How Einstein discovered (invented?) curved spacetime
It was a logical step
In 1905, Einstein published his theory about space and time that later became known as “Special Relativity Theory” (SRT). Ten years later he presented an alleged sequel he called “General Relativity Theory” (GRT). Obviously, the two have something (or more than one thing) in common, otherwise why should they bear the same surname?
There are vast differences between those two theories. Here are some of them:
- SRT deals with uniform movements without forces. GRT deals with non-uniform movements with forces.
- In SRT, every observer has his own space and his own time. In GRT, space and time are the same for all observers.
- In SRT, clocks must be synchronized individually. In GRT, all clocks are synchronized everywhere and always from the start.
- In SRT, there is length-contraction. There is none in GRT.
- SRT has explicitly abolished the ether. GRT has explicitly reintroduced the ether.
- In SRT, the speed of light is constant. In GRT, the speed of light is variable, dependent on gravity.
- In SRT there are special phenomena not present in GRT, e.g. the Thomas precession. In GRT there are special phenomena not present in SRT, e.g. frame dragging.
- In SRT space is always flat. In GRT it is always curved.
- The mathematics of SRT is very simple. The mathematics of GRT is very complicated and entirely different from SRT.
- In the limiting case of GRT (flat space, no forces) the formulas of SRT do not arise. In the limiting case of SRT (observer speed = 0) the formulas of GRT do not arise.
Then what moved Einstein to extend his theory? It was a rotating disc. His friend Paul Ehrenfest showed him the difficulty of treating it with (special) relativity. And here is how it works — or rather, how it can’t work. But first, an objection: You are not allowed to apply SRT, always dealing with uniform and linear motions, to a rotation. Well, you may. The master himself allowed the method, since Archimedes applied it 2000 years before when calculation the circumference of a circle: approximate the circumference by polygons. Then it is a case for SRT:
Einstein allowed this procedure in his seminal work “ Zur Elektrodynamik bewegter Körper (Annalen der Physik 17, 891–921, 1905)”. On pp. 904–905 he says:
Assuming that the result proved for a polygonal line also applies to a continuously curved curve …
Let’s see what happens to a rotating disc (see picture below). Due to the Lorentz-contraction, the circumference of the disc shrinks (red arrows), but the radii do not! Length-contraction only happens with transversal movements, not with radial ones.
After this asymmetric contraction, the disc looks like this: It got smaller, although it couldn’t:
Either it will be distorted like this (which is impossible with glass or concrete):
Or the radial part is lifted into the third dimension (green arrows):
so that this figure emerges (again impossible for rigid bodies):
Anyway, the figure is not distorted, and it pleased Einstein. Don’t forget: These changes are purely mathematical, “caused” by the Lorentz-contraction, which has no physical source. It is a mathematical phenomenon where people to this day aren’t sure whether it’s real or apparent. Einstein assumed it to be real, but he noticed one thing: The ratio of circumference to diameter of this distorted disc is no more equal to pi. Something like this is mathematically possible in a non-Euclidean space, where the ratio is either smaller or larger than pi. And that’s the reason Einstein switched to non-Euclidean (curved) spaces — to remedy a purely mathematical inconsistency with another purely mathematical trick. As he himself expressed it in his speech “Geometry and Experience” on January 27, 1921:
In a rotating reference system, the laws governing the positioning of rigid bodies do not correspond to the rules of Euclidean geometry because of the Lorentz contraction.
However, there can only be a so-called “elliptical” (finite, closed) world, similar to the surface of the earth, not a “hyperbolic” (infinite, open) world, which also causes problems for the boundary conditions of the GRT, which Einstein himself revealed in the same lecture:
The complete reduction of inertia from interaction between the masses — as Ernst Mach, for example, demanded — is only possible if the world is spatially finite.
This is how Einstein came to non-Euclidean geometry, i.e. curved spaces, which he tried to tame with the help of highly complex tensor calculus. Because there are no forces in SRT, Einstein tried to use curved spaces to replace forces — more precisely: gravity and inertial forces — with complicated movements (i.e. accelerations). This works with homogeneous (uniform) gravity conditions, but not with inhomogeneous fields or rotations.
Let’s return to Ehrenfest’s eerie enigma. Here are some suggested solutions (from Wikipedia):
1909, Paul Ehrenfest: There can be no rigid bodies. This not only is difficult to accept (think of glass or concrete), it contradicts Einsteins own explicit demand for “rigids rods” (starre Maßstäbe) to measure distances.
1910, Gustav Herglotz and Fritz Noether: A disc cannot be brought into a state of rotation. No acceleration from “rest” to “rotation” is possible. Nothing can spin — which contradicts everyday experience.
1910, Max Planck: You have to consider elasticity (which doesn’t exist in SRT).
1910, Theodor Kaluza: There is no paradox when you assume that the disc is rotating in hyperbolic geometry.
1911, Vladimir Varićak: Lorentz-contraction is purely subjective (apparent?), an opinion that Einstein rejected.
1911, Max von Laue: No rigid bodies can exist, due to special relativity.
1935, Paul Langevin: It’s the metric, again. If you apply the Langevin-Landau-Lifschitz metric, the paradox vanishes.
1937, Jan Weyssenhoff, in my opinion, delivers the most interesting explanation (sorry for the sarcasm): The Langevin observers are not hypersurface orthogonal. Therefore, the Langevin-Landau-Lifschitz metric is defined, not on some hyperslice of Minkowski spacetime, but on the quotient space obtained by replacing each world line with a point. This gives a three-dimensional smooth manifold which becomes a Riemannian manifold when we add the metric structure. Got it?
1946: Nathan Rosen showed that inertial observers instantaneously comoving with Langevin observers also measure small distances given by Langevin-Landau-Lifschitz metric.
1946, E. L. Hill: Centrifugal forces help (which do not exist in SR).
1977, Grünbaum and Janis: Bodies are “non-rigid”, which is not physically realistic for real materials from which one might make a disk, but it is useful for thought experiments.
2000, Hrvoje Nikolić: The solution lies in general theory.
A lot of “explanations” for so simple a phenomenon (and there are many more). Which one is the right one? Or is there no real (realistic) solution?
