# getEnergyTensor

## Description

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

<details>

<summary>Einstein Field Equation</summary>

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:

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

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

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

</details>

## Method

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

{% hint style="warning" %}
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.**
{% endhint %}

## Syntax

`[`<mark style="color:green;">`energy`</mark>`] = getEnergyTensor(`<mark style="color:blue;">`metric`</mark>`,`` `<mark style="color:orange;">`tryGPU`</mark>`,`` `<mark style="color:orange;">`diffOrder`</mark>`)`

### Input Arguments

{% hint style="info" %} <mark style="color:blue;">blue</mark> are required inputs.

<mark style="color:orange;">orange</mark> are optional inputs with native default values.
{% endhint %}

<table><thead><tr><th width="232">Inputs</th><th width="94">Format</th><th width="92">Type</th><th>Description</th></tr></thead><tbody><tr><td><mark style="color:blue;"><code>metric</code></mark> </td><td>struct</td><td>object</td><td>Input metric tensor. </td></tr><tr><td><mark style="color:orange;"><code>tryGPU</code></mark> </td><td>1x1 array</td><td>integer</td><td>Use the GPU for evaluations, input either 1 for true or 0 for false. <strong>The default value is 0.</strong></td></tr><tr><td><mark style="color:orange;"><code>diffOrder</code></mark></td><td>1x1 array</td><td>string</td><td>Sets the order of the finite difference method. Accepts either "<em>second</em>" to use second-order or "<em>fourth</em>" to use fourth-order methods. <strong>The default value is fourth order.</strong></td></tr></tbody></table>

### Output Arguments

<table><thead><tr><th width="192">Outputs</th><th width="94.33333333333331">Format</th><th width="96">Type</th><th>Description</th></tr></thead><tbody><tr><td><mark style="color:green;"><code>energy</code></mark></td><td>struct</td><td>object</td><td>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 <code>.index</code> field of the struct.</td></tr></tbody></table>