# How are the comoving coordinates NOT a prefered reference frame?

Physics me this:

The equivalence principle has rigorous physical definitions that say, for one, that the laws of physics are the same in all inertial reference frames. This is to say that the behavior of the universe is the same for any system regardless of velocity $(\vec{v})$ and position $(\vec{r})$ according to special relativity. General relativity even establishes some sorts of parallels for acceleration $(\vec{a})$ and spacetime curvature, although it still leaves some reference frames with an obviously more complicated geometry of spacetime.

The Copernican principle is the "notion" or "philosophy" that neither humans or any given group examining the universe has a privileged view. The idea has been generalized to extend from the original intention applying to the Earth to extend inward as far as a human individual, or outward as far as our local group of galaxies. At the largest scale we have found the distribution and characteristic of matter throughout the universe to be roughly homogenous, as sort of the ultimate example of this principle, leaving no location or collection of matter privileged.

Hubble's constant correlates $\vec{r}~$ and $\vec{v}$ between any two points in space, which defines the comoving coordinates, which identifies a point within the constantly ballooning and accelerating mesh of matter that fills the universe. Without loss of generality I'll say that $\vec{r}$ is truly independent and non-privileged, but $\vec{v}$ has a clearly preferred value, which is that which matches the average flow of matter around $\vec{r}$. Reflect for a second to note that $\vec{v}$ is a privileged view of the universe even though it is non-privileged regarding the physical laws of the universe. Why why why?!

I expect quick dismissals of the "problem" due to the fact that the Copernican principle is not a physical principle and can thus be comfortably wrong. But it still seems like the implications would be non-trivial and would keep physicists up at night. Ultimately if we truly find the "Grant Unified Theory" it should fully allow the Lorentz Invariance AND provide some way for the big bang to create matter that is roughly stationary to other (close-by) matter. If not, why didn't the big bang create matter with $\vec{v}=-c\dots c$ and momentum spanning $\vec{p}=\infty \dots \infty$ for a given $\vec{r}$? That would obviously be nonsensical since collisions would be releasing infinite energy. But wouldn't it be more consistent with the Copernican principle while at the same time being allowable by consistent physical laws to have galaxies flying by at $0.99999c$, allowed by some kind of reduced probability of interaction? Then matter would could uniformly occupy both $\vec{v}$ and $\vec{r}$. I have never heard this even mentioned as an issue.

Why did the big bang create matter along the tapestry of the comoving coordinates? Can any physical theories be said to predict or deny this?

The universe of today is not at all like the universe just after the big bang. What was actually "created by" the big bang (or rather, what our current physical theories predict existed very shortly after the big bang) was a plasma of elementary particles at an exceedingly high temperature (billions, trillions, etc. of Kelvins depending on how far back you go). This high-temperature sea of particles would have filled phase space, just as you're suggesting: the particles would be in random positions undergoing random motions, and many of them would indeed be passing each other at high speeds.

But as space expanded, the particles spread out and the energy density of the universe decreased. Once the energy density dropped enough, the particles started interacting, first via the strong force, then the electromagnetic force, then gravity. These interactions, which are necessary for the formation of large structures like galaxies, tend to slow down the high-velocity massive particles and transfer their energy to photons (the CMB) and gravitons. To put it another way, it's basically impossible to put together the number of particles required to form something like a galaxy (or a cluster, or a universe) if they're all passing each other at high speed.

• You bring up a good point about the high temperature sea of particles and I had not previously appreciated just how severe that was, and I put this in another answer. It turns out that the such a state of the universe was filled with insanely highly relativistic particles. That doesn't fully explain the fact that a conglomeration of such high energy particles resulted in a net-zero momentum state, but perhaps those with high enough energies "moved neighborhoods" until they were in a region with much more similar momentum vectors. Jun 27 '11 at 20:55

In order to gain more specificity in this discussion, I want to entertain a certain case, and hoping that it may provide "part" of the answer, I will put it here. Let's say we are a particle in the electroweak epoch, where temperatures were $10^{28} K$. You might have guessed it, but this would put particle speeds close to the speed of light. But how close? I will try that here. Firstly, by the simple definition of temperature we find the following for kinetic energy.

$$KE=\frac{3}{2} k T$$

With a high $T$ we will have relativistic speeds and it will be sufficient to set the above value, found for the electroweak epoch, to just $E$.

$$E\approx KE = 207,000 J$$

Yes, these are units of Joules... for a single particle. We are pretty thoroughly relativistic. Now let's find velocity, assuming it's an electron.

$$E^2=(pc)^2+(m c^2)^2$$ $$p=\gamma \beta c m$$ $$\beta = \frac{ \sqrt{ E^2-(m c^2)^2}}{E}$$

For highly relativistic particles you will find:

$$\beta \approx 1- \frac{(m c^2)^2}{2 E^2}$$

numerically, I can only write this slightly indirectly.

$$\beta = 1 - 10^{-37}$$

Now, this is only to demonstrate that the particles in the high-energy soup of the early universe were going so fast that they were effectively indistinguishable from the speed of light, although small differences in speed make a large difference in the energetics of a collision at this speed.

Of course, the funny thing about this is that the early universe actually fits rather well my concept of matter traveling at almost all possible speeds at all locations (almost). In such a early universe plasma, it would be difficult to distinguish which particle was closest to the net-zero momentum vector. The greater philosophical question, however, seems to be what dynamics could possibly create such a balancing of momentum. Could the universe somehow figure out a way to create a universe where no $\vec{p}$ was preferred, and then later things balance out to make it so? If we consider the universe coming from a true singularity, then it maybe it could be that originally all possible $\vec{p}$ existed and this then evolved into all possible $\vec{r}$ existing. I have just as much trouble imaging that as I do imagining an infinite universe in the first place.

• I think that it is strongly advisable to split the matter creation problem (bariogenesis) from the universe evolution problem (measured expansion). Jun 28 '11 at 13:18

Cosmological expansion is entirely "Copernican" in this sense: any observers in any far away galaxy will see our galaxy moving at an opposite $\vec{v}$ approximately symmetric (not exact because the model and real universe have inhomogeinity discrepancies) but at a cosmological scale the law is fully covariant in different points of space

any rough variations in a presumably entirely covariant momentum density of matter near the planck era should have been able to thermalize after the inflation stage ended, Although we certainly don't know how was matter and energy distributed near this planck era

i hope i'm clearing the confusion, or maybe i misunderstood your question?

• I've heard that the inhomogeinity discrepancies are caused by quantum effects in the early universe, or else the universe would be totally uniform. It's interesting to hear that we can't know the matter momentum distribution in the plank era, and that could get us closer to answering my question (although I admit it may be unanswerable at some level), since it reduces the problem to requiring that the big bang produce something that can average out to net-zero momentum in our comoving location. $\vec{v}$ could have varied quite violently before that. Jun 27 '11 at 19:45

You ask: why there are no stars or galaxies moving relative each other at speeds close to the speed of light?

The answer is simple: if such stars and galaxies existed now, they would be quite quickly slowed down by the light pressure of the CMB, the light from stars and by the viscosity of interstellar medium. This would be accompanied with enormous heating.

But still remains a question, why there could not be matter moving at speeds close to the speed of light previously, just after the Big Bang?

The answer is simple as well: such matter actually existed. The speed at which particles of matter move relative to each other determines temperature. Temperature is just statistical characteristic of kinetic energy and the speed. The closer time moment we consider to the Big Bang the higher was temperature, and the greater was kinetic energy of the particles and their speeds relative to each other.

Comoving coords are relative to a universe were matter shrinks.
Any measure we do is a ratio between a quantity and a chosen standard.
Any measure is a dimensionless quantity. If we measure an expansion it may signal that the unit of measure is shortening.
All system of units we use are atomic units.
But we can not only squeeze the atom length, also its mass has to decrease as time goes by. Also the time unit vary with atom size, provided that c, G, $\varepsilon$, properties of space (and fields) are really constants.
Now we have to decide: is space expanding or matter shrinking?
It happens that we are anthropocentric and we think that the atom is invariant.
Do we prefer to admit that 'space' expands or, like I do, that 'matter shrinks' ?

We have no basis, a priori, to assert one or the other.
A study must be done to study how can the laws of physics hold in a system of units, the comoving one, that are not derived from 'our' atomic standards were laws hold.

A preliminary study was done that shows that laws of physics hold good in that scenario:
A relativistic time variation of matter/space fits both local and cosmic data
No one dared to say a single argument (until now) against that novel theory. In my viewpoint there are no arguments to deny the correctness of the study.

In the very near future a definitive study will be published, revealing a scaling universe, self-similar.

But 'prefered' reference frame is strong, like absolute, and physics dont like this notion. In a surprising way no one realized that the notion of an invariant atom is also an absolute reference.

The Sun and the stars are the beacons (references) of the navigators. Now lets suppose that all the stars from the sky are erased. Are we lost? NO.

Fortunately there are still a reference available, like no other, common to all observers, which is : The Referential of the Light, the CMB. It is not an Einsteinien referential because it it not attached to any observer. With this referential alone we can say: the Earth is moving at 1 rev/day (sideral, local time, i.e. defined with our atoms) and at 369Km/s (+- see the WP) in direction to Leo constellation. Why is this CMB referential so special, so absolute? It is the only one where a released photon propagates isotropically. The CMB can provide a length and time unit common to all observers. Even if we dont like the word absolute, it is a prefered reference frame. I've no doubt.

• Well that is a very exciting answer and an exciting paper. Indeed, the CMB would also seem to also form a reference point to establish the co-moving coordinates. Although I am still confused by the part where atomic sizes are relative, the question of whether space is expanding or matter is shrinking is quite intuitive, and could potentially alleviate some of my issue with matter forming a "preferred" reference frame. Jun 28 '11 at 11:56
• @Zassounotsukushi, if you read it you saw why the velocity (and 'c' speed) is a preserved quantity in a system. And that the comoving coords are AS priviledged as the usual 'atomic units'. And that is good because it paves the way to understand the nature of dark energy (its an artifact of the model) and the nature of time. Jun 28 '11 at 13:12