What does Differential Geometry lack in order to "become Relativity" - References

Question

Given a regular curve $\gamma \colon I \to \mathbb{R}^n$, if we consider the variable $t \in I \subset \mathbb{R}$ as the time, then we have the usual interpretation of $\gamma'(t)$ as the (instantaneous) velocity vector at the position $\gamma(t)$ and $\lvert\gamma'(t)\rvert$ as the speed (or scalar velocity).

From this point of view, we know from differential geometry (very classical examples, indeed) that there are curves that "go through infinite length in a bounded amount of time". Of course, at these examples, the speed of these curves increase infinitely ("goes to infinity").

I have very rough ideas concerning Relativity Theory (I'm a mathematician, not a physicist) but I know that, for example, the speed a object can reach is bounded by the speed of light, and that mass, lengths and even the time are distorted at very high speed (near the speed of light).

As I said, I'm not an expert on this subject, so maybe my question even make sense, but what axioms do I insert to my models in order to get such results, from a purely mathematical/axiomatic point of view? Is the answer to this "the Einstein postulates"?

I want to know also if there is an area which study this kind of "differential geometry + relativity" (or even a "Riemannian geometry + relativity"). Would the answer to this question be simply "Relativity"? (as I said, I don't have a deep knowledge on that).

(Is there) Do you recommend any references at this subject?

Answer 1

In (special) relativity $\mathbb{R}^n$ comes with a different metric, with signature $(-,+, \ldots +)$.

The mathematical study of manifolds with a Lorentzian metric is called lorentzian geometry, or pseudo-Riemannian geometry or semi-Riemannian geometry (the latter are more general as it refers to any metric which is not definite positive).

A generic Lorentzian manifold is still not a spacetime model in General Relativity, for that you also need it to satisfy Einstein equations $G=8\pi T$.

Mathematical references are O'Neill "Semi-Riemannian Geometry With Applications to Relativity" Beem, Ehrlich "Global Lorentzian geometry"

Answer 2

Initially ignore the effects of spacetime curvature (general relativity). Consider a homogeneous (flat) real four-dimensional space. If the distance between a pair of points is given by a definite (Euclidean) quadratic form, the symmetries of space include the usual orthogonal symmetry group $\mathrm{O}(4)$ and translations (producing the Euclidean group $\mathrm{E}(4)$). The only difference in going to special relativity is that the quadratic form is indefinite (Lorentzian), which results in the indefinite orthogonal symmetry group $\mathrm{O}(1,3)$ and translations (producing the Poincaré group). Rotations just behave a little differently, but should still be thought of as a symmetry group of four dimensional space.

Everything about parameterization of curves and velocities follows directly from this - indeed, all of the geometry of special relativity follows. A pitfall to avoid is thinking of time as universal; it is specific to the coordinate system. If you extend this local picture to differential manifolds, and you add Einstein's equation that relates the curvature of spacetime to its content (specifically the stress-energy tensor), you have general relativity.

As an aside, just as dropping the parallel postulate of Euclidean geometry produces elliptic and hyperbolic geometries, Einstein's postulates for special relativity allow for "non-flat" (de Sitter and anti-de Sitter) geometries, with symmetry groups $\mathrm{O}(1,4)$ and $\mathrm{O}(2,3)$ respectively.

Answer 3

This question is more about physics than about math, so I hope you will bear with a physicist for posting an answer. The OP expressed the following doubt:

...Of course, at these examples, the speed of these curves increase infinitely ("goes to infinity"). ... what axioms do I insert to my models in order to get such results, from a purely mathematical/axiomatic point of view? Is the answer to this "the Einstein postulates"?

or, in other words, is there an extra requirement that space-time manifolds ought to satisfy, as compared to generic manifolds with the same signature as the Lorenzian metric, in order to prevent the existence of superluminal motions?

If this is indeed your worry, then no, there are no other requirements which distinguish a space-time manifold from another manifold (with the same signature): it is a matter of interpretation.

Since the metric is not positive-definite, there will be geodesics with $ds^2 = 0$, $ds^2 > 0$, $ds^2 < 0$. The geodesics with $ds^2 = 0$ (called null geodesics) describe the motion of photons. An interval $ds^2 < 0$ can always be rewritten, thru a change of coordinates, as $-dt^2 < 0$, which shows the separation to be time-like, and intervals with $ds^2 > 0$ can likewise always be expressed as $dx^2 > 0$, which shows the interval to be space-like. Now, the superluminal curves you refer to do exist in a space-time manifold: they are the curves on which, at least somewhere, $ds^2 > 0$, but they are not interpreted as the space-time motion of any physical observer, exactly because they are superluminal. Only time-like geodesics, i.e. those with $ds^2 < 0$, are possible paths for physical, subluminal observers.

A graphical way to represent this is to consider the bundle of null geodesics (both future-oriented and past-oriented) going thru a given space-time point $P$. This bundle is composed of two hypercones (the future-oriented null geodesics, and the past-oriented null geodesics) with vertices joining at $P$. All points within the cones may be joined to $P$ by a time-like (hence, subluminal) curve, thus providing a nice, coordinate independent characterization of the past and future of $P$.

Points lying outside both cones are not causally related to $P$,because they require superluminal motions to be connected to it. But they do play an important role in physics: a 3-dimensional hypersurface thru $P$ but lying otherwise wholly outside the null cones of all of its points provides an adequate surface for setting up boundary conditions for a time-dependent problem (like the evolution of the Universe) for obvious reasons of causality.

EDIT:

As for reading suggestions, there are at least two books by physicists which I consider eminently suitable to mathematicians: first, The large scale structure of spacetime by Hawking and Ellis, and then General Relativity by R. Wald. Both are classic texts; that by H&E focuses on on the study of singularities, while the one by Wald is wider in scope. But H&E also contains very interesting discussions about the properties of special solutions (like Taub-NUT space, or Godel's rotating Universe) which will at some point enjoy a comeback, IMHO. Basically the two, which have a large amount of material in common, complement each other very well in the remaining parts.