Chapter 4: Christoffel Symbols, Geodesic Equations and Killing Vectors Susan Larsen 29 October 2018
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1 Content 4 Chistoffel Symbols, Geodesic Equtions nd illing Vectos Chistoffel symbols Definitions Popeties The Chistoffel Symbols of digonl metic in Thee Dimensions Cylindicl coodintes The non-zeo Chistoffel symbols The geodesic eqution fo cylindicl coodintes The plne in Ctesin coodintes Solve the geodesics equtions of the plne in Ctesin coodintes The Hypebolic Plne Geodesics in the Hypebolic Plne The Flt Spce-time in two dimension The time-like geodesic X(T of the Flt Spce-time metic in two dimensions Ae these geodesics spce-like o time-like Is the wold-line XT Acosh(T time-like o spce-like: Thee-dimensionl flt spce-time Null geodesics in thee-dimensionl flt spce-time The wom-hole geomety Volume in the Womhole geomety Equtions fo geodesics in Womhole Geomety The tvel time though womhole Wp spce-time symmety: How much ship time, Δτ, elpses on tip between sttions tht tkes coodinte time t ΔT? Clssiclly Anti-de Sitte Spcetime Clssiclly Anti-de Sitte Spce-time is confomlly elted to the Einstein cylinde The Pth of light y the null geodesics in the Clssic Anti de Sitte spce-time... 6 Refeence... 8
2 4 Chistoffel Symbols, Geodesic Equtions nd illing Vectos 4. Chistoffel symbols. 4.. Definitions The Chistoffel symbols of fist kind Γ bc ( g bc + b g c c g b The Chistoffel symbols of second kind Γ bc gd ( c g db + b g dc d g bc The connection between the Chistoffel symbols of the fist nd the second kind Γ bc g d Γ bcd Notice: The thid subscipt is ised 4.. b Popeties Γ bc ( g bc + b g c c g b ( bg c + g cb c g b Γ bc Γ ( g + g g b ( g Γ b ( g b + g b b g ( g b b g Γ bb Γ bb ( g bb + b g b b g b ( g bb Γ bc + Γ cb ( g bc + b g c c g b + ( g bc + c g b b g c g bc Γ bc gd ( c g db + b g dc d g bc gd ( b g dc + c g db d g cb Γ cb Γ 3 g d Γ d Γ bb g d Γ bbd Γ b Γ b 4 g d Γ bd 4..3 The Chistoffel Symbols of digonl metic in Thee Dimensions The line element ds g xx dx + g yy dy + g zz dz The metic tenso nd its invese g b {g xx g yy g zz } Symmetic in the fist two indices: Γ bc Γ bc Symmetic in the lst two indices: Γ bc Γ cb 3 Only sum ove d 4 Only sum ove d
3 g b g xx g yy { g zz } The Chistoffel symbols of fist kind Chistoffel symbols of the second kind Γ bc ( g bc + b g c c g b Γ bc g d Γ bcd z y Γ xyz Γ xzy Γ yzx 0 Γ x xy Γ xz Γ yz 0 Γ g Γ 5 g d Γ d Γ xxx x xg xx Γ xx g xx Γ xxx Γ yyy y yg yy Γ yy g yy Γ yyy Γ zzz z zg zz Γ zz g zz Γ zzz Γ b ( g b b g b bg Γ g bd Γ d Γ xxy y yg xx Γ xx g yy Γ xxy Γ xxz z zg xx Γ xx g zz Γ xxz Γ yyx x xg yy Γ yy g xx Γ yyx Γ yyz z zg yy Γ yy g zz Γ yyz Γ zzx x xg zz Γ zz g xx Γ zzx Γ zzy y yg zz Γ zz g yy Γ zzy Γ bb Γ bb b g bb Γ b b Γ b 6 g bd Γ bd Γ xyy Γ yxy y xg yy Γ xy y Γ yx g yy Γ xyy Γ xzz Γ zxz z z xg zz Γ xz Γ zx g zz Γ xzz Γ yxx Γ xyx x x yg xx Γ yx Γ xy g xx Γ yxx Γ yzz Γ zyz z z yg zz Γ yz Γ zy g zz Γ yzz Γ zxx Γ xzx x x zg xx Γ zx Γ xz g xx Γ zxx Γ zyy Γ yzy y zg yy Γ zy y Γ yz g yy Γ zyy 4. c Cylindicl coodintes. The line element: 5 Only sum ove d 6 Only sum ove d 3
4 ds d + dφ + dz The metic tenso nd its invese: g b { } g b { } 4.. The non-zeo Chistoffel symbols The Chistoffel symbols of fist kind Chistoffel symbols of the second kind Γ bc ( g bc + b g c c g b Γ bc g d Γ bcd Γ φφ g φφ ( Γ φφ Γ φφ Γ φφ g φφ ( φ Γ φ g Γ φφ Γ φ φ g φφ Γ φφ 4.. The geodesic eqution fo cylindicl coodintes 7 The geodesics eqution: d x ds + Γ dx b dx c bc ds ds 0 x : d ds + Γ dx b dx c bc ds ds 0 d (dφ ds ds 0 x d φ: φ ds + Γ φ dx b dx c 0 bc ds ds d φ ds + Γ φ d dφ φ ds ds + Γ φ dφ d 0 φ ds ds d φ ds + d dφ 0 ds ds x z: z bc d z ds + Γ dx ds dx ds 0 d z ds 0 8 The Chistoffel symbols fom the geodesic equtions We hve Now we need x x : g bx x b ( + (φ + (z d ds ( x d ds ( φ d ( ds 7 In this cse we know the Chistoffel symbols nd wnt to find the geodesic equtions 8 In this cse we know the geodesic equtions nd wnt to find the Chistoffel symbols 4
5 0 φ x φ: φ d ds ( φ 0 d ds ( φ φ + φ 0 φ + φ + φ x z: z d ds ( z 0 d (z z ds Collecting the esults 0 φ 0 φ + φ + φ 0 z We cn now find the Chistoffel symbols fom the geodesic eqution: Γ φφ φ Γ φ Γ φ φ 4.3 The plne in Ctesin coodintes 4.3. d Solve the geodesics equtions of the plne in Ctesin coodintes. We use the Eule-Lgnge method. 0 d ds ( x x F g bx x b The line element ds dx + dy F x + y x x: x 0 x x x 0 x y: 0 y y y y 0 Collecting the esults x 0 y 0 The solution is obviously stight lines: x d x ds 0 5
6 x dx ds k 0 x k 0 S + k y d y ds 0 y dy ds c 0 y c 0 S + c y 0 x The Hypebolic Plne 4.4. e Geodesics in the Hypebolic Plne The line element: ds y (dx + dy y 0 To find the geodesics we need few integls which we cn solve:. illing vecto: Becuse the metic is independent of x illing vecto is ξ (ξ x, ξ y (,0 Accoding to (8.3 ξ u is conseved quntity long geodesic, whee u (u x, u y ( dx ds, dy ds ξ u ξ u g b ξ b u g x ξ x u g xx ξ x u x y u x dx ds y (I b. The line-element ds y (dx + dy y (( dx ds Substituting (I into (II y ((y + ( dy ds ( dy ds Combining (I nd (III y ( y + ( dy ds (II dy ds ±y y (III dx dy ± y y dx ± ydy y 6
7 x x 0 ± ydy 9 ± y y (x x 0 + y y 0 If y 0 the geodesics e the veticl lines x x 0 ±. If y > 0 the geodesics e semicicles centeed on the x-xis in (x 0, 0 with dius. x nd y s function of S: dy ds ds ±y y ± dy y y S S 0 ± exp(±(s S 0 dy y y 0 ± ln + y 0 (y exp(±(s S 0 y ( + y y + y y exp(±(s S 0 y exp(±(s S 0 + y y (y( + exp(±(s S 0 exp(±(s S 0 y 0 y exp(s S 0 ( + exp((s S 0 cosh(s S 0 y 3 exp( (S S 0 ( + exp( (S S 0 cosh(s S 0 y dx ds y ( cosh(s S 0 cosh (S S 0 9 (Spiegel, 990 (4.38 d 0 dy (Spiegel, 990 (4.4 ln y y y (+ y ex e x +e x e x (e x +e x e x +e x cosh x e x +e x e x e x (e x +e x e x +e x cosh x 7
8 x x 0 cosh (S S 0 ds 3 tnh(s S 0 tnh(s S 0 Rescling nd collecting the esults 4 y cosh(s x tnh(s y sinh(s 4.5 The Flt Spce-time in two dimension 4.5. f The time-like geodesic X(T of the Flt Spce-time metic in two dimensions To find X(T we need few integls which we cn solve:.the line element: ds X dt + dx dτ ds X dt dx X ( dt dτ ( dx dτ (I b. illing vectos: Becuse the metic is independent of T illing vecto is ξ (ξ T, ξ X (,0 ξ u is conseved quntity long geodesic, whee u (u T, u X ( dt ξ u dτ dτ ξ u g b ξ b u g TT ξ T u T + g XX ξ X u X X dτ + 0 dτ dt X dt dτ X dt dτ constnt (II Substituting (II into (I dx dτ If dx > 0: dτ Dividing (III by (II dx dτ X dt dτ X ( X 5 ± X X ( dx dτ X (dx dτ (III 3 (Spiegel, 990 (4.57 dx tnh x cosh (x 4 Checking: y ((dx ds + ( dy ds cosh (S (( d( ( cosh(s tnh(s + cosh (S tnh (S 5 > X tnh(s ds + ( d( cosh(s ds cosh S (( cosh (S + 8
9 dx dt X X X X dt X X dx T T dx X X 6 7 ln ( + X X Isolting X ( + X X exp( (T T ( X exp( (T T X ( X (exp( (T T X X And we find the geodesics exp( (T T + ( X X exp( (T T exp( (T T + exp( (T T exp( (T T + exp(t T cosh(t T X(T 8 cosh(t T 4.5. Ae these geodesics spce-like o time-like ds X dt + dx ( X + ( dx dt dt ( X + ( X X X4 dt cosh 4 (T T dt < 0 So these geodesics e time-like dt g Is the wold-line X(T A cosh(t time-like o spce-like: ds X dt + dx ( X + ( dx dt dt 6 dx x x 7 ( + X X x ln(+ (Spiegel, 990 (4.4 x > 0 if > X 8 Notice: If dx < 0: X(T dτ cosh(t T cosh(t T 9
10 ( X d(a cosh T + ( dt dt A ( cosh T + sinh TdT A dt < 0 i.e. the wold-line is time-like. 4.6 Thee-dimensionl flt spce-time h Null geodesics in thee-dimensionl flt spce-time. The line element: ds dt + d + dφ To find the null-geodesics we need few integls which we cn solve:. illing vectos: Becuse the metic is independent of t illing vecto is ξ (ξ t, ξ, ξ φ (,0,0 Accoding to (8.3 ξ u is conseved quntity long geodesic, whee u (u t, u, u φ ( dt ds, d ds, dφ ds ξ u ξ u g b ξ b u g t ξ t u g tt ξ t u t + g t ξ t u + g φt ξ t u φ u t dt (I ds Becuse the metic is independent of φ illing vecto is ζ (ζ t, ζ, ζ φ (0,0, ζ u ζ u g b ζ b u g φ ζ φ u g tφ ζ φ u t + g φ ζ φ u + g φφ ζ φ u φ u φ dφ (II ds b. Consevtion of the fou-velocity fo light y u u 0 u u g b u b u g tt u t u t + g u u + g φφ u φ u φ ( dt ds + ( d ds + ( dφ ds 0 ( dt ds + ( d ds + ( dφ ds Inseting (I nd (II into (III 0 ( + ( d ds + ( ( d ds Combining (I nd (IV d ds d dt ( ( ( ( (IV ( ( ( (III 0
11 dt d ( d ( t t 0 d ( Notice we cn ewite this into hypeboloid. 9 ( ( (t t 0 (V Combining (I, (II nd (V dφ dt dφ dt φ φ 0 (t t 0 + ( t t 0 tn(φ φ 0 Combining (V nd (VI ( dt (t t 0 + ( 0 tn [ (t t 0 ] tn(φ φ 0 ± tn (φ φ 0 + Collecting the esults we find the null-geodesics: ( (V (VI (VII (t t 0 (V t t 0 tn(φ φ 0 ± tn (φ φ 0 + (VII Light ys moves on stight lines in cuved spce. Fom ou point of view the tip of the light cone (t, moves long hypebolic pth eq. (V. 4.7 The wom-hole geomety 4.7. i Volume in the Womhole geomety The thee-dimensionl volume on t constnt slice of the womhole geomety bounded by two sphees of coodinte dius R on ech side of the thot. The line-element ds dt + d + (b + (dθ + sin θ dφ (VI 9 (Spiegel, 990 (4.0 d 0 (Spiegel, 990 (4.5 dt t + tn t
12 The volume V dl dl dl 3 R (b + d R R 4π (b + d R 4π [b ] R π sin θ dθ 0 R π dφ 0 4π ((Rb + 3 R3 (( Rb + 3 ( R3 4π 3 R (3b + R 4.7. j Equtions fo geodesics in Womhole Geomety We use the Eule-Lgnge method. 0 d ds ( x x F g bx x b The line element ds dt + d + (b + (dθ + sin θ dφ F t + + (b + θ + (b + sin θ φ x t: 0 t t t t 0 x : (θ + sin θ φ θ + sin θ φ x θ: θ (b + sin θ cos θ φ (b θ + θ d ds ( θ + (b θ + θ θ sin θ cos θ φ x φ: φ 0 φ (b + sin θ φ (b + θ 0 d ds ((b + sin θ φ sin θ φ + (b + sin θ cos θ θ φ + (b + sin θ φ
13 φ (b + φ cot θ θ φ Collecting the esults t 0 θ + sin θ φ θ sin θ cos θ φ (b + θ φ (b + φ cot θ θ φ k The non-zeo Chistoffel symbols e Γ θθ Γ φφ sin θ θ Γ φφ φ Γ φ θ sin θ cos θ Γ θ Γ φ φ (b + φ Γ θφ θ Γ θ φ Γ φθ (b + cot θ l The tvel time though womhole Use the geodesic equtions to clculte the pope tvel time of n stonut tvelling though womhole thot long the coodinte dius fom R to R. The initil dil fou-velocity is u U, nd becuse of spheiclly symmety u θ u φ 0 The fou-velocity is u (u t, u, u θ, u φ ( dt dτ, d dτ, dθ dτ, dφ dτ ( + U, U, 0, 0 we will only look t u. We use the geodesic eqution θ + sin θ φ which we cn ewite d θ + sin θ φ 0 dτ d d dτ dτ (d dτ du dτ 0 which implies tht u U is constnt long the stonuts wold-line. So we cn solve u d dτ U dτ U d R Δτ U d R U [] R R U (R ( R R U So the tvel time though the womhole Δτ R is vey much like the usul time/speed clcultion: U distncetime*velocity except with the velocity eplced by the fou velocity Is the tjectoy time-like o spce-like? The line-element dτ dt d (b + (dθ + sin θ dφ u t is found fom the fct tht the u u is conseved quntity: u u u η ij u (u t + (u + (u θ + (u φ (u t + U u t + U, whee + U > 0 If we insted used the line-element dτ dt + d + (b + (dθ + sin θ dφ we would find dt ( +U < 0, which in this cse lso is time-like. You lwys hve to ceful whethe you choose positive o negtive signtue. 3
14 dt ( ( d dt (b + (( dθ dt + sin θ ( dφ dt 3 dt ( ( d dτ dτ dt (b + ((0 + sin θ (0 dt ( (U + U dt ( + U U + U dt ( + U > 0 i.e. the tjectoy is time-like. 4.8 m Wp spce-time symmety: The line-element ds dt + [dx V s (tf( s dt] + dy + dz dτ ds dt [dx V s (tf( s dt] dy dz ( V s (t f( s dt + V s (tf( s dxdt dx dy dz Notice: The line-element is dependent on the velocity of the spceship V s (t. If the velocity is zeo the line-element educes to flt Minkowsky spce-time. The metic + V s (t f( s V s (tf( s 0 0 g b { V s (tf( s 0 0 } nd f( s is ny smooth positive function tht stisfies f(0 nd deceses wy Whee V s (t dx s (t dt fom the oigin to vnish fo s > R fo some R, s [(x x s (t + y + z ] A spceship tvels long cuve (t, x, y, z (t, x s (t, 0,0 With the fou-velocity u (u t, u x, u y, u z ( dt dτ, dx s(t, 0,0 dτ Mnipulting the metic we get dτ dt [dx V s (tf( s dt] dy dz ( [ dx dt V s(tf( s ] ( [ dx s(t V dt s (tf( s ] ( dy dt ( dz dt dt dt ( [ f( s ] V s (t dt > 0 Which mens the tjectoy is time-like nd t evey point long the cuve (t, x, y, z (t, x s (t, 0,0 the fou-velocity of the spceship lies inside the light cone if V s (t <, i.e. smlle thn the velocity of light. 3 Becuse ll the diffeentils with espect to θ nd φ e zeo, we cn use the chin ule in this simple mnne. 4
15 4.8. How much ship time, Δτ, elpses on tip between sttions tht tkes coodinte time t ΔT? The metic dτ dt [dx V s (tf( s dt] dy dz ( dτ dt [ dx dt V s(tf( s ] ( dy dt ( dz dt The ship moves on cuve in the (x, y-plne with the coodintes (t, x s (t, 0,0 ( dτ dt [ dx s(t V dt s (tf( s ] [V s (t V s (tf( s ] V s (t [ f( s ] dτ V s (t [ f( s ] dt If we ssume the spceship hs constnt velocity V s (t V s (0 Δτ V s (0 [ f( s ] ΔT In detil Δτ fo s 0 ΔT { Δτ V s (0 > Δτ fo s > R The light cone: The line-element is depending on the velocity of light. This hs the peculi effect. 0 dt + [dx V s (tf( s dt] ( + [ dx dt V s(tf( s ] dt dx dt ± + V s (tf( s Now, s you cn see thee e es whee dx is lge the thn one. This mens tht fom ou point of dt view, thee e es whee the spceship seems to move with velocity lge thn the speed of light. Thee is no contdiction hee though, becuse loclly the velocity of the spceship V s (t is smlle thn the speed of light. 4.9 n Clssiclly Anti-de Sitte Spcetime 4.9. Clssiclly Anti-de Sitte Spce-time is confomlly elted to the Einstein cylinde The line element ds cosh ( dt + d + sinh ( dθ + sinh ( sin θ dφ We use the tnsfomtion cosh( cos ψ sinh ( cos ψ tn ψ d(cosh( d ( cos ψ sinh( d sin ψ cos ψ dψ d sin ψ sinh ( cos 4 ψ dψ cos ψ dψ ds cos ψ dt + cos ψ dψ + tn ψ dθ + tn ψ sin θ dφ cos ψ ( dt + dψ + sin ψ dθ + sin ψ sin θ dφ Which is confomlly elted to the Einstein cylinde ds dt + ( 0 (dθ + sin θ (dφ + sin φ dψ 5
16 4.9. The Pth of light y the null geodesics in the Clssic Anti de Sitte spce-time To find the null-geodesics we need some integls which we cn solve:. illing vectos: Becuse the metic is independent of t illing vecto is ξ (ξ t, ξ, ξ θ, ξ φ (,0,0,0 ξ u is conseved quntity long geodesic, whee u (u t, u, u θ, u φ ( dt ds, d ds, dθ ds, dφ ds ξ u ξ u g b ξ b u g t ξ t u g tt ξ t u t + g t ξ t u + g θt ξ t u θ + g φt ξ t u φ cosh ( t t cosh ( Becuse the metic is independent of φ illing vecto is ζ (ζ t, ζ, ζ θ, ζ φ (0,0,0, ζ u ζ u g b ζ b u g φ ζ φ u g tφ ζ φ u t + g φ ζ φ u + +g θφ ζ φ u θ + g φφ ζ φ u φ sinh ( sin θ u φ φ sinh ( sin θ (II b. Consevtion of the fou-velocity fo light y u u 0 u u g b u b u g tt u t u t + g u u + g θθ u θ u θ + g φφ u φ u φ cosh ( t + + sinh ( θ + sinh ( sin θ φ 0 cosh ( t + + sinh ( θ + sinh ( sin θ φ (III c. The geodesic equtions. We use the Eule-Lgnge method. 0 d ds ( x x F g bx x b cosh ( t + + sinh ( θ + sinh ( sin θ φ x t: 0 t cosh ( t t d ds ( cosh ( t sinh( t t 0 cosh ( t + sinh( t (IV x : sinh( t + cosh( θ + cosh( sin θ φ (I 6
17 d ds ( 0 + sinh( t cosh( θ cosh( sin θ φ (V x θ: θ sinh ( cos θ φ θ sinh ( θ d ds ( cosh( θ + sinh θ ( θ 0 cosh( θ + sinh ( θ sinh ( cos θ φ (VI x φ: φ 0 φ sinh ( sin θ φ d ds ( φ cosh( sin θ φ + sinh ( cos θ φ + sinh ( sin θ φ 0 cosh( sin θ φ + sinh ( cos θ φ + sinh ( sin θ φ (VII Collecting the esults t cosh ( φ sinh ( sin θ (II 0 cosh ( t + + sinh ( θ + sinh ( sin θ φ (III 0 cosh ( t + sinh( t (IV 0 + sinh( t cosh( θ cosh( sin θ φ (V 0 cosh( θ + sinh ( θ sinh ( cos θ φ (VI 0 cosh( sin θ φ + sinh ( cos θ φ + sinh ( sin θ φ (VII The coodintes t nd : We need t sinh( cosh( cosh 4 ( (I Substituting (I nd (I into (IV 0 cosh sinh( cosh( ( ( cosh 4 ( + sinh( ( cosh ( 0 cosh( + cosh( (VIII Dividing (I with (VIII dt t t dt t 0 d cosh 3 ( cosh 3 ( d 0 (I 7
18 t t 0 4 [ sinh( cosh ( ] 0 + d cosh( 0 5 [ sinh( cosh ( + tn (e ] 0 sinh( cosh ( + tn (e ( sinh( 0 cosh ( 0 + tn (e 0 sinh( cosh ( + tn (e Intepeting this mens, tht no mtte how f the light tvels in this spcetime fom 0 to this hppens within limited time 6. As n execise we will look t the othe coodintes s well. The coodintes nd θ : We need cosh( sinh( cos θ sin θ φ ( sinh 4 ( sin + θ sinh ( sin 4 θ θ cosh( sinh( cos θ sin θ ( sinh 4 ( sin cosh( + θ sinh ( sin 4 θ θ (II Mnipulte eq. (VII nd substitute (II nd (II 0 cosh( sin θ φ + sinh ( cos θ φ + sinh ( sin θ φ θ 7 [ cosh( cos θ sinh + ( sin θ cosh ( sinh( cosh( tn θ sinh ( + sin θ cosh ( tn θ sinh( cos θ sin θ θ ] Dividing (IX with (VIII: θ dθ d tn θ sinh ( + cosh( sin θ tn θ tnh( This illusttes how difficult it is to solve the geodesic equtions in GR. (IX Refeence Choquet-Buht, Y. (05. Intoduction to Genel Reltivity, Blck Holes nd Cosmology. Oxfod: Oxfod Univesity Pess. Htle, J. B. (003. An Intoduction to Einstein's Genel Reltivity. Sn Fncisco: Peson Eduction. Htle, J. B. (003. Gvity - An intoduction to Einstein's Genel Reltivity. Addison Wesley. y, D. C. (988. Tenso Clculus. McGw-Hill. McMhon, D. (006. Reltivity Demystified. McGw-Hill. 4 d sinh( + (n cosh n ( (n cosh n ( (n d cosh n ( (4.588 (Spiegel, d cosh( tn (e (4.567 (Spiegel, Notice: We hven t used eq. (III so this esult lso vlid fo n object with mss. cosh( sinh( 7 cosh( sin θ + sinh ( sin θ sinh ( cos θ + sinh ( sin θ sinh ( sin θ [ ( cosh( + sinh 4 ( sin θ cos θ sin θ θ ] sinh ( sin 4 θ 8
19 Spiegel, M. R. (990. SCHAUM'S OUTLINE SERIES: Mthemticl Hndbook of FORMULAS nd TABLES. McGw-Hill Publishing Compny. (McMhon, 006, s. 7-73, (d'inveno, 99, p. 83, (y, 988, s , b (McMhon, 006, s. 34, (y, 988, s , c (McMhon, 006, s. 83 d (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 83 e (Htle, An Intoduction to Einstein's Genel Reltivity, 003, s. 84 f (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 84 g (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 43 h (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 84 i (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 66 j (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 7 k (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 74 l (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, p. 75 m (Htle, Gvity - An intoduction to Einstein's Genel Reltivity, 003, pp. 44, 66 n (Choquet-Buht, 05, s. 97 9
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