1 Introduction

The Tolman-Oppenheimer-Volkoff solution is a static perfect fluid “star”. It is frequently used as a test of relativistic hydro codes. Here it is intended for use without evolving the matter terms. This provides a compact strong field solution which is static but does not contain singularities.

2 Equations

The equations for a TOV star [1, 2, 3] are usually derived in Schwarzschild coordinates. In these coordinates, the metric can be brought into the form \begin {equation} ds^2 = -e^{2\phi }dt^2 + \left (1-\dfrac {2m}{r}\right )^{-1}dr^2 + r^2 d\Omega ^2 \ . \end {equation}

This thorn is based on the notes of Thomas Baumgarte [4] that have been partially included in this documentation. However the notation for the fluid quantities follows [5]. Here we are assuming that the stress energy tensor is given by \begin {equation} \label {eq:Tmunu} T^{\mu \nu } = (\mu + P)u^{\mu }u^{\nu } + Pg^{\mu \nu }, \end {equation} where \(\mu \) is the total energy, \(P\) the pressure, \(u^{\mu }\) the fluid four velocity, \(\rho \) the rest-mass density, \(\epsilon \) the specific internal energy, and

This enforces a polytropic equation of state. We note that in Cactus the units are \(c = G = M_{\odot } = 1\).

The equations to give the initial data are solved (as usual) in the Schwarzschild-like coordinates with the areal radius labelled \(r\). The equations of the relativistic hydrostatic equilibrium are

Here \(m\) is the gravitational mass inside the sphere radius \(r\), and \(\phi \) the logarithm of the lapse. Once the integration is done for the interior of the star we match to the exterior (see below). In the exterior we have \begin {align} \label {eq:TOVexterior} P & = {\tt TOV\_atmosphere}, \\ m & = M, \\ \phi & = \dfrac {1}{2} \log (1-2M / r). \end {align}

In order to impose initial data in cartesian coordinates, we want to transform this solution to isotropic coordinates, in which the metric takes the form \begin {equation} \label {eq:metr_iso} ds^2 = -e^{2\phi }dt^2+e^{2\Lambda }\left (d\bar {r}^2+\bar {r}^2d\Omega ^2\right ) \ . \end {equation} Here \(\bar {r}\) denotes the isotropic radius. Matching the two metrics, one obviously finds \begin {align} r^2 &= e^{2\Lambda }\bar {r}^2 \ , \\ \left (1-\dfrac {2m}{r}\right )^{-1} dr^2 &= e^{2\Lambda }d\bar {r}^2 \ . \end {align}

As a result, we have an additional differential equation to solve in order to have \(\bar {r}(r)\), that is \begin {equation} \label {eq:rbar} \frac {d (\log (\bar {r} / r))}{\partial r} = \frac {r^{1/2} - (r-2m)^{1/2}}{r(r-2m)^{1/2}} \ . \end {equation} Given such a solution, the missing metric potential is simply given by \begin {equation} e^{\Lambda } = \dfrac {r}{\bar {r}} \ . \end {equation} In the following section we concentrate on solving Eq. (??) in the exterior and in the interior of the star.

Then, given these one-dimensional data we interpolate to get data on the three-dimensional Cactus grid; that is, we interpolate on the three dimensional r given by the x, y, z variables the physical hydro and spacetime quantities that are function of the isotropic radius \(\bar {r}\) computed above. Only linear interpolation is used. This avoids problems at the surface of the star, and does not cause problems if the number of points in the one dimensional array is sufficient (\(1\times 10^5\) is the default, which should be sufficient for medium-sized grids).

2.1 Exterior

In the exterior of the star, \(r>R\), the mass \(M\equiv m(R)\) is constant, and Eq. (??) can be solved analytically up to a constant of integration. Fixing this constant such that \(r\) and \(\bar {r}\) agree at infinity, we find \begin {equation} \bar {r} = \dfrac {1}{2}\left (\sqrt {r^2-2Mr}+r -M\right ) \ , \end {equation} or, solving for \(r\) [cfr. Exercise 31.7 of MTW [3]] \begin {equation} r=\bar {r}\left (1+\dfrac {M}{2\bar {r}}\right )^2 \ . \end {equation} The metric potential as a function of \(\bar {r}\) is obviously \begin {equation} e^{\Lambda } = \left (1+\dfrac {M}{2\bar {r}}\right )^2 \ . \end {equation}

2.2 Interior

In the interior, Eq, (??) can not be integrated analytically, because \(m\) is now a function of \(r\). Instead, we have to integrate \begin {equation} \int _0^{\bar {r}} \dfrac {d\bar {r}}{\bar {r}} = \int _0^r\left (1-\dfrac {2m}{r}\right )^{-1/2}\dfrac {dr}{r} \ . \end {equation} The left hand side can be integrated analytically, and has a singular point at \(\bar {r}=0\). The right hand side cannot be integrated analytically, but will also be singular at \(r=0\), which poses problems when trying to integrate the equation numerically. We therefore rewrite the right hand side by adding and subracting a term \(1/r\), which yields \begin {equation} \int _0^r\dfrac {1}{r(1-2m/r)^{1/2}}dr = \int _0^r\dfrac {1-(1-2m/r)^{1/2}}{r(1-2m/r)^{1/2}}dr+\int _0^r\dfrac {dr}{r} \ . \end {equation} Since \(m\sim r^3\) close to the origin, the first term on the right hand side is now regular and the second one can be integrated analytically. As a result, we find \begin {equation} \int _0^{\bar {r}}d\ln \bar {r}-\int _0^rd\ln r=\int _0^r\dfrac {1-(1-2m/r)^{1/2}}{r(1-2m/r)^{1/2}}dr \ . \end {equation} Replacing the lower limits (\(r=\bar {r}=0\)) temporarily with \(r_0\) and \(\bar {r}_0\), we can integrate the right hand side and find \begin {equation} \ln \left (\dfrac {\bar {r}}{r}\right )-\ln \left (\dfrac {\bar {r}_0}{r_0}\right )=\int _0^r\dfrac {1-(1-2m/r)^{1/2}}{r(1-2m/r)^{1/2}}dr \ , \end {equation} or \begin {equation} \bar {r} = C r \exp \left [\int _0^r\dfrac {1-(1-2m/r)^{1/2}}{r(1-2m/r)^{1/2}}dr\right ] \ . \end {equation} Here the constant of integration \(C\) is related to the ratio \(\bar {r}_0/r_0\) evaluated at the origin (which is perfectly regular). It can be chosen such that the interior solution matches the exterior solution at the surface of the star. This requirement implies \begin {equation} C = \dfrac {1}{2R}\left (\sqrt {R^2-2MR}+R-M\right )\exp \left [-\int _0^R\dfrac {1-(1-2m/r)^{1/2}}{r(1-2m/r)^{1/2}}dr\right ] \ . \end {equation} In this respect, let us recall how we do initial data for the system of equations at \(r=0\). Given a value of the central density \(\rho _c\) (in Cactus units) we pose \(P_0 = K\rho _c^{\Gamma }\), \(m_0=0\), \(\phi _0=0\) and \(\bar {r}_0 = {\tt TOV\_Tiny}\), \(r_0 = {\tt TOV\_Tiny}\). The TOV_Tiny number is hardwired into the code to avoid divide by zero errors; it is \(10^{-20}\). Also the default parameters will give the TOV star used for the long term evolutions in [6]. That is, a nonrotating \(N=1\) (\(\gamma =1+1/N=2\)) polytropic star with gravitational mass \(M=1.4M_{\odot }\), circumferential radius \(R=14.15\)km, central rest-mass density \(\rho _c=1.28 \times 10^{-3}\) and \(K=100\).

3 Use of this thorn

To use this thorn to provide initial data for the ADMBase variables \(\alpha \), \(\beta \), \(g\) and \(K\) just activate the thorn and set ADMBase:initial_data = ‘‘TOV’’.

There are two ways of coupling the matter sources to the thorn that evolves the Einstein equations. One is to use the CalcTmunu interface. This will give the components of the stress energy tensor pointwise across the grid. For an example of this, see thorn ADM in CactusEinstein.

To use the CalcTmunu interface you should

• put the lines

       friend: ADMCoupling
       
       USES INCLUDE: CalcTmunu.inc
       USES INCLUDE: CalcTmunu_temps.inc
       USES INCLUDE: CalcTmunu_rfr.inc
  

  in your interface.ccl

• In any routine requiring the matter terms, put

  • #include ‘‘CalcTmunu_temps.inc’’ in the variable declarations

  • declare CCTK_REALs Ttt, Ttx, Tty, Ttz, Txx, Txy, Txz, Tyy, Tyz, Tzz.

  • Inside an i,j,k loop put #include ‘‘CalcTmunu.inc’’. This must be a Fortran routine (We could probably fix this if requested).

• You then use the real numbers Ttt etc. as the stress energy tensor at a point.

As an alternative you can use the grid functions StressEnergytt, StressEnergytx, etc. directly to have the stress energy tensor over the entire grid. To do this you just need the line friend: ADMCoupling in your interface.ccl. Although this seems much simpler, you will now only get the contributions from the TOVSolver thorn. If you want to use other matter sources, most of the current thorns (CosmologicalConstant, the hydro code, the scalar field code) all use the CalcTmunu interface.

You also have the possibility to use a parameter GRHydrotovsolver::TOV_Separation to obtain a spacetime consisting of one TOV-system for \(x>0\) and a second (similar) for \(x<0\). This parameter sets the separation of the centers of two neutron stars, has to be positive and should be larger than twice the radius of one star.
Be aware that the spacetime obtained by this is no physical spacetime and no solution of Einsteins Equations and therefore an IVP-run has to follow. This parameter was only introduced for testing purposes of the IVP-Solver and should only be considered as such. There would be better (and also easy) ways to obtain initial data for two TOVs than that.

References

4 Parameters

5 Interfaces

General

Implements:

tovsolver

Inherits:

admbase

hydrobase

constants

staticconformal

Uses header:

constants.h

Provides:

Set_Rho_ADM to

Set_Momentum_Source to

Set_Initial_Guess_for_u to

Rescale_Sources to

6 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/TOVSolver. Storage can either last for the duration of the run (Always means that if this thorn is activated storage will be assigned, Conditional means that if this thorn is activated storage will be assigned for the duration of the run if some condition is met), or can be turned on for the duration of a schedule function.

Storage

NONE

Scheduled Functions

CCTK_PARAMCHECK

  tov_c_paramcheck

  check parameters

CCTK_WRAGH

  tov_c_allocatememory

  allocate memory for tovsolver_c

CCTK_POSTPOSTINITIAL (conditional)

  tov_c_freememory

  free memory from tovsolver_c

HydroBase_Initial (conditional)

  tov_initial_data

  group for the tov initial data

TOV_Initial_Data (conditional)

  tov_c_integrate_rhs

  integrate the 1d equations for the tov star

TOV_Initial_Data (conditional)

  tov_write_1d_datafile

  save data to file and exit

TOV_Initial_Data (conditional)

  tov_set_properpositions

  steer ns position parameters according to proper distance

TOV_Initial_Data (conditional)

  tov_c_exact

  set up the 3d quantities for the tov star

(conditional)

  tov_prepare_fake_evolution

  prepare for fake evolution

MoL_PostStep (conditional)

  tov_c_exact

  use fake evolution