Schwarzschild black hole in Cartesian coordinates with harmonic gauge. More...

#include <HarmonicSchwarzschild.hpp>

Classes
struct	Center

class	IntermediateComputer
	Computes the intermediates and quantities that we do not want to recompute across the solution's implementation. More...

class	IntermediateVars
	Computes and returns spacetime quantities of interest. More...

struct	internal_tags
	Tags defined for intermediates specific to the harmonic Schwarzschild solution. More...

struct	Mass

Public Types
using	options = implementation defined

template<typename DataType , typename Frame = ::Frame::Inertial>
using	CachedBuffer = CachedTempBuffer< internal_tags::x_minus_center< DataType, Frame >, internal_tags::r< DataType >, internal_tags::one_over_r< DataType >, internal_tags::x_over_r< DataType, Frame >, internal_tags::m_over_r< DataType >, internal_tags::sqrt_f_0< DataType >, internal_tags::f_0< DataType >, internal_tags::two_m_over_m_plus_r< DataType >, internal_tags::two_m_over_m_plus_r_squared< DataType >, internal_tags::two_m_over_m_plus_r_cubed< DataType >, internal_tags::spatial_metric_rr< DataType >, internal_tags::one_over_spatial_metric_rr< DataType >, internal_tags::spatial_metric_rr_minus_f_0< DataType >, internal_tags::d_spatial_metric_rr< DataType >, internal_tags::d_f_0< DataType >, internal_tags::d_f_0_times_x_over_r< DataType, Frame >, internal_tags::f_1< DataType >, internal_tags::f_1_times_x_over_r< DataType, Frame >, internal_tags::f_2< DataType >, internal_tags::f_2_times_xxx_over_r_cubed< DataType, Frame >, internal_tags::f_3< DataType >, internal_tags::f_4< DataType >, gr::Tags::Lapse< DataType >, internal_tags::neg_half_lapse_cubed_times_d_spatial_metric_rr< DataType >, gr::Tags::Shift< DataType, 3, Frame >, DerivShift< DataType, Frame >, gr::Tags::SpatialMetric< DataType, 3, Frame >, DerivSpatialMetric< DataType, Frame >, ::Tags::dt< gr::Tags::SpatialMetric< DataType, 3, Frame > >, gr::Tags::DetSpatialMetric< DataType >, internal_tags::one_over_det_spatial_metric< DataType > >
	Buffer for caching computed intermediates and quantities that we do not want to recompute across the solution's implementation. More...

Public Types inherited from gr::AnalyticSolution< 3_st >
using	DerivLapse = ::Tags::deriv< gr::Tags::Lapse< DataType >, tmpl::size_t< volume_dim >, Frame >

using	DerivShift = ::Tags::deriv< gr::Tags::Shift< DataType, volume_dim, Frame >, tmpl::size_t< volume_dim >, Frame >

using	DerivSpatialMetric = ::Tags::deriv< gr::Tags::SpatialMetric< DataType, volume_dim, Frame >, tmpl::size_t< volume_dim >, Frame >

using	tags = implementation defined

Public Member Functions
	HarmonicSchwarzschild (double mass, const std::array< double, volume_dim > &center, const Options::Context &context={})

	HarmonicSchwarzschild (const HarmonicSchwarzschild &)=default

HarmonicSchwarzschild &	operator= (const HarmonicSchwarzschild &)=default

	HarmonicSchwarzschild (HarmonicSchwarzschild &&)=default

HarmonicSchwarzschild &	operator= (HarmonicSchwarzschild &&)=default

	HarmonicSchwarzschild (CkMigrateMessage *)

template<typename DataType , typename Frame , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, volume_dim, Frame > &x, double, tmpl::list< Tags... >) const
	Computes and returns spacetime quantities for a Schwarzschild black hole with harmonic coordinates at a specific Cartesian position. More...

void	pup (PUP::er &p)

double	mass () const
	Return the mass of the black hole.

const std::array< double, volume_dim > &	center () const
	Return the center of the black hole.

Static Public Attributes
static constexpr Options::String	help

Static Public Attributes inherited from gr::AnalyticSolution< 3_st >
static constexpr size_t	volume_dim

Detailed Description

Schwarzschild black hole in Cartesian coordinates with harmonic gauge.

Details

This solution represents a Schwarzschild black hole in coordinates that are harmonic in both space and time, as well as horizon-penetrating. Therefore, this solution fulfills the harmonic coordinate conditions Eq. (4.42), (4.44), and (4.45) in [14] :

$\begin{aligned} (1) & ^{(4)} Γ^{a} & = 0 \end{aligned}$

$\begin{aligned} (2) & (\partial_{t} - β^{j} \partial_{j}) α & = - α^{2} K \end{aligned}$

$\begin{aligned} (3) & (\partial_{t} - β^{j} \partial_{j}) β^{i} & = - α^{2} (γ^{i j} \partial_{j} \ln α - γ^{j k} Γ_{j k}^{i}) \end{aligned}$

(Note that Eq. 4.45 in [14] is missing a minus sign in front of the $Γ$ -contraction term)

We implement Eqs. (45)–(50) in [42] , which represent the zero-spin limit of the time-harmonic and horizon-penetrating slices of Kerr spacetime presented in the paper. We add the radial transformation $r \to r + M$ to make the spatial coordinates harmonic as well (see Eq. (43) in [42] ), so the coordinates remain harmonic under boosts.

Consider a Schwarzschild black hole of mass $M$ and center $C^{i}$ . The spacetime will be specified using Cartesian coordinates $x^{i}$ that are spatially and temporally harmonic. A radius centered on the black hole is

$\begin{aligned} (4) & r & = \sqrt{δ_{i j} (x^{i} - C^{i}) (x^{j} - C^{j})} \end{aligned}$

For computing the spatial metric, we define the following quantities:

$\begin{aligned} (5) & γ_{r r} & = 1 + \frac{2 M}{M + r} + {(\frac{2 M}{M + r})}^{2} + {(\frac{2 M}{M + r})}^{3}, \\ (6) & \partial_{r} γ_{r r} & = - \frac{1}{2 M} {(\frac{2 M}{M + r})}^{2} - \frac{1}{M} {(\frac{2 M}{M + r})}^{3} - \frac{3}{2 M} {(\frac{2 M}{M + r})}^{4}, \\ (7) & \frac{X^{i}}{r} & = \frac{x^{i} - C^{i}}{r}, \\ (8) & X_{j} & = X^{i} δ_{i j} \end{aligned}$

From these quantities, the spatial metric and its time derivative are computed as

$\begin{aligned} (9) & γ_{i j} & = (γ_{r r} - {(1 + \frac{M}{r})}^{2}) \frac{X_{i}}{r} \frac{X_{j}}{r} + δ_{i j} {(1 + \frac{M}{r})}^{2}, \\ (10) & \partial_{t} γ_{i j} & = 0 \end{aligned}$

The spatial derivative is given in terms of the following quantities:

$\begin{aligned} (11) & f_{0} & = {(1 + \frac{M}{r})}^{2} \\ (12) & \partial_{r} f_{0} & = 2 (1 + \frac{M}{r}) (- \frac{M}{r^{2}}), \\ (13) & f_{1} & = \frac{1}{r} (γ_{r r} - f_{0}), \\ (14) & f_{2} & = \partial_{r} γ_{r r} - \partial_{r} f_{0} - 2 f_{1} \end{aligned}$

In terms of these, the spatial metric's spatial derivative is

$\begin{aligned} (15) & \partial_{k} γ_{i j} & = f_{2} \frac{X_{i}}{r} \frac{X_{j}}{r} \frac{X_{k}}{r} + f_{1} \frac{X_{j}}{r} δ_{i k} + f_{1} \frac{X_{i}}{r} δ_{j k} + \partial_{r} f_{0} \frac{X_{k}}{r} δ_{i j} \end{aligned}$

The lapse and its derivatives are

$\begin{aligned} (16) & α & = γ_{r r}^{- 1 / 2}, \\ (17) & \partial_{t} α & = 0, \\ (18) & \partial_{i} α & = - \frac{1}{2} γ_{r r}^{- 3 / 2} \partial_{r} γ_{r r} \frac{X_{i}}{r} \end{aligned}$

The shift and its time derivative are

$\begin{aligned} (19) & β^{i} & = {(\frac{2 M}{M + r})}^{2} \frac{X^{i}}{r} \frac{1}{γ_{r r}}, \\ (20) & \partial_{t} β^{i} & = 0 \end{aligned}$

The spatial derivative of the shift is computed in terms of the following quantities:

$\begin{aligned} (21) & f_{3} & = \frac{1}{r} \frac{1}{γ_{r r}} {(\frac{2 M}{M + r})}^{2}, \\ (22) & f_{4} & = - f_{3} - \frac{1}{M} \frac{1}{γ_{r r}} {(\frac{2 M}{M + r})}^{3} - \partial_{r} γ_{r r} {(\frac{2 M}{M + r} \frac{1}{γ_{r r}})}^{2} \end{aligned}$

In terms of these, the shift's spatial derivative is

$\begin{aligned} (23) & \partial_{k} β^{i} & = f_{4} \frac{X^{i}}{r} \frac{X_{k}}{r} + δ_{k}^{i} f_{3} \end{aligned}$

Member Typedef Documentation

◆ CachedBuffer

template<typename DataType , typename Frame = ::Frame::Inertial>

Buffer for caching computed intermediates and quantities that we do not want to recompute across the solution's implementation.

Details

See internal_tags documentation for details on what quantities the internal tags represent

Member Function Documentation

◆ variables()

template<typename DataType , typename Frame , typename... Tags>

tuples::TaggedTuple< Tags... > gr::Solutions::HarmonicSchwarzschild::variables	(	const tnsr::I< DataType, volume_dim, Frame > &	x,
		double	,
		tmpl::list< Tags... >
	)		const

inline

Computes and returns spacetime quantities for a Schwarzschild black hole with harmonic coordinates at a specific Cartesian position.

Parameters

x	Cartesian coordinates of the position at which to compute spacetime quantities

Member Data Documentation

◆ help

constexpr Options::String gr::Solutions::HarmonicSchwarzschild::help

staticconstexpr

Initial value:

{

"Schwarzschild black hole in Cartesian coordinates with harmonic gauge"}

The documentation for this class was generated from the following file:

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/HarmonicSchwarzschild.hpp

Classes

Public Types

Public Member Functions

Static Public Attributes

Detailed Description

Details

Member Typedef Documentation

◆ CachedBuffer

Details

Member Function Documentation

◆ variables()

Member Data Documentation

◆ help