Why do particles around us have finite momenta?

[TGlad]

Since there is no privileged inertial frame, I would have expected the first particles in the universe to have no particular bias in their momenta. Relative to an observer I would expect the distribution to be uniform and unbounded. The mean momentum of the initial particles relative to an observer would therefore be infinite.

If this were the case, then what is the reason that the mean momentum of particles in any sphere around us appears to be finite?

Obviously gravity holds things together, friction between interacting particles may have acted to reduce relative speeds, as did the expansion of space and the period of acceleration.

But in all cases I would expect the particle momentum to reduce by a finite factor, so the unbounded distribution of momenta should still be unbounded.

So I am confused as to why we are not being bombarded by ultra-high velocity particles all the time.

Thanks for any clarification.

 

[Dale]

Relative to an observer I would expect the distribution to be uniform and unbounded. The mean momentum of the initial particles relative to an observer would therefore be infinite.

Since that is not observed, clearly your expectation is wrong. The distribution is not uniform and unbounded. Why would you expect that? You provided no derivation that would indicate that.

 

[jbriggs444]

I would expect the distribution to be uniform and unbounded.

It is well known that no such distribution exists. In addition, you seem to envision integrating across a distribution of reference frames. That does not seem to be a very sensible thing to do if one is after a meaningful physical quantity.

 

[TGlad]

Maybe it is clearer to use proper velocities, as I didn't make it clear I was talking about particles of the same rest mass.

What would a very simple toy universe with particles in it be like? Any unbiased distribution of particles proper velocities would be unbounded because there is no privileged inertial frame and there is no upper limit on proper velocities.

If I am wrong that the first particles had an unbounded range of proper velocities then what caused these particles to all be comoving (within a finite range of velocities)? How did the big bang produce a biased set of proper velocities?

I'm trying to work out why we are in a comoving universe (within a finite range of velocities) when an unbiased start to the universe should have an unbounded range of them. I think it might be due to expansion.

 

[jbriggs444]

Any unbiased distribution of particles proper velocities would be unbounded because there is no privileged inertial frame and there is no upper limit on proper velocities.

But no physical prediction actually depends on one's choice of reference frame. Pick an actual physical quantity (e.g. the relative velocity of two chosen particles) and you'll find that it is independent of one's choice of reference frame.

 

[Orodruin]

The basic assumption is wrong. While you can use any coordinates you want without problems, there is a cosmologically preferred frame - the rest frame of the CMB. Our observation of the CMB is not uniform, it has a significant dipole component due to our motion relative to the CMB frame.

 

[Dale]

Any unbiased distribution of particles proper velocities would be unbounded because there is no privileged inertial frame and there is no upper limit on proper velocities.

This is not true, but I am not sure what line of reasoning makes you believe this. Can you explain how you get from the premises "unbiased distribution of proper velocities" and "no privileged inertial frame" to the conclusion "the distribution is unbounded" ("no upper limit on proper velocities" follows from the definition of "proper velocities" so I leave it out as a separate premise, but feel free to use it or any other similar concepts as needed).

If @Orodruin 's remark is helpful then perhaps the problem is a misunderstanding of what "no privileged inertial frame" means. It means that the laws of physics are the same in all inertial frames. It does not mean that a given distribution of matter is the same in all inertial frames. The laws of physics will be the same regardless of the motion of any matter, but a given system of matter will have only one unique center of momentum frame.

Suppose that the distribution of proper velocities is normal. That distribution is unbounded, but there is only one frame where the mean is 0. On the other hand, suppose that the distribution of proper velocities is uniform. That distribution is bounded, but again there will be only one frame where the mean is 0.

 

[TGlad]

Yes I agree that 'no privileged inertial frame' does not decide what a particular distribution of proper velocities should be.

But I would assume that the very early universe was unbiased. If its particles all had proper velocities within 1000m/s of mine then that is not an arbitrary starting universe, it is quite specific and you have to ask, why are these velocities in a bounded cluster?

Imagine you had a simple toy universe, perhaps constant expansion, or even just empty Minkowski space-time, and you wanted to seed the universe with particles randomly. If you don't want to bias the initial state of the universe, then (by Lorentz invariance) you should be able to apply any boost and get the same distribution of proper velocities. That's what I mean by unbiased.

Then you get a problem, as @jbriggs444 pointed out, you would need to sample the velocities from the whole of R^3, which would basically give infinitely large proper velocities. And as @jbriggs444 pointed out, it isn't really a valid initial state for massive particles (but it would work for photons), and @Orodruin pointed out, this is not not what we see in CMB anyway.

So my question is, what gave the universe its bias towards similar velocities? If the CMB does not look the same under Lorentz transformations, then where did this extra information come from? Maybe this is about symmetry breaking... did the universe start in a big froth that was statistically symmetric to Lorentz transformations? and if so what caused this 'symmetry breaking' so that it now has a cosmic rest frame?

 

[Dale]

But I would assume that the very early universe was unbiased.

At each location there is one unique reference frame where that is true. This is called the comoving frame. That is not true in any other frame; in all other frames the mean proper momentum is non zero, ie biased.

If you don't want to bias the initial state of the universe, then (by Lorentz invariance) you should be able to apply any boost and get the same distribution of proper velocities.

There is, to my knowledge, no distribution that meets this criteria.

 

[Orodruin]

If you don't want to bias the initial state of the universe, then (by Lorentz invariance) you should be able to apply any boost and get the same distribution of proper velocities. That's what I mean by unbiased.

There is no such thing as a global Lorentz transformation and therefore no such thing as a global Lorentz invariance. You cannot apply special relativity to cosmology. It is not even the case that the relative motion of two particles is well defined. For that you need speed of separation, not actual differences of velocities.

The cosmological principle states that the universe is homogeneous and isotropic taken as a time-slice in the comoving frame. This does not mean that it will be homogeneous and isotropic in all frames.

 

[TGlad]

At each location there is one unique reference frame where that is true. This is called the comoving frame. That is not true in any other frame; in all other frames the mean proper momentum is non zero, ie biased.

There is, to my knowledge, no distribution that meets this criteria.

Good point, even for the massless case, a spherical uniform distribution of light directions becomes non-uniform under a Lorentz transformation.

The cosmological principle states that the universe is homogeneous and isotropic taken as a time-slice in the comoving frame. This does not mean that it will be homogeneous and isotropic in all frames.

So there is a comoving frame for this part of the universe. It sounds like (from reading this) that the comoving frame would be different if I moved 1000 galaxies to the left. In this sense, the comoving frame is not special or a single frame that needs explaining, but varies with location, is this right?

Hmm, thanks to everyone, your help was appreciated. I know my question was poorly asked, but you are helping to lift me out of my ignorance, so thanks.

 

[phinds]

So there is a comoving frame for this part of the universe. It sounds like (from reading this) that the comoving frame would be different if I moved 1000 galaxies to the left.

No, the point of the co-moving frame is that it would NOT be different if you were to move 1000 galaxies to the left or in any other direction. Our galaxy has a particular proper motion relative to the CMB. When that motion is subtracted out, you have the co-moving frame for our galaxy. When you move 1000 galaxies to the left, that galaxy will have a particular proper motion relative to the CMB (and it will almost certainly be different from out motion relative to the CMB). When you subtract that out you will have the co-moving frame for that galaxy. Those two co-moving frames are the same frame. In both of those co-moving frames the age of the universe will be the same.

 

[Dale]

Those two co-moving frames are the same frame.

Not really. In curved spacetime each event has its own set of frames called the tangent space. The tangent space at one event is not the same as the tangent space at another event, although there is a connection between neighboring tangent spaces.

 

[Orodruin]

So there is a comoving frame for this part of the universe. It sounds like (from reading this) that the comoving frame would be different if I moved 1000 galaxies to the left.

You really cannot think of the Universe in terms of the frames of SR. See #13.

 

[kent davidge]

In curved spacetime each event has its own set of frames called the tangent space

Is this also true for accelerated frames in special relativity?

 

[Orodruin]

Is it also true for accelerated frames in special relativity?

It has nothing to do with what coordinates you put on the spacetime. Minkowski spacetime is flat whatever coordinates you put on it and therefore there is a trivial connection that relates different tangent spaces and let's you identify them.

 

[phinds]

Not really. In curved spacetime each event has its own set of frames called the tangent space. The tangent space at one event is not the same as the tangent space at another event, although there is a connection between neighboring tangent spaces.

Thanks Dale. I was not aware of that. My point really was that both of those co-moving frames would see the same age of the universe, so in that sense I thought they could be considered the same frame.

 

[sweet springs]

Since there is no privileged inertial frame, I would have expected the first particles in the universe to have no particular bias in their momenta. Relative to an observer I would expect the distribution to be uniform and unbounded. The mean momentum of the initial particles relative to an observer would therefore be infinite.

GR cosmology says in some assumption momentum of all the universe is zero. (cf. Landau Lifshitz classical theory of fields section 111)

ps1
"some assumptions" : The universe is closed. Referring 4-momentum includes that of matter and gravitational field.
Then 4-momentum flux is closed within the universe.

ps2
Uncertainty relation in QM tells a similar case as yours that a timid observation of particle position makes momentum uncertainty almost infinite. The particle would almost have speed c with statistically isotropic direction.

 

[jbriggs444]

GR cosmology says in some assumption momentum of all the universe is zero. (cf. Landau Lifshitz classical theory of fields section 111)

Uncertainty relation in QM tells a similar case that a timid observation of particle position makes momentum uncertainty almost infinite. The particle would almost have speed c with statistically homogeneous direction.

There are several leaps of faith in this argument, one of which is that if the universe has zero momentum it follows that every particle in the universe has zero momentum. The fact that cars move along highways flies in the face of such a supposition.

 

[sweet springs]

If this were the case, then what is the reason that the mean momentum of particles in any sphere around us appears to be finite?

Let us think about our universe that is filled with cosmic background radiation which is ,for the Earth, almost isotropic. In loca IFR near the Earth moving very fast against Earth, cosmic background radiation is unisotropic and inhomogeneous. I assume numbers of such fast moving IFRs with existing celestial bodies at rest have Maxwell-Boltzman like Gaussian distribution that IFR of isotropic cosmic background radiation lies at center. This may answer your question in probability or statistics.

 

[Orodruin]

GR cosmology says in some assumption momentum of all the universe is zero.

This is incorrect. It is not even clear what such a statement would mean.

 

[Dale]

Is this also true for accelerated frames in special relativity?

Thanks Dale. I was not aware of that. My point really was that both of those co-moving frames would see the same age of the universe, so in that sense I thought they could be considered the same frame.

So there is a connection between neighboring tangent spaces. If you have two distant events then you can draw a path between those events, and you can successively connect each neighbor along the path and you can take a vector from one event and map it to the connected vector in the neighboring tangent space. This is called parallel transport.

The issue is that in a curved manifold the parallel transport from one event to another depends on the path. In flat spacetime the parallel transport does not depend on the path, so you can get away with thinking of all of the events as sharing the same tangent space. That does not change if you are using a non inertial frame in flat spacetime.