Warp Factory
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      • M1 - First Metric
      • M2 - Default Metrics
      • M3 - Building a Metric
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      • W1 Warp Shell Comoving
  • Modules
    • Metrics Module
      • Metric Library
        • metricGet_Minkowski
        • metricGet_Alcubierre
        • metricGet_Lentz
        • metricGet_VanDenBroeck
        • metricGet_WarpShellComoving
        • metricGet_ModifiedTime
        • metricGet_Schwarzschild
      • Metric Functions
        • setMinkowski
        • setMinkowskiThreePlusOne
        • threePlusOneBuilder
        • threePlusOneDecomposer
    • Analyzer Module
      • getMomentumFlowLines
      • doFrameTransfer
      • changeTensorIndex
      • getScalars
      • getEnergyConditions
      • evalMetric
    • Solver Module
      • getEnergyTensor
      • verifyTensor
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      • plotThreePlusOne
      • plotTensor
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  1. Modules
  2. Metrics Module
  3. Metric Functions

threePlusOneBuilder

Constructing a metric tensor from its 3+1 components.

PrevioussetMinkowskiThreePlusOneNextthreePlusOneDecomposer

Last updated 1 year ago

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 γ\gammaγ:

gij=γij ,  gij=γijg_{ij} = \gamma_{ij} \ , \ \ g^{ij} = \gamma^{ij}gij​=γij​ ,  gij=γij

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

g0i=gi0=βig_{0i} = g_{i0} = \beta_ig0i​=gi0​=βi​

The lapse rate α\alphaα and shift vector β\betaβ determine the time component of the metric:

g00=−α2+βiβig_{00} = -\alpha^2 + \beta^i \beta_ig00​=−α2+βiβi​

where βi=γijβi\beta^i = \gamma^{ij}\beta_iβi=γijβi​

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

Method

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

Syntax

[Metric] = threePlusOneBuilder(alpha, beta, gamma)

Input Arguments

blue are required inputs.

Inputs
Format
Type
Description

alpha

4D array

double

Lapse rate.

beta

1x3 cell of 4D arrays

double

Shift vector.

gamma

3x3 cell of 4D arrays

double

Spatial terms.

Output Arguments

Outputs
Format
Type
Description

Metric

4x4 cell of 4D arrays

double

Returns the standard metric tensor constructed from the 3+1 components.

Logo3+1 Formalism and Bases of Numerical RelativityarXiv.org