Mathematical Formulation of Classical Spacetime

Question

I have seen two formulations of Classical Mechanics:

• Newtonian spacetime (learned it from the lectures of Professor Frederic P. Schuller): Definition: A Newtonian spacetime is a quintuple $(M, \mathcal{O}, \mathcal{A}, \nabla, t)$ where $(M, \mathcal{O}, \mathcal{A}, \nabla)$ is a differentiable manifold with a torsion free connection, and $t\in C^{\infty}(M)$ is such that $dt\neq 0$ and $\nabla dt=0$
• Galilean spacetime (a popular approach used most famously in Arnold's Mathematical Methods of Classical Mechanics) Definition: A Galilean spacetime is a quadruple $(\mathcal{E}, V, g, \tau)$ where $\mathcal{E}$ is an affine space modeled on a four dimesional real vector space $V$, $\tau\in V^*$ and $g$ is an inner product on $\text{ker} \tau$

I find the first approach much more beautiful since it readily extends to relativistic spacetimes. Non the less I have trouble understanding its practical use since I haven't been able to find sources regarding it apart from a series of lectures Klassischen Mechanik by Schuller. On the other hand I've found at least three books that deal with the second approach.

I was wondering two things. Since my German is very bad, does anybody have a source in Spanish or English which deals with the first approach? Moreover, the first approach seems to be more general than the second one. Is there any way to get to the second approach from the first one by introducing some additional structures? I am particularly interested in a claim Schuller made in one of his classes. He said that the idea of having a metric on Newtonian spacetimes is an artifact of inertial reference frames. That's why in the first definition there is no metric, but instead a much weaker object, a connection. Since the second approach has a metric induced by the inner product maybe through a clever choice of charts in the first approach one could arrive to the second one.

Thanks!

Answer 1

Prof. Schuller is apparently referring to the Axiomatic formulation of the Newtonian theory of gravity, of which two versions I'm aware, one by Andrzej Trautman, described in:

• Trautman, Andrzej (1963). “Sur la théorie newtonienne de la gravitation”. In: Comptes rendus hebdomadaires des séances de l’ Académie des sciences 257, pp. 617–620,
• – (1967). “Comparison of Newtonian and relativistic theories of space-time”. In: Perspectives in Geometry and Relativity. Essays in honor of V. Hlavat’y. Ed. by B. Hoffman. Bloomington: Indiana University Press, pp. 413–425,
• section 5.5 of Trautman, A., F. A. E. Pirani, and H. Bondi (1965). Lectures on general relativity. Prentice-Hall,

and another introduced by Künzle–Ehlers. Both succeed the Newton-Cartan theory. Its main practical use is to show that such a theory of gravity has many of the problems general relativity has.

According to the development of this theory, the metric of the hypersurfaces of constant $t$ is the part of the degenerate matrix $g^{ab}$ that is non-singular. In order to fix the time coordinate and conclude that $g^{ab}$ is constant, we need to do the calculations using the flat connection $\overset{0}{\Gamma}{}_{bc}^a$, which corresponds to the inertial coordinate system.

One simple way to conclude this, even though it is outside the context of the aforementioned axiomatic formulation, is if you write the lagrangian: $$L = \frac{1}{2} h_{ab}(x) \dot{x}^a \dot{x}^b - V(x),$$ and derive the equations of motion, which will yield the geodesics. Newton's gravitational law will emerge in the familiar form only if the connection vanishes, which implies that the coordinate system is inertial, and leads to the correct interpretation of the metric $h_{ab}$.

Answer 2

Concerning your first question about similar approaches to the construction of Classical Space-Time in English, I would suggest you to look at the book by Marcelo Epstein: The geometrical language of continuum mechanics (Cambridge University Press). I am not quite sure about all your symbols (I do not have a background in general relativistics nor am I a specialist for differential geometry) but I think his construction is at least similar and might be of interest to you.

Answer 3

I am particularly interested in a claim Schuller made in one of his classes. He said that the idea of having a metric on Newtonian spacetimes is an artifact of inertial reference frames. That's why in the first definition there is no metric, but instead a much weaker object, a connection.

So, I've now watched parts of the video you posted in the comments, including the part I'm assuming you're basing this statement on. It's not about the derivation of a metric from the connection (ie metrizability), but the existence of a spatial metric.

Quoting the video:

Having a metric of space is an inertial system fiction

Emphasis mine. What Schuller wanted to point out is that in non-inertial systems, the fact that the metric lives on spacetime becomes relevant.

Further details can be found in lecture 9 of his German series. It introduces the Newtonian spacetime with absolute space, defined as a Newtonian spacetime with a spatial spacetime metric (a symmetric bilinear form on spacetime that ignores the time component of its arguments) that is compatible with the connection. It is then shown that thanks to the compatibility condition, such a form induces an actual spatial metric on the leaves at equal time which is time-independent, which allows us to pick any of the leaves as a representative of absolute space.

Note that such a spacetime is still more general than the second definition you gave for Galilean spacetime as the underlying manifold is not resticted to the affine space.