> For the complete documentation index, see [llms.txt](https://applied-physics.gitbook.io/warp-factory/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://applied-physics.gitbook.io/warp-factory/modules/metrics-module/metric-functions/threeplusonebuilder.md).

# threePlusOneBuilder

## Description

The solver in Warp Factory requires the metric tensor defined in the standard way, but constructing the spacetime in terms of its 3+1 foliation terms is often helpful. The builder function takes in the 3+1 components and constructs the metric for use in evaluating the stress-energy tensor and metric scalars.

<details>

<summary>Metric in 3+1</summary>

In this section, we will use Latin indices as summing from 1 to 3 and Greek indices as summing from 0 to 3. The spatial components of the metric map directly to the spatial terms $$\gamma$$:

$$g\_{ij} = \gamma\_{ij} \ , \ \ g^{ij} = \gamma^{ij}$$

The shift vector maps directly to the time-space cross terms of the metric:

$$g\_{0i} = g\_{i0} = \beta\_i$$

The lapse rate $$\alpha$$ and shift vector $$\beta$$ determine the time component of the metric:

$$g\_{00} = -\alpha^2 + \beta^i \beta\_i$$

where $$\beta^i = \gamma^{ij}\beta\_i$$

</details>

For more general background on 3+1 formalism please read:

{% embed url="<https://arxiv.org/abs/gr-qc/0703035>" %}

## Method

The metric is constructed from the 3+1 terms as defined in the spacetime grid.

## Syntax

`[`<mark style="color:green;">`Metric`</mark>`] = threePlusOneBuilder(`<mark style="color:blue;">`alpha`</mark>`,`` `<mark style="color:blue;">`beta`</mark>`,`` `<mark style="color:blue;">`gamma`</mark>`)`

### Input Arguments

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

<table><thead><tr><th width="125">Inputs</th><th width="125">Format</th><th width="169">Type</th><th>Description</th></tr></thead><tbody><tr><td><mark style="color:blue;"><code>alpha</code></mark></td><td>4D array</td><td>double</td><td>Lapse rate.</td></tr><tr><td><mark style="color:blue;"><code>beta</code></mark> </td><td>1x3 cell of 4D arrays</td><td>double</td><td>Shift vector. </td></tr><tr><td><mark style="color:blue;"><code>gamma</code></mark></td><td>3x3 cell of 4D arrays</td><td>double</td><td>Spatial terms.</td></tr></tbody></table>

### Output Arguments

<table><thead><tr><th width="205">Outputs</th><th width="144.33333333333331">Format</th><th width="93">Type</th><th>Description</th></tr></thead><tbody><tr><td><mark style="color:green;"><code>Metric</code></mark></td><td>4x4 cell of 4D arrays</td><td>double</td><td>Returns the standard metric tensor constructed from the 3+1 components.</td></tr></tbody></table>


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