metricGet_Schwarzschild

Description

Create the Schwarzschild spacetime in Cartesian coordinates.

Schwarzschild Metric

The classic Schwarzschild metric given in pseudo-Cartesian coordinates is:

$\begin{aligned} d s^2= & -\left(1-\frac{r_s}{r}\right)^2 d t^2+\left(\frac{x^2}{1-r_s / r}+y^2+z^2\right) \frac{d x^2}{r^2}+\left(x^2+\frac{y^2}{1-r_s / r}+z^2\right) \frac{d y^2}{r^2} \\ & +\left(x^2+y^2+\frac{z^2}{1-r_s / r}\right) \frac{d z^2}{r^2}+\frac{2 r_s}{r^2\left(r-r_s\right)}(x y d x d y+x z d x d z+y z d y d z), \end{aligned}$

where $r_s$ is the Schwarzschild radius.

For more details on the Schwarzschild metric please read the following reference (section 2.2.2) or any standard general relativity textbook:

Catalogue of SpacetimesarXiv.org

Method

The metric is constructed using the Schwarzschild radius as a unitless pass-in value in Cartesian coordinates.

Syntax

[metric] = metricGet_Schwarzschild(gridSize,worldCenter,rs, gridScale)

Input Arguments

blue are required inputs.

orange are optional inputs with native default values.

blue are required inputs

Inputs	Format	Type	Description
gridSize	1x4 array	integer	The size of the world specified as: $[t, x, y, z]$
worldCenter	1x4 array	double	The center of the world, which defines the center of $r_s$ as a 4-vector, specified as: $[t,x,y,z]$
rs	1x1 array	double	Schwarzschild radius, in unitless form.
gridScale	1x4 array	double	Unit scaling factor of the grid dimensions defined relative to gridSize. This determines the resolution of the grid along each dimension. Specified as: $[t_{scale}, x_{scale}, y_{scale}, z_{scale}]$ The default value is [1, 1, 1, 1].

Output Arguments

Outputs	Format	Type	Description
metric	struct	object	Schwarzschild solution returned as the metric tensor object.