My previous Blog post about Conformal Cyclic Cosmology (CCC) has made me ponder one of the the imponderables: does spacetime have an inherent structure, or is it just something that we have created to make things more ponderable? Or is it just a quantum foam with no structure?

Sir Roger's CCC postulates that there is no scale to empty space, despite the existence of a cosmological constant that 'looks' just like the energy of empty space. And energy is quantized, not so?

Ethan Siegel, a Ph.D. astrophysicist, concludes an article in Forbes.com with:
"In Einstein's relativity, space and time are still treated as two linked parts of a continuous fabric. In quantum field theory, spacetime is the continuous stage on which the dance of the quanta takes place. But there ought to be a quantum theory of gravity at the core of it all. The question of "discrete or continuous?" contains some fascinating possibilities, including the possibility that we cannot know below a certain scale. The question of "discrete or continuous?" contains some fascinating possibilities, including the possibility that we cannot know below a certain scale. Although many assume one answer or another, at this point, we need more information before we truly know what our Universe is up to at a fundamental level."

The quantum theory of gravity has not been found yet and maybe never will. So where does it leave the structure of spacetime? In a universe devoid of matter, but with photons, other massless particles and even a cosmological constant, there cannot be structure, even if it is spatially infinite, as Roger Penrose has shown.
But add massive particles into the mix and structure appears and with that the spacetime structure. This structure depends on both the spacing and the movements (energies) of such particles. When we put observers into the mix, we typically treat them as just another particle with negligible mass, so that it does not distort the spacetime,

Such observers do have an influence on the observed spacetime structure thought there locations and velocities, but intelligent observers may create a unique spacetime structure around themselves, one which simplifies their physics greatly.

You guessed it, that structure is the inertial reference frame in spacetime, centered on themselves. Each observer will have a slightly different inertial reference frame, except those permanently co-located. Those that are not permanently co-located, but static relative to each other, have identical inertial frames, just with a translational offset. Those that are not static relative to each other, may have a translational and rotational offsets.

The usual depiction of such coordinates is by the Minkowski Spacetime Diagram (STD or more fully MSTD) as pictured below, shown for only one space and one time dimension.

Fig. 1 Minkowski STD

One can clearly see the hyperbolic structure of this view of spacetime, satisfying the timelike, lightlike and spacelike interval concept, where the spacetime interval squared is given by
Δs² = c²Δt² - Δs² for the three cases respectively: timelike if if cΔt > Δx, lightlike if cΔt = Δx and spacelike if cΔt < Δx. The spacetime interval is the elapsed time as measured b clock that travel inertially between the two events.

Expanded to all 3 space dimensions, this is the most general form of diagramming special relativity and has the best utility in flat spacetime. Because of the hyperbolic distortion of scales between objects in relative motion, it can sometimes be difficult to understand for beginners in relativity.

There is another, less general way of depicting flat spacetime, somewhat limited in utility, but without the hyperbolic distortions - the Epstein SpacePropertime Diagram (ESPD). If used within its validity regime, it is generally easier to understand than the MSTD.

Fig. 2 Epstein ESPD (red) vs. Minkowski MSTD (black)

Because of its Euclidean nature, the ESPD uses circles in place of the Minkowski hyperbolas to show spacetime intervals (Δs). Note that the red and black worldlines were not drawn for the same relative speed - it was just to show the different construction clearly. Most noticeable is that the 45° line of lightcone of the MSTD is a 90° line in ESPD, so it is really a light-sphere in place of the lightcone.

Fig. 3 An Epstein SPD on its own, with more detail

Apart from the Euclidean structure, the most striking feature is the fact that the Lorenz factor √(1-v²/c²) is immediately visible as a simple Euclidean projection from red the coordinate points to the blue coordinate points (and also vice-versa). This is in accordance with special relativity's notion that each observer will reckon the others clock to tick too slowly and with distances contracted by the Lorenz factor.

It is also extremely easy to graphically find the correct angle of the red wordline, Φ = asin(v/c), without even needing a calculator. Just draw a circle at the origin with a radius (r) equal to the maximum time that you need and mark a point on that circle that is r v/c from the propertime axis, where v is the desired relative velocity.

This brings us yo the point where we can examine what spacetime structure looks like in Epstein's SpacePropertime.

Fig. 4 Structure in ESPD for a relative velocity between red and blue of v=0.6c, giving a dilation/contraction factor of 0.8

Beautiful! What can be easier than an Euclidean structure that is just rotated around the origin? And it does not care which one of the two structures are viewed as the reference - they are completely symmetrical.

I'm a supporter of the view that empty space has no structure, except that which has been imposed it by an observer or observers, private to each of them. The Epstein structure makes converting coordinate points from the one structure to the other exceptionally easy. It implements the correct Lorentz transformations in a very intuitive way.

There are a number of caveats pertaining to this simplicity though, i.e.

(1) One must take one of the structures as the reference and rotate all other structures, sharing the same origin, relative to it.

(2) It does not work for spacelike intervals, because observers and/or clocks cannot move between events that are separated by a spacelike interval.

(3) Obeying special relativity, no observers or clocks can reach a Φ = 90º worldline, because that would have required infinite energy. Only massless particles can achieve that and they do so without recording any proper time.

An flash of electromagnetic energy at the origin will propagate in all directions through space and time and can be detected by any observer at a time t =x/c after the flash, as measured in the indivudual's own Space-Propertime structure, using own stationary clocks and rods that were carried along.

I'll give some time for this to sink in (and for me to understand it better) and then show some very interesting extensions and applications using Epstein diagrams.

-J