I’m willing to bet on it! You don’t even have to look as far as our universe. Anything with a non-constant index of refraction will give rise to a geometry different from Euclidean geometry.

The geometry of spacetime has no meaning. This notion comes from the indefinite metric. This metric comes from a generalization of one that results from Lorentz transforms, i.e., the difference of the transformed time less the sum of the squares of the transformed space coordinates shall be a constant for inertial motion. There is nothing nonlinear about it and it works for Euclidean spaces and for Maxwell’s equations. However, the generalization where the coefficients of the square terms are allowed to be functions of space, and possibly of time runs into conceptual difficulties.

The metric distinguishes between two times: the laboratory time and the local, or “proper”, time which is what a clock on a moving body would record. Both times are meaningless by themselves; what is relevant, however, are the corresponding velocities.

Take for instance the usual expression for the outer Schwarzschild metric.

ds^2=B dt^2-dr^2/B+ the element of a 2-sphere,

where B=1–2m/r, and s is supposedly the proper time and t the laboratory time.

However, the two times are not independent but, rather, are related by

ds/dt=B.

The history of such a relation goes back to Levi-Civita (around 1920) who was studying the property of “regularization.” It is known that Newton’s inverse square law and Hooke’s harmonic oscillator are duals of one another (see for example, *Seeing Gravity*). Since the former has a singularity while the latter does not, it was ( and still is) advantageous to get rid of the singularity by considering its dual. This was shown to be related to a stereographic projection of curves on a sphere to elliptic curves in the plane by Moser in 1970. The arc lengths of curves on the sphere have time s while those on the plane have time t. The time s was referred to as “fictitious” time by Levi-Civita. It is, in fact, what relativists would refer to proper time.

Now, if we use the second relation to eliminate proper time in the first, we get

v^2= (dr/dt)^2=(1-B)B^2,

for radial motion. Differentiating it with respect to time t, gives

dv/dt=-g(B-3 v^2/B).

In the weak field limit you get back Newton’s law, dv/dt=g=m/r^2. But, if you don’t go to that limit, you get something people have referred to a gravitational repulsion because of the second term. Hilbert (1917), Droste and others thought that at speeds v >c/sqrt 3, gravitational attraction transforms into gravitational repulsion, like Jekyll and Hyde.

Now, use the second relation to eliminate dt from the metric. We then find

u^2=(dr/ds)^2=(1-B)=2m/r.

Differentiating with respect to s, then gives Newton’s law,

du/dt=-g.

Gravitational repulsion has completely disappeared. To the best of my knowledge this result was derived by Paul Drumaux in 1936, and who was criticized for it. Miraculously, gravitational repulsion has disappeared!

It is the reasoning from the metric that has led to all the confusion!

What is relevant are the lagrangians

L=(1/2)[v^2+(1-B)B^2]

and

L’=(1/2)[u^2+(1-B)].

They are related conformally by L=B^2L’, and their corresponding hamiltonians both vanish. This is because we are dealing with an escaped velocity rather than a regular Keplerian one, u^2=2m/r and not m/r.

All the non-Euclidean effects are contained in taking into consideration to the 2-sphere element. In terms of time t, the lagrangian becomes

L’=(1/2)[v^2/B^2+h^2/rB] - m/r

where h is the angular momentum. When the mass m is replaced by its density, the terms in the square brackets become the Beltrami metric of hyperbolic geometry of constant negative curvature. (This corresponds to the inner Schwarzschild solution.) Obviously, the same two terms correspond to the metric for the outer solution, but the curvature is no longer constant.

It was apparent all along that the “correct” results of general relativity had to make contact with non-Euclidean geometries, whether they have constant curvature or not. But such results have nothing whatsoever to do with spacetime.

For more details see *A New Perspective on Relativity: An Odyssey in Non-Euclidean Geometries*.

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