity vecto must be nomalized, u µ u ν g µν =, which means that u t u t = g tt = e 2F, so we have T tt = ρe 2F, T = pe 2H, T θθ = p 2, T ϕϕ = p 2 sin 2 θ. The density and pessue of the fluid can depend only on due to the spheical symmety, and must be zeo fo > R, ie. outside the ball. Finally, afte substituting all these esults into Einstein equations, G µν = 8π T µν, we obtain thee equations: the t-t equation: the - equation: d [ 2 e2f e 2H )] = 8πe 2F ρ), d 2 + 2F e 2H) = 8πe 2H p), the θ-θ and ϕ-ϕ equations which ae identical): 2 e [F 2H + F + ) ] F H ) = 8π 2 p). All othe equations ae vacuous, 0 = 0, and povide no infomation. Fist discuss the t-t equation. Staightfowad integation gives H) = 2 ln 2m) ), whee m) 4π d 2 ρ). Choosing the initial condition m0) = 0, we can intepet m) as the total mass inside adius, since it is defined as an integal of mass density ρ ove the volume of a ball of adius. 7

9 Next discuss the - equation. Solve it fo F to obtain F) = d m) + 4π3 p). [ 2m)] Fo < R, the integal is complicated and we shall not discuss it. Howeve, fo > R we have m) = M total mass of the ball) and p) = 0 zeo pessue in vacuum), so F) can be easily integated. The esult is F) = 2 ln 2M ), whee the constant of integation has been chosen so that in the limit the line element ecoves its Minkowski fom fa away fom the ball spacetime should be flat). Finally, discuss the θ-θ equation. Substitute all pevious esults and afte a tedious calculation) obtain the following esult: p ) + F )ρ) + p)) = 0. This is a diffeential equation that detemines the adial pessue distibution of matte within the ball. This distibution is such that the epulsive pessue balances attactive gavity eveywhee, theeby maintaining static configuation of matte inside the ball. In what follows, we shall not be inteested in the geomety within the ball, but athe only the geomety outside fo > R). In this case, the line element has the fom: ds 2 = 2M ) dt 2 + d dθ sin 2 θdϕ 2. 2M This is the famous Schwazschild solution of Einstein equations, and defines the so-called Schwazschild geomety. 8

10 Chapte 3 Physical intepetation and consequences 3. Gavitational time dilatation Conside an obseve sitting at the suface of the Eath, say at coodinates E = R, θ E = const, ϕ E = const. Assume that the obseve caies a clock with him. He can constuct the local Minkowski coodinates in the immediate neighbohood, descibed by the line element ds 2 = dτ 2 E + dx2 + dy 2 + dz 2. Since the obseve is at est in his own coodinates, we have dx = dy = dz = 0, and the coodinate τ E measues the pope time of that obseve, ie. it epesents the eadout of his clock. Given that the line element ds is invaiant, it must be the same in both local Minkowski coodinates and in Schwazschild coodinates, so since d E = dθ E = dϕ E = 0) we obtain: dτ E = dt 2M R. Next conside anothe obseve, sitting in geosynchonous obit at height h > 0 above the gound, ie. at coodinates O = R + h, θ O = const, ϕ O = const. He also caies a clock which measues his pope time, ds 2 = dτ 2 O. Equating this to the Schwazschild line element, and emembeing that d O = dθ O = dϕ O = 0, we obtain a simila elation: dτ O = dt 2M R + h. Now assume that both obseves measue the time of some physical pocess that begins at coodinate time t and ends at coodinate time t+dt. Since the coodinate time is the same fo both obseves, using above fomulas we obtain: 2M R dτ E = dτ O 2M. R+h Note that the expession unde the squae oot is always stictly less than one, so we have dτ Eath < dτ Obit, ie. the clocks on Eath tick slowe than clocks in obit! This effect is called gavitational time dilatation. 9

11 The gavitational time dilatation effect is not only measuable on Eath, but it is downight impotant in pecision measuements, fo example in the Global Positioning System. The GPS system is the vey fist eal-wold industial application of geneal elativity. 3.2 Gavitational edshift Conside an atom at est at the suface of the Sun, undegoing deexcitation and emitting a pulse of light in the pocess. In the atom s est fame it takes time τ S to emit one wavelength. So given this pope time between two equal phases of the wave, and given that velocity of light is equal to c = in est fame of the atom, the wavelength of the light pulse is given as λ S = c τ S τ S. Let the wave go adially upwads into obit, and conside an obseve on Eath, at a distance D fom the suface of the Sun. The obseve uses his est fame to measue the time between two equal phases of the incoming wave. Since velocity of light is again c =, he measues the wavelength as λ E = c τ E τ E. But as we have shown in the pevious section, the time inteval τ S on the suface of the Sun and the time inteval τ E in the Sun s obit whee Eath esides ae elated via 2M R τ S = τ E 2M, R+D which tanslates into 2M R λ S = λ E 2M. R+D We see that the wavelength of the light pulse emitted fom the suface of the Sun is always stictly less than the wavelength of the same pulse measued by the obseve on Eath: λ S < λ E, ie. the wavelength of the outgoing pulse is being shifted towads lage values. This effect is called gavitational edshift. Note that the enegy is invesely popotional to the wavelength, which means that the obseved photon has less enegy than it had when it was emitted. So the enegy of the photon is not conseved, due to the inteaction with the gavitational field. In the tems of Newtonian mechanics, the photon needs to climb up the gavitational potential well, and loses enegy in the pocess. 3.3 Newton s law of gavity Conside the nonelativistic motion of a test paticle in Schwazschild geomety. Nonelativistic means that the paticle s velocity can be consideed much less than the speed of light, as measued elative to the oigin of Schwazschild coodinates. It also means that the souce of the gavitational field is weak. Othewise a nonelativistic test paticle would obtain elativistic velocity duing its motion, due to the stong field that acts on it. Fist, we ewite the Schwazschild line element in SI units suitable fo nonelativistic analysis), ds 2 = 2M ) γ c 2 c 2 dt 2 + 2M 0 d γ dθ sin 2 θdϕ 2. c 2

12 and lineaize in M, in ode to implement the weak field equiement: ds 2 = 2M ) γ c 2 c 2 dt M ) γ c 2 d dθ sin 2 θdϕ 2. To get a pope feeling of the magnitude of this appoximation, note that this is easonably valid fo the gavitational field of the Eath and othe planets, but not fo the field nea the suface of the Sun, fo example. Second, implement the nonelativistic velocity equiement by taking the limit v/c 0. Note that v c 0 c dx µ ds 0 fo µ =, θ, ϕ. This essentially means that we ae to discad the coection tem which multiplies d 2, so we obtain the following line element: ds 2 = 2M ) γ c 2 c 2 dt 2 + d dθ sin 2 θdϕ 2. Now note that the spatial pat of the metic is Euclidean, and that we can easily tansfom fom spheical coodinates back to Catesian ones: ds 2 = 2M ) γ c 2 c 2 dt 2 + dx 2 + dy 2 + dz 2. So we see that in nonelativistic appoximation time is cuved, while space is flat. In ode to study this geomety futhe, we need the equations of motion fo the test paticle in cuved space. Those equations can be deived fom Einstein equations, but we omit the deivation hee. Neglecting the gavitational field poduced by the paticle itself the test paticle appoximation), the esulting tajectoy is in geneal called a geodesic line, and is detemined by the following diffeential equations: d 2 x λ dτ 2 + Γ λ dx µ dx ν µν dτ dτ = 0. The affine paamete τ counts the points of the tajectoy. The nonzeo Chistoffel symbols calculated fom the above line element ead emembe that x 2 + y 2 + z 2 ): Γ t tx = M 3 γ c 2 x, Γ t ty = M 3 γ c 2 y, Γ t tz = M 3 γ c 2 z, Γ x tt = γ M 3 x, Γ y tt = γ M 3 y, Γ z tt = γ M 3 z. Now witing down explicitly the fou components λ = t, x, y, z of the geodesic equation, we obtain: d 2 t t : dτ 2 + 2M γ dt 3 c 2 x dx dτ dτ + y dy dτ + z dz ) = 0, dτ x : y : z : d 2 x dτ 2 + γ M ) 2 dt 3 x = 0, dτ d 2 y dτ 2 + γ M ) 2 dt 2 y = 0, dτ d 2 z dτ 2 + γ M 2 z ) 2 dt = 0. dτ

13 Conside fist the t equation. Due to the nonelativistic velocity appoximation, we can neglect all the tems in the paentheses. The emaining fist tem is easily integated to: t = τ using a suitable initial conditions the equal scale and equal oigin fo t and τ). The emaining x, y and z equations can be multiplied with othonomal Catesian basis vectos e x, e y and e z espectively, and collected togethe to obtain d 2 dt 2 + γ M 2 e = 0. Note that e / and x e x + y e y + z e z.) Finally, if we multiply this with the mass m of the test paticle, we can ewite the esult as m a = F g, which is the familia Newton s second law of dynamics, while the foce tem F g = γ mm 2 e is the Newton s law of gavitation fo a body with spheical symmety and mass M. 3.4 Embedding diagam In ode to visually demonstate that Schwazschild spacetime is actually cuved, it is instuctive to epesent it as a cuved suface embedded in a bigge Euclidean space. Given that we cannot actually daw moe than 3 Euclidean dimensions faithfully, the usual example of this kind of embedding deals with a 2-dimensional suface in Schwazschild geomety it is a suface defined by t = const and θ = π/2. It is a spatial slice though a coodinate oigin, and has the line element ds 2 = 2m) d dϕ 2. 3.) Note that we have witten m) instead of M, in ode to include egion inside of the ball of matte. This kind of geomety can be epesented in 3-dimensional Euclidean space. It is convenient to use cylindical cuvilinea coodinates in this space, defined as which tansfom the Euclidean line element into the fom x = cosϕ, y = sin ϕ, ds 2 = dx 2 + dy 2 + dz 2 ds 2 = d dϕ 2 + dz 2. The pocedue of embedding now goes as follows epesent the Schwazschild equatoial suface as a suface in Euclidean space, given via paametic equations =, ϕ = ϕ, z = z), whee and ϕ ae both coodinates and paametes, while z) is the elevation function, which actually detemines the shape of the suface. Due to axial symmety, it does not depend on ϕ. Now constuct the line element fo this suface in Euclidean space, [ ) ] 2 dz ds 2 = + d dϕ 2, d 2

14 and equie it to be identical to the line element of the equatoial plane in Schwazschild geomety, 3.), in ode to epesent equivalent intenal geometies. Theeby obtain + dz d 2 = 2m), which can be integated to give z): z) = Z 0 d q 2m). The function m) is equal to M fo > R, while it depends on the detailed desciption of matte inside the ball, fo < R. Below ae two pictues, one fo some easonable homogeneous density model of m) left pictue), and the othe fo a black hole ight pictue). Note that fo the case of black hole the egion < 2M cannot be embedded into Euclidean space, due to its Minkowski signatue we shall give a physical explanation fo this the next chapte). Also note that this 3-dimensional Euclidean embedding space is not physical, and exists only as an abstaction fo the puposes of visualization of the suface. The only thing that physically exists is the suface itself, as pat of the Schwazschild geomety. Moeove, in Schwazschild geomety this same suface looks like a plane, the equatoial plane, only with distoted definition of adial distances. This distotion manifests itself in Euclidean embedding as cuvatue of the suface. 3

15 Chapte 4 Black holes The static configuation of the body of mass M and adius R is maintained due to the epulsive pessue of matte balancing the gavitational attactive foce. While the gavitational foce gets eve stonge with inceasing M, the pessue within the body has an uppe limit, based on the stuctue of paticles that fom the body, and the inteactions between them. This means that fo a lage enough total mass M, the pessue will become insufficient to balance the gavitational attaction. The bounday value of M when this happens is called the Chandaseka limit, and is oughly equal to thee to five Sola masses. If the mass of the body exceeds the Chandaseka limit, thee will be no equilibium to maintain, and it stats contacting all the way into a ball of Planck adius R l p 0 35 m). At those scales the effects of quantum gavity kick in, and Einstein s theoy of gavity ceases to be valid. But outside that small sphee geneal elativity is an accuate desciption of spacetime geomety. The minuscule ball of matte is idealized as a single point with divegent enegy density and spacetime cuvatue, and is called a singulaity. Thus, the geomety aound the singulaity has the Schwazschild fom all the way up to Planck scales, and such a configuation of matte and its suounding geomety is called a black hole. The above analysis suggests that we should conside the Schwazschild line element ds 2 = 2M ) dt 2 + 2M d dθ sin 2 θdϕ 2. fo all possible values of the adial coodinate geate than l p, while keeping M lage the stong field limit). If we do so, we easily see that some vey peculia effects stat happening aound and inside the sphee of adius = 2M. This sphee is called the event hoizon. Thee ae seveal impotant popeties of the geomety inside and at the event hoizon of a black hole. 4. Time stops at event hoizon Conside fist the time dilatation equation deived above, ewitten in the fom: 2M dτ = dτ O 2M. +h Hee τ is the pope time of the obseve at est at adius fom the cente, while τ O is the pope time of the obseve at est in obit at adius + h fom the cente. If the fist obseve is at est on the hoizon, = 2M, its pope time stops. Note, though, that this actually means that no obseve can keep himself at est thee. 4

16 4.2 Finite pope time to fall though the hoizon Next conside the obseve who stats fom some adius 0 = 2M + h outside the event hoizon, and stats falling adially towads the cente. He caies with himself a clock that measues his pope time dτ. Dividing the Schwazschild line element by ds 2 and using the elation ds 2 = dτ 2, we obtain: = 2M ) ) 2 dt + dτ 2M d dτ ) 2 ) 2 dθ sin 2 θ dτ ) 2 dϕ. dτ Since the obseve is falling adially inwads, we have θ = const and ϕ = const, so the above equation educes to: = 2M ) ) 2 ) 2 dt d + dτ 2M. 4.) dτ Next we need the time component of the geodesic equation fo the obseve, which in these coodinates has the fom d 2 t dτ 2 + 2M dt d 2 2M dτ dτ = 0. Multiply this equation with 2M/) and integate to find that dt dτ = C 2M, 4.2) whee C > 0 is the constant of integation. We want this constant to be positive in ode to ensue that both coodinate time t and pope time τ point to the futue. Now substitute this esult back into the equation 4.) to obtain d dτ = C 2 2M The choice of negative sign in font of the squae oot is because we conside the obseves motion towads the black hole, so his adial coodinate should decease with inceasing time. Finally, imagine that the obseve is falling fom the initial position 0 all the way down to the event hoizon = 2M. Calculate his pope time of flight, τ = 2M 2M+h dτ = and, using 4.2), the coodinate time of flight 2M+h 2M ). d C 2 ), 2M t = 2M dt = 2M+h 2M+h 2M C 2M d C 2 ). 2M It is easy to see, by diect evaluation of the integals, that the pope time of flight is finite, while the coodinate time of flight is infinite. Since the coodinate time is popotional to the pope time of the othe obseve the one at est in obit at = 2M + h), we conclude that the obit obseve will neve see the infalling obseve each the event hoizon, except asymptotically. Howeve, the infalling obseve eaches the event hoizon in finite pope time, and moeove if the appopiate integal is evaluated fom 2M + h not down to 2M but even down to = 0 the infalling obseve will pass though event hoizon and each the cente in finite pope time. 5

17 4.3 Black holes ae black Conside an excited atom going though deexcitation and emitting a photon just befoe it passes though the event hoizon. Imagine that the photon is emitted adially outwads, and detected by the stationay obseve in obit at adius R = 2M + h. The gavitational edshift that the photon undegoes is given by the fomula 2M R λ detected = λ emitted 2M, whee is the adial coodinate of the atom at the time of deexcitation. If the atom is at the event hoizon = 2M, the detected wavelength of the photon becomes infinite. If the atom was just above the event hoizon, the wavelength would be finite, but extemely lage. But lage wavelength means small fequency, which means small enegy, which will escape detection if it is small enough. Theefoe, no light can be emitted o eflected fom the event hoizon. The effect is the same also fo all othe foms of adiation and signals, since they all need to climb up an infinite gavitational potential well if they stat fom the hoizon. Theefoe, since no adiation o signals can eve come out of the event hoizon, black hole is indeed black. 4.4 Black holes ae holes Once the obseve has fallen though the event hoizon, he might ty to stat his ocket engines in ode to go back, o at least ty to send a light signal to the obseve outside the hoizon. We have aleady seen that it would take an infinite amount of enegy to climb back up, so the obseve is basically tapped inside the hoizon, unable to get out. By studying the obits of test paticles inside the event hoizon using geodesic equation) one can veify that it is impossible to climb up, no matte how much enegy one has at one s disposal fo, say, ocket boostes. In addition to the geodesic equations, thee is an intuitive geometic agument why this is so. Conside the obseve in Minkowski space. He has the feedom to tavel though space o to stand still. But he does not have the same choice with espect to time. Evey obseve invaiably tavels to the futue, without any ability to stop his motion though time. The eason fo this lies in the minus sign fo time coodinate in the Minkowski line element. Namely, we have ds 2 = dt 2 + dx 2 + dy 2 + dz 2, and if we divide by the pope time dτ = ds, we obtain: ) 2 ) 2 ) 2 ) 2 dt dx dy dz = = η µνu µ u ν. dτ dτ dτ dτ Fom hee we see that the magnitude of the 4-velocity u µ of the obseve is constant, and the 4-velocity vecto is always timelike ie. points to the futue, egadless of the choice of coodinate system. Thus the obseve can change his space velocity abitaily, but he cannot have zeo velocity though time. The eason fo this is that the coefficient in font of dt 2 in the line element is negative. The same effect exists in cuved spacetime. By looking at the Schwazschild line element, ds 2 = 2M ) dt 2 + 2M d dθ sin 2 θdϕ 2, we see that the coefficient in font of dt 2 is negative while othe thee ae positive), and thus conclude that the obseve must always have some nonzeo velocity though time. But this is only tue fo > 2M! Once the obseve falls though the event hoizon, the coefficients in font of dt 2 and d 2 change signs! This means that time coodinate and adial coodinate switch 6

18 thei oles, and consequently the obseve can adjust his obital velocity and velocity though time, but cannot have zeo adial velocity. So in the same sense that the obseve outside the hoizon can choose not to tavel though space, but invaiably must tavel though time, so the obseve inside the hoizon can choose not to tavel though time and can have zeo obital velocity, but invaiably must fall towad the cente. The eason fo this lies in the geomety of spacetime, and no amount of enegy, motion o ocket boostes can pevent it fom happening. Thus, once inside the hoizon, thee is no escape fom eaching the singulaity, o even maintaining a stable obit aound it. That is why black holes ae indeed holes. 7

19 Fo futhe eading [] Lev D. Landau, Evgeny M. Lifshitz, Couse on Theoetical Physics Vol. 2: The Classical Theoy of Fields, Buttewoth-Heinemann 975), ISBN [2] Chales W. Misne, Kip S. Thone, John Achibald Wheele, Gavitation, W. H. Feeman and Co. 973), ISBN