Metric of spacetime with zero speed of light

differential-geometrymetric-spaces

Despite references to physical spaces, this question is purely mathematical on differential geometry.

While deriving coordinate transformations based on common assumptions of homogeneity and uniformity of space and time (see A Primer on Special Relativity or Nothing but Relativity), typically three logical options are considered:

  1. Finite speed of light – Minkowski spacetime.

  2. Infinite speed of light – Galilean spacetime, ruled out by observation.

  3. Imaginary speed of light (negative square) – Euclidean spacetime, ruled out by causality.

There exist however the forth logical option:

  1. Zero speed of light.

At first it appears unreal, but in fact there exists a conceptual case asymptotically close to this scenario.

Imagine a thin spherical shell approaching its Schwarzschild radius. Spacetime inside this empty shell is locally flat Minkowski with the same time dilation as at the shell. As the shell approaches an infinite time dilation at the Schwarzschild radius, the speed of light at and inside the shell approaches zero.

What is the metric structure of this space in the limit of the speed of light being exactly zero?

Metrics for other three cases are well known. The Euclidean and Minkowski metrics don't require an introduction. The Galilean structure is described here: What is a mathematical definition of the Maxwellian spacetime?

Would the $c=0$ spacetime collapse simply to a 3D Euclidean space with no time or would it have two separate metrics for space and time like the Galilean spacetime?

Best Answer

From a mathematical perspective this is a question about signatures of quadratic forms. On $\mathbb{R}^4$ with space coordinates $(x, y, z)$ and time coordinate $t$ we can consider

  1. A Euclidean metric, corresponding to the quadratic form $x^2 + y^2 + z^2 + (ct)^2$, with signature $(+, +, +, +)$
  2. A Lorentzian / Minkowski metric, corresponding to the quadratic form $x^2 + y^2 + z^2 - (ct)^2$, with signature $(+, +, +, -)$ (or the negative of this, depending on your conventions)
  3. A Galilean metric ($c \to \infty$), corresponding to the quadratic form $-t^2$, with signature $(0, 0, 0, -)$; we get this by dividing the previous expression by $c^2$ and just taking the limit in the most obvious sense
  4. A $c = 0$ metric, corresponding to the quadratic form $x^2 + y^2 + z^2$, with signature $(+, +, +, 0)$.

The physical significance of this is not entirely clear to me; I don't have much experience thinking about relativity, but here are some speculations off the top of my head.

If we think of these quadratic forms as describing spacelike vs. timelike vs. lightlike directions, then in Galilean spacetime every direction is timelike or lightlike, while in $c = 0$ spacetime every direction is spacelike or lightlike. I guess we can think of $c$ as describing the "slope of the light cone," in which case the Galilean limit $c \to \infty$ corresponds to everything being in your light cone, while the $c \to 0$ limit corresponds to nothing being in your light cone.

I guess your conception of the Galilean limit $c \to \infty$ is that it describes a universe where "time is infinitely more important than space," so first we have the quadratic form $-t^2$ dividing up time slices but then in each timeslice we have the usual 3d Euclidean metric, reflecting the fact that an infinite speed of light means we can in principle reach any point in space from any other point in space at a given time, but we still can't e.g. travel backwards in time. If so, then the corresponding $c \to 0$ limit describes a universe where "space is infinitely more important than time," so we can think of it as divided up into isolated points of 3d space, each of which has a timeline with a 1d time metric, reflecting the fact that a zero speed of light means nothing can move.

Perhaps we should call the $c = 0$ case Zenoan spacetime.

Edit, 9/4/20: The $c \to 0$ limit is briefly discussed in Freeman Dyson's Missed Opportunities; he calls the corresponding automorphism group $G$ the "Carroll group," after Lewis Carroll:

"A slow sort of country," said the Queen, "Now, here, you see, it takes all the running you can do, to keep in the same place."

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