getEnergyTensor

Description

The input metric determines the stress-energy tensor by evaluating the field equations. This evaluation does not consider any additions from $\Lambda$.

Einstein Field Equation

The stress-energy tensor $T_{\mu\nu}$ is found by solving the field equation:

$T_{\mu\nu} = \frac{c^4}{8 \pi G} \left( R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu} \right)$

where the Ricci scalar$R$ is:

$R = R_{\mu\nu} g^{\mu\nu}$

and the Ricci tensor$R_{\mu\nu}$ is found directly in terms of derivatives of the metric:

$R_{\mu \nu} =-\frac{1}{2} \sum_{\alpha, \beta=0}^3\left(\frac{\partial^2 g_{\mu \nu}}{\partial x^\alpha \partial x^\beta}+\frac{\partial^2 g_{\alpha \beta}}{\partial x^\mu \partial x^\nu}-\frac{\partial^2 g_{\mu \beta}}{\partial x^\nu \partial x^\alpha}-\frac{\partial^2 g_{\nu \beta}}{\partial x^\mu \partial x^\alpha}\right) g^{\alpha \beta}$

$+\frac{1}{2} \sum_{\alpha, \beta, \gamma, \delta=0}^3\left(\frac{1}{2} \frac{\partial g_{\alpha \gamma}}{\partial x^\mu} \frac{\partial g_{\beta \delta}}{\partial x^\nu}+\frac{\partial g_{\mu \gamma}}{\partial x^\alpha} \frac{\partial g_{\nu \delta}}{\partial x^\beta}-\frac{\partial g_{\mu \gamma}}{\partial x^\alpha} \frac{\partial g_{\nu \beta}}{\partial x^\delta}\right) g^{\alpha \beta} g^{\gamma \delta}$

$-\frac{1}{4} \sum_{\alpha, \beta, \gamma, \delta=0}^3\left(\frac{\partial g_{\nu \gamma}}{\partial x^\mu}+\frac{\partial g_{\mu \gamma}}{\partial x^\nu}-\frac{\partial g_{\mu \nu}}{\partial x^\gamma}\right)\left(2 \frac{\partial g_{\beta \delta}}{\partial x^\alpha}-\frac{\partial g_{\alpha \beta}}{\partial x^\delta}\right) g^{\alpha \beta} g^{\gamma \delta}$

Method

The derivatives of the metric use finite central differencing methods on the spacetime grid.

The grid size defined in the metric must be 5 or greater in any direction for the finite difference to evaluate a derivative on the grid. If the size in a given direction is <5 then the derivative is assumed to be 0.

Syntax

[energy] = getEnergyTensor(metric, tryGPU, diffOrder)

Input Arguments

blue are required inputs.

orange are optional inputs with native default values.

Inputs	Format	Type	Description
metric	struct	object	Input metric tensor.
tryGPU	1x1 array	integer	Use the GPU for evaluations, input either 1 for true or 0 for false. The default value is 0.
diffOrder	1x1 array	string	Sets the order of the finite difference method. Accepts either "second" to use second-order or "fourth" to use fourth-order methods. The default value is fourth order.

Output Arguments

Outputs	Format	Type	Description
energy	struct	object	Returns the stress-energy tensor with the same grid and frame as the metric tensor. Note that it returns the contravariant form of the tensor, which can also be found by checking the .index field of the struct.