# metricGet\_ModifiedTime

## Description

Create the Modified Time warp solution from *Introducing Physical Warp Drives* in a defined spacetime grid.

<details>

<summary>Modified Time Metric</summary>

This solution modifies an Alcubierre geometry by the addition of an extra factor $$A$$ acting on the time component:

$$ds^2 = -((1-f(r\_s)+A^{-1}) dt)^2+ (dx^2 - v\_s(t) f(r\_s)^2 \ dt)^2 + dy^2 + dz^2$$

</details>

For more details on the Modified Time metric, please read (section 4.5):

{% embed url="<https://arxiv.org/abs/2102.06824>" %}

## Method

The shape function of $$A$$ takes on the same value as the shift vector in Alcubierre.

{% hint style="info" %}
The comoving version of this metric called `metricGet_ModifiedTimeComoving` has the same inputs but requires that the <mark style="color:blue;">`gridSize`</mark> along t = 1 and will return the metric in the Galilean comoving frame.
{% endhint %}

## Syntax

`[`<mark style="color:green;">`Metric`</mark>`] = metricGet_ModifiedTime(`<mark style="color:blue;">`gridSize`</mark>`,`<mark style="color:blue;">`worldCenter`</mark>`,`<mark style="color:blue;">`v`</mark>`,`<mark style="color:blue;">`R`</mark>`,`<mark style="color:blue;">`sigma`</mark>`,`<mark style="color:blue;">`A`</mark>`,`` `<mark style="color:orange;">`gridScale`</mark>`)`

`[`<mark style="color:green;">`Metric`</mark>`] = metricGet_ModifiedTimeComoving(`<mark style="color:blue;">`gridSize`</mark>`,`<mark style="color:blue;">`worldCenter`</mark>`,`<mark style="color:blue;">`v`</mark>`,`<mark style="color:blue;">`R`</mark>`,`<mark style="color:blue;">`sigma`</mark>`,`<mark style="color:blue;">`A`</mark>`,`` `<mark style="color:orange;">`gridScale`</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="213">Inputs</th><th width="109">Format</th><th width="93">Type</th><th>Description</th></tr></thead><tbody><tr><td><mark style="color:blue;"><code>gridSize</code></mark></td><td>1x4 array</td><td>integer</td><td><p>The size of the world specified as:</p><p></p><p><span class="math">[t, x, y, z]</span> </p></td></tr><tr><td><mark style="color:blue;"><code>worldCenter</code></mark></td><td>1x4 array</td><td>double</td><td><p>The center of the world, which defines the center of <span class="math">r_s</span> as a 4-vector, specified as:</p><p></p><p><span class="math">[t,x,y,z]</span></p></td></tr><tr><td><mark style="color:blue;"><code>v</code></mark></td><td>1x1 array</td><td>double</td><td>Speed of the warp drive, given as a factor of c.</td></tr><tr><td><mark style="color:blue;"><code>R</code></mark></td><td>1x1 array</td><td>double</td><td>Radius of the warp bubble.</td></tr><tr><td><mark style="color:blue;"><code>sigma</code></mark></td><td>1x1 array</td><td>double</td><td>Thickness factor for the warp bubble.</td></tr><tr><td><mark style="color:blue;"><code>A</code></mark></td><td>1x1 array</td><td>double</td><td>Lapse rate modification.</td></tr><tr><td><mark style="color:orange;"><code>gridScale</code></mark></td><td>1x4 array</td><td>double</td><td><p>Unit scaling factor of the grid dimensions defined relative to <mark style="color:blue;"><code>gridSize</code></mark>.  This determines the resolution of the grid along each dimension. Specified as:</p><p></p><p><span class="math">[t_{scale}, x_{scale}, y_{scale}, z_{scale}]</span></p><p></p><p> <strong>The default value is [1, 1, 1, 1].</strong></p></td></tr></tbody></table>

### Output Arguments

<table><thead><tr><th width="151">Outputs</th><th width="113.33333333333331">Format</th><th width="90">Type</th><th>Description</th></tr></thead><tbody><tr><td><mark style="color:green;"><code>metric</code></mark></td><td>struct</td><td>object</td><td>Modified Time solution returned as the metric tensor object.</td></tr></tbody></table>