Lorentzian Geometry
Lorentzian Manifold
A Lorentzian manifold is a smooth manifold
of dimension , equipped with a metric tensor of signature .
Spacelike, Null, Timelike
For a semi-Riemannian manifold
, we say that a tangent vector is
- spacelike if
or , - null (or lightlike) if
and , - timelike if
. If
is either timelike or null, then it is called causal.
Spacetime
A spacetimes is a 4-dimensional Lorentzian manifold
, endowed with an energy-momentum tensor that satisfies the Einstein field equations.
Arrow of Time & Causality
Time-Orientability
A spacetime
is time-orientable if it admits a continuous timelike vector field. Equivalently, if there is a continuous division of all non-spacelike tangent vectors at points in into two classes, called future-directed and past-directed.
Remark
This property ensures that the light cones at each point can be coherently oriented, so that all observers can agree on the direction of time’s flow. Time orientability is crucial for defining a consistent causal structure, which is fundamental for the formulation of physical laws and for understanding phenomena such as cause and effect.
However, it is possible to have time-orientable space-times which have closed timelike curves. An observer can return to an “event” in space-time at which he/she has already been present. Stephen Hawking conjectured that quantum effects prevent closed timelike curves formation (“chronology protection conjecture”), though no proof yet exists.