Warp Factory
  • What is Warp Factory?
  • Overview
    • Installing Warp Factory
    • Workflow
    • Codebase Structure
    • Object Types
    • Frames
  • Examples
    • Metrics
      • M1 - First Metric
      • M2 - Default Metrics
      • M3 - Building a Metric
    • Energy Tensor
      • T1 - First Energy Tensor
      • T2 - Cartoon Methods
      • T3 - GPU Computation
      • T4 - Solver Order
      • T5 - Errors
    • Analysis
      • A1 - Energy Conditions
      • A2 - Metric Scalars
      • A3 - Eval Metric
      • A4 - Momentum Flow
    • Warp Shell
      • 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
    • Visualizer Module
      • plotThreePlusOne
      • plotTensor
      • Plot Functions
    • Units Module
  • General
    • Contributing
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    • Citing Warp Factory
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  • Description
  • Method
  • Syntax
  • Input Arguments
  • Output Arguments
  1. Modules
  2. Metrics Module
  3. Metric Functions

threePlusOneDecomposer

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

PreviousthreePlusOneBuilderNextAnalyzer Module

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 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.

3+1 Formalism and Bases of Numerical RelativityarXiv.org
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