IDLinearWaves

Date

Abstract

Provides gravitational wave solutions to the linearized Einstein equations

1 Purpose

There are two diﬀerent linearized initial data sets provided, plane waves and Teukolsky waves.

2 Plane Waves

A full description of plane waves can be found in the PhD Thesis of Malcolm Tobias, The Numerical Evolution of Gravitational Waves, which can be found at http://wugrav.wustl.edu/Papers/Thesis97/Thesis97.html.

Plane waves travelling in arbitrary directions can be speciﬁed. For these plane waves the TT gauge is assumed (the metric perturbations are transverse to the direction of propagation, and the metric is traceless). In the case of waves travelling along the $z-$direction this would give the plus solution

${h}_{xx}=-{h}_{yy}=f\left(t±z\right),{h}_{xy}={h}_{xz}={h}_{yz}={h}_{zz}=0$

and the cross solution

${h}_{xy}={h}_{yx}=f\left(t±z\right),{h}_{yz}={h}_{xx}={h}_{yy}={h}_{zz}=0$

This thorn implements the plus solution, with the waveform $f\left(t±z\right)$ having the form of a Gaussian modulated sine function. Now working with a general direction of propagation $k$ we have the plane wave solution:

$f\left(t,x,y,z\right)={A}_{in}{e}^{-{\left({k}_{i}^{p}{x}^{i}+{\omega }_{p}\left(t-{r}_{a}\right)\right)}^{2}}cos\left({k}_{i}{x}^{i}+\omega t\right)+{A}_{out}{e}^{-{\left({k}_{i}^{p}{x}^{i}-{\omega }_{p}\left(t-{r}_{a}\right)\right)}^{2}}cos\left({k}_{i}{x}^{i}-\omega t\right)$

and

$\begin{array}{rcll}{g}_{xx}& =& 1+f\left[{cos}^{2}\varphi -{cos}^{𝜃}{sin}^{2}\varphi \right]& \text{}\\ {g}_{xy}& =& -f{sin}^{2}𝜃sin\varphi cos\varphi & \text{}\\ {g}_{xz}& =& fsin𝜃cos𝜃sin\varphi & \text{}\\ {g}_{yy}& =& 1+f\left[{sin}^{2}\varphi -co{s}^{2}𝜃{cos}^{2}\varphi \right]& \text{}\\ {g}_{yz}& =& fsin𝜃cos𝜃cos\varphi & \text{}\\ {g}_{zz}& =& 1-f{sin}^{2}𝜃& \text{}\end{array}$

The extrinsic curvature is then calculated from

 ${K}_{ij}=-\frac{1}{2\alpha }{ġ}_{ij}$ (1)

3 Teukolsky waves

Teukolsky waves are quadrupole wave solutions to the linearized Einstein equations. For a full description, see: PRD 26:745 (1982).

The extrinsic curvature is initialized assuming the initial lapse is one.

5 Parameters

 amplitude Scope: private REAL Description: Amplitude of the wave: both for teuk and plane Range Default: 0.001 0: positive amplitude

 mvalue Scope: private INT Description: m value for teukwaves waves: integer from -2 to 2 Range Default: (none) -2:2 implemented : m = -2..2

 packet Scope: private KEYWORD Description: Packet for teukwaves: eppley,evans,square Range Default: eppley eppley Eppley type evans Evans type square Square type

 parity Scope: private KEYWORD Description: Parity for teukwaves: even or odd Range Default: even even even parity odd odd parity

 teuk_no_vee Scope: private KEYWORD Description: Initialize Teuk. waves with V=0? Range Default: no no Bona Masso setting yes Bona Masso setting

 wavecenter Scope: private REAL Description: linears waves thingie Range Default: 0.0 :

 wavelength Scope: private REAL Description: linearwaves wave length Range Default: 2.0 0: positive wavelength

 wavephi Scope: private REAL Description: Phi angle for planewaves Range Default: 0.0 :

 wavepulse Scope: private REAL Description: planewaves thingy for the gaussian pulse Range Default: 1.0 0: positive pulse

 wavesgoing Scope: private KEYWORD Description: in and outgoing waves... Range Default: both in Ingoing wave out Outgoing wave both In and outgoing wave

 wavetheta Scope: private REAL Description: Theta angle for planewaves Range Default: 0.0 :

 conformal_storage Scope: shared from STATICCONFORMAL KEYWORD

Implements:

idlinearwaves

Inherits:

staticconformal

grid

7 Schedule

This section lists all the variables which are assigned storage by thorn EinsteinInitialData/IDLinearWaves. 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.

NONE

Scheduled Functions

CCTK_PARAMCHECK (conditional)

idlinearwaves_paramchecker

check that the metric_type is recognised

 Language: c Options: global Type: function

idlinearwaves_planewaves

construct linear planewave initial data

 Language: fortran Type: function

CCTK_PARAMCHECK (conditional)

idlinearwaves_paramchecker

check that the metric_type is recognised

 Language: c Options: global Type: function

idlinearwaves_standwaves

construct linear planewave initial data

 Language: fortran Type: function

CCTK_PARAMCHECK (conditional)

idlinearwaves_paramchecker

check that the metric_type is recognised

 Language: c Options: global Type: function

idlinearwaves_teukwaves

construct linear teukolsky wave initial data

 Language: fortran Type: function

CCTK_PARAMCHECK (conditional)

idlinearwaves_paramchecker

check that the metric_type is recognised

 Language: c Options: global Type: function