Lorentzian Manifold

The Lorentz manifold is a pseudo-Riemannian manifold. It is named after the Dutch physicist Hendrik Lorentz (1853–1928). The default LorentzMetric is the MinkowskiMetric named after the German mathematician Hermann Minkowski (1864–1909).

Within Manifolds.jl it is used as the embedding of the Hyperbolic space.

Manifolds.Lorentz — Type
Lorentz{T} = MetricManifold{Euclidean{T,ℝ},LorentzMetric}

The Lorentz manifold (or Lorentzian) is a pseudo-Riemannian manifold.


Lorentz(n[, metric=MinkowskiMetric()])

Generate the Lorentz manifold of dimension n with the LorentzMetric m, which is by default set to the MinkowskiMetric.

Manifolds.LorentzMetric — Type
LorentzMetric <: AbstractMetric

Abstract type for Lorentz metrics, which have a single time dimension. These metrics assume the spacelike convention with the time dimension being last, giving the signature $(++...+-)$.

Manifolds.minkowski_metric — Method
minkowski_metric(a, b)

Compute the minkowski metric on $\mathbb R^n$ is given by

\[⟨a,b⟩_{\mathrm{M}} = -a_{n}b_{n} + \displaystyle\sum_{k=1}^{n-1} a_kb_k.\]