Lorentzian Manifold
The Lorentz manifold is a pseudo-Riemannian manifold. It is named after the Dutch physicist Hendrik Lorentz (1853โ1928). The default LorentzMetric is the MinkowskiMetric named after the German mathematician Hermann Minkowski (1864โ1909).
Within Manifolds.jl it is used as the embedding of the Hyperbolic space.
Manifolds.Lorentz โ Type
Lorentz{T} = MetricManifold{โ,Euclidean{โ,T},<:LorentzMetric}The Lorentz manifold (or Lorentzian) is a pseudo-Riemannian manifold.
Constructor
Lorentz(n[, m=MinkowskiMetric()]; parameter::Symbol=:type)Generate the Lorentz manifold of dimension n with the LorentzMetric m, which is by default set to the MinkowskiMetric.
Manifolds.LorentzMetric โ Type
LorentzMetric <: AbstractMetricAbstract 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.MinkowskiMetric โ Type
MinkowskiMetric <: LorentzMetricAs a special metric of signature $(++...+-)$, i.e. a LorentzMetric, see minkowski_metric for the formula.
Manifolds.minkowski_metric โ Method
minkowski_metric(a, b)Compute the minkowski metric on $\mathbb R^n$ is given by
\[โจa,bโฉ_{\mathrm{M}} = -a_{n}b_{n} + \displaystyle\sum_{k=1}^{n-1} a_kb_k.\]
ManifoldsBase.is_flat โ Method
is_flat(M::Lorentz{<:Any, MinkowskiMetric})Return whether the Lorentz manifold M is flat. This is the case for the MinkowskiMetric, since it is the same matrix at every point.