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I’m learning general relativity recently and something confused me. if we return back to the original formulation of special relativity by Einstein, we need nothing but Lorentz transformation to complete the theory and it seems no motivation to study the geometrical structure of spacetime. no matter what metric tensor we define in a 4D Riemann space, the minkowski line element is invariant under Lorentz transformation. does the geometrization of spacetime in SR lead to some mathematical unnecessaries in the physical theory or not?

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  • $\begingroup$ Studying SR with only space is like studying Newtonian mechanics with only the plane. $\endgroup$
    – J.G.
    Commented Nov 1, 2022 at 16:37

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In terms of transforming a set of measurements in one inertial frame of reference to another, the Lorentz Transformations are what are needed. But Special Relativity is so much more than those transformation equations. (Similarly, Euclidean geometry is so much more than a rotation matrix. How well can you analyze a triangle with only formulas and a rotation matrix?)

Note: in general relativity, the tangent space at a point-event has the structure of a Minkowski vector space. (Similarly in Riemannian geometry, the tangent space at a point has the structure of a Euclidean vector space.)

So, one important value of the structure of Minkowski spacetime is that it can be used in more general situations than it was initially conceived for.

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Special relativity is a theory of inertial reference frames. The idea underlying it is that the laws of physics are the same in all inertial reference frames, and we can use the Lorentz transformations to transform observations in one reference frame to observations in another reference frame. Importantly, at the level of special relativity, we have to ignore the effects of gravity.

Einstein's big idea in developing general relativity was the equivalence principle: the idea that a freely falling reference frame in a gravitational field should be equivalent to an inertial reference frame in the absence of gravity. This stems from the idea that "a freely falling observer does not feel their own weight".

The problem is that freely falling reference frames can only be equivalent locally to an inertial reference frame in deep space, since gravitational fields are generally non-uniform. If I'm falling towards the Equator of the Earth, and another person is falling towards the North Pole, we are in different freely falling reference frames; if we each try to extend our frames to include the other, we would each conclude that the other person has forces acting on them (causing them to accelerate) while we personally have no forces acting on us.

So if we buy the idea of the equivalence principle, then we have to give up on the idea that an inertial observer (redefined as a freely-falling observer) can construct a global inertial reference frame. Rather, the best any observer can do is to construct a local inertial reference frame which is equivalent a neighborhood of Minkowski space. And a space in which there's a neighborhood of every point which is equivalent to $\mathbb{R}^4$ is precisely the idea of a manifold, which leads naturally into the study of metrics, parallel transport, curvature, etc.

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We don't need Minkowski spacetime, but it's a convenient mathematical abstraction. Poincaré's refinement of Lorentz's ether theory is experimentally indistinguishable from Special Relativity even though it does not employ Minkowski spacetime.

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Einstein originally conceived SR in terms of Lorentz transformations. Minkowski later pointed out that the theory could thought of as the geometry of spacetime. Einstein originally wasn't fond of this idea. But later in the development of General Relativity, he realised that the geometry idea was much preferable.

Basically in General Relativity, Lorentz transformations no longer hold a special place. Technically, they don't hold a special place in SR either. Laws of physics can be written in any co-ordinate system: you just have to specify the metric. In SR, there exist nice co-ordinate systems where the metric is the diagonal one. Lorenrz transformations are the way to switch between these nice co-ordinate systems with the diagonal metric. In GR, there is curvature, so we must explicitly specify the metric. Lorentz transformations are no longer special.

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