# A preferred frame?

1. Sep 29, 2004

### Garth

The Principle of Relativity, (PR) may be summarised as the doctrine of no preferred frames of reference. It not only underlies SR in the absence of gravitational fields but GR with its gravitational fields as well.

It leads to the conservation of four-momentum and the concept of coordinate independent geometrodynamic objects.

As four-momentum, or energy-momentum, is conserved one frame dependent component of it, energy, is not in general conserved.

However it might be argued that once matter and its associated gravitational fields are introduced in GR, then a preferred frame of reference might be discerned. It is that co-moving with the Centre of Mass or centroid of that matter. Cosmologically it might be identified with that frame in which the CMB is globally isotropic co-moving with the surface of last emission.

But should such a frame of reference be preferred, except as a convenient frame in which to establish a coordinate system when solving the field equation?

I would humbly suggest that in the inertial Centre of Mass frame as energy is conserved, at least for a static system, then that frame should be preferred .

In other words whether the Centroid frame is considered preferred or not depends on whether the conservation of energy is considered important or not.

Just food for thought (and debate!)

Garth

Last edited: Sep 29, 2004
2. Sep 29, 2004

### blue_sky

Interesting. How you can mathematically define that special ref. of frame?

3. Sep 29, 2004

### jcsd

It's interesting subject certainly, but one thing your summary of the PR is little to vague, aftre all what do we mean by preferred? We can always accept the possibilty of extreme anistropy in a given region of space, for example if we take say the Earth and moon system and the space around it, we see a fairly extreme anistropy in the distribution of matter, but no-one is or should be horrified by this. The PR is more explicit as it's really about the invaraince of the laws of physics

Are talking SR here? Cos we shouldn't be to bothered about the non0-conservation of enrgy when it occurs in SR as it is always conserved in any given inertial frame and even in Newtonian physics energy is not generally conserved for non-inertial observers.

Well we should always expect such a frame to exist, shouldn't we, just as we would equally expect to find an extremely anisotropic frame (lets ignore any issues about homegenity as that really isn't frame dependent, so we might as well assume total homogentity). When Friedmann et al started to formjulate big bang theory, they assumed total isotropy and were therfore assuming a certain frame of refrence, so it's no suprise that these equations are 'biased' towards a single frame.

Well it probably should be prefered, but not in the snese of the PR, but for convience's sake. It's really no trouble ofor GR and the PR that we have this convient frmae, but it is ceratinly interesting that this frame exists.

4. Sep 29, 2004

### Garth

If you link to my eprint 'The derivation of the coupling constant in the new Self Creation Cosmology' http://arXiv:gr-qc/0302088 page 24 equation 120 you will see the mathematical definition of a centroid; the co-moving centroid associated with the rest frame of the system is defined to be its Centre of Mass (CoM). It is this frame that I consider to be "preferred".
Garth

5. Sep 29, 2004

### Garth

I am talking about inertial frames in GR.
Garth

6. Sep 29, 2004

### jcsd

My post wasn't about SR only the sentence after the one you quoted (cos your next sentence talks about bringing in GR).

Okay in refernce to the conservation of energ,y obviosuly it's slightly worrying because of the equality of all frames, but on the tother hand energy has never been conserevd in all frames in any theory anyway.

7. Sep 29, 2004

### Chronos

I tend to agree with jcsd. It is convenient, but, I'm not sure it is reliable in a mathematical model [where theory must ultimately stand]. In my mind, using the CMB as a preferred reference frame is the same as making it a universal constant. It would, however, be a constant that changes over time, hence not truly fundamental. The mathematics would be complicated and carry many potential complications. That does not make it wrong, just more complicated. That tends to rub most scientists the wrong way because the models we already have are complicated enough as is.

8. Sep 29, 2004

### pervect

Staff Emeritus
Actually, I'd like to see a little more background on what causes the conservation of four momentum. Saying that the "principle of relativity" leads to it seems a little vague. Perhaps you could supply some references and/or commentary.

Since it the length of the 4-momentum vector is a conserved quantity, it should be associated with a symmetry by Noether's theorem. It's not clear to me exactly what symmetry that is. Is it as simple as the invariance of physical law with the Lorentz group? I'm not positive.

: It's still not entirely clear to me what the symmetry is, but it's not the Lorentz group, that has too many parameters. Energy is time-translation, momentum is space translation, but what is the symmetry associated with the invariant mass of the energy-momentum 4-vector?
[end edit]

It certainly isn't time-translation symmetry, that's what gives us (more or less) energy conservation of energy in GR. You don't have time translation symmetry in general, but you can get it asymptotically, so asymptotically flat space-times conserve energy.

Last edited: Sep 29, 2004
9. Sep 30, 2004

### Garth

Thank you for your considered replies; my responses are as follows:

First, rather than be enthralled by GR, here I am questioning its basis principally Einstein’s Principle of Equivalence (EEP) so I do expect a lot of flak!

Jcsd – “but on the other hand energy has never been conserved in all frames in any theory anyway.”

– But in classical physics energy has been in conserved in inertial frames – it was the fundamental principle right up to the moment Einstein ditched it – and then he worried about the loss!
In GR, because of space-time curvature, energy is not conserved except in the centre of mass frame when the gravitational field is static and then only in a limited and rather forced sense. [the covariant time component of energy-momentum is conserved – but often it is the contravariant component that is defined as energy – these two are only equal in Minkowski space-time]

Chronos – “In my mind, using the CMB as a preferred reference frame is the same as making it a universal constant. It would, however, be a constant that changes over time, hence not truly fundamental.”

1. - In order to choose the CMB isotropic frame as preferred some other principle is required, actually one such principle already exists and was one of Einstein’s guiding principles – it is Mach’s Principle. In SR there is nothing ‘to hang a frame of reference on’ yet once matter is introduced with its corresponding gravitational field there is – the centre of mass or rather momentum of that matter.

2. - As we are talking about the motion of that surface of last emission, the motion of the isotropic CMB frame: How do you expect it to change? How would you detect any such change and with what standard inertial frame would you compare it?”

pervect – “I'd like to see a little more background on what causes the conservation of four momentum”

Consider the Principle of Least Action. The key characteristic of this method/principle is the fact that it works for generalised coordinates. Therefore it is ideal when there is no preferred frame, as it works for all frames. From Action in 3D we obtain the conservation of energy and the conservation of momentum separately, in Minkowski 4D space-time we obtain the conservation of energy-momentum.
This is because of the signature of the metric. The relativistic mass and the momentum are frame dependent, however in taking the norm of the energy-momentum vector the momentum (squared) is subtracted from the relativistic mass (squared) leaving just the rest mass (squared) in all frames of reference. Hence this can be called simply mass or norm of the energy-momentum vector defined by the 4-momentum equation and is invariant under coordinate transformations. The problem is that neither energy nor momentum on their own need be conserved, only the root difference between their squares – the norm of the 4-vector. In GR if I lift a body the energy expended disappears from the accounts, it is this that I question.

2 “Since it the length of the 4-momentum vector is a conserved quantity, it should be associated with a symmetry by Noether's theorem.”

You make this statement and then wonder about which symmetry it is that mass is associated with. The problem is the first postulate. The length of the 4-momentum is not a conserved quantity in GR over space-time curvature, because the vector cannot be parallel transported unless a killing vector exists and generally it does not. This was Weyl’s point that led to conformal gravitational theory. The norm of the 4-momentum is defined to be invariant in GR. It is a measurement problem; how do you know that units of mass, length and time at the far end of the universe are the same as in your laboratory? You do not but have to define them to be so. However, for example as an alternative, you might define the energy of a photon to be invariant instead of the mass of an atom. In such a conformal theory (SCC) cosmological and gravitational red shift would be interpreted as the apparatus gaining energy rather than the photon losing it.

Just food for thought!
Garth

Last edited: Sep 30, 2004
10. Sep 30, 2004

### blue_sky

It looks the link doent work, can you check pls?

blue

11. Sep 30, 2004

### Garth

12. Oct 1, 2004

### pervect

Staff Emeritus
I still have to disagree with this - in my books, the energy does not dissappear, it goes into the gravitational field.

The sort-of odd thing about GR is that you can't localize the energy to any specific region of space-time. As long as you have asymptotic flatness, though, you can calculate the total energy in the gravitational field, even if you can't assign it a definite location.

As I recall, you can parallel transport a vector as long as you have a derivative operator, or a connection - (either one). You don't need any sort of symmetry (Killing vector) to do parallel transport.

The derivative operator, though, is designed to preserve the length. So I suppose it's not terribly suprising that the length is indeed preserved, since that's how the deriviatve operator was picked.

13. Oct 1, 2004

### Garth

The energy does disappear from the account though, the explanation is that it "goes into the gravitational field", and will pop right out again when the process is reversed. The total energy of the whole system, defined at a null inifinity with asyptotic flatness remains the same (conserved) all along, however it is the local conservation of energy that I worry about.

Garth