1 ScalarEvolve

This thorn provides the tools to evolve a complex scalar field with nonlinear potential in arbitrary background as first described in [1]. Additionally, it is possible to add an external periodic forcing to the right-hand side of the Klein-Gordon equation for studies of parametric resonances.

ScalarEvolve interfaces with the ScalarBase thorn, which defines the evolution variables phi1, phi2, Kphi1 and Kphi2. This thorn merely takes care of setting the scalar field stress-energy tensor and evolving (with MoL) the scalar field itself. The evolution of the metric sector needs to be done elsewhere.

2 Physical System

Conventions used here are as in [1] and as follows: \begin {align} G_{\mu \nu } & = 8 \pi G T_{\mu \nu } \\ \square \Phi & = \mu ^2 \Phi \end {align}

where \(\mu \) is the mass parameter defined in ScalarBase and \begin {equation} T_{\mu \nu } = \bar \Phi _{,\mu } \Phi _{,\nu } + \Phi _{,\mu } \bar \Phi _{,\nu } - g_{\mu \nu } [ \bar \Phi ^{,\sigma } \Phi _{,\sigma } + \mu ^2 \bar \Phi \Phi ] \end {equation} and the gridfunctions phi1 and phi2 used in the code (defined in ScalarBase) are the real and imaginary part of the complex field \(\Phi = \phi _1 + i \phi _2\), which for reasons of convenience are evolved as separate, independent, variables.

Let us briefly present the \(3+1\) decomposed equations of motion in the Einstein-Klein-Gordon case. To complete the characterization of the full spacetime we define the extrinsic curvature \begin {equation} \label {eq:KijDef} K_{ij} = - \frac {1}{2\alpha } \left ( \partial _{t} - \mathcal {L}_{\beta } \right ) \gamma _{ij} \ , \end {equation} and analogously introduce the “canonical momentum” of the complex scalar field \(\Phi \) \begin {equation} \label {eq:Kphi} K_{\Phi } = -\frac {1}{2\alpha } \left ( \partial _{t} - \mathcal {L}_{\beta } \right ) \Phi \,, \end {equation} where \(\mathcal {L}\) denotes the Lie derivative. Our evolution system can then be written in the form \begin {align} \partial _{t} \gamma _{ij} & = - 2 \alpha K_{ij} + \mathcal {L}_{\beta } \gamma _{ij} \,, \label {eq:dtgamma} \\ \partial _{t} K_{ij} & = - D_{i} \partial _{j} \alpha + \alpha \left ( R_{ij} - 2 K_{ik} K^{k}{}_{j} + K K_{ij} \right ) \nonumber \\ & \quad + \mathcal {L}_{\beta } K_{ij} + 4\pi \alpha \left [ (S-\rho ) \gamma _{ij} - 2 S_{ij} \right ] \,, \label {eq:dtKij} \\ \partial _{t} \Phi & = - 2 \alpha K_\Phi + \mathcal {L}_{\beta } \Phi \ \,, \label {eq:dtPhi} \\ \partial _{t} K_\Phi & = \alpha \left ( K K_{\Phi } - \frac {1}{2} \gamma ^{ij} D_i \partial _j \Phi + \frac {1}{2} \mu ^2 \Phi \right ) \nonumber \\ & \quad - \frac {1}{2} \gamma ^{ij} \partial _i \alpha \partial _j \Phi + \mathcal {L}_{\beta } K_\Phi \,, \label {eq:dtKphi} \end {align}

where \(D_i\) is the covariant derivative with respect to the \(3\)-metric.

For completeness, we present also the full evolution equations re-written in the BSSN scheme [2, 3]. The full system of evolution equations is \begin {align} \partial _t \tilde {\gamma }_{ij} & = \beta ^k \partial _k \tilde {\gamma }_{ij} + 2\tilde {\gamma }_{k(i} \partial _{j)} \beta ^k - \frac {2}{3} \tilde {\gamma }_{ij} \partial _k \beta ^k -2\alpha \tilde {A}_{ij}, \\ \partial _t \chi & = \beta ^k \partial _k \chi + \frac {2}{3} \chi (\alpha K - \partial _k \beta ^k), \\ \partial _t \tilde {A}_{ij} & = \beta ^k \partial _k \tilde {A}_{ij} + 2\tilde {A}_{k(i} \partial _{j)} \beta ^k - \frac {2}{3} \tilde {A}_{ij} \partial _k \beta ^k \notag \\ & \quad + \chi \left ( \alpha R_{ij} - D_i \partial _j \alpha \right )^{\rm TF} + \alpha \left ( K\,\tilde {A}_{ij} - 2 \tilde {A}_i{}^k \tilde {A}_{kj} \right ) \notag \\ & \quad - 8 \pi \alpha \left ( \chi S_{ij} - \frac {S}{3} \tilde \gamma _{ij} \right ), \\ \partial _t K & = \beta ^k \partial _k K - D^k \partial _k \alpha + \alpha \left ( \tilde {A}^{ij} \tilde {A}_{ij} + \frac {1}{3} K^2 \right ) \notag \\ & \quad + 4 \pi \alpha (\rho + S), \\ \partial _t \tilde {\Gamma }^i & = \beta ^k \partial _k \tilde {\Gamma }^i - \tilde {\Gamma }^k \partial _k \beta ^i + \frac {2}{3} \tilde {\Gamma }^i \partial _k \beta ^k + 2 \alpha \tilde {\Gamma }^i_{jk} \tilde {A}^{jk} \notag \\ & \quad + \frac {1}{3} \tilde {\gamma }^{ij}\partial _j \partial _k \beta ^k + \tilde {\gamma }^{jk} \partial _j \partial _k \beta ^i \nonumber \\ & \quad - \frac {4}{3} \alpha \tilde {\gamma }^{ij} \partial _j K - \tilde {A}^{ij} \left ( 3 \alpha \chi ^{-1} \partial _j \chi + 2\partial _j \alpha \right ) \notag \\ & \quad - 16 \pi \alpha \chi ^{-1} j^i \label {eq:tilde-Gamma-evol} \,, \\ \partial _{t} \Phi & = - 2 \alpha K_\Phi + \mathcal {L}_{\beta } \Phi \ \,, \label {eq:dtPhi-BSSN} \\ \partial _{t} K_\Phi & = \alpha \left ( K K_{\Phi } - \frac {1}{2} \gamma ^{ij} D_i \partial _j \Phi + \frac {1}{2} \mu ^2 \Phi \right ) \nonumber \\ & \quad - \frac {1}{2} \gamma ^{ij} \partial _i \alpha \partial _j \Phi + \mathcal {L}_{\beta } K_\Phi \,, \label {eq:dtKphi-BSSN} \end {align}

with the source terms given by \begin {align*} \rho & \equiv T^{\mu \nu }n_{\mu }n_{\nu } \,,\\ j_i &\equiv -\gamma _{i\mu } T^{\mu \nu }n_{\nu } \,, \\ S_{ij} &\equiv \gamma ^{\mu }{}_i \gamma ^{\nu }{}_j T_{\mu \nu } \,, \\ S & \equiv \gamma ^{ij}S_{ij} \,. \end {align*}

Note however, as mentioned above, that the evolution of the metric sector is not done in this thorn, and thus needs to be done elsewhere (ie, LeanBSSNMoL or McLachlan).

3 Obtaining This Thorn

This thorn is included in the Einstein Toolkit and can also be obtained through the Canuda numerical relativity library [4].

References

4 Parameters

5 Interfaces

General

Implements:

scalarevolve

Inherits:

admbase

tmunubase

scalarbase

boundary

sphericalsurface

Grid Variables

5.0.1 PRIVATE GROUPS

6 Schedule

This section lists all the variables which are assigned storage by thorn Scalar/ScalarEvolve. 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

Scheduled Functions

SetTmunu (conditional)

  scalar_zero_densities

  initialise the sf densities to zero

CCTK_PARAMCHECK (conditional)

  scalar_init

  initialise flags and boundary condition

AddToTmunu (conditional)

  scalar_calc_tmunu

  compute the energy-momentum tensor

CCTK_BASEGRID (conditional)

  scalar_initsymbound

  schedule symmetries

CCTK_BASEGRID (conditional)

  scalar_zero_rhs

  set all rhs functions to zero to prevent spurious nans

MoL_Register (conditional)

  scalar_registervars

  register variables for mol

MoL_CalcRHS (conditional)

  scalar_ord4_calc_rhs

  mol rhs calculation for scalar variables

MoL_CalcRHS (conditional)

  scalar_ord4_calc_rhs_bdry

  mol boundary rhs calculation for scalar variables

MoL_RHSBoundaries (conditional)

  scalar_ord4_calc_rhs_bdry_sph

  mol boundary rhs calculation for scalar variables for the multipatch system

MoL_PostStep (conditional)

  scalar_boundaries

  mol boundary enforcement for scalar variables

MoL_PostStep (conditional)

  applybcs

  apply boundary conditions

Aliased Functions