Question

Is General Relativityy described by Pseudo-Riemannian manifold or Riemannian manifold? I cannot understand the vast difference between the two manifolds. In books, General Relativity is looked as a pseudo-Riemannian manifold, though I am not sure after reading some threads on the web which confused me. Now checking wikipedia, it says here:

After Riemannian manifolds, Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds. They are important in applications of general relativity. A principal basis of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). Unlike Riemannian manifolds with positive-definite metrics, a signature of (p, 1) or (1, q) allows tangent vectors to be classified into timelike, null or spacelike.

This is the best I've gotten searching for an answer and it is still confusing and not clear.
A) Pseudo-Riemannian manifold
B) Riemannian manifold
C) Both depending on the specific context
D) Neither of the above

Answer 1

General Relativity utilizes a Pseudo-Riemannian manifold, specifically a Lorentzian manifold, to account for the geometry of space-time, where the mixed signature of the metric accommodates both spatial and temporal dimensions.

Step-by-step explanation:

General Relativity, as proposed by Einstein, is a theory that describes gravity as a consequence of the curvature of space-time. In this framework, the mathematical model used to represent the geometry of space-time is a Pseudo-Riemannian manifold, specifically a Lorentzian manifold.

This type of manifold allows for the classification of tangent vectors into timelike, null, or spacelike, which is necessary for the proper description of relativistic phenomena such as time dilation, length contraction, and the behavior of light near massive objects like black holes.

While both Riemannian and Pseudo-Riemannian manifolds deal with curved spaces, the key difference lies in the metric signature. Riemannian manifolds have a purely positive-definite metric, indicating all dimensions are spatial, whereas a Pseudo-Riemannian manifold like the one used in General Relativity has a metric of mixed signature (3,1) or (1,3), reflecting the inclusion of time as a dimension that possesses different qualities from space.

The answer to whether General Relativity is described by a Pseudo-Riemannian manifold or a Riemannian manifold is A) Pseudo-Riemannian manifold.