Lorentzian manifold

A Lorentzian manifold or Lorentz manifold (after the Dutch mathematician and physicist Hendrik Antoon Lorentz ) is a manifold with a Lorentz metric . It is a special case of a pseudo-Riemannian manifold with the metric signature (-, +, +, +, ...). Lorentz manifolds are of decisive importance for the general theory of relativity , since there space-time is modeled as a four-dimensional Lorentz manifold.

Point relations and structure of the manifold

Since the Lorentzian metric, in contrast to the Riemannian metric, is not positive definite , three different types of tangential vectors appear on the manifold: ${\displaystyle g|_{p}\colon T_{p}M\times T_{p}M\rightarrow \mathbb {R} }$ ${\displaystyle v}$

• temporal vectors with , ${\displaystyle g(v,v)<0}$
• space-like vectors with , ${\displaystyle g(v,v)>0}$
• light-like vectors with , therefore also called zero vectors. ${\displaystyle g(v,v)=0}$

Non-space-like vectors (i.e. those with ) are also called causal vectors. Curves in the manifold are called time-like, space-like, light-like, causal if the tangential vectors on the curve belong to the corresponding categories over the entire length of the curve. ${\displaystyle g(v,v)\leq 0}$

One can now assign their relation to pairs of points in the manifold. If a piecewise smooth time-like curve exists between the points, one point lies in the future of the other. The time-like future or the content of the light cone of a point is the set of all points that can be reached from with a future-oriented piece-wise smooth time-like curve. It is designated with . The causal future is analogously the set of all points that can be reached with piecewise smooth causal curves. According to define the time-like and causal past and . ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle I^{+}(p)}$ ${\displaystyle J^{+}(p)}$ ${\displaystyle I^{-}(p)}$ ${\displaystyle J^{-}(p)}$

Lorentzian length

Is the Lorentz length of a smooth causal curve ${\displaystyle \lambda \colon [a,b]\to M}$

${\displaystyle L(\lambda )=\int _{a}^{b}{\sqrt {-g(\lambda '(t),\lambda '(t))}}\,dt.}$

t is any curve parameter, not necessarily the time.

In contrast to Riemannian geometry, the infimum of the Lorentzian length of all smooth curves between two time-like spaced points is always zero. However, the temporal geodesic between these two points, if it exists, has the greatest Lorentzian length of all causal curves between these two points.

Lorentzian distance

As the Lorentzian distance between two points and the supremum of the Lorentzian length over all causal curves from to is chosen, if in lies, otherwise one defines . ${\displaystyle d(p,q)}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle J^{+}(p)}$ ${\displaystyle d(p,q)=0}$

See also

• Minkowski room

literature

• John K. Beem, Paul E. Ehrlich, Kevin L. Easley: Global Lorentzian Geometry (= Monographs and Textbooks in Pure and Applied Mathematics 202). 2nd edition. Marcel Dekker Inc., New York NY et al. 1996, ISBN 0-8247-9324-2 .