# Coordinate System for Minkowskian Spacetime Relative to Event

• I
• Halc
In summary: This is the Milne universe. It is a special case of a Robertson-Walker universe with a linearly growing scale factor and is a prime example of how the initial singularity of a RW universe with a(0)=0 may be a coordinate singularity depending on the explicit form.Thanks all. That's kinda what I was looking for.
Halc
Gold Member
TL;DR Summary
Looking for material on a coordinate system relative to an event rather than to an inertial frame
I have been using a coordinate system that is anchored on an event (rather than a speed reference) in Minkowskian spacetime. This makes it sort of a special case (no gravity or dark energy, just like special relativity) of the cosmological (or CMB-isotropic) coordinate system used to foliate the entire universe, with the big bang being the reference event.

The reference event can be any event. It becomes the origin of the coordinate system.
So imagine the event in question being an explosion in existing space (boundless in all 4 dimensions) with infinite material expelled at evenly spaced speeds. By evenly spaced, I mean the typical separation between a piece of inertial ejecta and its neighboring piece is a uniform amount of proper velocity difference. In this way, from the viewpoint of any bit of ejecta, it appears to be the center of material that is stationary in expanding spacetime. This is the special-relativity version of expanding space coordinates. Is there a name for such a coordinate system? I would appreciate a link to some material on it.

I'm not speculating the possibility of the big bang being an explosion of material in space. If that were so, the concentrated mass would have gravity that would never allow any expansion at all, but this is just a mathematical model with no consideration of gravitational mass.

Some properties that I've worked out:

The coordinate system only foliates spacetime within the light cones of the reference event, sort of similar to Rindler coordinates only foliating a light-cone of spacetime. I think the coordinates work fine with past events, but events without time-like separation from the reference event are not part of the coordinate system.

Any inertial object whose worldline intersects the origin of the coordinate system is said to be stationary. Any motion relative to this stationary line is peculiar velocity, a vector. Velocities add the relativistic way, but recession rates (which are not vectors), being proper speeds, add linearly, hence can exceed c. Peculiar velocity decreases over time, so absent proper acceleration, all objects will eventually approach being stationary.

A pair of objects that are stationary relative to each other in an inertial frame (say at either end of a rigid object with no proper acceleration) are, in this coordinate system, always moving apart. In other words, rigid objects are always growing towards (but never reaching) their proper length. Likewise, the time it takes for light to travel from one end to the other is always decreasing, approaching but never reaching the time it takes light to travel the object's proper length. I've not worked out formulas for the above effects.

The rate at which clocks run is a function of the clock's peculiar velocity and not a function of any observer frame. No value seems observer dependent. There are thus no frame rotations, but there are still conversions to inertial coordinates relative to any selected inertial worldline.

Light will eventually reach any point in space from any other point in space. This is another big difference with the actual universe where the acceleration of expansion due to dark energy forms event horizons similar to Rindler horizons, beyond which emitted light can never reach portions of space.I am mostly asking for references since lacking a name, all my searches have turned up nada, and also feedback if any of the properties I've listed are nonsense. I'm speculating no new physics, just assigning different abstract coordinates to ordinary Minkowski spacetime.

Look up Milne coordinates. It is the closest to what you are seeking.

Halc and Orodruin
As PAllen has already said, this is the Milne universe. It is a special case of a Robertson-Walker universe with a linearly growing scale factor and is a prime example of how the initial singularity of a RW universe with a(0)=0 may be a coordinate singularity depending on the explicit form.

vanhees71 and Halc
Thanks all. That's kind of what I wanted.

Not sure if it's the right thing. The diagram on the right of the wiki page puts the observer at the reference event. It speaks of the cones being the observable universe from that event, where what I was doing was putting the 'bang' at the reference event, and noting that any observer in its future light cone would observe something like what we actually see.

Last edited:
PeterDonis said:
This wiki page?
Assuming that it is, the term "observable universe" is a misnomer. The Milne universe is the future light cone of the "bang" event, in hyperbolic coordinates. But "future light cone" is not the same as "observable universe"; the term "observable universe" properly used means the past light cone of some event of significance to an observer (such as us here on the Earth now).

Halc said:
The diagram on the right of the wiki page puts the observer at the reference event. It speaks of the cones being the observable universe from that event
Which, as noted above, is bad terminology. Your view, where the "bang" event is the reference event and the entire universe (not just the part of it we can observe) is the future light cone of the "bang" event, is correct. We here on Earth now (if this model were a correct description of our actual universe, which, as noted, it isn't) would be located at an event somewhere way up in the future light cone of the "bang" event, and our "now" surface would be a hyperboloid through that event that asymptoted to the future light cone.

vanhees71
Dredging this up again. Disclaimer: This is not a speculation. I know very much that the Milne model contradicts empirical measurements. For one, it predicts a linear scalefactor and not the decelerating-then-accelerating one that has been measured.

PeterDonis said:
Yes, from that page. I have a question:

"Incompatibility with observation:
...
In particular it makes no prediction of the cosmic microwave background radiation[citation needed] ... "

Other than the fact that it is a zero-energy model and thus would seemingly have no plasma to undergo the recombination event, I do very much get a CMBR in my model. Why does the model not predict it? I mean if the zero-energy part is the problem, then our mere existence as observers sort of sinks it. They must have an additional reason why the model does not predict the CMBR.

Citation needed indeed. One would have been nice.

Halc said:
Other than the fact that it is a zero-energy model and thus would seemingly have no plasma to undergo the recombination event, I do very much get a CMBR in my model.
You can't have a CMBR in a "zero-energy model", which the Milne universe is (the stress-energy tensor vanishes everywhere). A CMBR is nonzero energy density (nonzero stress-energy tensor). If you are getting a CMBR in a Milne universe model, you are doing something wrong.

Halc said:
Why does the model not predict it?
You are correct that the past lightcone of any event inside the Milne region intersects a hyperbola of constant small proper time since the explosion. That implies that everyone can see the immediate aftermath of the explosion, increasingly redshifted, at any future time.

However, the fundamental problem with Milne cosmology is that the energy density must be everywhere zero. Look at it in an Einstein frame and you'll see that near the surface of the future lightcone of the explosion the density must tend to infinity if it's uniform in Milne space and non-zero anywhere. So the density has to be exactly zero everywhere and hence no CMBR (or anything else). No infinite family of test particles allowed.

## 1. What is a coordinate system for Minkowskian spacetime?

A coordinate system for Minkowskian spacetime is a mathematical framework used to describe the relationship between events in the four-dimensional spacetime of special relativity. It consists of four coordinates, representing time and three spatial dimensions, and is based on the principles of the theory of relativity.

## 2. How is Minkowskian spacetime different from Euclidean space?

Minkowskian spacetime is different from Euclidean space in that it incorporates the concept of time as a fourth dimension. This means that the distance between two points in spacetime is not just measured in terms of spatial coordinates, but also in terms of time. Additionally, Minkowskian spacetime follows the principles of special relativity, which includes the idea that the speed of light is constant for all observers.

## 3. What is the role of the event in Minkowskian spacetime?

An event in Minkowskian spacetime is a specific point in spacetime that represents a physical occurrence, such as the location and time of a particular event. It is the fundamental building block of the coordinate system and is used to describe the relationship between different events and their coordinates.

## 4. How does the coordinate system account for the relativity of simultaneity?

The coordinate system for Minkowskian spacetime accounts for the relativity of simultaneity by allowing for different observers to have different measures of time and space for the same event. This is due to the fact that the speed of light is constant for all observers, and therefore, the perception of time and space can vary depending on the observer's relative motion.

## 5. Can the coordinate system be used for objects in motion?

Yes, the coordinate system for Minkowskian spacetime can be used to describe the motion of objects. It allows for the measurement of an object's position and velocity in both space and time, and can be used to calculate the effects of special relativity, such as time dilation and length contraction, on moving objects.

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