Silly question about "full" abstract structure of a Spacetime

Question

My question is simple. Consider then a vector space:

$$\mathfrak {V} \equiv [\mathcal{V},(\mathbb{K},+_{\mathbb{K}},\cdot_{\mathbb{K}}),\boxplus_{\mathfrak{V}},\boxdot_{\mathfrak{V}}]$$

Where $\mathcal{V}$ is a non empty set of elements, $(\mathbb{K},+_{\mathbb{K}},\cdot_{\mathbb{K}})$ another algebric structure called a Field; $\boxplus_{\mathfrak{V}}$ and $\boxdot_{\mathfrak{V}}$ are two binary operations called, respectively, sum of vectors and scalar multiplication. Right.

Now, it's quite common in elementary courses to define a spacetime as "the set of all events". Introductory/advanced courses the one might encounter definitions for a spacetime like from Naber's (Spacetime and singularities):

A spacetime is a 4-dimensional real vector space $\mathfrak{V}$ on which is defined a symmetric billinear form $g$

Then, a generalization is:

$$\Big(\mathfrak{M}, g \Big)$$

Where $\mathfrak{M}$ is the (curved) manifold and $g$ is the metric tensor field defined on it.

But maybe this definition is a shortly one, I mean, we have maybe a few other fields defined on the spacetime. Like:

$$\Big(\mathfrak{M}, g, \nabla \Big)$$

where $\nabla$ is the truly notion of "rate of change": the connection (levi-civita conection mostly) defined. I suspect that exists some other fields and I would like to know which ones to see the whole mathematical structure of a space time (like the vector space is a 4-uple and stops on this). An some sort of structure like:

$$\Big(\mathfrak{M}, g, \nabla, S, N,B... \Big)$$

Where $S$,$N$,$B$... are?

Answer 1

A spacetime in its most general form is usually written as the multiplet $(\mathcal{M}, \mathfrak{A}, g, \nabla)$, with

• $\mathcal{M}$ is a paracompact, Hausdorff manifold, of dimension $n \geq 2$, that can admit a Lorentz metric
• A smooth structure $\mathfrak{A}$ on that manifold (this is generally not really important since the smooth structure is usually either unique or there exists a "standard" one)
• A Lorentzian metric $g$
• A connection, which is generally the Levi-Civitta connection (torsion-free and metric-compatible)

There are many structures you can additionally add to a spacetime, but those either stem from the first four, will not generally hold for all spacetimes or will depend on the theory concerned. Here's a few notable ones :

• A wide variety of fiber bundles are useful in GR, such as the tangent bundle ($T\mathcal{M}$), cotangent bundle ($T^*\mathcal{M}$), tensor bundle, Grassmann bundle, frame bundle ($L\mathcal{M}$ or $F\mathcal{M}$), orthonormal frame bundle ($O\mathcal{M}$), metric bundle, Clifford bundle, spin bundle, etc etc.
• A time-orientation (usually $t$ or $\tau$ or some variant), which is usually defined as a timelike vector field. Not all spacetimes admit one, but most reasonable ones do.
• A spacetime orientation ($\eta$ or $\varepsilon$), which is a nowhere vanishing continuous $n$-form, defined if the spacetime manifold is orientable.
• A causal structure, which stems from the metric tensor. If we consider the manifold $\mathcal{M}$ as a set $X$, then the causal structure is a partial order $(X, \ll, \leq, \to)$
• A spin structure, if all conditions are met.
• A variety of matter fields expressed by vector bundles, gauge fields from principal bundle, as well as the jet bundle and Legendre bundle for those fields to perform calculations upon.
• Extensions of the spacetime to include its boundaries. There are tons of structures you may use for it, such as the GKP method.

I could go on and on, from the various topologies you can impose on the spacetime (such as the Alexandrov topology or $C^0$ topology), the loop spaces for curves or timelike curves, time functions, foliations, etc etc. But the first four things (and matter fields if we consider them) are enough to derive all of this later on.

Answer 2

I am sorry but this is a large topic to cover. I can tell you this much: first of all, spacetime is not a vector space. It is a manifold, a 4d manifold which is equiped with topology. It is also equiped with a set of coordinate maps called charts. Complete set of such maps that covers the manifold is called an atlas. It is also equiped with a connection. Because of our choice of definition of a shortest or how to say it, stationary curve, which coincides with the definition of autoparallel curve, this is equivalent to a choice of metric, g. Also, in adition to that, this manifold we call spacetime has to be torsion-free. So, it is 4-d manifold (M, o, A, g) where o is topology and A the atlas. Small g is the metric but in place of that we can write connection. M stands for manifold. To talk about physics further on, you need to define a tangent vector space in each point of your spacetime, or, to define more generaly, a tangent bundle. After that, a cotangent bundle is needed also. Then you can define a vector field as a section of a tangent bundle. To talk about symmetry you have to define push-forward and pull-back maps, a flow of a vector field and a notion of a completness of a vector field. Also, you need to define a Lie algebra of vector fields and a Lie derivative.