Tolman-Oppenheimer-Volkoff equations

Transcription

1 Tolman-Oppenheime-Volkoff equations This Jupyte/SageMath woksheet is elative to the lectues Geneal elativity computations with SageManifolds given at the NewCompSta School 216 (Coimba, Potugal). These computations ae based on SageManifolds (vesion 1., as included in SageMath 7.5). Click hee to download the woksheet file (ipynb fomat). To un it, you must stat SageMath with the Jupyte notebook, with the command sage -n jupyte This woksheet is divided in two pats: 1. Deiving the TOV system fom the Einstein equation 2. Solving the TOV system to get stella models NB: a vesion of SageMath at least equal to 7.5 is equied to un this woksheet: In [1]: Out[1]: vesion() 'SageMath vesion 7.5.1, Release Date: ' Fist we set up the notebook to display mathematical objects using LaTeX endeing: In [2]: %display latex 1. Deiving the TOV system fom the Einstein equation Spacetime M We declae the spacetime manifold : In [3]: M Manifold(4, 'M') pint(m) 4-dimensional diffeentiable manifold M To get some infomation about the object M, we use the question mak: In [4]: M? Using a double question mak, we get the Python souce code (SageMath is open souce, isn't it?): In [5]: M?? We declae the chat of spheical coodinates how to use it, we again use the question mak: (t,, θ, ϕ), via the method chat acting on M; to see In [6]: M.chat? 1

2 In [7]: Out[7]: X.<t,,th,ph> M.chat('t :(,+oo) th:(,pi):\theta ph:(,2*pi):\phi ') X (M, (t,, θ, ϕ)) Metic tenso The static and spheically symmetic metic ansatz, with the unknown functions and : ν() m() In [8]: Out[8]: g M.loentzian_metic('g') nu function('nu') m function('m') g[,] -exp(2*nu()) g[1,1] 1/(1-2*m()/) g[2,2] ^2 g[3,3] (*sin(th))^2 g.display() g e(2 ν()) 1 dt dt + ( 2 m() d d + 2 dθ dθ + 2 sin (θ) 2 dϕ 1 ) dϕ One can display the metic components as a list: In [9]: Out[9]: g.display_comp() g t t g g θ θ g ϕ ϕ (2 ν()) e m() 2 2 sin (θ) 2 By default, only the nonzeo components ae shown; to get all the components, set the option only_nonzeo to False: 2

3 In [1]: Out[1]: g.display_comp(only_nonzeofalse) g t t g t g t θ g t ϕ g t g g θ g ϕ g θ t g θ g θ θ g θ ϕ g ϕ t g ϕ g ϕ θ g ϕ ϕ (2 ν()) e m() 2 sin (θ) 2 We can also display the metic components as a matix, via the [] opeato: In [11]: g[:] Out[11]: e (2 ν()) m() 2 2 sin (θ) 2 The [] opeato can also be used to access to individual elements: In [12]: Out[12]: g[,] (2 ν()) e Einstein equation Let us stat by evaluating the Ricci tenso of : g In [13]: Ric g.icci() pint(ric) Field of symmetic bilinea foms Ric(g) on the 4-dimensional diffeent iable manifold M 3

4 In [14]: Out[14]: Ric.display() ( 2 e (2 ν()) 2 e (2 ν()) ν 2 m ())( ) (2 ν()) m (e 2 e (2 ν()) + 3 e m ()) + ( 2 e (2 ν()) (2 ν()) 2 e m ()) 2 ν Ric (g) 2 2 (2 ν()) ν dt dt ( ν 2 m m ())( ) 2 ( 2 m ν m ()) + ( m ()) 2 ν m () + d m () d + ( m 2 ν ( 2 m ()) + m () m 2 ν (θ) 2 ) dθ dθ ( ( 2 m ()) + m ()) sin + dϕ dϕ The Ricci scala is natually obtained by the method icci_scala(): In [15]: g.icci_scala? In [16]: Out[16]: (g) : M g.icci_scala().display() (t,, θ, ϕ) R As a check we can also compute it by taking the tace of the Ricci tenso with espect to : ν 2 m 2 (( 2 2 m()) ( ( 2 +3 m()) +( 2 m()) 2 ) 2 ) 2 ν 2 g 2 ν m In [17]: Out[17]: g.icci_scala() g.invese()['^{ab}']*ric['_{ab}'] Tue The Einstein tenso o in index-fee notation: 1 G ab : R ab R, 2 g ab G : Ric(g) 1 (g) g 2 In [18]: G Ric - 1/2*g.icci_scala() * g G.set_name('G') pint(g) Field of symmetic bilinea foms G on the 4-dimensional diffeentiable manifold M 4

5 In [19]: Out[19]: G.display() (2 ν()) m 2 e 2 (( 2 2 m ()) m ()) G dt dt + d d 2 ( m () ) ( ν 2 m m ())( ) ( 2 m 2 + m ()) ν + ( m ()) ν + m () dθ dθ ν 2 m (( m ()) ( ( + m ) ()) ν 2 m 2 ν sin (θ) 2 + ( m ()) 2 ν + m () ) + 2 dϕ dϕ The enegy-momentum tenso We conside a pefect fluid matte model. Let us fist defined the fluid 4-velocity : u In [2]: Out[2]: u M.vecto_field('u') u[] exp(-nu()) u.display() u e ( ν()) t In [21]: u[:] Out[21]: [ e ( ν()),,, ] In [22]: pint(u.paent()) Fee module X(M) of vecto fields on the 4-dimensional diffeentiable m anifold M u g(u, u) 1 Let us check that is a nomalized timelike vecto, i.e. that, o, in index-fee notation, : g ab u a u b 1 In [23]: g(u,u) Out[23]: g (u, u) In [24]: pint(g(u,u)) Scala field g(u,u) on the 4-dimensional diffeentiable manifold M In [25]: g(u,u).display() Out[25]: g (u, u) : M (t,, θ, ϕ) R 1 In [26]: Out[26]: g(u,u).paent() C (M) 5

6 In [27]: pint(g(u,u).paent()) Algeba of diffeentiable scala fields on the 4-dimensional diffeenti able manifold M In [28]: g(u,u) -1 Out[28]: Tue To fom the enegy-momentum tenso, we need the 1-fom u a g ab u b : u that is metic-dual to the vecto, i.e. u In [29]: u_fom u.down(g) pint(u_fom) 1-fom on the 4-dimensional diffeentiable manifold M In [3]: Out[3]: e ν() dt u_fom.display() The enegy-momentum tenso is then o in index-fee notation: Since the tenso poduct T ab (ρ + p) u a u b + p g ab, T (ρ + p) u u + p g is taken with the * opeato, we wite: In [31]: ho function('ho') p function('p') T (ho()+p())* (u_fom * u_fom) + p() * g T.set_name('T') pint(t) Field of symmetic bilinea foms T on the 4-dimensional diffeentiable manifold M In [32]: Out[32]: T.display() (2 ν()) p () T e ρ () dt dt + ( ) d d + 2 p () dθ dθ 2 m () + 2 p () sin (θ) 2 dϕ dϕ In [33]: T(u,u) Out[33]: T (u, u) In [34]: pint(t(u,u)) Scala field T(u,u) on the 4-dimensional diffeentiable manifold M In [35]: T(u,u).display() Out[35]: T (u, u) : M (t,, θ, ϕ) R ρ () 6

7 The Einstein equation The Einstein equation is E G 8πT E : G 8πT We ewite it as with : In [36]: E G - 8*pi*T E.set_name('E') pint(e) Field of symmetic bilinea foms E on the 4-dimensional diffeentiable manifold M In [37]: Out[37]: In [38]: E.display() + ( E.display_comp() (2 ν()) e 2 (4 π 2 e ρ () ) E dt dt 2 (2 ν()) m 2 (4 π 3p () ( 2 2 m ()) ν + m ()) m () d d ) 8 π 3 p () ( ν 2 m m ())( ) + + ( 2 m 2 + m ()) ν ( m ()) ν m () dθ ν dθ 2 m ( 8 π3p () ( m ()) ( + + ( + m ) ()) ( m ()) 2 ν m () ) 2 dϕ 2 m 2 ν sin (θ) 2 dϕ Out[38]: E t t E E θ θ E ϕ ϕ 2 (4 π 2 (2 ν()) e ρ() e ) 2 (2 ν()) m 2 (4 π 3 p() ( 2 ν 2 m()) +m()) m() 8 π 3p() ( m()) ( ν ) 2 m + +( 2 m 2 ν +m()) ( m()) 2 ν m() 2 (8 π 3p() ( m()) ( ν ) 2 m + +( 2 m 2 ν +m()) ( m()) 2 ν m() ) sin 2 ( In [39]: Out[39]: In [4]: Out[4]: E[,] 2 (4 π 2 (2 ν()) e ρ () e ) 2 (2 ν()) m EE_sol solve(e[,].exp(), diff(m(),)) EE_sol [ m () 4 π 2 ρ () ] 7

8 In [41]: Out[41]: EE EE_sol[] EE m () 4 π 2 ρ () In [42]: Out[42]: In [43]: Out[43]: In [44]: Out[44]: In [45]: Out[45]: E[1,1] 2 (4 π 3 p () ( 2 2 m ()) ν + m ()) m () EE1_sol solve(e[1,1].exp(), diff(nu(),)) EE1 EE1_sol[] EE1 ν () E[3,3] E[2,2]*sin(th)^2 Tue E[2,2] 4 π 3 p () + m () 2 2 m () 8 π 3 p () ( m ())( ) + + ( + m ()) ( m ()) 2 ν m () 2 ν 2 m 2 m 2 ν The enegy-momentum consevation equation The enegy-momentum tenso must obey b T b a We fist fom the tenso T b a by aising the fist index of T ab : In [46]: Tu T.up(g, ) pint(tu) Tenso field of type (1,1) on the 4-dimensional diffeentiable manifold M We get the Levi-Civita connection associated with the metic : g In [47]: nabla g.connection() pint(nabla) Levi-Civita connection nabla_g associated with the Loentzian metic g on the 4-dimensional diffeentiable manifold M 8

9 In [48]: Out[48]: nabla.display() Γ t t Γ t t Γ t t Γ Γ θ θ Γ ϕ ϕ Γ θ θ Γ θ θ Γ θ ϕ ϕ Γ ϕ ϕ Γ ϕ θ ϕ Γ ϕ ϕ Γ ϕ ϕ θ ν ν (e (2 ν()) 2 e (2 ν()) ν m()) m m() 2 2 m() + 2 m () ( 2 m ()) sin (θ) cos(θ) sin(θ) 1 cos(θ) sin(θ) 1 cos(θ) sin(θ) We apply nabla to Tu to get the tenso ( T) b ac c T b a (MTW index convention): In [49]: dtu nabla(tu) pint(dtu) Tenso field of type (1,2) on the 4-dimensional diffeentiable manifold M b T b a ( T) b ac b c The divegence is then computed as the tace of the tenso on the fist index (, position ) and last index (, position 2): In [5]: divt dtu.tace(,2) pint(divt) 1-fom on the 4-dimensional diffeentiable manifold M We can also take the tace by using the index notation: In [51]: Out[51]: In [52]: divt dtu['^b_{ab}'] Tue pint(divt) 1-fom on the 4-dimensional diffeentiable manifold M In [53]: Out[53]: divt.display() ( (p () + ρ ()) ν + p ) d 9

10 In [54]: Out[54]: divt[:] [, (p () + ρ ()) ν + p,, ] The only non tivially vanishing components is thus In [55]: divt[1] Out[55]: ν (p () + ρ ()) + p Hence the enegy-momentum consevation equation b T b a educes to In [56]: Out[56]: EE2_sol solve(divt[1].exp(), diff(p(),)) EE2 EE2_sol[] EE2 p () (p () + ρ ()) ν () The TOV system Let us collect all the independent equations obtained so fa: In [57]: fo eq in [EE, EE1, EE2]: show(eq) m () 4 π ν () 2 ρ () 4 π 3 p () + m () 2 2 m () p () (p () + ρ ()) ν () 2. Solving the TOV system In ode to solve the TOV system, we need to specify a fluid equation of state. Fo simplicity, we select a polytopic one: p kρ 2 In [58]: Out[58]: kρ() 2 va('k', domain'eal') p_eos() k*ho()^2 p_eos() We substitute this expession fo p in the TOV equations: In [59]: Out[59]: EE1_ho EE1.substitute_function(p, p_eos) EE1_ho ν () 4 πk 3 ρ () 2 + m () 2 2 m () 1

11 In [6]: EE2_ho EE2.substitute_function(p, p_eos) EE2_ho Out[6]: 2 kρ () ρ () (kρ() 2 + ρ ()) ν () In [61]: Out[61]: EE2_ho (EE2_ho / (2*k*ho())).simplify_full() EE2_ho (kρ () + 1) ρ () 2 k ν () We subsitute the expession of equation EE1_ho fo the ight-hand sides: ν/, in ode to get id of any deivative in In [62]: Out[62]: EE2_ho EE2_ho.subs({diff(nu(),): EE1_ho.hs()}).simplify_full() EE2_ho ρ () 4 πk 2 3 ρ () πk 3 ρ () 2 + km () ρ () + m () 2 (k 2 2 km ()) The system to solve fo (m(), ν(), ρ()) is thus In [63]: fo eq in [EE, EE1_ho, EE2_ho]: show(eq) m () 4 π ν () ρ () 2 ρ () 4 πk 3 ρ () 2 + m () 2 2 m () 4 πk 2 3 ρ () πk 3 ρ () 2 + km () ρ () + m () 2 (k 2 2 km ()) Numeical esolution Let us use a standad 4th-ode Runge-Kutta method: In [64]: desolve_system_k4? We gathe all equations in a list fo the ease of manipulation: In [65]: eqs [EE, EE1_ho, EE2_ho] To get a numeical solution, we have of couse to specify some numeical value fo the EOS constant k k 1/4 ; let us choose : In [66]: k 1/4 hs [eq.hs().subs(kk) fo eq in eqs] hs Out[66]: [ 4 π 2 ρ (), π 3 ρ () 2 + m (), π 3 ρ () π 3 ρ () 2 + m () ρ () + 4 m () 2 2 m () 2 ( 2 2 m ()) ] 11

12 m() ρ() ρ() The integation fo and has to stop as soon as become negative. An easy way to ensue this is to use the Heaviside function (actually SageMath's unit_step; fo some eason, heaviside does not wok fo the RK4 numeical esolution): In [67]: plot(unit_step(x), (x,-4,4), thickness2, aspect_atio1) Out[67]: dm/d dρ/d h(ρ) and to multiply the.h.s. of the and equations by : In [68]: Out[68]: hs[] hs[] * unit_step(ho()) hs[2] hs[2] * unit_step(ho()) hs π 3 ρ () 2 + m () 4 π ρ () u (ρ ()),, [ m () (π 3 ρ () π 3 ρ () 2 + m () ρ () + 4 m ())u (ρ ()) 2 ( 2 2 m ()) ] Let us add an exta equation, fo the pupose of getting the sta's adius as an output of the dr/d 1 integation, via the equation, again multiplied by the Heaviside function of : ρ In [69]: Out[69]: hs.append(1 * unit_step(ho())) hs π 3 ρ () 2 + m () 4 π ρ () u (ρ ()),, [ m () (π 3 ρ () π 3 ρ () 2 + m () ρ () + 4 m ())u (ρ ()), u (ρ ()) 2 ( 2 2 m ()) ] Fo the pupose of the numeical integation via desolve_system_k4, we have to eplace the m() ν() ρ() m 1 ν 1 ρ 1 symbolic functions, and by some symbolic vaiables,, and, say: In [7]: Out[7]: va('m_1 nu_1 ho_1 _1') hs [y.subs({m(): m_1, nu(): nu_1, ho(): ho_1}) fo y in hs] hs [ 4 π2ρ 1 u ( ρ ), π 3 ρ m 1 1, (π 3 ρ π 3 ρ m 1 ρ m 1 )u ( ρ 1 ) 2 m, (2 m 1 2 ) u ( ρ 1 ) ] The integation paametes: 12

13 In [71]: ho_c 1 _min 1e-8 _max 1 np 2 delta_ (_max - _min) / (np-1) The numeical esolution, with the initial conditions set in the paamete ics: ( min, m( min ), ν( min ), ρ( min ), R( min )) ( min,,, ρ c, min ) In [72]: sol desolve_system_k4(hs, vas(m_1, nu_1, ho_1, _1), iva, ics[_min,,, ho_c, _min], end_points(_min, _max), stepdelta_) The solution is etuned as a list, the fist 1 elements of which being: In [73]: Out[73]: sol[:1] [[1. 1 8,,, 1, ], [ , , , , ], [ , , , , ], [ , , , , ], [ , , , , ], [ , , , , ], [ , , , , ], [ , , , , ], [ , , , , ], [ , , , , ]] Each element is of the fom. So to get the list of values, we wite [, m(), ν(), ρ(), R()] (, ρ()) 13

14 In [74]: Out[74]: ho_sol [(s[], s[3]) fo s in sol] ho_sol[:1] [(1. 1 8, 1), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )] We may then use this list to have some plot of list into a gaphical object: ρ(), thanks to the function line, which tansfoms a In [75]: gaph line(ho_sol, axes_labels['$$', '$\ho$'], gidlinestue) gaph Out[75]: m() The solution fo : 14

15 In [76]: m_sol [(s[], s[1]) fo s in sol] m_sol[:1] Out[76]: [(1. 1 8, ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )] In [77]: gaph line(m_sol, axes_labels['$$', '$m$'], gidlinestue) gaph Out[77]: The solution fo ν() (has to be escaled by adding a constant to ensue ν(+ ) 1): 15

16 In [78]: Out[78]: nu_sol [(s[], s[2]) fo s in sol] nu_sol[:1] [(1. 1 8, ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , ), ( , )] In [79]: gaph line(nu_sol, axes_labels['$$', '$\tilde{\nu}$'], gidlines Tue) gaph Out[79]: The solution fo R() : 16

17 In [8]: Out[8]: _sol [(s[], s[4]) fo s in sol] line(_sol) The total gavitational mass of the sta is obtained via the last element (index: -1) of the list of (, m()) values: In [81]: M_gav m_sol[-1][1] M_gav Out[81]: Similaly, the stella adius is obtained though the last element of the list of (, R()) values: In [82]: R _sol[-1][1] R Out[82]: The sta's compactness: In [83]: M_gav/R Out[83]: Sequence of stella models Let us pefom a loop on the cental density. Fist we set up a list of values fo ρ c : 17

18 In [84]: ho_c_min.1 ho_c_max 3 n_conf 4 ho_c_list [ho_c_min + i * (ho_c_max-ho_c_min)/(n_conf-1) fo i in ange(n_conf)] ho_c_list Out[84]: [.1, , ,.24, , ,.47, , ,.7, , ,.93, , , 1.16, , , 1.39, , , 1.62, , , 1.85, , , 2.8, , , 2.31, , , 2.54, , , 2.77, , , 3. ] The loop: In [85]: M_list list() R_list list() fo ho_c in ho_c_list: sol desolve_system_k4(hs, vas(m_1, nu_1, ho_1, _1), iva, ics[_min,,, ho_c, _min], end_points(_min, _max), stepdelta_) M_list.append( sol[-1][1] ) R_list.append( sol[-1][4] ) The mass along the sequence: In [86]: M_list Out[86]: [ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ] The adius along the sequence: 18

19 In [87]: R_list Out[87]: [ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ] To daw M as a function of ρ c, we use the Python function zip to constuct a list of ( ρ c, M) values: 19

20 In [88]: zip(ho_c_list, M_list) Out[88]: [(.1, ), ( , ), ( , ), (.24, ), ( , ), ( , ), (.47, ), ( , ), ( , ), (.7, ), ( , ), ( , ), (.93, ), ( , ), ( , ), (1.16, ), ( , ), ( , ), (1.39, ), ( , ), ( , ), (1.62, ), ( , ), ( , ), (1.85, ), ( , ), ( , ), (2.8, ), ( , ), ( , ), (2.31, ), ( , ), ( , ), (2.54, ), ( , ), ( , ), (2.77, ), ( , ), ( , ), (3., )] 2

21 In [89]: Out[89]: gaph line(zip(ho_c_list, M_list), axes_labels['$\ho_c$', '$M$'], gidlinestue) gaph M Similaly, we daw as a function of : R In [9]: gaph line(zip(r_list, M_list), axes_labels['$r$', '$M$'], gidlin estue) gaph Out[9]: 21

22 and we save the plot in a pdf file to use it in ou next publication ;-) In [91]: gaph.save('plot_m_r.pdf') 22