I Definition of "Physical Frame": All You Need To Know

[Vincentius]

TL;DR Summary

Definition of “physical frame”

My question is about the precise definition of what is being referred to as “physical frame”, in particular in the context of cosmology. Is it simply the observational frame in which physical units are held constant? Is the FLRW frame physical? A good reference would also be helpful. Thanks for answering!

 

[George Jones]

Summary: Definition of “physical frame”

My question is about the precise definition of what is being referred to as “physical frame”, in particular in the context of cosmology. Is it simply the observational frame in which physical units are held constant? Is the FLRW frame physical? A good reference would also be helpful. Thanks for answering!

Do you have a cosmological reference where the term "physical frame" is used? A common definition is that a frame at an event $p$ of any spacetime is an orthonormal basis for the tangent space at $p$ for which the timelike unit vector is future-directed. A frame field in a (region) of spacetime is a set of four orthonormal vector field. Particular spacetimes, e.g., FLRW spacetimes, often have specially adapted frames.

 

[Vincentius]

I do not have such a reference. But https://arxiv.org/abs/1407.6874 is about equivalence of the Jordan and Einstein frames in FLRW context, and takes the position that since dimensionfull quantities depend on (potentially varying) physical units, only dimensionless quantities are “really physical”, and conformally invariant.

https://arxiv.org/abs/gr-qc/9612053v1 finds that when transforming to conformal time “the conformal equivalence between Jordan and Einstein frame in the cosmological case turns out to be very simple to verify.”

The definition you gave, is that defining a physical frame? Can you explain what is physical about it? Or is it just a matter of postulating constant units, e.g., defining proton rest mass is the same in all events, etc.?

 

[PeterDonis]

I do not have such a reference.

Then this question...

The definition you gave, is that defining a physical frame?

...is unanswerable, because we have nothing that tells us what "physical frame" means. Since the only person we know of who even uses the term "physical frame" is you, since you say you can give no reference, then it's up to you to explain what "physical frame" means and why you think it's important.

 

[Vincentius]

The term “physical frame” is likely best known of the (long) debat on whether the Einstein or Jordan frame is the physical frame. So I gave two references in that context. But so far I have not found a definition of what exactly makes a frame a “physical frame”. Therefore my question.

So far I found no clear reference on the subject in the specific context of cosmology, but I suppose the term has a general meaning in GR.

 

[George Jones]

The term “physical frame” is likely best known of the (long) debat on whether the Einstein or Jordan frame is the physical frame. So I gave two references in that context. But so far I have not found a definition of what exactly makes a frame a “physical frame”. Therefore my question.

So far I found no clear reference on the subject in the specific context of cosmology, but I suppose the term has a general meaning in GR.

But the references you gave were not for GR, they were for alternate theories of gravity.

 

[Vincentius]

Well the Einstein frame is. But indeed, this forum is on GR, so you’re right. Please forget the references, since not important to the question. I still like to know what physical frame means in the context of GR. The statement that the Einstein frame is physical, or not, supposes a definition of what a physical frame is. So how is it defined?

 

[PeterDonis]

The statement that the Einstein frame is physical, or not, supposes a definition of what a physical frame is. So how is it defined?

If you can't find a standard GR reference that even uses the term "physical frame", that should tell you that that term is not a standard GR term and does not have a standard GR definition. So what you are asking for does not exist.

 

[Fra]

Well the Einstein frame is. But indeed, this forum is on GR, so you’re right. Please forget the references, since not important to the question. I still like to know what physical frame means in the context of GR. The statement that the Einstein frame is physical, or not, supposes a definition of what a physical frame is. So how is it defined?

I think it's "definition" is more a conceptual one, and rather reflects how choose to understand/interpret inference within GR and this you may think of GR within a hypothetical QG theory.

I think the physical frame is supposedly the "observational frame" that is justified and natural from the perpective of the physical observer(ie. agent), via some principle.

Identifying and characterinsing the "physical observer frames" and it's possible equivalence classes is exactly what you need to do when seeking an agent-centere theory of inference, but I see this as an open extremely complex question and it seems impossible to separate this question without unifying the forces, as it decides how spacetime relations separate from the other fields and it can't be addressed withing regular GR I think.

/Fredrik

 

[Vincentius]

I think the physical frame is supposedly the "observational frame" that is justified and natural from the perpective of the physical observer(ie. agent), via some principle.

Yes, that’s what I gathered, the physical frame simply being the (observational) frame where observables are measured using rods and clocks. This seems evident. Though, a complication arises in remote observation in standard cosmology. Given the cosmological principle, in constant time slices, the units of the comoving observer do match the units of a hypothetical comoving observer at arbitrary distance. But remote observation is on the past light cone, not in the present time slice, and only the perfect cosmological principle guarantees that physical units are constant over intervals of time. So in standard cosmology it is not evident that physical units do not evolve with the scale factor. This is where the meaning of a “physical frame” (i.e., defined by present units) becomes troubled, even while there seems no fundamental objection to hold units constant: one is free to define units, so that the “physical frame” is clearly defined nonetheless. But evolution of physical units - while considered constant - could make remote observation physically incomprehensible.

 

[PeterDonis]

only the perfect cosmological principle guarantees that physical units are constant over intervals of time

Where are you getting this from? We can test for variation in the physical constants, such as the fine structure constant, that form the basis for our system of physical units. All such tests that we have done so far have shown no such variation.

 

[Fra]

Yes, that’s what I gathered, the physical frame simply being the (observational) frame where observables are measured using rods and clocks. This seems evident. Though, a complication arises in remote observation in standard cosmology.

If you think a lot about this, an especially if you add in how "measurements" are done (event detection, post-processsing, storage etc), there are plenty of complications indeed. So the question is: how much does it make sense to think of this, confined to a context (regular GR and in this subforum) that seems unsuitable to handle these issues?

My personal motivation in elaborating the classical GR, and classical cosmological models are grossly numbed by the frustration from thinking about the things that by definition leads to thinking about reconstructions and fix the missing parts.

At the heart of the issue I see this:

I think of a physical observer is not just just a coordinate frame of reference (ie diffemorhpism generated), a real observer probably has limited capacity to detect, process and encode/store events. A real agent can gain and loose information, and a real agent can itself evolve and change, which requires deformation of any "maps" that is has about the rest of the world. This includes any "clocks" or "rods" which would have to be "subsystems" of the agent that are used as internal references to parameterise of relative changes. So the GR notion of "physical frame" is IMO unsatisfactory in the first place, and so is the QM notion. But both have pros and cons, and we need to keep the good stuff from both and get rid of what doesn't belong there.

/Fredrik

 

[robphy]

https://scholar.google.com/scholar?hl=en&as_sdt=0,24&q="physical+frame"+cosmolog*&btnG=
https://scholar.google.com/scholar?hl=en&as_sdt=0,24&q="physical+frame"+relativity&btnG=

At this point, since it seems that "physical frame" is likely a non-standard term,
it's probably better to focus on
understanding the associated ideas and their context in the references you are reading.

Side comment:
recently I stumbled on an answer to
https://matheducators.stackexchange.com/questions/25348/tangent-line-to-a-curve
which made this comment about "definitions"

definitions in mathematics are carefully crafted to balance several competing aesthetic and practical needs.
...
A definition is a work of art, not a commandment set down by our mathematical grandparents.

Thus, I think it is more useful to understand what
your references or whatever you are reading
are trying to say
rather than what they call it.

 

[Vincentius]

Where are you getting this from? We can test for variation in the physical constants, such as the fine structure constant, that form the basis for our system of physical units. All such tests that we have done so far have shown no such variation.

Full spacetime symmetry by the perfect cosmological principle implies a universe in stationary state, where all global features, including physical units, are intrinsically constant. The standard cosmological principle does not provide time translation symmetry, so does not guarantee general conservation of (dimensionfull) physical quantities, like, e.g., proton mass. Is there any observational evidence of conservation of mass in cosmology? Of course, c and other ratios can still be constant.

 

[phinds]

Is there any observational evidence of conservation of mass in cosmology?

That seems like an odd question since it's been known for a very long time that mass can be converted to radiation.

 

[Vincentius]

Meaning conservation of mass/energy

 

[PeterDonis]

Full spacetime symmetry by the perfect cosmological principle implies a universe in stationary state

Even if we accept that this is true for the sake of argument, it does not mean that the perfect cosmological principle is the only possible way that, for example, the fine structure constant could be constant, not changing with time.

The standard cosmological principle does not provide time translation symmetry

Indeed not, since our actual observations of our actual universe on cosmological scales do not show such symmetry. So of course our standard model of the universe does not have it either.

so does not guarantee general conservation of (dimensionfull) physical quantities

Dimensionful quantities are the wrong ones to focus on, since they depend on your choice of units. The quantities to focus on are the dimensionless ones, like the fine structure constant. Our standard cosmological model does not "guarantee" that such quantities are constant, but we don't need it to; we can test for that directly by observations, as I said before; and so far we have not found any variation in such quantities.

Is there any observational evidence of conservation of mass in cosmology?

No, and we would not expect any, because mass is not a conserved quantity.

There is abundant evidence for local conservation of stress-energy, i.e., that the covariant divergence of the stress-energy tensor is zero, as GR predicts it should be.

 

[phinds]

Meaning conservation of mass/energy

Mass/energy is not conserved on cosmological scales. See
http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

 

[vanhees71]

Meaning conservation of mass/energy

In relativity nowadays mass is defined as the invariant mass and only the invariant mass, and it is not a conserved quantity in general. What's conserved (in special relativity) is the total energy of a closed system due to the application of Noether's theorem to temporal translation invariance. Also the total momentum of a closed system is conserved due to spatial translation invariance. The total energy together with momentum are the components of a four-vector, $(p^{\mu})=(E/c,\vec{p})$. Then the conservation of the mass follows from energy-momentum conservation, because of course also $m^2=p_{\mu} p^{\mu}/c^2$ is conserved.

That's one more subtle distinction between special-relativistic and Newtonian physics: In non-relativistic quantum theory the mass is a socalled "central charge" of the Galilei Lie algebra. It thus must be conserved, and there cannot be superpositions of states of particles with different mass. The Poincare group has no non-trivial central charges and thus there's no extra mass-conservation law. All that's conserved are energy and momentum and thus also the invariant mass $m=\sqrt{E^2/c^2-\vec{p}^2}/c$ of a closed system is of course conserved, but it's not an independent conservation law but follows from energy and momentum conservation.

 

[Vincentius]

About whether there is observational evidence of conservation of mass:

No, and we would not expect any, because mass is not a conserved quantity.

Do you mean (particle) mass is not conserved in expanding space?

 

[PeterDonis]

Do you mean (particle) mass is not conserved in expanding space?

If by "particle mass" you mean, for example, the mass of the electron, that is not a "conserved quantity" in the sense you are using the term. All of the Standard Model particles that have nonzero rest mass get that rest mass from some variant of the Higgs mechanism, meaning that before electroweak symmetry breaking, they were all massless, and after electroweak symmetry breaking, their masses are constant because the Higgs mechanism that produces them doesn't change. The expansion of the universe does not affect any of this.

As I have already said, the relevant quantity in GR that corresponds to your intuitive notion of "conservation of mass" (and "conservation of energy" and "conservation of momentum") is local conservation of stress-energy--that the covariant divergence of the stress-energy tensor is zero. This has nothing whatever to do with "particle mass". Also, it is only local; there is no global conserved "energy" in a general curved spacetime (although in special classes of spacetimes there can be).

 

[Vincentius]

If by "particle mass" you mean, for example, the mass of the electron, that is not a "conserved quantity" in the sense you are using the term. All of the Standard Model particles that have nonzero rest mass get that rest mass from some variant of the Higgs mechanism, meaning that before electroweak symmetry breaking, they were all massless, and after electroweak symmetry breaking, their masses are constant because the Higgs mechanism that produces them doesn't change. The expansion of the universe does not affect any of this.

This is useful information, thanks!

To be sure: after electroweak symmetry breaking, particle mass is (or is considered?) constant in the evolving universe. Correct?

It seems it has to be like that, for baryon density is (considered?) proportional to a^-3.

My understanding is that particle conversion itself preserves energy. Correct?

 

[PeterDonis]

after electroweak symmetry breaking, particle mass is (or is considered?) constant in the evolving universe. Correct?

See the last clause of the second last sentence in what you quoted.

My understanding is that particle conversion itself preserves energy. Correct?

In the sense of local conservation of stress-energy, yes.

 

[vanhees71]

There's of course one qualification often forgotten. The most of the mass of the matter surrounding us is not due to the Higgs mechanism but is dynamically generated by the strong force and is mostly due to the anomalous breaking of the dilation symmetry ("trace anomaly") and somewhat due to the spontaneous breaking of the approximate chiral symmetry ("$\sigma$ term"). Of course, also here the expansion of the universe has (almost) no influence on the very strong binding of these bound states of quarks and gluons, and that's why the mass of the hadrons are the same everywhere in the universe.