General “relativity” without special relativity

Question

I was wondering if there could be a way of using the principle of equivalence and develop a theory of warped space-time without using special relativity, ie, without assuming any maximum possible speed. Would such a theory be possible in principle? Or does GR specifically depend on the absoluteness of $c$ in order to make sense as a theory of gravity with warped space-time? Any thoughts?

PD: I have seen this post, but for what I understand, the question (and answers) are concerned with the possibility of having GR as we know it, instead of a modified GR, without a limit speed (which is what I’m asking).

Answer 1

There's a really nice version of Newtonian gravity that talks about curvature in space-time and I think it might be interesting to you.

It begins with the idea that, since all things fall at the same rate, this might be explained by a curved space. However, when you try to get the equations, they fail: the geodesic equation looks like $\frac{dv^i}{dt}=-\Gamma^{i}_{jk}v^kv^j$ Where i,j,k, go from 1 to 3, since they represent only time direction.

Why do they fail? Well, as you can see, the RHS clearly depends on the velocity, while Newtonian gravity does not. However, there's a really nice way to go around this. What if, instead of space, we worked with space-time? In this case, our velocity vector would have components $(1, v_x,v_y,v_z)$ instead of just the last three. And that one could really come in handy. Using it, we can make the geodesic equation look like the newtonian one just by making $\Gamma^i_{jk}=0$ for $j=0$ or $k=0$.

In that case, it would just look like: $\frac{dv^i}{dt}=-\Gamma^{i}_{00}v^0v^0=-\Gamma^{i}_{00}$ (where i goes from 0 to 3 this time)

Which is way more sensible now.

Answer 2

I was wondering if there could be a way of using the principle of equivalence and develop a theory of warped space-time without using special relativity, ie, without assuming any maximum possible speed. Would such a theory be possible in principle?

I would say that the answer is basically no, although it may be necessary to highlight the relevant assumptions more clearly.

The existence of a maximum speed is not an assumption in special relativity. Depending on what axiomatization you use for SR, there could be an assumption that there's an invariant speed (Einstein's 1905 axiomatization), which is a different thing. Actually Einstein's axiomatization is probably not one that anyone would pick today. You just see it in popularizationa dn undergraduate textbooks because it's become a traditional way of presenting it. For other styles of presentation, see, e.g., Pal, "Nothing but relativity," https://arxiv.org/abs/physics/0302045 , or Bertel Laurent, Introduction to spacetime: a first course on relativity.

The machinery of general relativity, in the formalism that is most often used, depends on the existence of a metric. Without a metric, you can't do things like raising and lower indices. In particular, you need this metric to be nondegenerate. If the equivalence principle holds, then spacetime is locally flat, and SR applies locally, where you can choose coordinates such that this metric has the form $\operatorname{diag}(1,-1,-1,-1)$.

The Pal paper explains why, under certain reasonable basic assumptions, a flat spacetime has to be either Galilean or Lorentzian. If you want spacetime to be locally Galilean, then you don't have a metric (or you could think of the metric as being degenerate or split into separate time and space parts). This is not compatible with the machinery of GR as outlined above.

Answer 3

The point is that when you have the space-time metric $g_{\mu\nu}$ and a geodesic in the said metric, then it will always conserve the norm $g_{\mu\nu} u^{\mu} u^{\nu}$ of the tangent vector $u^\mu$ of the geodesic parametrized by the affine parameter $\lambda$. The affine parameter makes the geodesic equation look like $$\frac{d^2 x^\mu}{d \lambda^2} + \Gamma^\mu_{\nu}{}_{\nu \kappa} \frac{d x^\nu}{d \lambda} \frac{d x^\kappa}{d \lambda} = 0 $$

How this is interpreted in General relativity (and in metric gravities in general), is that the affine parameter is proportional to proper time (measured by strictly local physical experiments such as atomic clocks etc.), $u^\mu \propto dx^\mu/ d\tau$. When you use this interpretation, there is no avoiding special relativity in local frames, since there is always a frame (Riemann normal coordinates) where the metric is locally $\eta_{\mu\nu} + ...$ and in these frames physics will work exactly like in special relativity.

There are ways to modify this, e.g., by stating that the affine parameter of the geodesic is not proportional to time measured in physical experiments. For instance, you can postulate that there is a time-like vector field $t^\mu$ at every point such that physical time $d t$ measured in local experiments along the geodesic fulfills $$\frac{d x^\mu}{d t} t_\mu = -1$$ Note that this also adds new terms to the geodesic equations when using this parameter, and the affine parameter looses any physical meaning. The foliation of space-time by $t^\mu$ plays a role similar to a privileged global time coordinate and there is no limitation on the components of the vector $v^\mu = d x^\mu/dt$ which are space-time orthogonal to it. You can see this by defining $$v^\mu_{\perp} \equiv v^\nu (\delta^\mu_\nu - \frac{1}{t_\alpha t^\alpha}t^\mu t_\nu)$$ Then it is easy to see that $v^\mu_{\perp}v_{\perp \mu}$ is unbounded. When you think about it in detail, this implies there is no limit of velocities of particles physically.

Of course, to make this work you need to introduce this new privileged field $t^\mu$. Also, you have to find a way to make this work alongside the Michelson-Morley experiment and so on and so on. But in principle this can be done.