180 Lorentzian Manifold
A smooth domain with a metric of one negative and three positive signature directions that is used to model curved spacetime.
definition [d] (Lorentzian Manifold = Lorentz Manifold) A smooth manifold \(M\) equipped with a smooth, symmetric, nondegenerate metric tensor field \(g\) of Lorentzian signature:
- \((M, g)\) with signature \((-+++)\) .
where
- \(M\) is a smooth manifold.
- \(g\) is the metric tensor field.
- \(g_{p}\) is the value of \(g\) at the point \(p\), a symmetric nondegenerate bilinear form on \(T_{p}M\).
- \(T_{p}M\) is the tangent space at \(p\).
Note:
- the opposite Lorentzian signature is \((+---)\).
- exactly one eigenvalue of \(g\) has opposite sign to the others.
- unlike a Riemannian metric, \(g\) is not positive-definite.
- the flat prototype is Minkowski spacetime with metric \(\eta_{\mu\nu}\).
definition [d] (Lorentzian Manifold = Lorentz Manifold) A pseudo-Riemannian manifold \((M, g)\) whose metric has Lorentzian signature \((-+++)\):
- \(g\) is smooth, symmetric, and nondegenerate on each \(T_{p}M\) .
where
- \(M\) is a smooth manifold.
- \(g\) is the Lorentzian metric tensor field.
- \(T_{p}M\) is the tangent space at \(p\).
Note:
- the opposite Lorentzian signature is \((+---)\).
180.1 Elementary Example
180.1.1 Simple
A Lorentzian metric has one negative direction. Let \(\eta\) be the \(1+1\) Minkowski metric.
\[ \eta = \operatorname{diag}(-1,1) \]
\[ \eta(e_{0},e_{0}) = -1,\quad \eta(e_{1},e_{1}) = 1 \]
where
- \(\eta\) is the Minkowski metric tensor.
- \(e_{0}, e_{1}\) are basis vectors in the two signature directions.
180.1.2 General
In \(1+3\) dimensions the flat Lorentzian metric is the \(4 \times 4\) diagonal matrix of signature \((-+++)\).
\[ \eta = \operatorname{diag}(-1,1,1,1) \]
\[ \{ e_{0},\ e_{1},\ e_{2},\ e_{3} \} \]
\[ \eta(e_{0},e_{0}) = -1,\quad \eta(e_{i},e_{i}) = 1\ \text{for } i = 1,2,3 \]
where
- \((M,\eta)\) with this \(\eta\) is flat Minkowski spacetime.
- signature \((-+++)\) means one negative and three positive diagonal entries.
180.2 References
- Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — Lorentzian manifold and metric signature.
- Nakahara, M. Geometry, Topology and Physics. IOP, 2003. — Lorentz manifold.
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