Gravity. Tom Charnock

Transcription

1 Gavity Tom Chanock

2 Contents 1 Beyond Newtonian Gavity and Geomety Inconsistency of Newtonian Gavity Geometies Shapes on a Euclidean Plane Shapes on a Two-Sphee Line Elements R S Vatiational Pinciples Classical Mechanics Geodesics Special Relativity Notation Spacetime Cuves Paticle Motion Massive Paticles Massless Paticles Spacetime Geodesics Gavitation Gavitational and Inetial Mass Einstein s Equivalence Pinciple Local Inetial Fames Ceating Gavity With Acceleation Bending Light With Gavity Gavitational Red Shift Cuved Spacetime Newtonian Gavity fom Geneal Relativity Pound-Rebka Expeiment Paticle Motion Coodinates, Metic Components and Geometies ds vs. g αβ Local Inetial Fames Geneal Stuctue of ds Chistoffel Symbols R 2 in Catesian Coodinates R 2 in Pola Coodinates Chistoffel and Newton Diffeentiating Vectos Paallel Tanspot Paallel Tanspot and Geodesics Riemann Cuvatue Tenso

3 7 Schwazschild Geomety Einstein s Equations Schwazschild Solution Weak Field Visualising Schwazschild Gavitational Redshift Paticle Obits Keple s Law in Geneal Relativity Pecession of the Peihelion fo a Massive Body Obits of Massless Paticle Ambiguity in λ Photon Obits Deflection of Light Shapio Time Delay Gavitational Collapse Coodinate and Physical Singulaities Eddington-Finkelstein Coodinates Radial Null-Cuves Collapse of Pessueless Matte Black Holes Gyoscopes Newtonian Gyoscope Gyoscopes in Obit Ke Solution The Singulaity Theoems Geneal Stuctue of the Ke Solution Black Hole Mechanics Themodynamics of Black Holes Hawkin Radiation/Tempeatue

4 Chapte 1 Beyond Newtonian Gavity and Geomety 1.1 Inconsistency of Newtonian Gavity Newtonian Gavity conflicts with special elativity. To see why, fist it is useful to see why Coulombs law is incoect. Coulombs foce law is: F C = 1 4πε o q 1 q 2 1t) 2t) Hee the instantaneous distance between points is needed, and because thee is no notion of simultaneity in special elativity then this law beaks down. This can be coected by Maxwell s field equations since they ae elativistically covaiant and contain within them the Loentz symmety of special elativity. The action at a distance of Coulombs law is eplaced by popagating waves. Newtons Gavity is also inconsistent with natues laws because it again acts via action at a distance. m 1 m 2 G = G F 1t) 2t) This cannot be solved by using a Maxwellian Gavity because mass is not invaiant unde Loentz tansfomations unlike chage in Coulombs law. The esolution is geneal elativity. Fom Galileo s pinciple of equivalence then objects ae known to fall with the same acceleation in a gavitational field iespective of thei mass. This leads to Einsteins statement that the pesence of mass causes spacetime to cuve and paticles tomove on thei shotest tajectoy. 1.2 Geometies The two geometies concened in this chapte ae the Euclidean plane, R 2 and the two-sphee, S 2 defined by x 2 + y 2 + z 2 = a Shapes on a Euclidean Plane A tiangle on R 2 has: Cicles have cicumfeence of: These ae pincipal obsevables. inteio angles = π C R = 2π 4

5 1.2.2 Shapes on a Two-Sphee A tiangle is a moe subtle object on a sphee. This is because a staight line is now a segment of a geat cicle athe than being staight as imagined on a 2D plane. Fo example: Figue 1.1: This tiangle has inteio angles = 3 2 π In geneal then: inteio angles = π + A a 2 A is the aea of the tiangle and a is the adius of the sphee. When A a 2 then the flat space case is ecoveed. This is why measuing a sensible sized tiangle on Eath gives π locally athe than a value geate than π. Cicles on S 2 no longe have the adius expected when on R 2 since all the popeties of the cicle must be descibed by points and lengths on the sphee. ϑ Figue 1.2: Cicle on a Two-Sphee As the adius of the cicle is now a sin ϑ and the ac-length is ϑ = R a then: C sin R/a = 2π R R/a The cicumfeence is affect by the geomety of the backgound suface. In the limit whee R a then C/R 2π as expected fo R 2. 5

6 Chapte 2 Line Elements Intinsic stuctue is detemined solely by measuements within the geomety, in paticula the distance between neighbouing points, ds, the line element. 2.1 R 2 The distance between points on R 2 in Catesian coodinates is: ds 2 = dx 2 + dy 2 This is in the limit when ds 0 and uses simple Pythagous to find. It can also be expessed in pola coodinates o any othe possible coodinate system): ds 2 = d dϑ 2 To calculate finite distances o quantities using the line element then it needs to be integated between the two points. Fo cicles on R 2 then in Catesian coodinates the line element is ds 2 = dx 2 + dy 2 whee x = a cos ϑ and y = a sin ϑ so that: ds 2 = da cos ϑ) 2 + da sin ϑ) 2 = a sin ϑdϑ) 2 + a cos ϑdϑ) 2 = a 2 dϑ 2 sin 2 ϑ + cos 2 ϑ) = a 2 dϑ 2 The cicumfeence of a cicle is obtained fom: C = ds = a 2π 0 = a [ϑ] 2π 0 = 2πa Using pola coodinates fo the line element instead simplifies this calculation since the equation fo a cicle of adius a is = a so d = 0 leaving the line element: ds = adϑ dϑ 6

7 This can now be integated: 2.2 S 2 ds = a 2π 0 = a [ϑ] 2π 0 = 2πa dϑ The line element on the two-sphee can be found easily by embedding it in R 3. This gives the elations: x = a sin ϑ cos φ y = a sin ϑ sin φ z = a cos ϑ These can be veified to be the coect equations by using the equation of a sphee and substituting in: The line element in R 3 is: x 2 + y 2 + z 2 = a 2 sin 2 ϑ cos 2 φ + a 2 sin 2 ϑ sin 2 φ + a 2 cos 2 ϑ = a 2 sin 2 ϑcos 2 φ + sin 2 φ) + cos 2 ϑ ) = a 2 sin 2 ϑ + cos 2 ϑ) = a 2 ds 2 = dx 2 + dy 2 + dz 2 The line element of the two-sphee geomety can then be constucted: ds 2 = da sin ϑ cos φ) 2 + da sin ϑ sin φ) 2 + da cos ϑ) 2 = a cos ϑ cos φdϑ + a sin ϑ sin φdφ) 2 + a cos ϑ sin φdϑ a sin ϑ cos φdφ) 2 +a sin ϑdϑ) 2 = a 2 cos 2 ϑ cos 2 φdϑ 2 + a 2 sin 2 ϑ sin 2 φdφ 2 2a 2 cos ϑ sin ϑ cos φ sin φdϑdφ +a 2 cos 2 ϑ sin 2 φdϑ 2 + a 2 sin 2 ϑ cos 2 φdφ 2 + 2a 2 cos ϑ sin ϑ sin φ cos φdϑdφ +a 2 sin 2 ϑdϑ 2 = a 2 [ cos 2 ϑcos 2 φ + sin 2 φ)dϑ 2 + sin 2 ϑsin 2 φ + cos 2 φ)dφ 2 + sin 2 ϑdϑ 2] = a 2 [ cos 2 ϑ + sin 2 ϑ)dϑ 2 + sin 2 ϑdφ 2] = a 2 [ dϑ 2 + sin 2 ϑdφ 2] Now when calculating the cicumfeence of a cicle it is easiest to stat at the noth pole and move though an angle ϑ = ϑ 0 towads the equato so that dϑ = 0 meaning: C = ds 2π = a sin ϑ 0 dφ 0 = a sin ϑ 0 [φ] 2π 0 = 2πa sin ϑ 0 7

8 The adius is obtained when dφ = 0 so that: Reaanging this so that ϑ 0 = R/a then: ϑ0 R = a dϑ 0 = aϑ 0 C sinr/a) = 2π R R/a This is witten in tems of the physical quantities, a and R instead of quantities that depend on the coodinate system. 8

9 Chapte 3 Vatiational Pinciples The extema is geneally thought of as f = 0 which can equivalently be witten as: x The fist tem vanishes. An action is given by: δf = 0 + Oδx 2 ) = df dx δx + Oδx2 ) S[x] = tb t A dtlx, ẋ) The action is minimised when δs = 0 and eveals the xt) extema, xt) xt) + δxt), whee any petubations vanish at the end points due to the bounday conditions, δxt A ) = δxt B ) = 0: δs[x] = S[x + δx] S[x] = = = tb t A tb t A tb t A dt [Lx + δx, ẋ + δẋ) Lx, ẋ)] [ ] L L dt δx + x ẋ δẋ [ L dt x δx + d ) L dt ẋ δx d dt ) ] L δx ẋ An integal of a total deivative evaluated at the end points, such that δx vanishes, means that: tb t A [ )] d L dt dt ẋ δx = 0 This means that: δs[x] = tb t A [ L dt x d ] L δx dt ẋ As vaiation must allow δs[x] = 0 δx then fo this to be tue then: These ae the Eule-Lagange equations. L x d L dt ẋ = 0 9

10 3.1 Classical Mechanics The Lagangian is given by the diffeence between the kinetic and the potential enegy, L = T V x). If T = 1 2 mẋ2 then the Eule-Lagange equation is: d dt L ẋ L x ) = V x = d dt mẋ) = mẍ L x d L dt ẋ = V x mẍ This is Newton s 2 nd Law. 3.2 Geodesics The path which extemises the distance between two points on any suface can be calculated using vaiational pinciples. Fo R 2 with a paametisation given by σ, so that xσ), yσ)) descibes the cuve, then stat with the line element: ds 2 = dx 2 + dy 2 This can be witten as: The action is theefoe: ) 2 ds = dσ ) 2 dx + dσ ) 2 dy dσ S = = = B A B A B A ds ds dσ dσ ) 2 dx dσ + dσ ) 2 dy dσ The Lagangian is theefoe: Lẋ, ẏ) = ) 2 dx + dσ ) 2 dy dσ Extemising this action gives the path taken between A and B. The Eule-Lagange equations ae: d L dσ ẋ = 0 d L dσ ẏ = 0 ẏ L L ẋ = ẏ ẋ L ẋ = constant = ẋ L L ẏ = constant = dy dx = ẏ = constant L This is the equation fo a staight line as expected. On S 2 then the geodesics coespond to geat cicles. 10

11 Chapte 4 Special Relativity Flat spacetime, R 1,3 = M, is descibed by: ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 This is the Minkowski line element. The otation goup of R 3 is now expanded into Loentz tansfomation fo R 1,3 and the line element is invaiant unde this goup. ds efes to the pope time, τ, and so fo timelike sepaation, ds 2 < 0 then: c 2 2 = ds 2 Clocks measue. 4.1 Notation The coodinates ae defined as x 0 = ct, x 1 = x, x 2 = y, x 3 = z. Vecto components ae denated by v α o v α. The inne poduct is constucted as: R 3 ṽ, w ) = ṽ w = v 1 w 1 + v 2 w 2 + v 3 w 3 = δ ij v i w j α,β = δ ij v i w j = v i w i R 1,3 A, B) = A 0 B 0 + A 1 B 1 + A 2 B 2 + A 3 B 3 = η αβ A α B β α,β = η αβ A α B β = A α B α η αβ is the Minkowski metic whee η 00 = 1 and when i = j then η ii = 1 and when i j then η ij = 0. The Minkowski metic and its invese have the popety that: η αβ η βγ = δ γ α In tems of matices then ηη 1 = I. Raising and loweing indices now can be achieved using these matices: v α = η αβ v β, v α = η αβ v β 11

12 The line element can be defined now as: ds 2 = η αβ dx α dx β Due to the stuctue of η αβ then the inne poduct of vectos can be descibed in thee diffeent ways. ds 2 < 0 timelike ds 2 = 0 null o lightlike ds 2 > 0 spacelike This teminology is boowed fom: ds 2 < 0 timelike points ds 2 = 0 null o lightlike points ds 2 > 0 spacelike points 4.2 Spacetime Cuves Fo a cuve x α σ) the tangent has components T α = dxα. Fo a unit cicle xσ) = cos σ and yσ) = sin σ dσ then the tangent components ae: T x = sin σ, T y = cos σ A timelike cuve has: η αβ T α T β < 0 = η αβ dx α dσ dx β dσ < 0 = ) 2 ds < 0 dσ 4.3 Paticle Motion Massive Paticles Massive paticles have a pope time τ and as such it can be used to quantify the paamete σ. The components of the tangent vecto ae: u α = dxα A fee massive paticle satisfies: du α = d2 x α 2 = 0 Thee is no acceleation if thee is no foce applied Massless Paticles Massless paticles do not have a pope time and as such they cannot be paameteised by τ. The position of the photon at paamete λ is: x α λ) = A α + v α λ 12

13 A α is a constant fou-vecto which denotes the position at λ = 0 and v α is any constant null vecto. This type of paametisation is called affine. Fom this it can be seen that: dx α dλ dv α dλ = v α = d2 x α dλ 2 = 0 This has the same fom as fo massive paticles. 4.4 Spacetime Geodesics The woldlines of fee paticles between two timelike sepaated points extemises the pope time between them. Taking c = 1 then: τ AB = = = B A σb σ A σb dσ dσ σ A dσlx, ẋ) The pope time inteval is 2 = dt 2 dx 2 dy 2 dz 2 and so: L = dσ = dt ) 2 2 ) dx 2 2 ) dy 2 2 dz 2 dσ dσ dσ dσ ) 2 Using the Eule-Lagange equations then the equations of motions ae: ) d L dσ ẋ α = 0 d dσ ẋα ) L = L d = L d = 0 ẋα ) L ) dx α Fo massive paticles then: d 2 x α 2 = 0 These equations of motion aise due to the paticle tavelling along geodesics. 13

14 Chapte 5 Gavitation 5.1 Gavitational and Inetial Mass The inetial mass, m i, aises as a eaction to acceleation, as seen fom Newton s 2 nd Law: a = F mi m i is a measue of the tendency to esist foces. The gavitational mass, m g, appeas in Newton s Law of Gavitation: = m g g F m g gives the eaction of a paticle to a gavitational field g. Togethe these show that the acceleation of a paticle in a gavitational field is: a = m g g m i It seems to be tue that m g = m i always and is attibuted to Galileo in Galileo s pinciple of equivalence. Suppose that a paticle p is acted on by gavity and an inetial foce, F. p 0 g Since the laboatoy is in fee-fall then 0 = g. It is also known that = + mg. Fom the efeence m F fame of the laboatoy being stationay then needs to be found. Since = 0 then: m = m m 0 = F + mg mg = F In Newtonian gavity then an expeiment in the laboatoy appeas to have non-gavitational foces only fom the obsevations in the laboatoy. Hee it has been assumed that m i = m g = m. 14

15 5.2 Einstein s Equivalence Pinciple Einstein s equivalence pinciple states that an expeiment in a sufficiently small, fee-falling laboatoy ove a sufficiently small time gives esults indistinguishable fom those in Minkowski spacetime. These constaints on the laboatoy being small and expeiments taking place ove shot times must exist since inetial foces aise othewise. d p a eff a eff q O Both p and q acceleate towads O so an obseve in the fee-falling laboatoy would see an effective attactive foce between the paticles. This is the tidal foce: a q ã p = GM R 3 q p) Whee M is the mass of the gavitating object. Since q p R then: a T GM R 3 d As d gets smalle then the tidal foce gets smalle. Even if the laboatoy is small, if the expeiment is done ove a long time then thee will appea to be a non-existent macoscopic attaction between the two paticles. 5.3 Local Inetial Fames In a small laboatoy, ove a shot timescale inetial coodinates t, x, y, z) can be set up as they can in Minkowski space. This defines a local inetial fame. The coodinates of a local inetial fame only cove a small egion aound a point p. Thee ae infinitely many local inetial fames at p, each distinguished by its velocity. These fames ae elated by Loentz tansfomations. In each of these local inetial fames special elativity holds Ceating Gavity With Acceleation An inetial fame can be identified by, fo example, dopping it and fom this an obseve in the laboatoy can tell that acceleation is occuing with espect to the local inetial fame. This means that a laboatoy in a gavitational field on the eath acts exactly the same as a laboatoy which is being acceleated in open space at ate g. This can be used to pove that light is bent by gavity. 15

16 5.3.2 Bending Light With Gavity g As the feely falling fame is locally flat Minkowski spacetime then a photon tavels in staight lines and so emains paallel to the floo. Fo an obseve on the gound then the laboatoy is acceleating downwads with ate g and since the photon emains paallel to the floo then that too must acceleate downwads at ate g. 5.4 Gavitational Red Shift As well as affecting matte, photons ae affected by gavity both by changing thei path and thei fequency. h A g 0 B The fequency at A is lowe than at B which tanslates to a lowe enegy, E = hν, due to climbing out of the gavitational field. Using the equivalence pinciple the gavitational expeiment can be eplaced with an acceleating system. The local inetial fame in the gavitational expeiment is identified as acceleating up at ate g and so the ocket needs to be acceleating up at ate g with espect to a local inetial fame. The height of A in this acceleating fame is: y A t) = h gt2 And the bottom of the laboatoy at B has a height: y B t) = 1 2 gt2 Hee the velocity of the laboatoy is small, v c, so that only gavitational effects ae seen and not special elativistic. Also h is small. Two signals ae sent fom A in the fom of photons. These ae sepaated in time by a pope time inteval at A, τ A. These signals ae eceived at B with a sepaation of τ B. The fist pulse leaves A at t = 0 and is eceived at t = t 1. The second pulse theefoe is sent at t = τ A and eceived at t = t 1 + τ B. Since photons tavel at the same speed then the fist photon tavels y A t = 0) y B t = t 1 ) = ct 1. The second photon tavels y A t = τ A ) y B t = t 1 + τ B ) = c[t 1 + τ B τ A ]. Combining these to linea ode gives: τ A 1 + gt 1 c 16 ) τ B

17 Since t 1 = hc then: τ B 1 gh ) c 2 τ A This is a puely gavitational effect which states that A ages faste than B. Conveting this back into gavitational quantities, Φ = Φ A Φ B = gh then: τ B 1 Φ ) c 2 τ A If instead of the pulses being sepaate photons they ae now sepaate neighbouing peaks of an electomagnetic wave then using τ A = λ A c and τ B = λ B c 1 + Z = λ B λ A the ed shift is: = τ B τ A 1 Φ ) c 2 When λ B < λ A so that Z < 0 then the signals sent fom A to B ae blue shifted and signals fom B to A ae ed shifted. Photons gain lose) enegy by falling into climbing out of) a gavitational field. The ed shift fomula holds also fo non-linea potentials. 17

18 Chapte 6 Cuved Spacetime Supposing τ measued = 1 Φ c 2 ) τ Minkowski then τ B = 1 Φ use a flat geomety when the undelying geomety is eally cuved. 6.1 Newtonian Gavity fom Geneal Relativity Set c = 1. Conside a line element: ds 2 = [1 + 2Φx, y, z)]dt 2 + [1 2Φx, y, z)]dx 2 + dy 2 + dz 2 ) c 2 ) τ A woks, but it is difficult to Thee ae conditions on Φ such that Φx, y, z) 1 and ) 0. This means that Minkowski spacetime is ecoveed when Φ = 0 and that this geomety Φx is asymptotically flat. t, x, y, z) ae the coodinates in which the souce is at est and so they ae not local inetial fame coodinates. t is the pope time fo a stationay obseve dx = dy = dz = 0) when fa away fom the gavitational souce. This geomety is the weak field solution to geneal elativity Pound-Rebka Expeiment A is at x, y, z) = 0, 0, h) and B is at x, y, z) = 0, 0, 0). A photon is sent fom p 1 at A and is eceived by q 1 at B. A second photon is sent fom p 2 at A and is eceived by q 2 at B. Since the geomety is static independent of t) then the woldline of each photon has an identical shape. B A t B q 2 q 1 p 2 p 1 0 h ta The coodinate time diffeence, t A = t B = t. Using the line element then t can be elated to the pope times at A and B, τ A and τ B. Since A and B ae stationay then thei line element is: This means that the pope times ae: τ 2 = s 2 = 1 + 2Φ) t 2 τ AB = 1 + 2Φ AB t AB 1 + Φ AB ) t AB 18

19 The Taylo expansion of the squae oot is used in the second line. τ B 1 Φ) τ A, the equivalence pinciple is ecoveed again. This is only tue fo small Φ Paticle Motion In the weak field limit with cuves paameteised by t then: 2 = dt 2 [ 1 + 2Φ) 1 2Φ)ẋ 2 + ẏ 2 + ż 2 ) ] Extemising the pope time between points A and B shows: τ AB = B A dt 1 + 2Φ) 1 2Φ)ẋ 2 + ẏ 2 + ż 2 ) Hee Φ 1 and ẋ 1 so that the backets can be expanded out and the φ ẋ tem ignoed so: Binomially expanding this gives: τ AB = τ AB = B A B A dt 1 + 2Φ ẋ + ẏ + ż) dt 1 + Φ 12 ) ẋ + ẏ + ż) The Eule-Lagange equations can then be used to get the equations of motion: ẍ = Φ This is Newton s 2 nd Law. In the weak field limit it is clea that not only is Φ the gavitational potential but also that Newtonian gavity is ecoveed in the appopiate limit. 6.2 Coodinates, Metic Components and Geometies Fom Minkowski space the line element is: ds 2 = dt 2 + dx 2 + dy 2 + dz 2 In lightcone coodinates then t = 1 2 v u) and x = 1 2 v + u) with y and z emaining the same. This gives a line element: The metic components ae: ds 2 = 2dudv + dy 2 + dz g αβ = ds vs. g αβ If x α is known as a function of x β then the metic components g αβ can be found. Since the line element is invaiant, ds 2 = ds 2. This is tue because the geomety doesn t change. As g αβ dx α dx β = ds 2 then: g αβ dx α dx β = g µνdx µ dx ν Since x α is known as a function of x β then: ) x α x β g αβ dx µ x µ dx ν x ν ) = g µνdx µ dx ν This is tue fo all dx µ and dx ν so that: g αβ x α x µ x β x ν = g µν 19

20 These ae how the components tansfom in geneal. In Minkowski space using lightcone coodinates then: g uv = xα u x β v η αβ = t t x x 1) + u v u v 1) = = 1 This shows that the lightcone coodinate metic components ae the same as found befoe Local Inetial Fames Thee exists coodinates whee g αβ = η αβ always. This is tue because g αβ is a symmetic matix which implies that it can be diagonalised. In the new coodinate system then the diagonalised matix can be escaled via the coodinates to get η αβ. This is only tue in a small patch. Futhemoe, the deivative of the metic can always be made to vanish by choice of coodinates fo a point p: g αβ x µ = 0 p 2 g αβ x µ x ν geneally cannot be made to vanish since they ae a measue of cuvatue of tidal foces Geneal Stuctue of ds 2 ds 2 = g αβ dx α dx β is a geneal fomula fo the line element. Othe tems such as v α dx α and w αβγ dx α dx β dx γ ae unphysical since they do not look like Euclidean space. Thee is a type of geomety called Finsle geomety which has K αβµν dx α dx β dx µ dx ν which eveals anisotopic stuctue, but this is not included in geneal elativity. 6.3 Chistoffel Symbols A test paticle has sufficiently small mass that it does not distot the spacetime geomety, it poduces negligible spacetime cuvatue on the backgound. Test paticles follow geodesics. The pope time between between two points is: τ AB = B The Eule-Lagange equations can always be witten as: d 2 x α A dσ + Γ α µνx) dxµ g αβ x) dxα dσ x ν = 0 As a sketch, ignoing indices then this can be shown stating with the Lagangian: The Eule-Lagange equations ae: L = gẋẋ dx β dσ d L dσ ẋ L ) x = d dσ ) gẋ + L g xẋẋ L = 0 20

21 Make σ pope time τ so that σ = τ then L = 1. This leaves: d Simply put, the Chistoffel symbols ae: g g gẋ) + = ẍ + g 1 xẋẋ xẋẋ = 0 1 g Γ = g x In a local inetial fame then g must be zeo and so the Chistoffel symbols vanish. They ae symmetic x in thei lowe indices Γ α βγ = Γα γβ R 2 in Catesian Coodinates The line element is ds 2 = dx 2 + dy 2. The geodesic equations of motion ae: d 2 x ds 2 = 0 and d2 y ds 2 = 0 The Chistoffel symbols ae obviously all zeo since thee ae no R 2 in Pola Coodinates ) 2 dx, ds dy ds ) 2 o dx dy ds ds tems. The line element is now ds 2 = d dϑ 2. The geodesic equations of motion ae found fom the Eule-Lagange equations and ae: d 2 ds 2 dϑ dϑ ds ds d 2 ϑ ds d dϑ ds ds The Chistoffel symbols ae theefoe Γ ϑϑ = and Γϑ ϑ = Γϑ ϑ = 1/. It is seen hee that the Chistoffel symbols ae explicitly not tensos since they do not disappea in all coodinate systems if they have disappeaed in any Chistoffel and Newton In Catesian coodinates a fee paticle has: d 2 x dt 2 = 0 = 0 = 0 Now in pola coodinates then x = cos ϑ and y = sin ϑ so that: d 2 ) 2 dϑ dt 2 = d2 dϑ dt dt 2 + Γ ϑϑ dt ) 2 d 2 ϑ dt dϑ dt ) d dt = 0 ) = d2 ϑ d dt 2 + dϑ 2Γ ϑ dt dt = 0 Although the Γ α βγ belong on the left-hand side of the equation as pat of the stuctue, it can also be seen as an imaginay foce, like the Coiolis foce. The imaginay foce aises due to the choice of coodinates. Geneally the Newtonian equations of motion ae: d 2 x i dt 2 + dx j dx k Γi jk dt dt = 0 21

22 6.3.4 Diffeentiating Vectos A vecto field which is constant and points in a diection given by α in Catesian coodinates is: v y) = l cos α, l sin α) x, Fo this to be constant then vi = 0. In pola coodinates then the vecto field is witten as: xj The deivatives ae now given by: v = l cos α cos ϑ + l sin α sin ϑ v ϑ = l sin α cos ϑ l cos α sin ϑ i) v = 0 ii) v ϑ = 1 vϑ i v + 1 vϑ = 0 iii) ϑ v = v ϑ ϑ v vϑ = 0 iv) ϑ v ϑ = 1 v ϑ v ϑ + 1 v = 0 The vecto field is now chaacteised by these fou equations. The geneal way this can be witten is as the covaiant deivative: i v j = i v j + Γ j ik vk The covaiant deivative tansfoms as a tenso, even though i and Γ j ik do not on thei own Paallel Tanspot Since tanspoting vectos is non-tivial then covaiant deivatives need to be used. Without indices, then a sketch of the deivative is: ṽx + δx) ṽx) ṽ = δx The subtaction does not make sense unless the vectos ae at the same point. x) v x ṽ x + δx) x + δx v + δx) x 22

23 Tanspoting in some sensible way gives: Since α v β = α v β + Γ β αγv γ then: ṽx + δx) ṽ x + δx) ṽ = δx ṽ α x + δx) = v α x) Γ α βγv β δx γ Γ contols the movement of vectos in a paallel way. This means that paallel tanspot is path dependent. If a vecto moves along two diffeent paths, I and II then the diffeence between ṽ I and ṽ II is a measue of the cuvatue. If ṽ is paallel tanspoted along a cuve with tangent T α then: T α α v β = 0 Fo R 2 then in Catesian coodinates Γ = 0) then tangent vecto is: The covaiant deivative is simple since Γ = 0): T α = dxβ dσ Paallel tanspot equies: dx α dσ α v β = vβ x α + 0 v β x α = dvβ dσ = Paallel Tanspot and Geodesics Geodesics can be defined by consideing: T α α T β = dxβ dσ [ ] T β x α + Γβ αδ T δ = dt β dσ + dx α Γβ αδ dσ T δ = d2 x β dσ 2 + dx α dx δ Γβ αδ dσ dσ = 0 A tangent vecto is paallel tanspoted along a geodesic. This definition of geodesics also applies to null cuves as well as timelike geodesics. The null cuve is paameteised by λ and so the geodesic equation is: d 2 x α dλ 2 + dx β dx γ Γα βγ dλ dλ = 0 Both massive and massless paticles have the same equations of motion but diffeent initial conditions: Massive: ds 2 = c 2 2 g αβ dx α dx β = c 2 Massless: ds 2 = 0 g αβ dx α dλ dx β dλ = 0 23

24 6.3.7 Riemann Cuvatue Tenso The Riemann cuvatue tenso is: [ α, β ]v α = α β v γ β α v γ = R γ δαβ vδ The Riemann cuvatue tenso has deivatives of the connection coefficients which ae themselves deivatives of the metic so the Riemann cuvatue tenso contains 2 g tems at least. 24

25 Chapte 7 Schwazschild Geomety 7.1 Einstein s Equations If matte is pesent then spacetime is cuved. It is in fact also tue that if a egion contains no matte then the vacuum equations need to be solved, but cuvatue can still be pesent. One solution to the Einstein equations is Minkowski space, g αβ = η αβ. Anothe solution to the matte fee equations is gavitational adiation, the analogue of photons. 7.2 Schwazschild Solution The Schwazschild solution has spheical symmety which gives ise to the S 2 line element ds 2 = dϑ 2 + sin 2 ϑdφ 2 which has no coss tems, ddϑ o ddφ. As well as this it is also time-independent so that it is static. This means that thee ae no coss tems such as dtdx. The ansatz fo the line element is: It is convenient to set W R) = 2 so that: ds 2 = UR)dt 2 + V R)dR 2 + W R)[dϑ 2 + sin 2 dφ 2 ] ds 2 = g tt dt 2 + g )d [dϑ 2 + sin 2 ϑdφ 2 ] The ten functions, g αβ, of fou coodinates has been educed to two functions of one coodinate. Using Einstein s field equations fo a matte fee univese then it can be seen that the solution is: ds 2 = 1 2GM ) c 2 c 2 )dt GM ) 1 ) c 2 d dϑ 2 + sin 2 ϑdφ 2 ) ) Hee M is constant and G is Newton s Gavitational constant. Bikhoff s theoem states that spheical systems in a vacuum ae always time-independent and so it is not a equiement to state that the Schwazschild solution is time-independent. In fact, Bikhoff s theoem states that the Schwazschild solution is the unique spheically symmetic vacuum solution to geneal elativity. t,, ϑ, φ ae the Schwazschild coodinates. Sufaces at constant t and have: ds 2 t,=constant = 2 dϑ 2 + sin 2 ϑdφ 2 ) The aea of such a sphee is 4π 2 since it is the same as the line element fo S 2. is not the distance to the cente of the sta since this is given by: ds d = t,ϑ,φ=constant 0 1 2GM As then the line element becomes: ds 2 dt 2 + d dϑ 2 + sin 2 ϑdφ 2 ) This is just Minkowski space in spheical pola coodinates and as such Schwazschild geomety is asymptotically flat. 25

26 7.2.1 Weak Field When 2GM then a Taylo expansion of the d 2 coefficient can be made: 1 2GM ) GM Defining a new coodinate, R, whee = R + GM then if 2GM then R GM so that GM Also: GM R. 2 = R + GM) 2 R GM ) R The esult of this is : ds 2 = 1 2GM ) dt GM ) dx 2 + dy 2 + dz 2 ) R R This last pat can be ecognised because the spheical coodinates wee just the line element of R 3. The gavitational potential fo this geomety can be ead off by compaison to the geneal weak field geomety to get: GM Φ = x2 + y 2 + z 2 This is just the Newtonian potential fo a souce of mass M. Something odd happens at the Schwazschild adius, = 2GM = S because the t coodinate vanishes and the coodinate blows up. Since S is inside the massive body of most systems, such as the sun o the eath, then the singulaity is avoided as the Schwazschild solution is only valid in the vacuum. Radius S Eath = m SEath = m Sun = m SSun = m Neuton = m SNeuton = m Neuton stas get much close to the Schwazschild adius and so geneal elativity needs to be consideed a majo facto in these egions Visualising Schwazschild Consideing t and ϑ = π/2 constant such that: ds 2 = 1 S ) d dφ 2 It can be seen that this is just the geomety of z) = 2 S S ) embedded in R 3, d dφ 2 +dz 2 ). 7.3 Gavitational Redshift An obseve, A, is at coodinate position A and an obseve, B is at B whee B > A > S and both obseves ae stationay and ae at the same ϑ and φ. Two signals sepaated by τ A is sent fom A and eceived at B sepaated by τ B. 26

27 t t B t A S A B The pope time between the fist and second signals being sent can be calculated. Since d = dϑ = dφ = 0 then the line element is: ds 2 = 1 2GM ) dt 2 The pope time is theefoe: τ A = 1 2GM t A The pope time between the eceived signals can be calculated simply as: Equating these gives: τ B = τ B = 1 2GM t B 1 2GM A 1 2GM B This is the exact esult. To calculate the edshift then the signals ae taken to be successive wave peaks so that, using τ = λ/c, then: 1 + Z = λ B λ A = 1 2GM A 1 2GM B τ A As A S then the edshift gets geate and geate, 1 + Z. 7.4 Paticle Obits The geodesic equations fo the Schwazschild geomety can be found using the Eule-Lagange equations and fo t, ϑ and φ: 1 2GM ) dt = e = constant 2 sin 2 ϑ dφ = l d 2 dϑ ) l2 cos ϑ 2 sin 3 ϑ = 0 27 = constant

28 By choosing ϑ = π then these equations ae simplified geatly. This can be done since the coodinates 2 can be abitaily otated to be witten as such with no loss of geneality. O P ϑ = π 2 Fistly the coodinates ae otated such that ϑτ 0 ) = π/2 which is the equatoial plane. Next the coodinates ae otated about OP such that the paticle velocity is in the plane. This means that dϑ = 0 so that the equations of motion become: 1 2GM ) dt = e = constant 2 dφ = l = constant The thid equation is solved with these initial conditions. The physical significance of e and l can be analysed in the classical egime whee S so that the geomety is that of Minkowski and whee the velocities ae small so that dt. e is the enegy pe unit mass as seen by: e dt E m = p0 m and l 2 dφ dt = U 0 = dx0 = dt l has the fom the angula momentum pe unit mass. When not at lage and at elativistic speeds then e and l ae the geneal elativistic analogues of enegy and angula momentum pe unit mass of the paticle which ae conseved. The adial geodesic can be calculated using the Eule-Lagange equations also, but it can be found moe easily since massive paticles have ds 2 = 2 : ) 2 ds = 1 2GM ) ) 2 dt + 1 2GM ) 1 ) 2 dϑ ) 2 ) ) 2 d dφ sin 2 ϑ = 1 2GM ) e 2 1 2GM ) GM = 1 2GM ) 1 e GM = 1 ) 1 ) 2 d + l2 2 ) 1 ) 2 d + 2 l

29 e 2 = 1 2GM ) 1 + l2 2 ) + ) 2 d = 1 2GM + l2 2 2GMl2 3 + ) 2 d 1 2 e2 1) = 1 2 E = 1 2 ) 2 d GM + l2 2 2 GMl2 3 ) 2 d + V ) This equation is the consevation of enegy, T + V = E. The only diffeence between this and the Newtonian calculation is the GMl2 tem and the fact that t τ. It is this tem which causes the 3 pecession of the peihelion. By looking at V ) then the solutions can be analysed. The potential takes the fom: V ) S The extema of V ) ae: max min = l2 2GM ) 2 GM 1 ± 1 12 l 0 < l < 12GM The squae oot is imaginay and so thee ae no extema bottom line). This is diffeent to the Newtonian case since the mass dominates and causes the paticle to spial into the gavitating body athe than tavelling in closed obits. All inwad diected paticles each S and all outwad diected paticles with E < 0 also each S but if E > 0 then they can escape to infinity. In the case when E < 0 then the tuning point has d = 0 so that V ) = E. 29

30 Inwad Tajectoy Initially Outwads Tajectoy E > 0 Initially Outwads Tajectoy E < 0 l = 0 When thee is no angula momentum then the potential cuve still looks like the bottom line but thee is no spialling since thee is no angula velocity. This is called adial plunge. If the initial conditions ae d = 0 then E = 0 and so e = 1. Since E = 0 fo the whole tajectoy then: = This can be solved analytically and it is found that: τ) = 1 2 ) 2 d = GM ) GM) 3 τ τ0 ) Whee τ 0 is a constant of integation. To find out how the time component changes with espect to distance fom the gavitating body then: dt d = dt d This gives a diffeential equation: 2GM = ) 1 t = t GM GM 3 2GM Plotting vs. τ and t vs. τ gives: 2 1 2GM ) 1 [ ln 1 + ) ln 1 )] 2GM 2GM τ) tτ) S τ τ 0 τ τ τ An obseve moving in the est fame of an in-falling paticle has the same pope time as the paticle which takes a finite time to hit the Schwazschild adius. Fo an obseve at spatial infinity then the pope time is τ = t. Since t diveges at τ then it appeas to take an infinite time to each S accoding to this obseve. l > 12GM The potential now looks like the uppe thee cuves of the gaph on page 28. The top line is l > 4GM, the next is l = 4GM and the thid one down is l = 12GM. The cuve when l > 4GM is: 30

31 V ) E > V min 0 < E < V min S min max V max < E < 0 When E > V min then the paticles can ovecome the baie and so each S. When 0 < E < V min then the paticle can come in and each a minimum value of befoe leaving to spatial infinity. When V max < E < 0 then the obits ae bound, but it is not yet known if these obits ae closed. 7.5 Keple s Law in Geneal Relativity Ω 2 Newton = GM 3 Ω is the ate of change of azimuthal angle with espect to t, whee t is the pope time of someone sat at infinity: Ω = dφ dt = dφ dt 1 ) S = l 2 e If the paticle is in the minimum of the potential then the situation is esticted to cicula obits, = max. max has aleady been calculated and it can be eaanged as: l 2 = Smax 2 2 max 3 S 31

32 Also, since the obits ae cicula then d = 0 so that V max) = E and because E = 1 2 e2 1) then: ) e 2 = 1 + l2 1 ) S max This can be used to define the atio: 2 max l e = s max 2 1 ) 1 S 2 max This theefoe means that the fequency is: 1 ) S S max max 2 Ω = max 2 1 S max = 2GM 3 max So at = max then: Ω 2 = GM 3 max This is exactly the same elation as in the Newtonian case, even with no appoximation. But thee ae two subtleties: the obseve is at spatial infinity; and is not the distance to the cente of the sta, athe it is a coodinate in this spacetime. 7.6 Pecession of the Peihelion fo a Massive Body The amount by which a point of minimum adius moves ove subsequent obits can be calculated using geneal elativity. If the obit is closed then φ changes by 2π. The pecession angle is δφ and can be defined by φ = 2π + δφ. φ is the total change in φ ove one obit. δφ just measues the deviation fom closue. The total deviation can be calculated using: φ = 2 1 dφ d d + 2 dφ = 2 1 d d 1 2 dφ d d Hee 1 and 2 ae the minimum and maximum distances of the obit. This elation is tue because thee is complete symmety in the obit. Using the chain ule then: dφ d = dφ d = l [ 2 e 2 1 S ) 1 + l2 2 )] 1 2 The integal is theefoe: 2 [ d φ = 2l 1 2 e 2 1 S ) 1 + l2 2 )] 1 2 This can be solved exactly but it is easie to do petubatively since S δφ = φ 2π = 3π S ) ) 2 S 1 ˆε a + O a 32 is small and as such:

33 a is the semi-majo axis of the obit and ˆε is the eccenticity of the obit: a = ) and ˆε = Fo mecuy then S = 3km, a = km, ˆε = 0.2 so that: The obit of Mecuy takes 88 days so: δφ ad obit 1 δφ 43 pe centuy 7.7 Obits of Massless Paticle Fo null geodesics then τ λ in the Eule-Lagange equations: 1 S ) ṫ = e, 2 sin 2 ϑ φ = l, d ) 2 ϑ l2 cos ϑ dλ 2 sin 2 ϑ = 0 Again ϑ = π/2 can be used, which simplifies the equations with no loss of geneality. Now in a simila way to using ds 2 = 2 fo massive paticles, ds 2 = 0 fo massless paticles. This means that: 1 l 2 ) 2 d + W R) = 1 dλ b Whee b is the atio of the angula momentum to the enegy b = l e and W R) = S ) Ambiguity in λ The Eule-Lagange equations and the null equation ae invaiant unde tansfomations λ κλ d 2 x α dλ 2 + dx β dx γ Γα βγ dλ dλ = 0 = d2 x α dκλ) 2 + Γα βγ dx β dx γ dκλ dκλ = 0 ds dλ = 0 = ds dκλ = 0 This is not tue fo the massive case whee a facto of κ 2 would aise on the ight-hand side in the Eule-Lagange equations. Since the paameteisation is affine then λ can be multiplied by a facto of κ abitaily. Since is physical then W cannot be escaled which means that the escaling must occu in l l/κ and e e/κ. This leaves the paamete b as a physical object. The line element fa fom the souce is: ds 2 dt 2 + d dϑ 2 + sin 2 ϑdφ 2 ) φ φ d Since φ is small then φ d/ whee d is the impact paamete. The path is null so ds = 0 which implies 33

34 that d dt. Witing b out shows that: b = l e = 2 dφ ) ) dλ dλ dt = 2 dφ dt = 2 dφ d d dt 2 d ) 1) d So b is just the geneal elativistic vesion of the impact paamete. 7.8 Photon Obits The point of closest appoach 0 ) has d dλ W ) 0 = 0 which means that W 0 ) = 1. W ) has the fom: b2 S 3 2 S Hee it can be seen that W S ) = 0, W ) 2 and W 0) S /. The stable point of this potential occus when W = 3 S /2) = 0. At = 3 S /2 then: ) 3S 4 W = 2 27 S ) 2 1 b 2 > S ) 2 The total enegy, b 2 ove the baie and heads fo = 0: is geate than the peak of the potential and so the photon goes 1 b 2 < 4 27 S ) 2 Then d dλ changes sign at 0 and so the photons path is deflected by the gavitating souce. 34

35 0 1 b 2 = 4 27 S ) 2 Thee is an unstable cicula obit at = 3 2 S Deflection of Light 0 Sta Image t) φt) b Actual Sta The deflection angle can be defined as dφ def by: φ = = d dφ d d dφ dλ dλ d = 2 d 1 [ ] b 2 W ) 2 When b S 2 then: φ b ) S 2 π + 2 S b Fo the case when light just gazes the suface adius of the sun the S 3km and b km so: dφ def sun) Shapio Time Delay In flat spacetime a signal is sent fom Eath to a eflecto satellite and back and the time taken measued. 35

36 Reflecto Satellite Sun R 0 E Eath ] t flat = 2 [ 2E 20 R Now the geomety of the sta is taken to be Schwazschild and so the path the adio waves takes is: Reflecto Satellite Sun R 0 E Eath is not a physical distance hee but athe the coodinate, unlike in the flat case. The adio signals paths ae calculated fom: dt d = dt dλ dλ d = ± 1 b 1 S ) 1 1 b 2 W ) ) 1 2 = ±F) Fo inceasing then +F) is used and F) is used fo deceasing. Fo the whole path then: [ E R ] t = 2 df) + df) 0 0 At the point of neaest appoach then d dλ = 0 and so W 0 ) = 1 0 b 2 whee b 1 + ) S since b is 2 0 taken to be the impact paamete with a coection due to the geomety. If 0 R and 0 E then: )] 4R E t GR = t flat + 2 S [1 + ln In 1970 the Maine 6 Mission caied out this expeiment and had an ageement with the geneal elativistic coection to a high level of accuacy

37 Maine Delay 250 µs Eath 1 Jan Jun Gavitational Collapse While a sta is buning nuclea fuel, themal pessue balances gavitational contaction. The buning of nuclea fuel follows the ode: Hydogen fuses to Helium Helium fuses to Cabon Cabon fuses to Neon So on This caies on until Ion at which point the nuclea fuel uns out. Thee things can then happen depending on the mass of the sta. If M 1.4M then gavity balances out with electon degeneacy and so leaves a white dwaf. If 1.4M M 3M then gavity is balanced by neuton pessue and so a neuton sta is ceated. Finally is M > 3M then thee is complete collapse beyond S Coodinate and Physical Singulaities A line element such as ds 2 = d dϑ 2 is degeneate at = 0 so the full ange of ϑ can be used, 0 < ϑ 2π but is esticted to > 0. Using a coodinate tansfomation, x = cos ϑ and y = sin ϑ, gives the new line element: ds 2 = dx 2 + dy 2 This also has the condition that x 2 + y 2 > 0 although it can actually be seen that nothing peculia happens at x = y = 0 and so these coodinates can cove moe of the manifold, in fact the entie manifold. The manifold has not been extended at all, but moe is able to be descibed by these coodinates Eddington-Finkelstein Coodinates A new coodinate, v can be intoduced though: t = v S ln 1 S This can be substituted into the line element to give: ds 2 = 1 S ) dv 2 + 2dvd + 2 dϑ 2 + sin 2 ϑdφ 2 ) This is well behaved at = S so that the Schwazschild adius is pat of the geomety. Howeve, = 0 is a physical singulaity whee the cuvatue diveges. The coodinate ange of is 0 < < which still does not cove the whole manifold. 37

38 7.11 Radial Null-Cuves Fo adial motion dϑ = dφ = 0 and fo null-cuves then ds = 0. The line element in Eddington-Finkelstein theefoe becomes: 0 = 1 ) S dv 2 + 2dvd dv factos out and so the adial null cuves can be witten as: [ 1 ) S dv dλ + 2 d ] dv dλ dλ = 0 Type i) cuves have constant v so dv = 0. Fo lage then: dλ t = v S ln 1 S t v This means that as t inceases deceases and so the adial cuves ae ingoing. Type ii) cuves have: d dλ = 1 1 ) S dv 2 dλ [ This can be solved to give v 2 + ln ] S 1 = constant. Fo lage then is dominant ove S ln S 1 so v 2 +. Substituting this into the definition of v then it can be seen that: t + So as t inceases also inceases and thus the cuves ae outgoing. Now intoducing t = v then: t Type ii) Cuves Type i) Cuves = 0 Singulaity S Fo > S then thee ae both outgoing and ingoing null-cuves, but fo < S then both of the cuves ae ingoing Collapse of Pessueless Matte A sta is modelled by pessueless dust so that the suface paticles follow geodesics of the Schwazschild geomety. The adial plunge can be calculated as in section 7.4. l = 0 and E = 0 so that: τ) = ) S 2 τ 0 τ)

39 If τ 1 ) = S and τ 2 ) = 0 then: τ 2 τ 1 = 2 S 3 Fo the sun with 3km then τ 10µs. This adial plunge is vey quick so that the singulaity is eached soon afte passing the Schwazschild adius. By changing τ) into the v coodinate then: t Path of Stas Suface S Sta Anything outside the stas matte so paticles on the line o above) ae in Schwazschild geomety since it is a vacuum. This means that the paticle passes though a eal Schwazschild adius whee its lightcone always points to the singulaity. If the Schwazschild coodinates ae enamed T and t R and the fist two tems of the line element ae swapped then: ds 2 S ) = T 1 dt 2 S ) + T 1 dr 2 + T 2 dϑ 2 + sin 2 ϑdφ 2 ) Fo < S T < S ) then T ) is the time coodinate. The line element wholly depends on T ) the time coodinate and so this is a homogeneous cosmology Black Holes A black hole is defined as a egion of spacetime fom which it is impossible to send a signal to infinity. The bounday of this egion is the event hoizon Gyoscopes In geneal, objects ae descibed by moe than just the mass and fou velocity. The spin fou-vecto, S α, is one such exta popety. In the est fame the spin fou-vecto has no time component, S α = 0, S ) and the fou velocity vanishes, U α = 1, 0 ). This means that: U α S α = 0 Although this is constucted in the est fame, it is a scala equation and so holds in all fames. Minkowski spacetime S α is constant if no toque acts: d Sα = 0 In 39

40 This is theefoe tue in a local inetial fame since S α is a vecto and tansfoms as: S α = x α x β Sβ Tansfoming the pope time deivative of S α gives: d S α = d x α x β Sβ This is not a vecto tansfomation and so the pope time deivative of the spin fou-vecto vanishing is not valid in all fames of efeence. The coect tansfomation can be found by compaing it to the tansfomation of the pope time deivative of the fou-velocity. In a local inetial efeence this is: In geneal the tansfomation is: Whee D is the covaiant deivative. fou-vecto: d U α = 0 D U α = δγ α d U γ + Γ α βγ U β U γ = 0 ) This can now be used to find the tansfomation fo the spin D Sα = d Sα + Γ α βγ U β S γ = 0 It is known that Γ α βγ = 0 in the local inetial fame so D Sα = which holds in all fames Newtonian Gyoscope The coodinate tansfomation fom Catesian to pola coodinates is given by: The spinning top has components: x = cos φ y = sin φ z = z S = S x cos φ + S y sin φ S φ = S x sin φ + S y cos φ The Newtonian equations with no toque in Catesian coodinates ae: Fistly taking S y = S z = 0 then: Hee S x sin φ d dt S = 0 d dt S S x sin φ ) dφ dt d dt S = d dt Sx cos φ) = S x sin φ dφ dt = 0 = S φ and = Γ φφ. Fo abitay then: S d dt Si + Γ i jks j v k = 0 d Sα = 0. This is a tenso equation 40

41 Gyoscopes in Obit The geneal elativistic vesion of the the Newtonian gyoscope can be used to calculate the effect of gyoscopes in obit. Taking cicula obits whee = max then: ) 2 dφ = GM dt max 3 Fo U = U ϑ = 0 and: U φ = dφ = dφ dt dt = ΩU t This means that U α = U t 1, 0, 0, Ω). This scala equation is U α S α = 0 so g αβ U α S β = g tt U t S t + g φφ U φ S φ = 0. This eveals: S t = 1 ) 1 S Ω maxs 2 φ max This means that: d dt S = This is a simple hamonic oscillato since: Hee ω = Ω 1 3 S d dt Sφ = Ω S max max 3 ) 2 S ΩS φ d 2 dt 2 S = ω 2 S. It can be seen hee that the geomety effects the diection of the spin vecto 2 max 1 3 S. 2 max since its angula fequency is diffeent to the obital angula velocity by a small facto of 41

42 Chapte 8 Ke Solution 8.1 The Singulaity Theoems Thee ae fou statements about singulaities. Hawkins and Penose showed that singulaities ae inevitable once collapse has poceeded fa enough. Thee is a cosmic censoship conjectue which states that all singulaities which aise fom physical collapses ae hidden behind hoizons. The black hole uniqueness theoem states that thee is only one vacuum black hole solution, that of a otating black hole. This is the Ke solution. Physical black holes settle to Ke on time scales of ode GM 10µs fo M ). 8.2 Geneal Stuctue of the Ke Solution Thee ae two hoizons fo otating black holes. These ae: φ e ϑ) ϑ + The outside of the egosphee, e, is defined when g tt = 0 so that: [ ] a 2 e ϑ) = GM 1 + M 2 cos ϑ a = J/M whee J is the angula momentum of the black hole. The oute hoizon, +, occus when g : [ ] ± = GM 1 ± 1 a2 M 2 An obseve can escape the egosphee but not the hoizon. Futhemoe, obseves cannot be stationay inside e because they ae foced to otate since the lightcone point aound the black hole. 42

43 Chapte 9 Black Hole Mechanics 9.1 Themodynamics of Black Holes Zeoth Law The hoizon has a constant suface gavity, κ. This is defined as the local pope acceleation multiplied by the ed shift, both of which when taken at infinity ae infinite on thei own. Fo Schwazschild geomety then: Fist Law κ S = c4 4GM If a small amount of mass is dopped into a black hole then its aea gows by a small amount: dm = κ da + ΩdJ 8πG Hee Ω is the obital angula fequency and J is the angula momentum. Fo Schwazschild geomety, whee J = 0 and S = 2GM/c 2 and A S = 4π 2 S so: dm S = κ S 8πG da S Second Law Thid Law The aea of a black hole neve deceases: da dt 0 Black holes cannot be constucted such that: κ = Hawkin Radiation/Tempeatue Beckenstein stated that the black hole entopy should be popotional to the aea. Hawking s esponse was to state that this was not tue because black holes ae black and so have zeo tempeatue and as such the entopy cannot be popotional to the aea. Beckensteins agument comes fom: dm = κ 8πG da de = T ds And so T M 1. If black holes have tempeatue then they must emit adiation. As the black hole can be thought of as static then the only way that adiation can be emitted is though quantum mechanical pocesses. This means that thee must be a facto of in the elationship. Looking at the powe emitted: de dt dm dt 43

44 Since the powe emitted fom adiation is: This implies: Using dimensional analysis then: σat 4 = σ4π 2 S T 4 M 2 T 4 dm dt cα M β G γ dm dt c3 1 G M This indeed shows that T M 1 as Beckenstein had stated. The constant of popotionality cannot be fixed this way but caying out the full calculation eveals the Hawking Tempeatue to be: k B T H = c3 8πGM S + Pai Ceation Themal adiation comes fom pai ceation on the hoizon of the black hole. Although the mass of the antimatte is positive, the enegy is technically negative because of the definition of enegy changing within the event hoizon. This negative enegy paticle causes the black hole to dissipate. 44