Doubt about the ways to define Spacetimes

Question

My Question:

The most general way to define a Spacetime is by the aid of Differentiable Manifolds; therefore the underlying structure is of a topological manifold. Furthermore, we can talk about the geometrical properties of spacetime.

It seems, though, that the geometrical structure of spacetime runs through some meandering territories when we talk about Special Relativity. Of course that Special Relativity is, roughly speaking, a solution of Einstein's Equations and therefore, it's structure is just a particular pair:

$$ (\mathcal{M}, \eta), \tag{1}$$

where $\mathcal{M}$ is the underlying manifold and $\eta$ is the (Minkowski) metric tensor (Due to this structure (again) the "Spacetime of Special Relativity" have the structure of a Topological Manifold) but, the thing is, some authors $[2]$,$[3]$ prefer to introduce the geometrical structure of spacetime in terms of an Affine Space.

So I would like to know: why in the general picture (General Relativity) we define Spacetime as a Manifold, but when we aim to study just Special Relativity we define the fundamental stage (Spacetime again) as an affine space?$[*]$

Further Considerations:

An professor of mine said that the definition of a Spacetime, given by $[4]$, is incorrect. Again, he said that the right definition is that of using Affine Spaces. So if we stop to think about it, we can, more or less, say that we have a non-standard definition of a Spacetime; with Manifolds $[1]$ , Affine Spaces $[2], [3]$ and Vector Spaces $[4]$; this confuses me a little bit when we have to talk about generalizations of Minkowski spacetime and Special Relativity to General Relativity [concerning what physical facts (like proximity) we want to encode].

$$---\circ ---$$

$[1]$ HAWKING.S,ELLIS.G; Large-Scale Structure of Space-Time

$[2]$ KRIELE.M; Spacetime

$[3]$ VANZELLA.D; Special Relativity http://www.gradadm.ifsc.usp.br/dados/20192/7600028-1/Notas%20de%20aula%20RR%20Cap1.pdf

$[4]$ NABER.G; The Geometry of Minkowski Spacetime

$[*]$ Please, I know the differences between the structures mentioned above in my question. What I'm asking is why people use different structures sinces a Differentiable Manifold like $(1)$ is the most general one!

Answer 1

First, note that we are talking about a physical theory and, as in many cases, there are various different ways of putting this theory in a mathematical dress. What matters in the end are the predictions of your model and not the exact way you describe it. The "right" describtion does simply not exist on a natural basis. However, there are indeed more and less beautiful ways of describing a physical theory. And clearly, a model can be mathematically ill-defined or inconsistent. But there is no naturally right mathematical describtion of a physical theory.

Having said that, lets consider general relativity. Let's start with a definition of spacetime in terms of a differential geometry. Here, a spacetime is a tuple $(M,g,\epsilon,\mathfrak{t})$ consisting of:

• A set $M$ of spacetime points (sometimes called events) endowed with a differential structure (and thus a topological one) of dimension $m$
• A pseudo-Riemannian metric $g$ (symmetric, positive-definite, bilinear tensor field) of signiture (1,m-1)
• Often forgotten or not mentioned: A orientation or volume form $\epsilon\in\Omega^m(M)$ (I name it $\epsilon$ because it is the Levi-Civitas tensor but coordinate-free)
• Not Always necessary but anyway: a time orientation, call it $\mathfrak{t}$, deciding what we call future and what past

Again, there are different ways of actually defining this ingredients, so I leave it a bit hand waving. Now regard that a affine space $A$ modelled by a $m$-dimensional real vector space $V$ inherits a differentable structure from $V$. For $A$ with this smooth structure there is a isomorphism of vector spaces $T_pA\cong V$ for all $p\in A$ and a Minkowski metric $\eta:V\times V\rightarrow\mathbb{R}$ can by pushed to a globally flat pseudo-Riemannian metric tensor field on $A$ as manifold. Thus, the differential geometric describtion includes the affine one. On the differential geometric side you could define a Minkowski spacetime as globally flat spacetime endowed with a free and transitive action of the translation group $(\mathbb{R}^m,+)$. However, the later is precisely the definition of an affine space. So These definitions can be considered equal. However, in application an affine space seems to be more handy and Things get even better when passing to the vector space describtion by choosing a base point $\vec{0}\in A$.

So we have seen that the differention geomentric, the affine and the vector space definition of Minkowski spacetime all describe the "same object" to some extend. However, what exact definition you choose depends on what you want to use it for and on your preference (I clearly prefere the manifold). This is why different authors choose different paths. In the end, a spacetime modelled by an affine space corresponds to a specific type of spacetimes modelled by differential manifolds. However, the set of differential manifolds is much larger and there are a lot of spacetimes that cannot be modelled by an affine space (at least not without adding additional structure).

I hope this can help you finding your way through general relativity! Cheers!

Answer 2

Special relativity uses Minkowski spacetime, which is an affine space. This is correct as a local approximation to the differential manifold used in general relativity. Of course it is also much easier.

One can think of the differential manifold as a structure which can be approximated at every point by a Minkowski tangent space, just as one can think of a curved surface as being approximated by a plane in any small region