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Relativity and Cosmology

This is a Blog on relativity and cosmology for engineers and the like. My website "Relativity-4-Engineers" has more in-depth stuff.

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Spacetime Structure

Posted August 04, 2020 6:00 PM by Jorrie

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

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#1

Re: Spacetime Structure

08/05/2020 11:20 AM

The Epstein SPD reminds me of a simplified visualization of Special Relativity I ran across awhile back.

https://medium.com/predict/we-all-travel-through-spacetime-at-the-speed-of-light-d60cb389dfc2#

Basically, at rest (inertial frame), we all always travel through spacetime at lightspeed (c) in the time direction. Two observers would both be traveling in their own time direction. If they were in relative motion, their coordinate axes would be rotated with respect to each other, and they would drift apart like two airplanes traveling the same speed but in slightly different directions. To each plane, the other appears to fall behind (time dilation) and appears to be slightly foreshortened (space dilation).

https://medium.com/predict/we-all-travel-through-spacetime-at-the-speed-of-light-d60cb389dfc2#

"Your total speed through spacetime is constant and consists of both a movement through space and through time. When you move faster through spacetime, your time component will be smaller compared to someone standing still. Image credits: Wikipedia Commons."

"In total, we all move at the total speed of light, c, through spacetime, with the speed spread between space and time. We can’t go faster than light through space. And we neither can go faster nor slower than light through spacetime. It’s the constant speed of everything in the fabric of spacetime."

https://medium.com/predict/we-all-travel-through-spacetime-at-the-speed-of-light-d60cb389dfc2#

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#2
In reply to #1

Re: Spacetime Structure

08/06/2020 1:19 AM

Hi Rixter, your quote: "Your total speed through spacetime is constant and consists of both a movement through space and through time. When you move faster through spacetime, your time component will be smaller compared to someone standing still."

Yes, that's the Epstein diagram's idea, but one must be careful with the words "speed through spacetime", because it is misleading. It clashes with the standard meaning of speed and velocity, which are measured through space alone. What remains constant and can be scaled to the speed of light is the four-velocity vector through spacetime.

In a space-propertime diagram, it should rather be viewed as simply the propertime vector of the worldline. Coordinate time and distances covered is then simply a projection of this vector onto the reference axes.

But is a very useful concept, as I will show in follow-on Blog posts.

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#3
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Re: Spacetime Structure

08/06/2020 10:06 AM

Thanks, Jorrie. That's a very good point. "Speed through spacetime", where speed = distance/time, doesn't make any sense.

Einstein once said, "that everything must be made as simple as possible, but not one bit simpler." This might be an example of "simpler".

https://quoteinvestigator.com/2011/05/13/einstein-simple/

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#4

Re: Spacetime Structure

08/07/2020 1:55 PM

A graphic example of this rotation is the Andromeda Paradox. By walking across the room, I can change the "current time" in the Andromeda galaxy by hours or days.

My current time is a line (or plane, or 3 space volume, depending on the number of spatial dimensions) perpendicular to my time axis at the current time. Another observer in the Andromeda at rest with me at my current time would have space and time axes parallel with mine. If I start walking, by axes rotate very slightly so that my current-time "line" (or plane, etc.) crosses his time axis hours or days in his future or past.

Pretty strange stuff...

https://www.researchgate.net/publication/318016152_On_the_Andromeda_paradox

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#5
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Re: Spacetime Structure

08/08/2020 12:52 AM

Yes, those faraway"paradoxes" always sound strange, but are generally rather meaningless. At the core of it is the fact that how we view and represent our lines/planes of simultaneity is just a convenient convention - to make light propagate isotropically from our (private) perspective.

There are other conventions that are equally valid, but they make our calculations extremely difficult for local phenomena - hence we follow Einstein's method of clock synchronization. His choice "made it as simple as possible..."

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#6

Re: Spacetime Structure

08/10/2020 5:57 AM

Interesting concepts and beyond the edge of my current capabilities.

I must however share an observation that your described system with common origin and rotated cartesian reference planes immediately reminded me of my rotating machinery theory of energy/magnetism/force where "motoring" and "generating" would be the viewpoints of observers in each axis system.

I have for some years postulated that an entity in a single dimensional system represented by a line (forwards/backwards) might be able to imagine concepts requiring a second dimension (sideways) but with no means to describe it from their perspective other than to "bend" their datum.

similarly for an entity on a 2D environment to conceive a 3D world requires a bend/fold but still no means to actually understand it.

In our 3D world, we talk about "bending" space/time. Are we simply limited by our 3D understanding attempting to understand a 4D reality that exists around us? Could that 4th Dimension then correlate to the unexplainable observations that we have of "dark matter/energy" where we are able to observe the effect, but not individually participate in that experience?

From the motor example. If you were unaware of magnetism, then the observed movement of the armature relative to the field would seem to be lagging and then by different amounts dependant on the external load that it was driving, this being isolated from the observer by a second degree of separation.

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#7
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Re: Spacetime Structure

08/10/2020 10:48 AM

"In our 3D world, we talk about "bending" space/time. Are we simply limited by our 3D understanding attempting to understand a 4D reality that exists around us? Could that 4th Dimension then correlate to the unexplainable observations that we have of "dark matter/energy" where we are able to observe the effect, but not individually participate in that experience?"

In a way yes, but 4D spacetime is reasonably concrete to us, because we understand 3D space and 1D time fairly well and we know how to combine them mathematically. It is when that 4D spacetime becomes curved that we run into trouble. We do not understand velocity time dilation and gravity any better than we understand dark matter/energy.

All have to do with curved spacetime, for which we have a beautiful geometrical theory to predict the observational effects, but deep down we simply don't know what is going on. I guess for us engineers it is the normal state of affairs. We don't have to understand quantum mechanics to do electronics, or send a spaceship to Mars...

Anyway, isn't it a little comforting when one can draw a spacetime diagram and pretend that you understand?

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#8
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Re: Spacetime Structure

08/12/2020 5:36 AM

I had real problem for a while getting my mind to move beyond 3D, but as an EE once I understood "other attributes" were possible, moving to 6D, 12D and then 18D wasn't a problem, but that's where I hit my next roadblock. The maths simply continued, but I ran out of tangible "dimensions" to go beyond 18.

For me as an EE, every point in space has other attributes relating to physical reality. It was actually magnetism that got me over the 3D threshold by understanding that every point in our 3D space also has related magnetic vectors necessary to typify it.

The comfort comes in realisation that the 2D spacetime diagram represents a single plane within the modelled space and thus leads to comprehension if not understanding.

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