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A Lorentzian manifold is a type of pseudo-Riemannian manifold in which the metric tensor is a Lorentz metric. This means the signature of the metric is ($$+,-,-,\ldots$$) (or equivalently ($$-,+,+,\ldots$$)).

Manifolds
Main article: Manifold

In differential geometry a differentiable manifold is a space which is locally similar to a Euclidean space. In an $$n$$-dimensional Euclidean space any point can be specified by $$n$$ real numbers. These are called the coordinates of the point.

An $$n$$-dimensional differentiable manifold is a generalisation of $$n$$-dimensional Euclidean space. In a manifold it may only be possible to define coordinates locally. This is achieved by defining coordinate patches: subsets of the manifold which can be mapped into $$n$$-dimensional Euclidean space.

See Manifold, Differentiable manifold, Coordinate patch, Transition function for more details.

Tangent spaces and tensors
Main articles: Tangent space, Tensor

Associated with each point in an $$n$$-dimensional differentiable manifold is a tangent space. This is an $$n$$-dimensional vector space whose elements can be thought of as equivalence classes of curves passing through the point.

Functions which act linearly on $$r$$ tangent vectors to give a real number known as contravariant tensors of rank $$r$$.

Metric tensors and Metric signatures
A metric tensor is a rank 2 tensor defined on a manifold.

A Lorentz metric has a signature of (1,n-1).

Uses in physics
The term Lorentzian in honor of the physicist Hendrik Lorentz.