threePlusOneDecomposer

Finds the 3+1 foliation components from a metric tensor.

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.

Metric in 3+1

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 $\beta^i = \gamma^{ij}\beta_i$

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

3+1 Formalism and Bases of Numerical RelativityarXiv.org

Method

The metric terms are used to find the 3+1 terms and return them in the spacetime grid.

Syntax

[alpha,betaDown,gammaDown,betaUp,gammaUp] = threePlusOneDecomposer(metric)

Input Arguments

blue are required inputs.

Inputs	Format	Type	Description
metric	struct	object	Metric tensor object.

Output Arguments

Outputs	Format	Type	Description
alpha	4D array	double	lapse function across the spacetime grid.
betaDown	1x3 cell of 4D arrays	double	covariant shift function across the spacetime grid.
gammaDown	3x3 cell of 4D arrays	double	covariant spatial function across the spacetime grid.
gammaUp	3x3 cell of 4D arrays	double	contravariant spatial function across the spacetime grid.
betaUp	1x3 cell of 4D arrays	double	contravariant shift function across the spacetime grid.