Does Hoyle C field violate conservation of energy-momentum?

[bcrowell]

The Einstein field equation is inconsistent unless we demand a divergence-free stress-energy tensor. This makes me think that Hoyle's steady-state cosmology is inconsistent with general relativity.

But Hawking and Ellis has this at p. 90:

[The weak energy condition] will not hold for the 'C'-field proposed by Hoyle and Narlikar (1963). This again is a scalar field with m zero, only this time the energy-momentum tensor has the opposite sign and so the energy density is negative. This allows the simultaneous creation of quanta of positive energy fields and of the negative energy 'C'-field.

I had always imagined that the C field was just some vague hand-waving by Hoyle. Is it really a field theory?

Is everything perfectly OK in classical relativity if we allow such a field?

Quantum-mechanically, is there really a reasonable field theory with such a field? I'm imagining a Dirac sea that isn't full. Not sure what state you would refer to as the vacuum. Are we talking about doing quantum mechanics with a spectrum of energy states that isn't bounded below? Don't Bad Things happen then?

If we're creating negative-energy C-field quanta in the steady-state theory, where do they all end up? Can we detect them? Don't their gravitational fields cancel out the gravitational fields of the hydrogen atoms being created?

 

[martinbn]

I know almost nothing about it but my impression, from comments in books, is that the only problem within classical relativity is the violation of energy conditions.

http://arxiv.org/abs/astro-ph/0205064 (and some of the other papers by Narlikar)

On page 4 they have the stress energy tensor of the creation field

 

[bcrowell]

I know almost nothing about it but my impression, from comments in books, is that the only problem within classical relativity is the violation of energy conditions.

http://arxiv.org/abs/astro-ph/0205064 (and some of the other papers by Narlikar)

On page 4 they have the stress energy tensor of the creation field.

Thanks. Unfortunately, Narlikar is a kook, and that makes it more work to go through one of his papers and try to figure out what's not nonsense. Ned Wright has a nice discussion: http://www.astro.ucla.edu/~wright/stdystat.htm

 

[PeterDonis]

Quantum-mechanically, is there really a reasonable field theory with such a field?

The paper martinbn linked to proposes a scalar field with negative energy density. Mathematically, it works, and the covariant divergence of the resulting stress-energy tensor vanishes, so it does not violate conservation laws. Whether it's "reasonable" is a different question.

 

[martinbn]

There is a little about it in Weinberg's book.

 

[bcrowell]

There is a little about it in Weinberg's book.

Do you think you could find a specific reference? I have vol I of his field theory book, but I couldn't find anything in the index.

There's a discussion in Coles and Lucchin, Cosmology, 2002, p. 57. They describe the C-field in classical terms as a modification to the Einstein field equations:

$G_{ij}+C_{ij} = 8\pi T_{ij}$

They don't mention anything about negative-energy quanta. I suppose the idea is that both $C_{ij}$ and $T_{ij}$ have nonvanishing divergences. Presumaly there is no physical significance attached to the fact that they write the C piece and the T piece separately; Hawking and Ellis consider it as just another matter field.

Coles and Lucchin say, "Hoyle suggested that $C_{ij}$ should be given by $C_{ij}=C_{;i;j}$[...]" The right-hand side is a second derivative of a scalar field C, which they say is defined as $C=-8\pi/H_0(\rho_0+p_0)t$, where $\rho_0 = 3H_0^2/8\pi$.

They don't define $p_0$. Two possibilities would be that it's numerically equal to $\rho_0$ or that it equals the actual average pressure in our universe. Since the two parameters are simply added, it seems that expressing them this way is merely for notational convenience; there is really only one adjustable parameter involved.

Apparently t is just a time coordinate. (This seems to check out in terms of dimensional analysis.) If so, then the equation assumes some preferred coordinate system, and I guess it would have to be the one defined by the Killing vector. This seems like it would clearly violate Lorentz invariance, since the local laws of physics would allow us to use local experiments in order to determine our motion in relation to the Hubble flow, without actually having to observe the Hubble flow. (I suppose the scalar field C itself is not meant to be directly observable, so its linear dependence on time wouldn't be observable.)

 

[PAllen]

I think he means Weinberg's Gravitation and Cosmology book. There does appear to be a section on this in that book.

[Chapter 14, section 8, per contents in 'look in book' on Amazon]

 

[bcrowell]

Now that I think about it, any steady-state model of this flavor is bound to violate Lorentz invariance. For example, we could sit in a laboratory, observe a box full of vacuum, and wait for hydrogen atoms to appear. These atoms are in some state of motion, which must on the average be the motion of the Hubble flow. This makes the Hubble-flow frame a preferred frame.

 

[PAllen]

But violating energy conditions rather generally allows Lorentz violations - they codify what it means for energy and momentum to be locally consistent with SR. For example, a blob of negative energy may have a spacelike world line. This fact is mentioned, for example, the the Wald-Gralla derivation of geodesic motion for test bodies from the field equations - it is necessary to either:

- assume timelike motion as an axiom (then geodesic motion can be derived)
- assume the dominant energy condition in order to derive timelike motion from the field equations

The older Geroch-Ehlers derivation has the same logical requirement: dominant energy condition is necessary and sufficient for timelike geodesic motion of test bodies.

 

[bcrowell]

But violating energy conditions rather generally allows Lorentz violations - they codify what it means for energy and momentum to be locally consistent with SR. For example, a blob of negative energy may have a spacelike world line.

I'm not following you here. You seem to be arguing that a negative-energy particle would be a tachyon, and that tachyons imply Lorentz violation. Tachyons would have imaginary mass, but are normally assumed to have real, positive energy. (For example, people speculated for a long time that neutrinos could be tachyonic, but it was known that when they deposited energy in a detector, the energy was positive.) And in any case tachyons do not imply Lorentz violation.

I also don't think it can be correct to say that energy conditions "codify what it means for energy and momentum to be locally consistent with SR." Dark energy violates various energy conditions, but it's locally consistent with SR. If violating energy conditions meant violating SR, then we could say that all the energy conditions must automatically be true. That's not the case. In fact, there are no energy conditions that are currently believed to be true in all cases: http://arxiv.org/pdf/gr-qc/0205066

It's been a while since I looked at the Ehlers-Geroch derivation, but IIRC the reason they need to assume energy conditions is that you can get effects such as an interaction between the spin of a particle and the curvature of the spacetime. In order to prove that that interaction has a negligible effect in the limit of small size, you need an energy condition.

 

[PAllen]

A negative energy particle could move ftl, but its description is not the same as a tachyon. Anyway, I see that what I meant was violating SR rather than Lorentz invariance per se. What constitutes SR violations is more a matter of opinion, but I take it to include matter of any typefollows a timelike trajectory, and that causality is observed - no influence outside the light cone.

[aside: your argument doesn't necessarily show Lorentz violation either since a frame being preferred by be able to detect motion relative to a field is not a Lorentz violation. The created atoms have a momentum determined by the field that gives rise to them. No different than the CMB picking out frames for which it is isotropic.]

 

[PAllen]

I'm not following you here. You seem to be arguing that a negative-energy particle would be a tachyon, and that tachyons imply Lorentz violation. Tachyons would have imaginary mass, but are normally assumed to have real, positive energy. (For example, people speculated for a long time that neutrinos could be tachyonic, but it was known that when they deposited energy in a detector, the energy was positive.) And in any case tachyons do not imply Lorentz violation.

I also don't think it can be correct to say that energy conditions "codify what it means for energy and momentum to be locally consistent with SR." Dark energy violates various energy conditions, but it's locally consistent with SR. If violating energy conditions meant violating SR, then we could say that all the energy conditions must automatically be true. That's not the case. In fact, there are no energy conditions that are currently believed to be true in all cases: http://arxiv.org/pdf/gr-qc/0205066

It's been a while since I looked at the Ehlers-Geroch derivation, but IIRC the reason they need to assume energy conditions is that you can get effects such as an interaction between the spin of a particle and the curvature of the spacetime. In order to prove that that interaction has a negligible effect in the limit of small size, you need an energy condition.

No, there are papers on the necessity in Ehlers-Geroch that show spacelike motion is possible in the limit without the assumption the dominant energy condition.I am not aware of specific counter-example papers for Wald-Gralla, but their paper states the condition is necessary to ensure timelike motion.

The dominant energy condition amounts to saying that mass-energy cannot locally flow FTL. It is thus not surprising that assuming the converse allows ftl motion.

 

[bcrowell]

No, there are papers on the necessity in Ehlers-Geroch that show spacelike motion is possible in the limit without the assumption the dominant energy condition.I am not aware of specific counter-example papers for Wald-Gralla, but their paper states the condition is necessary to ensure timelike motion.

I don't dispute this. Certainly if you have matter that violates an energy condition, it may have spacelike world-lines.

aside: your argument doesn't necessarily show Lorentz violation either since a frame being preferred by be able to detect motion relative to a field is not a Lorentz violation. The created atoms have a momentum determined by the field that gives rise to them. No different than the CMB picking out frames for which it is isotropic.

I would agree if the field was one that followed Lorentz-invariant dynamics. But their massless, scalar C field doesn't follow the Lorentz-invariant dynamics you expect (massless Klein-Gordon equation). Instead, it simply evolves according to a prescribed equation, and that equation is coordinate-dependent. Their C field literally *is* the time coordinate (with a minus sign in front).

The CMB is different. It evolves according to Lorentz-invariant dynamics (Maxell's equations).

I wonder if their $C_{ij}$ has some simple interpretation, such as being the expansion tensor or something.

 

[PAllen]

I

I would agree if the field was one that followed Lorentz-invariant dynamics. But their massless, scalar C field doesn't follow the Lorentz-invariant dynamics you expect (massless Klein-Gordon equation). Instead, it simply evolves according to a prescribed equation, and that equation is coordinate-dependent. Their C field literally *is* the time coordinate (with a minus sign in front).

The CMB is different. It evolves according to Lorentz-invariant dynamics (Maxell's equations).

This I have no opinion on, since I've never read a formal treatment of any steady state model. There also appear to be many of them, and Narlikar seems to change the model as soon as someone points out a flaw he can't avoid in the prior model. I found Weinberg actually has two sections on steady state models - one on Bondi-Gold, one on Hoyle's model. From 'look in book' feature, the section I mentioned above was on the Bond-Gold model.

 

[bcrowell]

This I have no opinion on, since I've never read a formal treatment of any steady state model.

What I gave in #6 is the complete classical description according to Coles and Lucchin. There's really no more to it than that. You write down a field C which is basically just a time coordinate. You take its second covariant derivative and call that a stress-energy.

 

[PAllen]

I can still see a way to argue no Lorentz violation. The C field is given in some coordinates, but if you assume you get its description (locally) in other frames by Lorentz transform, then it is (arguably) Lorentz invariant.

 

[bcrowell]

I can still see a way to argue no Lorentz violation. The C field is given in some coordinates, but if you assume you get its description (locally) in other frames by Lorentz transform, then it is (arguably) Lorentz invariant.

Sorry, not buying it :-) That's much weaker than what Lorentz invariance really says.

 

[PAllen]

Doesn't seem so bad. Consider an isolated charge. It has a uniquely simple field description (static, Coulomb) in one frame, and you can (without specifying any specific law) derive what it is in other frame's by Lorentz transform. So, the C field is specified in standard cosmological coordinates, where it takes its simplest form. As a physical field, there is, nothing, per se, wrong with being able to detect (directly or indirectly) you motion 'relative to it' [ more precisely, relative to an observer who sees its simplest form - e.g. created atoms have no momentum, for example]. Just like for an isolated charge, even an observer who only sees the field can distinguish whether they are in motion relative to the charge.

 

[bcrowell]

Doesn't seem so bad. Consider an isolated charge. It has a uniquely simple field description (static, Coulomb) in one frame, and you can (without specifying any specific law) derive what it is in other frame's by Lorentz transform. So, the C field is specified in standard cosmological coordinates, where it takes its simplest form. As a physical field, there is, nothing, per se, wrong with being able to detect (directly or indirectly) you motion 'relative to it' [ more precisely, relative to an observer who sees its simplest form - e.g. created atoms have no momentum, for example]. Just like for an isolated charge, even an observer who only sees the field can distinguish whether they are in motion relative to the charge.

Your example is Lorentz-invariant because Maxwell's equations are Lorentz-invariant. The charge is a physical body, it has dynamics, and you're measuring your state of motion relative to it. That's completely different from simply prescribing the electric field to have some fixed behavior in some special coordinates.

You can make tensor-scalar theories that are Lorentz-invariant. Brans-Dicke gravity is an example. The C field isn't, at least according to the description given by Coles.

 

[PeterDonis]

the equation assumes some preferred coordinate system

That's not necessarily the same as violating Lorentz invariance. See below.

This seems like it would clearly violate Lorentz invariance

It depends on how the $C$ field, or the $t$ quantity, transforms under local Lorentz transformations. If it transforms in such a way as to leave the laws of physics invariant, then it's Lorentz invariant. In other words, $t$ would have to be a scalar field on spacetime with the dimensions of time. This is possible AFAIK, but I don't know enough about the Coles model to know if that's what they are specifying.

If $t$ is in fact a scalar field, then the fact that it can also function as a time coordinate in a "preferred" coordinate system does not break Lorentz invariance. It just means that there is a particular coordinate chart that is adapted to the underlying symmetry of the spacetime.

 

[bcrowell]

It depends on how the $C$ field, or the $t$ quantity, transforms under local Lorentz transformations. If it transforms in such a way as to leave the laws of physics invariant, then it's Lorentz invariant. In other words, $t$ would have to be a scalar field on spacetime with the dimensions of time.

I don't think that's what Coles is describing. I think he's simply defining t as the FLRW t coordinate (notation defined earlier in the book). If that wasn't what he meant, I think he'd say so.

If t were a scalar field, then it would have to have some Lorentz-invariant dynamics. No such dynamics are described. Maybe you could make a Lorentz-invariant tensor-scalar theory with C as the scalar field, but that isn't what Coles is talking about.

 

[PeterDonis]

I think he's simply defining t as the FLRW t coordinate

Technically, this could be viewed as a scalar field, since it qualifies as a global time function (it assigns a value to every point in spacetime, and its gradient is everywhere timelike and future directed), and any such function can be viewed as a scalar field. However...

If t were a scalar field, then it would have to have some Lorentz-invariant dynamics. No such dynamics are described.

...this seems to me to be a much more telling objection.

 

[bcrowell]

It may be that Coles is describing an earlier, simpler version of the theory, which was later elaborated on by Hoyle and Narlikar. This paper may be relevant: Hoyle and Narlikar, "The C-field as a direct particle field," Proceedings of the Royal Society of London, 282 (1964) 178, http://www.jstor.org/stable/2414803?origin=JSTOR-pdf

 

[martinbn]

Do you think you could find a specific reference? I have vol I of his field theory book, but I couldn't find anything in the index.

I meant the Gravitation and Cosmology book, chapter 16 section 3 has a couple of pages on the steady state. It seems he doesn't say more than what you've already found. The C-tensor is on the side of the Einstain tensor, it is the second derivative of a scalor and it is a constant times the t coordinate of the FRWL coordinates. The stress energy tensor is the same as in FRWL. The #\rho_0#, #p_0# and #H_0$ I believe are equal to the present values of these in the standard model. I am not sure, he is very sketchy.

 

[bcrowell]

Helge Kragh, Cosmology and Controversy, has this:

In the new [i.e., Hoyle-Narlikar as opposed to Bondi-Gold] formulation there was strict conservation of energy (and momentum), and this would, Hoyle and Narlikar believed, put an end to the criticism of steady-state theory as being based on an unexplained origin of matter.

It also seems that Hoyle and Narlikar were very proud of the fact that the existence of a C field would prevent the formation of singularities. This claim may be controversial ( http://www.nature.com/nature/journal/v204/n4961/pdf/204868a0.pdf ). Since the field has negative energy, it exhibits gravitational repulsion, and the hypotheses of the Hawking-Penrose singularity theorems don't apply.

There is a paper, Deser and Pirani, "Critique of a New Theory of Gravitation," that seems to have been an influential critique of the C field, but unfortunately I can only access the abstract. They claim that the theory is falsified by solar system tests.

If I'm understanding correctly, "The C-field as a direct particle field" refers to what's known as a "direct interaction," meaning action at a distance rather than an interaction mediated by a field. There are supposed to be no-go theorems like the Currie-Jordan-Sudarshan theorem that make such theories non-viable, so I don't know how they evaded that.

 

[bcrowell]

The original 1948 papers on the steady state are available online:

Bondi and Gold, "The Steady-State Theory of the Expanding Universe," MNRAS 108 (1948) 252 http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1948MNRAS.108..252B

Hoyle, "A New Model for the Expanding Universe," MNRAS 108 (1948) 372. http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1948MNRAS.108..372H

Both papers explicitly discuss the need for a preferred frame, which in Hoyle's notation is defined by a velocity vector $C_\mu$. This is the covariant gradient of the scalar C-field.

Bondi and Gold want to replace GR with a tensor-vector theory in which $C_\mu$ is dynamical and plays a fundamental role along with the metric. They say they're going to produce such a theory in a later paper. They explicitly say that their theory is not consistent with GR, in part because "... the strict conservation of mass implied by the [Einstein] field equations conflicts with the very basis of our theory."

Hoyle wants $C_\mu$ to be a fixed geometrical object in the background, and he defines it by fixing some event O in the distant past. At any later event P, you construct a geodesic from O to P, and $C_\mu$ is a velocity vector tangent to that geodesic. His goal is to modify GR as little as possible. At the end of the Bondi-Gold paper, they say that Hoyle has told them in a private communication that in his theory, the equivalence principle is violated, so that the newly created atoms are acted on by a force that makes them settle down, while the effect on an object like a star is negligible.

Re quantization, when we talk about a QFT, we mean something that combines relativity with quantum mechanics. Since the C-field violates Lorentz invariance, it's not really relativistic, so as a matter of definition it doesn't make sense to talk about making it into a QFT. You could worry about things like whether the C-field violates CPT, but that seems beside the point; the CPT theorem is based on Lorentz invariance, and it isn't really applicable if you're not trying to make a Lorentz-invariant theory.

So for these reasons it makes sense that they came up with a direct-interaction theory. Apparently the history is that in the 1960s a lot of physicists weren't sold on QFT, maybe because they felt uncomfortable with renormalization or didn't believe the weak and strong forces could be handled by QFT. This motivated them to consider theories involving direct particle interactions. I'm getting this history from this talk: http://streamer.perimeterinstitute.ca/Flash/1a7787fa-5478-49ca-82c2-4b7a342117c8/index.html

 

[bcrowell]

I think I understand now why there's no classical field theory for the C field. For a massless scalar field, we expect the wave equation to be $\nabla_a \nabla^a C=0$. But for the C field this is precisely the volume expansion scalar. Therefore if we had a classical field theory of the C field, the universe could not be expanding.

 

[ShayanJ]

So for these reasons it makes sense that they came up with a direct-interaction theory. Apparently the history is that in the 1960s a lot of physicists weren't sold on QFT, maybe because they felt uncomfortable with renormalization or didn't believe the weak and strong forces could be handled by QFT. This motivated them to consider theories involving direct particle interactions. I'm getting this history from this talk: http://streamer.perimeterinstitute.ca/Flash/1a7787fa-5478-49ca-82c2-4b7a342117c8/index.html

Direct interaction theories go back to Schwarzschild(1903), when he described electrodynamics with an action containing no field degree of freedoom. Then we have Tetrode(1922) and Fokker(1929) who obtain a general form of such interactions. Then we have Feynman's extensions to the Tetrode-Fokker action and the well known Wheeler-Feynman theory. I should also mention that such theories can be relativistic! In fact all of the theories I mentioned are relativistic. But the problem is, Wheeler couldn't quantize the theory. But I'm not sure, maybe people tried quantizing such theories after Wheeler too.

 

[bcrowell]

But the problem is, Wheeler couldn't quantize the theory. But I'm not sure, maybe people tried quantizing such theories after Wheeler too.

I think the Currie-Jordan-Sudarshan theorem basically tells you that you can't quantize these theories.

 

[ShayanJ]

I think the Currie-Jordan-Sudarshan theorem basically tells you that you can't quantize these theories.

That theorem forbids interaction in a Hamiltonian formulation which means you can't use canonical quantization. I'm not sure, but maybe you can quantize such theories using path integrals.

EDIT:
Yeah, I found it. But I haven't read it! The abstract says it uses a S-matrix.

 

[bcrowell]

That theorem forbids interaction in a Hamiltonian formulation which means you can't use canonical quantization. I'm not sure, but maybe you can quantize such theories using path integrals.

EDIT:
Yeah, I found it. But I haven't read it! The abstract says it uses a S-matrix.

I guess I'm really confused now. Isn't it true that E&M can also be quantized using a Hamiltonian? If it can also be expressed as a direct field, wouldn't that be a counterexample to CJS?

 

[ShayanJ]

I guess I'm really confused now. Isn't it true that E&M can also be quantized using a Hamiltonian? If it can also be expressed as a direct field, wouldn't that be a counterexample to CJS?

It seems to me your problem is with the CJS theorem itself. I quote a good statement of the theorem from Eric Gourgoulhon's special relativity in general frames:

according to a theorem established by D.G. Currie, J.T. Jordan and E.C.G. Sudarshan (1963), the conditions
(i) Invariance of the Hamiltonian structure under the action of the Poincare group.
(ii) Using the spacetime coordinates of the particles as canonical coordinates.
are not compatible, except if there is no interaction between the particles. This result has been called the no-interaction theorem.

I also should note that he explains this in the context of such direct interaction theories. The point is, the structure of the actions are very different for Maxwell EM+charged particles and charged particles with direct interaction. In the former the coordinates of the particles only couple to field variables while in the latter they should couple to each other. The CJS theorem says that such direct coupling causes an inconsistency. But there is nothing wrong with the coupling of coordinates to field variables.

 

[bcrowell]

I guess it's going to be hard (for me at least) to figure out what's going on without reading the Davies article, which I don't have access to.

 

[bcrowell]

I realized something that I hadn't noticed before. Hawking and Ellis have this:

"[The weak energy condition] will not hold for the 'C'-field proposed by Hoyle and Narlikar (1963). This again is a scalar field with m zero, only this time the energy-momentum tensor has the opposite sign and so the energy density is negative. This allows the simultaneous creation of quanta of positive energy fields and of the negative energy 'C'-field."

This looks wrong to me. The C field has the stress-energy tensor of a perfect fluid with *zero* energy density and negative pressure. Classically, it doesn't patch up conservation of energy-momentum by making enough negative energy to cancel out the positive energy of the newly created atoms. There is no negative energy. So my mental pictures of an unfilled Dirac sea were all wrong, I guess.

I think this sort of makes sense because if the C field compensated for the creation of atoms simply by canceling out their energy with negative energy, then every region of space at all times would have zero energy density, which is not what we observe.