Schwarzschild spacetime

SageManifolds. Schwazschild spacetime This woksheet demonstates a few capabilities of SageManifolds vesion., as included in SageMath 7.5) in computations egading Schwazschild spacetime. Click hee to download the woksheet file ipynb fomat). To un it, you must stat SageMath with the Jupyte notebook, via the command sage -n jupyte NB: a vesion of SageMath at least equal to 7.5 is equied to un this woksheet: In []: Out[]: vesion) 'SageMath vesion 7.5, Release Date: 27--' Fist we set up the notebook to display mathematical objects using LaTeX endeing: In [2]: %display latex Spacetime manifold We declae the Schwazschild spacetime as a 4-dimensional diffeentiable manifold: In [3]: Out[3]: In [4]: M Manifold4, 'M', '\mathcal{m}') ; M pintm) 4-dimensional diffeentiable manifold M The spacetime manifold can be split into 4 egions, coesponding to the 4 quadants in the I IV I III Kuskal diagam.let us denote by to the inteios of these 4 egions. and ae asymtotically flat egions outside the event hoizon; II is inside the futue event hoizon and IV is inside the past event hoizon. In [5]: egi M.open_subset'R_I', '\mathcal{r}_{\mathm{i}}') egii M.open_subset'R_II', '\mathcal{r}_{\mathm{ii}}') egiii M.open_subset'R_III', '\mathcal{r}_{\mathm{iii}}') egiv M.open_subset'R_IV', '\mathcal{r}_{\mathm{iv}}') egi, egii, egiii, egiv Out[5]: I, II, III, IV ) The paamete m of the Schwazschild spacetime is declaed as a symbolic vaiable: In [6]: m va'm') ; assumem>) Boye-Lindquist coodinates The standad Boye-Lindquist coodinates also called Schwazschild coodinates) ae defined on I II

SageManifolds. In [7]: Out[7]: In [8]: egi_ii egi.unionegii) ; egi_ii I II X.<t,,th,ph> egi_ii.chat't :,+oo) th:,pi):\theta ph:,2*pi ):\phi') pintx) Chat R_I_union_R_II, t,, th, ph)) In [9]: X Out[9]: I II, t,, θ, ϕ)) X I II We natually intoduce two subchats as the estictions of the chat to egions and espectively. Since, in tems of the Boye-Lindquist coodinates, I esp. II ) is defined by esp. ), we set > 2m < 2m In []: In []: X_I X.estictegI, >2*m) ; X_I Out[]: I, t,, θ, ϕ)) X_II X.estictegII, <2*m) ; X_II Out[]: II, t,, θ, ϕ)) At this stage, the manifold's atlas has 3 chats: In [2]: M.atlas) Out[2]: [ I II, t,, θ, ϕ)), I, t,, θ, ϕ)), II, t,, θ, ϕ))] In [3]: M.default_chat) Out[3]: I II, t,, θ, ϕ)) Thee vecto fames have been defined on the manifold: the thee coodinate fames: In [4]: Out[4]: In [5]: M.fames) [, )), I II t θ I ϕ t θ ϕ )), II, t θ ϕ ))] pintm.default_fame)) Coodinate fame R_I_union_R_II, d/dt,d/d,d/dth,d/dph)) In [6]: Out[6]: M.default_fame).domain) I II Metic tenso The metic tenso is defined as follows: 2

SageManifolds. In [7]: g M.loentzian_metic'g') pintg) Loentzian metic g on the 4-dimensional diffeentiable manifold M The metic tenso is set by its components in the coodinate fame associated with Schwazschild coodinates, which is the cuent manifold's default fame: In [8]: g[,], g[,] --2*m/), /-2*m/) g[2,2], g[3,3] ^2, *sinth))^2 In [9]: g.display) Out[9]: g 2 m ) dt dt + 2 m d d + 2 dθ dθ + 2 sin θ) 2 dϕ ) dϕ As an example, let us conside a vecto field defined only on I : In [2]: v egi.vecto_field'v') v[] v[] - 2*m/ # unset components ae zeo v.display) Out[2]: 2 m v + + t ) In [2]: Out[2]: v.domain) I In [22]: Out[22]: g.domain) I g v Since, it is possible to apply to : In [23]: s gv,v) ; pints) Scala field gv,v) on the Open subset R_I of the 4-dimensional diffee ntiable manifold M In [24]: s.display) # v is indeed a null vecto Out[24]: g v, v) : I t,, θ, ϕ) R Levi-Civita Connection The Levi-Civita connection associated with : g In [25]: nab g.connection) ; pintnab) Levi-Civita connection nabla_g associated with the Loentzian metic g on the 4-dimensional diffeentiable manifold M 3

SageManifolds. Let us veify that the covaiant deivative of with espect to vanishes identically: g In [26]: nabg) Out[26]: Tue In [27]: Out[27]: g g nabg).display) The nonzeo Chistoffel symbols of can be deduced by symmety: g with espect to Schwazschild coodinates, skipping those that In [28]: Out[28]: g.chistoffel_symbols_display) Γ t t Γ t t Γ Γ θ θ Γ ϕ ϕ Γ θ θ Γ θ ϕ ϕ Γ ϕ ϕ Γ ϕ θ ϕ 2 m 2 m 2 m2 3 m m 2 m 2 2 m 2 m ) sin θ) 2 cosθ) sinθ) cosθ) sinθ) Cuvatue The Riemann cuvatue tenso associated with : g In [29]: R g.iemann) ; pintr) Tenso field Riemg) of type,3) on the 4-dimensional diffeentiable manifold M The Weyl confomal tenso associated with : g In [3]: C g.weyl) ; pintc) Tenso field Cg) of type,3) on the 4-dimensional diffeentiable man ifold M 4

SageManifolds. In [3]: Out[3]: C.display) 2 2 C g) ) d dt d + ) d d 2 m 2 3 t 2 m 2 3 t m sin θ) 2 dt dθ dt dθ + dθ dθ dt dϕ t t t m sin θ) 2 2 2 m 2 m) dt dϕ + dϕ dϕ dt dt dt t 4 2 2 m 2 m) d + dt d dt dθ d dθ + 4 m sin θ) 2 m sin θ) 2 dθ dθ d dϕ d dϕ + dϕ dϕ 2 m 2 2 m 2 d + dt dt dθ + dt dθ 4 ) θ 4 ) θ dt + ) d d dθ + ) d dθ 2 m 2 3 θ 2 m 2 3 θ 2 m sin θ) 2 2 m sin θ) 2 d + dϕ dθ dϕ dϕ dϕ dθ θ θ 2 m 2 2 m 2 + dt dt dϕ + dt dϕ dt 4 ) ϕ 4 ) ϕ + ) d d dϕ + ) d dϕ d 2 m 2 3 ϕ 2 m 2 3 ϕ 2 2 dθ dθ dϕ + dθ dϕ dθ ϕ ϕ The Ricci tenso associated with : g In [32]: Ric g.icci) ; pintric) Field of symmetic bilinea foms Ricg) on the 4-dimensional diffeent iable manifold M Einstein equation Let us check that the Schwazschild metic is a solution of the vacuum Einstein equation: In [33]: Ric Out[33]: Tue In [34]: Out[34]: Ric g) Ric.display) # anothe view of the above Contay to the Ricci tenso, the Riemann tenso does not vanish: 5

SageManifolds. In [35]: R[:] Out[35]: [[ ], [ ], [ ], [ ]], [[ [[, 2 m,, ], 2 m [ ], [ ], [ ] ], 2 m 2 3 2 m 2 3 [[,, m, ], [ ], m [ ], [ ] ], [[ m sin θ) 2 ], [ ], [ ], m sin θ) 2, [ ]]] [[[, 2 2 m 2 m),, ], 2 2 m 2 m), [ ], [ ], 4 [ 4 ] ] [[ ], [ ], [ ], [ ]], [ [ ], [,, m, ], [, m,, ], [ ] ], [ [ ], [ m sin θ) 2 ], [ ], [, m sin θ) 2,, ]]], [[[,, 2 m 2 m, ], [ ], [ 2 m 2 m ], [ ] ], 4 4 [ [ ], [,, m, ], [, m 2 m,, ], [ ] ], 2 3 2 m 2 3 [[ ], [ ], [ ], [ ]], [ [ ], [ ], [ 2 m sin θ) 2 ], [,, 2 m sin θ) 2, ]]], [[[ 2 m 2 m ], [ ], [ ], [ 2 m 2 m ]], 4 4 [ [ ], [ m ], [ ], [, m 2 m,, ]], 2 3 2 m 2 3 [ [ ], [ ], [ 2 m ], [,, 2 m, ]], [[ ], [ ], [ ], [ ]] ] ] 6

SageManifolds. In [36]: Out[36]: R.display) 2 2 Riem g) ) d dt d + ) d 2 m 2 3 t 2 m 2 3 t m sin θ) 2 d dt dθ dt dθ + dθ dθ dt t t t m sin θ) 2 2 2 m 2 m) dϕ dt dϕ + dϕ dϕ dt dt t 4 2 2 m 2 m) dt d + dt d dt dθ d dθ 4 m sin θ) 2 m sin θ) 2 + dθ dθ d dϕ d dϕ + 2 m 2 2 m 2 dϕ dϕ d + dt dt dθ + 4 ) θ 4 ) θ dt dθ dt + ) d d dθ + 2 m 2 3 θ 2 m 2 3 ) θ 2 m sin θ) 2 2 m sin θ) 2 d dθ d + dϕ dθ dϕ dϕ θ θ 2 m 2 2 m 2 dϕ dθ + dt dt dϕ + dt 4 ) ϕ 4 ) ϕ dϕ dt + ) d d dϕ + ) d 2 m 2 3 ϕ 2 m 2 3 ϕ 2 2 dϕ d dθ dθ dϕ + dθ dϕ dθ ϕ ϕ The nonzeo components of the Riemann tenso, skipping those that can be deduced by antisymmety: In [37]: Out[37]: R.display_componly_nonedundantTue) Riemg) t t Riemg) t θ t θ m 2 m 2 m 2 3 Riemg) t ϕ t ϕ Riemg) t t Riemg) θ θ Riemg) ϕ ϕ Riemg) θ t t θ Riemg) θ θ Riemg) θ ϕ θ ϕ Riemg) ϕ t t ϕ Riemg) ϕ ϕ Riemg) ϕ θ θ ϕ m sin θ)2 2 2 m m 2 4 m sin θ)2 2 m 2 m 4 m 2 m 2 3 2 m sin θ) 2 2 m 2 m 4 m 2 m 2 3 2 m m) 7

SageManifolds. In [38]: Ric[:] Out[38]: Since the Ricci tenso is zeo, the Weyl tenso is of couse equal to the Riemann tenso: In [39]: Out[39]: C R Tue Bianchi identity p R i jkl + k R i jlp + l R i jpk Let us check the Bianchi identity : In [4]: DR nabr) ; pintdr) Tenso field nabla_griemg)) of type,4) on the 4-dimensional diffe entiable manifold M In [4]: fo i in M.iange): fo j in M.iange): fo k in M.iange): fo l in M.iange): fo p in M.iange): pint DR[i,j,k,l,p] + DR[i,j,l,p,k] + DR[i,j,p,k,l], + Let us check that if we tun the fist sign into a one, the Bianchi identity does no longe hold: 8

SageManifolds. In [42]: fo i in M.iange): fo j in M.iange): fo k in M.iange): fo l in M.iange): fo p in M.iange): pint DR[i,j,k,l,p] - DR[i,j,l,p,k] + DR[i,j,p,k,l], -2*m/2*m*^3 - ^4) 2*m/2*m*^3 - ^4) -6*m/^2 6*m/^2-6 *m*sinth)^2/^2 6*m*sinth)^2/^2-6*m/^2 6*m/^2-6*m/ ^2 6*m/^2-6*m*sinth)^2/^2 6*m*sinth)^2/^2-6*m*sinth)^ 2/^2 6*m*sinth)^2/^2-2*2*m^2 - m*)/^5 2*2*m^2 - m*)/^5-6*4*m^3-4*m^2* + m*^2)/^4 6*4*m^3-4* m^2* + m*^2)/^4-6*4*m^3-4*m^2* + m*^2)*sin th)^2/^4 6*4*m^3-4*m^2* + m*^2)*sinth)^2/^4-6*m/^2 6*m/^2 6*2*m^2 - m*)*sint h)^2/ -6*2*m^2 - m*)*sinth)^2/ -6*m*sinth)^2/^2 6*m*sinth)^2/^2-6*2*m^2 - m*)*sinth)^2/ 6*2*m^2 - m*)*sin th)^2/ 6*2*m^2 - m*)/^5-6*2*m^2 - m*)/^5 6*2*m^2 - m*)/^5-6*2*m^2 - m*)/^5-6*m/2*m*^3 - ^4) 6*m/2*m*^3 - ^4) 6*m*sinth)^2/^2-6*m *sinth)^2/^2 2*m*sinth)^2 /^2-2*m*sinth)^2/^2 6*m*sinth)^2/^2-6*m*sinth)^2/^2-6*m*sinth)^2/^2 6*m*sinth) ^2/^2 6*2*m^2 - m* )/^5-6*2*m^2 - m*)/^5 6*2*m^2 - m*)/^5-6*2*m^2 - m*)/^5-6*m/2*m*^3 - ^4) 6*m/2*m*^3 - ^4) -6*m/^2 6*m/^2-2*m/^2 2*m/^2-6*m/ ^2 6*m/^2 6*m/^2-6*m/^2 Ketschmann scala Let us fist intoduce tenso R, of components R ijkl : g ip R p jkl In [43]: dr R.downg) ; pintdr) Tenso field of type,4) on the 4-dimensional diffeentiable manifold M and tenso R, of components R ijkl : g jp g kq g l R i pq 9

SageManifolds. In [44]: ur R.upg) ; pintur) Tenso field of type 4,) on the 4-dimensional diffeentiable manifold M K : R ijkl R ijkl The Ketschmann scala is : In [45]: K fo i in M.iange): fo j in M.iange): fo k in M.iange): fo l in M.iange): K + ur[i,j,k,l]*dr[i,j,k,l] K Out[45]: 48 m 2 6 Instead of the above loops, the Ketschmann scala can also be computed by means of the contact) method, asking that the contaction takes place on all indices positions,, 2, 3): In [46]: K ur.contact,, 2, 3, dr,,, 2, 3) K.exp) Out[46]: 48 m 2 6 The contaction can also be pefomed by means of index notations: In [47]: Out[47]: 48 m 2 K ur['^{ijkl}']*dr['_{ijkl}'] K.exp) 6 Eddington-Finkelstein coodinates Let us intoduce new coodinates on the spacetime manifold: the ingoing Eddington-Finkelstein ones: In [48]: X_EF.<v,,th,ph> egi_ii.chat'v :,+oo) th:,pi):\theta ph:,2 *pi):\vaphi') pintx_ef) ; X_EF Chat R_I_union_R_II, v,, th, ph)) Out[48]: I II, v,, θ, φ)) The change fom Schwazschild Boye-Lindquist) coodinates to the ingoing Eddington-Finkelstein ones: In [49]: ch_bl_ef_i X_I.tansition_mapX_EF, [t++2*m*ln/2*m)-),, th, ph ], estictions2>2*m) In [5]: pintch_bl_ef_i) ; ch_bl_ef_i Change of coodinates fom Chat R_I, t,, th, ph)) to Chat R_I, v,, th, ph)) Out[5]: I, t,, θ, φ)) I, v,, θ, φ))

SageManifolds. In [5]: Out[5]: ch_bl_ef_i.display) v θ φ 2 m log ) + + t 2 m θ φ In [52]: X_EF_I X_EF.estictegI) ; X_EF_I Out[52]: I, v,, θ, φ)) In [53]: ch_bl_ef_ii X_II.tansition_mapX_EF, [t++2*m*ln-/2*m)),, th, ph], estictions2<2*m) In [54]: pintch_bl_ef_ii) ; ch_bl_ef_ii Change of coodinates fom Chat R_II, t,, th, ph)) to Chat R_II, v,, th, ph)) Out[54]: II, t,, θ, φ)) II, v,, θ, φ)) In [55]: Out[55]: ch_bl_ef_ii.display) v θ φ 2 m log + ) + + t 2 m θ φ In [56]: X_EF_II X_EF.estictegII) ; X_EF_II Out[56]: II, v,, θ, φ)) The manifold's atlas has now 6 chats: In [57]: M.atlas) Out[57]: [ I II, t,, θ, φ)), I, t,, θ, φ)), II, t,, θ, φ)), I II, v,, θ, φ)),, v,, θ, φ)),, v,, θ, φ))] I II The default chat is 'BL': In [58]: M.default_chat) Out[58]: I II, t,, θ, φ)) The change fom Eddington-Finkelstein coodinates to the Schwazschild Boye-Lindquist) ones, computed as the invese of ch_bl_ef: In [59]: ch_ef_bl_i ch_bl_ef_i.invese) ; pintch_ef_bl_i) Change of coodinates fom Chat R_I, v,, th, ph)) to Chat R_I, t,, th, ph))

SageManifolds. In [6]: Out[6]: In [6]: ch_ef_bl_i.display) t 2 m log2) + 2 m logm) 2 m log2 m + ) + v θ θ φ φ ch_ef_bl_ii ch_bl_ef_ii.invese) ; pintch_ef_bl_ii) Change of coodinates fom Chat R_II, v,, th, ph)) to Chat R_II, t,, th, ph)) In [62]: Out[62]: ch_ef_bl_ii.display) t 2 m log2) 2 m log2 m ) + 2 m logm) + v θ θ φ φ At this stage, 6 vecto fames have been defined on the manifold: the 6 coodinate fames associated with the vaious chats: In [63]: Out[63]: M.fames) [, )), I II t θ I ϕ t θ ϕ )),, )) II I II, t θ ϕ v θ φ )), )) I II, v θ φ v θ φ ))] The default fame is: In [64]: Out[64]: M.default_fame) I II, t θ ϕ )) The cofames ae the duals of the defined vecto fames: In [65]: M.cofames) Out[65]: [ I II, dt, d, dθ, dϕ)), I, dt, d, dθ, dϕ)), II, dt, d, dθ, dϕ)), I II, dv, d, dθ, dφ)), I, dv, d, dθ, dφ)), II, dv, d, dθ, dφ))] If not specified, tenso components ae assumed to efe to the manifold's default fame. Fo instance, fo the metic tenso: In [66]: Out[66]: g.display) g 2 m ) dt dt + 2 m d d + 2 dθ dθ + 2 sin θ) 2 dϕ ) dϕ 2

SageManifolds. In [67]: g[:] Out[67]: 2 m 2 m 2 2 sin θ) 2 The tenso components in the fame associated with Eddington-Finkelstein coodinates in Region I ae obtained by poviding the fame to the function display): In [68]: Out[68]: g.displayx_ef_i.fame)) 2 m g ) dv dv + dv d + d dv + 2 dθ dθ + 2 sin θ) 2 dφ dφ They ae also etuned by the method comp), with the fame as agument: In [69]: Out[69]: g.compx_ef_i.fame))[:] 2 m 2 2 sin θ) 2 o, as a schotcut, In [7]: Out[7]: g[x_ef_i.fame),:] 2 m 2 2 sin θ) 2 Similaly, the metic components in the fame associated with Eddington-Finkelstein coodinates in Region II ae obtained by In [7]: Out[7]: g.displayx_ef_ii.fame)) 2 m g ) dv dv + dv d + d dv + 2 dθ dθ + 2 sin θ) 2 dφ dφ Note that thei fom is identical to that in Region I. Plot of the Boye-Lindquist coodinates in tems of the Eddington- Finkelstein ones Let us pefom the plot in Region I: 3

SageManifolds. In [72]: Out[72]: X_I.plotX_EF_I, anges{t:,8), :2.,)}, fixed_coods{th:pi/2,ph :}, ambient_coods,v), style{t:'--', :'-'}, paametes{m:}) Tetad of the static obseve Let us intoduce the othonomal tetad e α ) associated with the static obseves in Schwazschild spacetime, i.e. the obseves whose woldlines ae paallel to the timelike Killing vecto in the Region I. The othonomal tetad is defined via a tangent-space automophism that elates it to the Boye- Lindquist coodinate fame in Region I: In [73]: Out[73]: ch_to_stat egi.automophism_field) ch_to_stat[,], ch_to_stat[,] /sqt-2*m/), sqt-2*m/) ch_to_stat[2,2], ch_to_stat[3,3] /, /*sinth)) ch_to_stat[:] 2 m + 2 m + sinθ) 4

SageManifolds. In [74]: e X_I.fame).new_famech_to_stat, 'e') ; pinte) Vecto fame R_I, e_,e_,e_2,e_3)) At this stage, 7 vecto fames have been defined on the manifold : In [75]: Out[75]: M.fames) [, )), I II t θ I ϕ t θ ϕ )),, )) II I II, t θ ϕ v θ φ )), )) I II, v θ )), I φ v, e θ φ, e, e 2, e 3 )) ] The fist vecto of the tetad is the static obseve 4-velocity: In [76]: pinte[]) Vecto field e_ on the Open subset R_I of the 4-dimensional diffeenti able manifold M In [77]: Out[77]: e[].display) e 2 m + t As any 4-velocity, it is a unit timelike vecto: In [78]: Out[78]: ge[],e[]).exp) Let us check that the tetad e α ) is othonomal: In [79]: fo i in M.iange): fo j in M.iange): pint ge[i],e[j]).exp), pint " " - Anothe view of the above esult: In [8]: g[e,:] Out[8]: o, equivalently, 5

SageManifolds. In [8]: g.displaye) Out[8]: g e e + e e + e 2 e 2 + e 3 e 3 The expession of the 4-velocity and the vecto in tems of the fame associated with Eddington-Finkelstein coodinates: e e In [82]: e[].displayx_ef_i.fame)) Out[82]: e 2 m + ) v In [83]: e[].displayx_ef_i.fame)) Out[83]: e 2 m + + 2 m + ) v ) Contay to vectos of a coodinate fame, the vectos of the tetad coefficients ae not identically zeo: e do not commute: thei stuctue In [84]: Out[84]: e.stuctue_coeff)[:] [[[, m 2 m +,, ], m 2 m +, [ ], [ ] 2 m 2 ) [ 2 m 2 ) ] ] [[ ], [ ], [ ], [ ]], [ [ ], [,, 2 m +, ], [, 2 m +,, ], [ ] ], 3 2 [ [ ], [ 2 m + ], [ cosθ) ], [, 2 m + sinθ), 3 2 3 2 3 2 cosθ) sinθ) ] Equivalently, the Lie deivative of one vecto along anothe one is not necessaily zeo: In [85]: Out[85]: e[].lie_dee[]).displaye) m 2 m + 2 m 2 ) ) e The cuvatue 2-fom Ω associated with the tetad e α ): In [86]: Out[86]: g.connection).cuvatue_fom,,e).displayx_i.fame)) Ω 2 m dt d 3 6

SageManifolds. Kuskal-Szekees coodinates Let us now intoduce the Kuskal-Szekees coodinates U, V, θ, φ) on the spacetime manifold, via t,, θ, φ) the standad tansfomation expessing them in tems of the Boye-Lindquist coodinates : In [87]: In [88]: In [89]: M egi.unionegii).unionegiii).unionegiv) ; M Out[87]: I II III IV X_KS.<U,V,th,ph> M.chat'U V th:,pi):\theta ph:,2*pi):\vaphi' ) X_KS.add_estictionsV^2 < + U^2) X_KS Out[88]: I II III IV, U, V, θ, φ)) X_KS_I X_KS.estictegI, [U>, V<U, V>-U]) ; X_KS_I Out[89]: I, U, V, θ, φ)) In [9]: Out[9]: ch_bl_ks_i X_I.tansition_mapX_KS_I, [sqt/2*m)-)*exp/4*m))*c osht/4*m)), sqt/2*m)-)*exp/4*m))*s inht/4*m)), th, ph]) pintch_bl_ks_i) ch_bl_ks_i.display) Change of coodinates fom Chat R_I, t,, th, ph)) to Chat R_I, U, V, th, ph)) U V θ φ t 2 m cosh ) 2 m θ φ e 4 m ) 4 m e 4 m ) sinh ) t 4 m In [9]: X_KS_II X_KS.estictegII, [V>, V>U, V>-U]) ; X_KS_II Out[9]: II, U, V, θ, φ)) In [92]: Out[92]: ch_bl_ks_ii X_II.tansition_mapX_KS_II, [sqt-/2*m))*exp/4*m) )*sinht/4*m)), sqt-/2*m))*exp/4*m) )*cosht/4*m)), th, ph]) pintch_bl_ks_ii) ch_bl_ks_ii.display) Change of coodinates fom Chat R_II, t,, th, ph)) to Chat R_II, U, V, th, ph)) U V θ φ + 2 m e 4 m ) + t 2 m cosh ) θ φ t 4 m sinh ) 4 m e 4 m ) 7

SageManifolds. Plot of the Boye-Lindquist coodinates in tems of the Kuskal ones We daw the Boye-Lindquist chat in Region I ed) and Region II geen), with lines of constant being dashed: In [93]: gaphi X_I.plotX_KS, anges{t:-2,2), :2.,5)}, numbe_values {t:7, :9}, fixed_coods{th:pi/2,ph:}, ambient_coodsu,v), style{t:'--', :'-'}, paametes{m:}) gaphii X_II.plotX_KS, anges{t:-2,2), :.,.999)}, numbe_ values{t:7, :9}, fixed_coods{th:pi/2,ph:}, ambient_coodsu,v), style{t:'--', :'-'}, colo'geen', paametes{m :}) showgaphi+gaphii, xmin-3, xmax3, ymin-3, ymax3, axes_labels['$u $', '$V$']), θ, ϕ), π/2, ) 2m, θ, ϕ) fixed at 2.m, π/2, ) : We may add to the gaph the singulaity as a Boye-Lindquist chat plot with fixed at. Similaly, we add the event hoizon as a Boye-Lindquist chat plot with 8

SageManifolds. In [94]: gaph_ X_II.plotX_KS, fixed_coods{:, th:pi/2, ph:}, ambient_c oodsu,v), colo'yellow', thickness3, paametes{m:}) gaph_2 X_I.plotX_KS, anges{t:-4,4)}, fixed_coods{:2., th:pi/2, ph:}, ambient_coodsu,v), colo'black', thickness2, p aametes{m:}) showgaphi+gaphii+gaph_+gaph_2, xmin-3, xmax3, ymin-3, ymax3, axes_labels['$u$', '$V$']) Plot of the Eddington-Finkelstein coodinates in tems of the Kuskal ones We fist get the change of coodinates v,, θ, ϕ) U, V, θ, ϕ) v,, θ, ϕ) t,, θ, ϕ) with t,, θ, ϕ) U, V, θ, ϕ) : by composing the change In [95]: ch_ef_ks_i ch_bl_ks_i * ch_ef_bl_i ch_ef_ks_i Out[95]: I, v,, θ, φ)) I, U, V, θ, φ)) 9

SageManifolds. In [96]: Out[96]: ch_ef_ks_i.display) U V θ φ 2 m log2)+2 m logm)2 m log2 m+)+v 2 2 m+ cosh ) 2 m 4 m e 4 m ) 2 2 m+e 4 m ) sinh ) θ φ 2 m log2)+2 m logm)2 m log2 m+)+v 2 m 4 m In [97]: ch_ef_ks_ii ch_bl_ks_ii * ch_ef_bl_ii ch_ef_ks_ii Out[97]: II, v,, θ, φ)) II, U, V, θ, φ)) In [98]: gaphi_ef X_EF_I.plotX_KS, anges{v:-2,2), :2.,5)}, numbe_ values{v:7, :9}, fixed_coods{th:pi/2,ph:}, ambient_coodsu, V), style{v:'--', :'-'}, paametes{m:}) gaphii_ef X_EF_II.plotX_KS, anges{v:-2,2), :.,.999)}, n umbe_values{v:7, :9}, fixed_coods{th:pi/2,ph:}, ambient_coods U,V), style{v:'--', :'-'}, colo'geen', paamet es{m:}) showgaphi_ef+gaphii_ef+gaph_+gaph_2, xmin-3, xmax3, ymin-3, ymax3, axes_labels['$u$', '$V$']) 2

SageManifolds. Thee ae now 9 chats defined on the spacetime manifold: In [99]: M.atlas) Out[99]: [ I II, t,, θ, φ)), I, t,, θ, φ)), II, t,, θ, φ)), I II, v,, θ, φ)), I, v,, θ, φ)), II, v,, θ, φ)), I II III IV, U, V, θ, φ)),, U, V, θ, φ)),, U, V, θ, φ))] I II In []: lenm.atlas)) Out[]: 9 Thee ae 8 explicit coodinate changes the coodinate change KS fom): BL is not known in explicit In []: M.cood_changes) Out[]: { II, v,, θ, φ)), II, U, V, θ, φ))) : II, v,, θ, φ)) II, U, V, θ, φ)), II, v,, θ, φ)), II, t,, θ, φ))) : II, v,, θ, φ)) II, t,, θ, φ)), II, t,, θ, φ)), II, U, V, θ, φ))) : II, t,, θ, φ)) II, U, V, θ, φ)), I, t,, θ, φ)), I, v,, θ, φ))) : I, t,, θ, φ)) I, v,, θ, φ)), I, v,, θ, φ)), I, t,, θ, φ))) : I, v,, θ, φ)) I, t,, θ, φ)), I, t,, θ, φ)), I, U, V, θ, φ))) : I, t,, θ, φ)) I, U, V, θ, φ)), I, v,, θ, φ)), I, U, V, θ, φ))) : I, v,, θ, φ)) I, U, V, θ, φ)), II, t,, θ, φ)), II, v,, θ, φ))) : II, t,, θ, φ)) II, v,, θ, φ))} In [2]: lenm.cood_changes)) Out[2]: 8 Thee ae vecto fames among which 9 coodinate fames): In [3]: Out[3]: In [4]: M.fames) [, )), I II t θ I ϕ t θ ϕ )),, )) II I II, t θ ϕ v θ φ )), )) I II, v θ )), I φ v, e θ φ, e, e 2, e 3 )),, )), I II III IV I U V θ φ U V θ φ )), II, U V θ φ ))] lenm.fames)) Out[4]: Thee ae 4 fields of tangent space automophisms expessing the changes of coodinate bases and tetad: 2

SageManifolds. In [5]: lenm.changes_of_fame)) Out[5]: 4 Thanks to these changes of fames, the components of the metic tenso with espect to the Kuskal- Szekees can be computed by the method display) and ae found to be: In [6]: Out[6]: g.displayx_ks_i.fame)) 32 m 3 e g 2 m ) 32 m 3 e du du 2 m ) dv dv + 2 dθ dθ + 2 sin θ) 2 dφ dφ In [7]: g[x_ks_i.fame),:] Out[7]: 32 m 3 e 2 m ) 32 m3 e 2 m ) 2 2 sin θ) 2 In [8]: g.displayx_ks_ii.fame)) Out[8]: 32 m 3 e g 2 m ) 32 m 3 e du du 2 m ) dv dv + 2 dθ dθ + 2 sin θ) 2 dφ dφ The fist vecto of the othonomal tetad e expessed on the Kuskal-Szekees fame: In [9]: Out[9]: e[].displayx_ks_i.fame)) 2 e 4 m ) sinh 4 m ) e + 8 m 3 U 2 t 2 cosh ) 8 m 3 2 t 4 m e 4 m ) V The Riemann cuvatue tenso in tems of the Kuskal-Szekees coodinates: 22

SageManifolds. In []: Out[]: g.iemann).displayx_ks_i.fame)) 64 m 4 e Riem g) 2 m ) 64 m 4 e dv du dv + 2 m ) dv 4 U 4 U dv du dθ du dθ + dθ dθ du U U m sin θ) 2 m sin θ) 2 dφ du dφ + dφ dφ du U U 64 m 4 e 2 m ) 64 m 4 e du du dv + 2 m ) du dv du 4 V 4 V m sin θ) 2 dθ dv dθ + dθ dθ dv dφ V V V m sin θ) 2 32 m 4 e dv dφ + dφ dφ dv + 2 m ) du V θ 32 m 4 e du dθ 2 m ) 32 m 4 e du dθ du 2 m ) dv θ θ 4 32 m 4 e dv dθ + 2 m ) 2 m sin θ) 2 dv dθ dv + dφ θ θ 4 2 m sin θ) 2 32 m 4 e dθ dφ dφ dφ dθ + 2 m ) du θ φ 32 m 4 e du dφ 2 m ) 32 m 4 e du dφ du 2 m ) dv φ φ 4 32 m 4 e dv dφ + 2 m ) 2 dv dφ dv dθ dθ dφ 4 φ φ 2 + dθ dφ dθ φ 4 4 4 4 23

SageManifolds. In []: g.iemann).display_compx_ks_i.fame), only_nonedundanttue) Out[]: Riemg) U V U V Riemg) U θ U θ 64 m4 e 2 m ) 4 m Riemg) U φ U φ Riemg) V U U V Riemg) V θ V θ m sin θ)2 64 m4 e 2 m ) 4 m Riemg) V φ V φ Riemg) θ U U θ Riemg) θ V V θ Riemg) θ φ θ φ Riemg) φ U U φ Riemg) φ V V φ Riemg) φ θ θ φ m sin θ)2 32 m 4 e 2 m ) 4 32 m4 e 2 m ) 4 2 m sin θ) 2 32 m 4 e 2 m ) 4 32 m4 e 2 m ) 4 2 m The cuvatue 2-fom Ω associated to the Kuskal-Szekees coodinate fame: In [2]: om g.connection).cuvatue_fom,, X_KS_I.fame)) ; pintom) 2-fom cuvatue,) of connection nabla_g w..t. Coodinate fame R _I, d/du,d/dv,d/dth,d/dph)) on the Open subset R_I of the 4-dimensiona l diffeentiable manifold M In [3]: om.displayx_ks_i.fame)) Out[3]: Ω 64 m 4 e 2 m ) du dv 4 Isotopic coodinates Let us now intoduce isotopic coodinates t,, θ, φ) on the spacetime manifold: In [4]: Out[4]: In [5]: egi_iii egi.unionegiii) ; egi_iii I III X_iso.<t,i,th,ph> egi_iii.chat't i:,+oo):\ba{} th:,pi):\t heta ph:,2*pi):\vaphi') pintx_iso) ; X_iso Chat R_I_union_R_III, t, i, th, ph)) Out[5]: I III, t,, θ, φ)) 24

SageManifolds. In [6]: X_iso_I X_iso.estictegI, i>m/2) ; X_iso_I Out[6]: I, t,, θ, φ)) The tansfomation fom the isotopic coodinates to the Boye-Lindquist ones: In [7]: Out[7]: In [8]: assume2*i>m) # we conside only egion I ch_iso_bl_i X_iso_I.tansition_mapX_I, [t, i*+m/2*i))^2, th, ph ]) pintch_iso_bl_i) ch_iso_bl_i.display) Change of coodinates fom Chat R_I, t, i, th, ph)) to Chat R_I, t,, th, ph)) t θ φ t m 4 + 2) θ φ 2 assume>2*m) # we conside only egion I ch_iso_bl_i.set_inveset, -m+sqt*-2*m)))/2, th, ph, vebosetu e) Check of the invese coodinate tansfomation: t t i i th th ph ph t t th th ph ph In [9]: Out[9]: ch_iso_bl_i.invese).display) t θ φ t 2 2 m + + θ φ 2 m ) 2 At this stage, chats have been defined on the manifold : In [2]: M.atlas) Out[2]: [ I II, t,, θ, φ)), I, t,, θ, φ)), II, t,, θ, φ)), I II, v,, θ, φ)), I, v,, θ, φ)), II, v,, θ, φ)), I II III IV, U, V, θ, φ)), I, U, V, θ, φ)), II, U, V, θ, φ)), I III, t,, θ, φ)), I, t,, θ, φ))] In [2]: lenm.atlas)) Out[2]: 2 vecto fames have been defined on : coodinate bases and the tetad e α ): 25

SageManifolds. In [22]: Out[22]: In [23]: M.fames) [, )), I II t θ I ϕ t θ ϕ )),, )) II I II, t θ ϕ v θ φ )), )) I II, v θ )), I φ v, e θ φ, e, e 2, e 3 )),, )), I II III IV I U V θ φ U V θ φ )),, )) II I III, U V θ φ t θ φ )), I t θ φ ))] lenm.fames)) Out[23]: 2 The components of the metic tenso in tems of the isotopic coodinates ae given by In [24]: Out[24]: g.displayx_iso_i.fame), X_iso_I) m 2 4 m + 4 2 g dt dt m 2 + 4 m + 4 2 ) m 4 + 8 m 3 + 24 m 2 2 + 32 m 3 + 6 4 + d d 6 4 ) m 4 + 8 m 3 + 24 m 2 2 + 32 m 3 + 6 4 + dθ dθ 6 2 ) m 4 + 8 m 3 + 24 m 2 2 + 32 m 3 + 6 4 ) sin θ) 2 + dφ dφ 6 2 The g component can be factoized: In [25]: In [26]: g[x_iso_i.fame),,, X_iso_I] Out[25]: m 2 4 m + 4 2 m 2 + 4 m + 4 2 g[x_iso_i.fame),,, X_iso_I].facto) Out[26]: m 2 ) 2 m + 2 ) 2 Let us also factoize the othe components: In [27]: fo i in ange,4): g[x_iso_i.fame), i,i, X_iso_I].facto) The output of the display) command looks nice: 26

SageManifolds. In [28]: Out[28]: g.displayx_iso_i.fame), X_iso_I) m 2 ) 2 m + 2 ) 4 m + 2 ) 4 g dt dt + d d + dθ dθ m + 2 ) 2 6 4 6 2 m + 2 ) 4 sin θ) 2 + dφ dφ 6 2 Expession of the tetad associated with the static obseve in tems of the isotopic coodinate basis: In [29]: Out[29]: In [3]: Out[3]: In [3]: Out[3]: In [32]: Out[32]: e[].displayx_iso_i.fame), X_iso_I) e m + 2 m 2 ) t e[].displayx_iso_i.fame), X_iso_I) e 4 2 m 2 + 4 m + 4 2 ) e[2].displayx_iso_i.fame), X_iso_I) e 2 4 m 2 + 4 m + 4 2 ) θ e[3].displayx_iso_i.fame), X_iso_I) e 3 4 m 2 + 4 m + 4 2 ) sinθ) φ In [ ]: 27