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This question asks how exactly Special Relativity (SR) emerges mathematically as a special case of General Relativity (GR).

In GR, spacetime is modeled as a pseudo-Riemannian manifold with generally non-zero curvature. Treating SR really as the special case of vanishing curvature, spacetime should still just be a manifold. However, spacetime in SR is often modeled as a vector space or an affine space: we speak about the length of a rod $|\mathbf{x}_1 - \mathbf{x}_2|$, spacetime intervals $c^2(t_1 - t_2)^2 - (\mathbf{x_1 - x_2})^2$ etc. (see, e.g., Jackson: Classical Electrodynamics, Third Edition, p. 527). This works, of course, but is unsatisfactory from a conceptual point of view since we introduce new structure in SR while it should really be just a special case of GR.

  1. Can we make sense of the length of a rod, spacetime intervals etc. in SR while sticking to the mathematical framework of GR and treating spacetime as a manifold?
  2. If yes, how exactly can this be done mathematically?
Figaro
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    A vector space is a special case of a smooth manifold, so your question does not make sense. – Danu May 18 '22 at 13:40
  • A vector space is indeed a manifold, with additional structure. The question is if we can do SR without this additional structure, on an equal footing with GR where the spacetime manifold does not carry a vector space or affine structure. – Figaro May 18 '22 at 13:47
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    The affine structure can be re-derived from the metric (and the connection), so you are not loosing anything by starting from GR. For example, the space-time intervals are equal to the geodesic distance. – SolubleFish May 18 '22 at 14:01
  • @SolubleFish, are you saying that flat spacetime of SR can be equipped with an affine structure derived from the metric and the connection, whereas curved spacetimes in GR cannot in general? – Figaro May 18 '22 at 14:21

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Let $(\mathcal M,\eta)$ be Minkowski space, viewed as a lorentzian manifold. From only this data, we can rebuild the affine structure and therefore derive the formalism of special relativity from general relativity (on a flat space-time).

Because the metric is flat, parallel transport allows us to identify every tangent space to $\mathcal M$ into one vector space $M$ (equipped with the lorentzian metric $\eta$). Then, the exponential map $T\mathcal M \simeq \mathcal M\times M \to \mathcal M$ gives us the affine structure. More explicitely, given a point $p\in\mathcal M$ and a vector $v\in M$, we can take $\gamma$ the geodesic whose tangent vector at $p$ is $v$. Then we see that the translated point $p+v$ (defined by the affine structure) is $\gamma(1)$.

In other words, Minkowski space has a unique lorentzian affine structure which is compatible with its structure of a lorentzian manifold.

SolubleFish
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  • Thank you. This means that the "extra structure" in SR (the affine structure of Minkowski space) is not an ad hoc additional structure but comes for free from the lorentzian manifold structure if spacetime is flat - correct? – Figaro May 18 '22 at 17:04
  • Yes, if space-time is flat, complete (maximally extended) and simply-connected to be completely precise. – SolubleFish May 18 '22 at 17:12
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General Relativity postulates that spacetime is locally Minkowski which means that the invariance of light velocity holds under gravity, too. According to Einstein [1], without matter or energy somewhere there is no space or time. Thus, a "flat" spacetime does not really exist. However, in case of vanishing Riemann curvature the local metric differences are negligible and one can use the same local metric on the whole manifold, which is then the spacetime of Special Relativity.

[1] https://medium.com/@rloldershaw/einstein-without-matter-there-is-no-space-or-time-c2357c75286b

JanG
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  • About the alleged Einstein's 1921 assertion: historians of science describe the following. During the development of GR Einstein was guided by a conviction that the existence of spacetime and matter/energy in it are mutually dependent. When GR was completed Einstein was convinced that the Einstein field equations do not have a solution that describes a universe with inertia, but without matter in it. The astronomer Willem de Sitter found such a solution (deSitter space) Einstein tried hard to disprove that solution but in the end he had to acknowledge its validity. Continuing... – Cleonis Jun 26 '22 at 18:55
  • Continued: in the wake of the deSitter space solution Einstein came to a profound reassessment. For instance, before: Einstein believed that Mach's principle was central to GR, but after: Einstein abandoned Mach's principle. In the light of that: an assertion that 'without matter/energy there is no space or time' seems very much overreaching. See also: https://www.ilorentz.org/research/vanbaal/DECEASED/ART/E-dS.pdf – Cleonis Jun 26 '22 at 18:55
  • @Cleonis, it is very interesting. I have to read it. What about reverse question? Can matter exist without spacetime? – JanG Jun 26 '22 at 19:27