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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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2 Answers2

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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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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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