Numerical relativity

Numerical relativity solves Einstein's equations as an initial-value problem, usually after decomposing spacetime into spatial hypersurfaces and evolution in time.

Table of contents
- 3+1 decomposition of spacetime Numerical relativity
- Harmonic coordinate Numerical relativity
  - Harmonic slicing Harmonic coordinate
    - Bona--Masso slicing condition Harmonic slicing
  - Hyperbolic reduction of Einstein's equations Harmonic coordinate

3+1 decomposition of spacetime

A 3+1 decomposition writes

d s^{2} = - α^{2} d t^{2} + γ_{ij} (d x^{i} + β^{i} d t) (d x^{j} + β^{j} d t)

in terms of lapse, shift, and spatial metric.

Lapse function ( $α$ )

The lapse function measures proper-time separation between neighboring spatial hypersurfaces along their unit normal.

Shift vector ( $β^{i}$ )

The shift vector describes how spatial coordinates move tangentially between neighboring hypersurfaces.

Extrinsic curvature of a spatial hypersurface ( $K_{ij}$ )

With future unit normal

n^{μ}

, the convention

K_{μν} = - ⊥^{ρ}_{μ} \nabla_{ρ} n_{ν}

measures how the hypersurface is embedded in spacetime.

3+1 Noether-current conservation law

For

J^{μ} = Q n^{μ} + Q^{μ}

with spatial

Q^{μ}

, current conservation gives

n^{μ} \nabla_{μ} Q = K Q - D_{μ} Q^{μ} - Q^{μ} D_{μ} lo g α

under the stated extrinsic-curvature convention.

Harmonic coordinate

A harmonic coordinate satisfies

□_{g} x^{λ} = 0

, equivalently

g^{μν} Γ_{μν}^{λ} = 0

or

\partial_{μ} (- g g^{λ μ}) = 0

.

Harmonic slicing

Harmonic slicing imposes the harmonic-coordinate condition only on time. In 3+1 form it obeys

(\partial_{t} - β^{i} \partial_{i}) α = - α^{2} K

.

Bona--Masso slicing condition

The Bona--Masso family is

(\partial_{t} - β^{i} \partial_{i}) α = - α^{2} f (α) K

. Harmonic slicing is the choice

f = 1

.

Hyperbolic reduction of Einstein's equations

In harmonic gauge, the vacuum Einstein equations have principal part

- g^{μν} \partial_{μ} \partial_{ν} g_{α β} /2

, forming a quasilinear wave system supplemented by constraints.

 Ancestors (4)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (1)

Numerical relativity by

Wikipedia Bot  1

Numerical relativity is a subfield of computational physics that focuses on solving the equations of general relativity using numerical methods. General relativity, formulated by Albert Einstein, describes the gravitational interaction as a curvature of spacetime caused by mass and energy. The equations governing this curvature, known as the Einstein field equations, are highly complex and often impossible to solve analytically in realistic scenarios, especially in dynamic situations like the collision of black holes or neutron stars.

 Read the full article

  See all articles in the same topic Create my own version