The Schwazschild Solution Johannes Schmude 1 Depatment of Physics Swansea Univesity, Swansea, SA2 8PP, United Kingdom Decembe 6, 2007 1 pyjs@swansea.ac.uk
Intoduction We use the following conventions: Geek coodinate indices un fom 0 to 3, while latin ones un fom 1 to 3. The signatue of the metic is mostly minus. Thee-vectos and thee-dimensional coodinates ae denoted as X, x, while thei fou-dimensional equivalents ae X, x. The mateial pesented hee is based on the books by Caoll and Weinbeg, [1, 2]. Deivation of the solution Recall the field equations of geneal elativity Contacting with g µν gives R µν 1 2 g µνr = 8πGT µν 1) R 2R = 8πGT µ µ 2) We shall seach fo a solution of 1) descibing a point-like mass at est. Physically such a solution can be used to descibe the gavitational field of objects with appoximate spheical symmety such as planets, stas, black holes. Ou solution should satisfy the following citeia 1. Fa away fom the mass the gavitational field should be tivial. I.e. the space should be asymptotically flat 2. The solution needs to espect the spheical symmety. Recall that the spheical symmety goup in thee dimensions is SO3) = { O GL3, R) O T = O 1} 3) Whee GL3, R) is the goup of invetible 3 3 matices. As an example of an element of SO3) ecall the fom of a otation aound the x 3 axis cos φ sin φ 0 O = sin φ cos φ 0 4) 0 0 1 One says that a solution especting otational symmety is isotopic. 3. The solution should be static; that is it should be invaiant with espect to tanslations in time. 4. The solution should beak the tanslational symmety. It is thus non-homogeneous. We choose a set of coodinates x 1, x 2, x 2, x 0 t. Note that these ae not the same as the coodinates of flat Minkowski space, yet as the space is supposed to be asymptotically flat we equie ds 2 asymptotically dx 02 + dx 12 + dx 22 + dx 32 5) 1
Unde otations, that is unde the action of O SO3), the x tansfom as x i O i jx j dx i O i jdx j 6) O moe succintly Now i xi x i = x.x tansfoms as x Ox dx Odx 7) x.x Ox).Ox) = xo T Ox = x.x 8) and is theefoe invaiant. The same holds fo x.dx and dx.dx. The popety that the gavitational field should be static means that g µν should be independent of the time-coodinate t = x 0. Note that this is highly heuistic howeve, as the notion of time in cuved spaces is a vey subtle issue ecall the definition of pope-time. Thee is anothe constaint on the metic which also follows fom the homogeneity with espect to time. We shall state it hee without poof: Thee ae no off-diagonal tems such as dtdx i. Howeve thee may easily be tems of the fom dx i dx j ). Witing = x.x, we may summaise ou discussion up to this point by making the following Ansatz fo the metic: ds 2 = F )dt 2 + D) x.dx) 2 + C)dx 2 9) It is appopiate to intoduce spheical coodinates One obtains x 1 = sin θ cos φ 10) x 2 = sin θ sin φ 11) x 3 = cos θ 12) dx 3 = cos θd sin θdθ 13) As an execise you might want to calculate dx 1 and dx 2 and veify that ds 2 = F dt + }{{} 2 Dd 2 + Cd 2 + C }{{} 2 dθ 2 + 2 sin 2 θdφ 2) }{{} 1) 2) 3) 14) To simplify the ansatz 9) futhe, we use the diffeomophism-invaiance 1 of GR to define a new adial coodinate 2 C) 2 15) We now look at the sepeate pats of 14) to investigate how they change unde this tansfomation. 1. F ) B ) 1 Diffeomophism-invaiance is just a fancy way of saying that we ae fee to use new coodinates x = x x) as long as the functions x x) ae diffeentiable, i.e. smooth. 2
2. Fist note that taking the total diffeential d on both sides of 15) leads to d 2 4 2 d 2 = 2C) + 2 C 16) ) Pluggint this into 2) in 14) gives 2 D) + C) ) d 2 = 2 D)/C) + 1 1 + /2C )) 2 d 2 A )d 2 17) }{{} A ) 3. C) 2 dθ 2 + sin 2 θdφ 2) = 2 dθ 2 + sin 2 θdφ 2) Dopping the pimes, the new metic eads ds 2 = B)dt 2 + A)d 2 + 2 dθ 2 + sin 2 θdφ 2) 18) To make life easie fo us we assume A) = B) 1. Thee is no physical o mathematical eason to do so, except that we will see that we get the coect esult. It will howeve make ou calculations a bit easie. Except fo the point-like mass we assume ou space to be empty. That is, thee ae no futhe masses o electomagnetic fields etc. It follows that Apat fom the point at = 0 whee we assume the piont-like mass to be, the enegy-momentum tenso vanishes, T µν = 0. As we explained ealie, it follows that R = 0 and theefoe the gavitational equations of motion ae R µν = 0 19) We shall calculate some Chistoffel symbols. Recall that Now as in with the Ansatz 18) Γ µ νρ = g µλ 1 2 νg λρ + ρ g λν λ g νρ ) 20) g tt = B g = 1 B g θθ = 2 g φφ = 2 sin 2 θ 21) it follows that Γ t µν = 1 2B = 1 2B t g tt g tt θ g tt φ g tt g tt t g 0 0 θ g tt 0 t g θθ 0 φ g tt 0 0 t g φφ 0 B 0 0 B 0 0 0 0 0 0 0 0 0 0 0 22) One may similialy compute the othe components of Γ µ νρ. See e.g. chapte 8 of [2]. Note howeve that Weinbeg s metic is of opposite signatue. Also he does not make the simplifying assumption A = B 1.) 3
One calculates the Ricci tenso to be ) 1 2 B 2B + B 0 0 0 R µν = 0 2B +B 2B 0 0 0 0 1 B B 0 0 0 0 sin 2 θ 1 + B + B ) 23) By staing at this fo a sufficiently long time, one sees that thee ae in fact two diffeential equations fo B) to satisfy so that R µν = 0 holds. These ae 0 = B + 2B 24) 0 = B + B 1 25) The second is only a fist-ode equation and theefoe efeed to as a constaint. Thus we ae looking fo a solution of the second-ode equation that also satisfies the fist-ode equation. Howeve taking the diffeential of the second equation with espect to, d d B + B 1) = B + 2B 26) we note that the fist-ode equation actually entails the second-ode one. It is sufficient fo us to find a solution to the easie fist-ode equation. While thee ae sophisticated methods of finding the solution, we simply use the wold-famous & convenient appoach known as guess & check. The tivial solution is B) = 1. The metic is now given by ds 2 = dt 2 + d 2 + 2 dθ 2 + sin 2 θdφ 2) 27) This howeve is simply the metic of flat Minkowski space witten in spatial-pola coodinates. It does not beak the tanslationally symmety we demanded initially. We continue theefoe to look fo a diffeent solution. Anothe solution is B) = c h 28) The constant of integation c is fixed by the fist-ode solution to be c = 1. Thee ae no constaints on the second constant of integation h. We shall assume it to be positive howeve. Theefoe the metic eads ds 2 = 1 h This is the famous Schwazschild solution. Asymptotics Recall the geodesic equation ) dt 2 + d2 1 h + 2 dθ 2 + sin 2 θdφ 2) 29) ẍ µ + Γ µ ρσẋ ρ ẋ σ = 0 30) 4
We want to find solutions of the fom x µ = τ, τ), π/2, 0) 31) Recalling that the only non-vanishing coefficients of Γ t µν ae Γ t t and Γ t t gives t 0 = ẗ = 2Γtṫṙ 32) Similialy fo, one obtains afte calculating the elevant Chistoffel symbols, = 1 ) BB ṫ 2 B 2 B ṙ2 33) Assuming that we ae vey fa away fom the mass, we want to conside to be lage. One needs to ask though lage in compaison to what? The only othe quantity with dimensions of length though is the constant h. Hence we conside h to be lage and theefoe pefom a Taylo expansion in h χ. We stat with BB BB = 1 ) h h 2 = 1 χ 1 χ = 1 χ + χ 2 + O χ 3)) = 1 ) h + 2 h 2 +... 34) Similialy B B = 1 ) 2 h 2 +... 35) Thus to zeoth ode in h / the geodesic equation 32) is tivially satisfied, while 33) becomes = 0 36) which is the equation of motion fo a paticle in empty space the paticle will continue to move with constant velocity. Hence vey fa away fom the mass, space is flat, as it should be, and masses moving vey fa away ae not influenced by gavitation. When moving close to the point mass at = 0, we need to conside fist ode effects in χ. Equation 32) is still tivially satisfied, while 33) becomes = h 2 2 37) This is actually a highly inteesting esult. To see why, conside odinay Newtonian gavitation, whee the gavitational foce on a test-paticle of mass m is F ) = GMm. Combining this with F ) = m gives 2 = GM 2 38) 5
Compaing the two esults we see that the Schwazschild solution epoduces classical Newtonian mechanics iff h = 2GM 39) whee M is the mass of the mass at = 0. So we finally e-wite the metic in it s most common fom ds 2 = 1 2GM ) dt 2 d 2 + 1 2GM + 2 dθ 2 + sin 2 θdφ 2) 40) The black hole 2 To undestand the elation between the Schwazschild solution and black holes - o athe to undestand black holes in geneal - we need to study null geodesics, that is cuves y µ τ) satisfying ẏ 2 = 0. Making again an Ansatz which is tivial in the the spheical coodinates, ẏ µ = tτ), τ), 0, 0) which leads us to 0 = 1 h dt dt d = dτ d dτ ) ṫ2 + 1 h = ṫ ṙ = ± 1 h ) 1 ṙ2 41) ) 1 42) As h, dt d ±. Ifft appeas as if null geodesics cannot coss fom > h into the aea defined by < h. As it is often the case in geneal elativity, the notion that null geodesics - and theefoe light ays - cannot each h is due to a bad choice of coodinates. The equation fo dt d can be integated to give [ )] t = ± + h ln 1 + const. 43) h If we define new coodinates t,, θ, φ), null geodesics ae now defined by t + = const.. infalling 44) t = const.. outgoing 45) Finally defining a coodinate v = t + and using it to eplace t in the oiginal Schwazschild solution, the metic becomes ds 2 = 1 ) h dv 2 + dvd + ddv) + 2 dθ 2 + sin 2 θdφ 2) 46) These ae the Eddington-Finkelstein coodinates. Infalling null geodesics ae chaacteised by v = const., which means dv d = 0. Outgoing ones ae defined by t = 0 v = 2 47) 2 See chapte 5.6 of [1]. 6
which leads to { ) dv 2 1 d = h 1 outgoing) 0 infalling) 48) The point is that fo < h all futue-diected null geodesics ae in the diection of deceasing. It follows that once past h, light cannot escape the egion < h. As the light-cone bounds the space of time-like cuves, the same holds tue fo massive objects. The hypesuface defined by = h denotes an event hoizon. Thee is one note of caution on black holes and event hoizons. Thee ae many othe solutions to the field equations descibing black objects than only the Schwazschild solution. 3 Fo most of these it is tue that the event hoizon is given by a hypesuface whee g tt = 0 and g = ±, as in the ogiginal Schwazschild metic. Howeve as the Eddington-Finkelstein coodinates show, this does not have to be the case. A moe appopiate definition of a black hole and its event hoizon is an aea fom whee null- & timelike-geodesics may not escape to spatial infinity. O in the wods of Sean Caoll, a black hole is simply a egion of spacetime sepaated fom infinity by an event hoizon. Refeences [1] S. M. Caoll, Spacetime and geomety: An intoduction to geneal elativity, San Fancisco, USA: Addison-Wesley 2004) 513 p [2] S. Weinbeg, Gavitation and cosmology: Pinciples and applications of the geneal theoy of elativity, New Yok, USA: John Wiley & Sons 1972) 657 p 3 See e.g. the Reissne-Nodstom descibing electically chaged black holes o the Ke solution, which descibes otating black holes. In sting theoy & supegvity one consides also extended objects such as black ings and banes. 7