Could GR's background independence be a theoretical artifact?

[ensabah6]

Could GR's "background independence" be a theoretical artifact?

==quote from Rovelli "Unfinished Revolution" (2006) page 2==

...Others, on the other hand, and in particular some hard–core particle physicists, do not accept the lesson of GR. They read GR as a field theory that can be consistently formulated in full on a fixed metric background, and treated within conventional QFT methods. They motivate this refusal by insisting than GR’s insight should not be taken too seriously, because GR is just a low–energy limit of a more fundamental theory. In doing so, they confuse the details of the Einstein’s equations (which might well be modified at high energy), with the new understanding of space and time brought by GR. This is coded in the background independence of the fundamental theory and expresses Einstein’s discovery that spacetime is not a fixed background, as it was assumed in special relativistic physics, but rather a dynamical field.

Nowadays this fact is finally being recognized even by those who have long refused to admit that GR forces a revolution in the way to think about space and time, such as some of the leading voices in string theory. In a recent interview [1], for instance, Nobel laureate David Gross says: “ [...] this revolution will likely change the way we think about space and time, maybe even eliminate them completely as a basis for our description of reality”. This is of course something that has been known since the 1930’s [2] by anybody who has taken seriously the problem of the implications of GR and QM. The problem of the conceptual novelty of GR, which the string approach has tried to throw out of the door, comes back by the window.
==endquote==
Would it be possible to develop a theory of gravity that is background dependent, a fluctuating field over a fixed metric, that reproduces all known GR results where GR has been tested? What about the spin-2 field on flat QFT?

I wonder if BI is a theoretical artifact that, while a valid conclusion based on GR"s field equations, are invalid when applied to nature. In otherwords, while GR succeeds as modeling gravity as encoding the geometry of spacetime, what gravity fundamentally is is some sort of emergent van der waals type interaction of something more fundamental.

 

[atyy]

Could one have both? In AdS/CFT, the CFT has a fixed background, but it describes an almost fully "dynamic" metric in AdS space, being fixed only at the boundary.

There is also a comment I don't understand in Hamber's http://arxiv.org/abs/0704.2895 "It is possible though to formulate quantum gravity on a flat hypercubic lattice, in analogy to Wilson’s discrete formulation for gauge theories, by putting the connection center stage. In this new set of theories the natural variables are then lattice versions of the spin connection and the vierbein." I do understand it enough to know that gravity is not emergent in the approaches he is discussing.

 

[Fra]

Would it be possible to develop a theory of gravity that is background dependent, a fluctuating field over a fixed metric, that reproduces all known GR results where GR has been tested? What about the spin-2 field on flat QFT?

I wonder if BI is a theoretical artifact that, while a valid conclusion based on GR"s field equations, are invalid when applied to nature. In otherwords, while GR succeeds as modeling gravity as encoding the geometry of spacetime, what gravity fundamentally is is some sort of emergent van der waals type interaction of something more fundamental.

I think that's more than possible, and I see arguments that it is in fact more plausible than the realist type of background indepdence in the sense of perfect observer independent diff-invarance symmetry.

However, there are different ways to think of this.

1) To replace one fundamental realist type of symmetry with another one (more fundamental), is generally suspect becuase it just replaces one fixed symmetry with another fixed symmetry (relative to which the prior one is emergent as a subsymmetry per a particular mechanism).

2) The other option, which I find most attractive, is to really find a new understanding of the physical basis and the status of symmetry that does NOT incorporate realist fictions. Usually symmetry is a constraint that allows us to make inferences. But if we are looking for a MEASUREMENT THEORY, and not a realist type of old style model, then we should wonder what is the origin of these symmetries? shouldn't we? Because all the effective symmetries we do now, are actually inferred from nature. We find the apparent symmetries of nature from experiment and measurement, to guide further measurements. They are not god given constraints as far as I see it. This detail are rarely acknowledged in many types of models. And it sure disturbs me, and I can't come to terms with this inconsistency.

/Fredrik

 

[tom.stoer]

Background independence, diffeomorphism invariance and gauge symmetry are in some sense always a theoretical artefacts. Assume you have a path integral Z which incorporates all relevant physical information (eigenstates, spectrum, scattering amplitudes, ...). Assume for a moment that it's possible to write it as

Z = ∫Da^phys exp iS[a^phys]

Here everything is written in terms of physical fields a(x).

Instead we usually start the development of a theory with an expression like

Z = ∫DA exp iS[A]

where A(x) and S refer to fields and action with some (local) symmetry. Therefore A(x) are no physical degrees of freedom (no observable quantities), but one must implement a local symmetry in order to get a well-defined path integral.

Usually this is done via

Da^phys = DA / symm

which means that we "factor out" the symmetry group of the action S.

If we were able to write down these expression directly in terms of physical quantities, then nobody would care about these local symmetries.

 

[ConradDJ]

==quote from Rovelli "Unfinished Revolution" (2006) page 2==

...Others, on the other hand, and in particular some hard–core particle physicists, do not accept the lesson of GR. They read GR as a field theory that can be consistently formulated in full on a fixed metric background...

The other option, which I find most attractive, is to really find a new understanding of the physical basis and the status of symmetry that does NOT incorporate realist fictions... if we are looking for a MEASUREMENT THEORY, and not a realist type of old style model, then we should wonder what is the origin of these symmetries.

Hi Fredrik –

What you’re calling a realist model is one that takes the world as consisting of given facts... so then the purpose of theory is to express as concisely as possible the order of those facts. Uncovering symmetries is a basic way to do that.

But since physics gives us many different kinds of symmetry – some exact and some “broken” or only approximate – I agree that we need to look for a deeper explanation of each symmetry. And in a realist model, where do we even look for that? The equations of GR express a remarkable kind of symmetry – but we don’t really have any basis for judging whether diffeomorphism invariance is fundamental, as Rovelli thinks, or emergent, as I think the OP is suggesting. And we don’t know how to relate this to the many other kinds of symmetry of the Standard Model.

In what you call a measurement theory, on the other hand, physics not only describes actual (measured) facts, but also structured systems of possible fact – I'm thinking of the quantum wave-function, describing the physical situation in which a certain type of measurement can be made, but where the outcome is not yet given.

If the quantum-type theory is fundamental, then symmetries have a natural explanation in terms of sequential levels of physical possibility-structure... i.e. where some aspects of physical structure become determinable before others.

Maybe at a very “primitive” level – e.g. in the quantum vacuum, or in the very early universe – the possibilities of interaction are essentially unconstrained by any law, and nothing is determinable.

At a very “high” level – that of our “macroscopic observation” – the physical measurement-context is so structured by physical law and so densely redundant, giving so many ways of measuring things, that the world looks like a body of precisely determinate fact, with no dependence on the measurement-context at all.

Symmetries would arise in between, in the intermediate levels where some aspects of physical structure are already given, other aspects are determinable through interactions (measurements), and some are not yet determinable at all.

Take a very simple, non-physical example of spatial symmetry –

> Suppose the topology of possible points on a 2-dimensional surface is already determined.
> Suppose the distance from any given point to another point is determinable, at this level –
> Suppose the direction between any two points is not yet determinable.
​

Say we randomly select (measure) points at a determined distance from a given point. If we then look at the result from a higher level – where directions as well as distances are determinate, and we can see the surface as a plane – the we see our selection has resulted in a symmetrical circle.

So – where in a “realist model” a symmetry is just a fact-pattern – in a “measurement model”, symmetry indicates the order in which different aspects of structure become measurable. From that standpoint, symmetries in the laws of physics indicate levels of interaction-structure in which some parameters are physically definable (measurable) and others are not.

GR would then be telling us that the aspects of physical interaction that make the smooth connectedness of the spacetime manifold determinable are “more primitive”, more basic than the aspects that let us measure spacetime intervals and coordinates – which supports Rovelli’s viewpoint.

As another example, the CPT symmetry of all known physical laws would indicate a very deep level of interaction-topology that “predates” the structures that let us determine the +/– charge, left/right parity or the forward/back direction of the time-arrow.

 

[Bob_for_short]

Usually this is done via

Da^phys = DA / symm

which means that we "factor out" the symmetry group of the action S.

If we were able to write down these expression directly in terms of physical quantities, then nobody would care about these local symmetries.

In QED in the Coulomb gauge (or in Dirac's gauge-invariant variables) the formulation is in fact a gauge-independent, i.e., only physical degrees of freedom (that carry and exchange the energy-momentum) are involved.

Why not to do the same thing for the gravitational field?

 

[atyy]

Background independence, diffeomorphism invariance and gauge symmetry are in some sense always a theoretical artefacts. Assume you have a path integral Z which incorporates all relevant physical information (eigenstates, spectrum, scattering amplitudes, ...). Assume for a moment that it's possible to write it as

Z = ∫Da^phys exp iS[a^phys]

Here everything is written in terms of physical fields a(x).

Instead we usually start the development of a theory with an expression like

Z = ∫DA exp iS[A]

where A(x) and S refer to fields and action with some (local) symmetry. Therefore A(x) are no physical degrees of freedom (no observable quantities), but one must implement a local symmetry in order to get a well-defined path integral.

Usually this is done via

Da^phys = DA / symm

which means that we "factor out" the symmetry group of the action S.

If we were able to write down these expression directly in terms of physical quantities, then nobody would care about these local symmetries.

Would it be better defined if the question were rephrased as "Does the classical limit of the quantum theory contain the full solution space of classical general relativity"?

In the case of Asymptotic Safety (if it exists) and string theory, I think the answer is almost all, and hopefully the missing topologies are not required. The caveat in string theory is that non-perturbative definitions are only known for some sectors, while other physically relevant sectors are still only perturbatively defined.

In cases where the quantum theory has a fixed non-relativistic background such as Horava-Lifgarbagez, the classical solutions are different, eg. no exact Schwarzschild, and a scalar graviton in addition to the usual spin-2 one:
http://arxiv.org/abs/0905.2798
http://arxiv.org/abs/0905.2579

 

[tom.stoer]

In QED in the Coulomb gauge (or in Dirac's gauge-invariant variables) the formulation is in fact a gauge-independent, i.e., only physical degrees of freedom (that carry and exchange the energy-momentum) are involved.

Why not to do the same thing for the gravitational field?

Yes, of course, it works for QED, and for QCD as well.

1) The main problem with QG is that the path integral seems to escape standard renormalization and therefore seems to be ill-defined.
2) The standard treatment of non-abelian gauge fields requires gauge-fixing via Fadeev-Popov ghost, which is not really a formulation in terms of physical degrees of freedom only. Something similar applies to QG as well.
3) In addition there's the problem that one must never use any reference to a background metric, as this metric must not be used as input (e.g. in order to define polarizations, positive and negative ferquency conditions etc.)

I agree that in principle this approach should be possible, but as far as I know it never worked due to technical inconsistencies.

 

[tom.stoer]

Would it be better defined if the question were rephrased as "Does the classical limit of the quantum theory contain the full solution space of classical general relativity"?

I am not sure what you mean. The question was if these symmetries are theoretical artefacts. I tried to explain why this is - in a certain sense - always true. Gauge symmetries and diffeomorphism invariance are coordinate symmetries only.

It may not be true if a more fundamental theory from which spacetime emerges is correct. Then it may be possible that these symmetries are only valid in a low-energy / long distance limit.

But my argument remains valid if you exclude certain sectors of GR. If e.g. spacetime is restricted (for some reason yet to be discovered) to one topological sector, then factoring away these symmetries simply means that the group you factor away is smaller.

 

[atyy]

I am not sure what you mean. The question was if these symmetries are theoretical artefacts. I tried to explain why this is - in a certain sense - true.

It may not be true if other theories are correct; then it may be possible that the spacetime symmetries emerge from more fundamental principles, or that the they are only asymptotic symmetries, e.g. in a low-energy / long distance limit.

My argument remains valid if you exclude certain sectors. If e.g. spacetime is restricted (for some reason yet to be discovcred) to one topological sector, then factoring away simply means that the group you factor away is smaller.

I agree with your argument. Gauge invariance is an "artifact" even of classical general relativity. I was rephrasing the original question in such a way that "background independence" is not an artifact in classical general relativity, ie. let's only consider the physically allowable metric geometries, and take background independence to mean that classical general relativity has no geometry that is common to all allowable physical configurations, and the geometrical configurations and matter configurations specify each other, ie. let's define "background independence" not as "general covariance", but as "no prior metric geometry".

 

[marcus]

Tom, I'd be interested to learn what you think of the following line of reasoning.

A fixed metric on a manifold imposes a causal structure----it decides for us which events could cause or influence which other events.

Suppose one takes a manifold and fixes a metric g, and then one introduces a disturbance "on top of that" so that the real metric is g+h.

In a kind of naive or braindead way, g is the fixed background and h is the "gravitational field" defined on top of the fixed geometry specified by g.

This will typically change the logical (or lightcone) structure. So the logical structure originally specified by the background is no longer being followed by nature. Nature goes by the full g+h geometry.

So not only the gravitational field but all of the field theory that one defines on top of the fixed background is generically invalid. This includes all the matter fields, as well as the phony (because only partial) "gravitational field" called h, which one has added to the picture. They do not take into account the real causal structure.

Because the logic of causality and locality is one, the gravitational field is one. It cannot be divided into a main or fixed background part and an additional field of "forces" or "gravitons" which are supposed to be empty of logical effect. That kind of picture is a useful mathematical fiction with a limited applicability, but can't be taken as in any sense fundamental.

That's one way to look at it. Do you see anything obviously wrong with that point of view?

 

[tom.stoer]

I fully agree. Splitting g = g°+h is plainly wrong.

 

[Bob_for_short]

I fully agree. Splitting g = g°+h is plainly wrong.

Well, there is the A.A. Logunov's relativistic theory of gravity (RTG) where splitting g=η+h is meaningful because the equations for matter contain the entire g (equivalence principle) but the gravitational field equations contain η - the Minkowsky metric. The main space-time is flat, its curvature R is equal to zero, and the gravitational field η is as physical as the electromagnetic one. In such a formulation the inertia forces are different from the gravitational ones since transition into an accelerating reference frame does not change R=0.
This theory describes well all observable effects but works in the frame of a flat space-time.

I think such a formulation is very promising since after removing the self-action from interaction this theory will not need renormalizations at all. The calculations will turn into calculations of compound systems, like in atom-atomic scattering, i.e., with no conceptual and divergence problems.

 

[marcus]

Logunov's 1984 RTG (relativ. theor. grav.) is a good illustration of the difficulty.
It was formulated by a highly respected older physicist (he was born in 1927) and yet it never caught on.

It has some problems like predicting that black holes cannot exist, because there is a fundamental fixed Minkowski flat. It is hard to see how a universe with that basis could expand a thousand-fold, and do all the other neat stuff that our universe evidently does.

Logunov formulation seems kind of arbitrary and unnatural, but it certainly is good for doing a certain limited class of calculations.

Here is the main 1984 paper:
http://www.slac.stanford.edu/spires/find/hep/www?irn=1280147

It is listed as having been cited only 7 times.

People like James Hartle and Gary Gibbons have cited Logunov's RTG---there is awareness of it, it has not been overlooked.
But the citation I checked out was to illustrate why something doesn't work. It is used as an example of a flawed or inadequate approach.
Like "half a dozen different authors have tried it but if you want to take way seriously you have to deal with the...[lightcones]..." or words to that effect.

The idea is say call the Minkowski metric g and the additional field h. So the total is g+h. Then the artificial assumption is that the other fields all know about g+h, and live on the full g+h geometry. But the gravitational field h only knows about Minkowski geometry g and lives on the partial geometry g.
Why should the gravitational field be treated differently? And if you do this then you are permanently stuck with this clumsy stiff Minkowski space and cannot do things like black holes and big bangs and expansion and all the other really curved stuff. Like wearing a wooden shirt.

However I gather that Logunov is well respected for his excellent work in other areas of physics such as nuclear and particle physics. He has received many honors. Only this slightly odd idea of his about gravity so far at least until now did not fly.

 

[atyy]

Well, there is the A.A. Logunov's relativistic theory of gravity (RTG) where splitting g=η+h is meaningful because the equations for matter contain the entire g (equivalence principle) but the gravitational field equations contain η - the Minkowsky metric. The main space-time is flat, its curvature R is equal to zero, and the gravitational field η is as physical as the electromagnetic one. In such a formulation the inertia forces are different from the gravitational ones since transition into an accelerating reference frame does not change R=0.
This theory describes well all observable effects but works in the frame of a flat space-time.

I think such a formulation is very promising since after removing the self-action from interaction this theory will not need renormalizations at all. The calculations will turn into calculations of compound systems, like in atom-atomic scattering, i.e., with no conceptual and divergence problems.

Logunov's 1984 RTG (relativ. theor. grav.) is a good illustration of the difficulty.
It was formulated by a highly respected older physicist (he was born in 1927) and yet it never caught on.

It has some problems like predicting that black holes cannot exist, because there is a fundamental fixed Minkowski flat. It is hard to see how a universe with that basis could expand.

Also it seems kind of arbitrary and unnatural. But it certainly is good for doing a certain limited class of calculations.

Here is the main 1984 paper:
http://www.slac.stanford.edu/spires/find/hep/www?irn=1280147

Well, maybe Logunov had a good idea nonetheless - do you think these are in the same spirit?

http://arxiv.org/abs/hep-th/0009230
External Fields as Intrinsic Geometry
John Madore, Stefan Schraml, Peter Schupp, Julius Wess
"There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one."

http://arxiv.org/abs/hep-th/0212262
Noncommutative Field Theories and Gravity
Victor O. Rivelles
"We show that after the Seiberg-Witten map is performed the action for noncommutative field theories can be regarded as a coupling to a field dependent gravitational background. ... This shows that noncommutative field theories can be seen as ordinary theories in a gravitational background produced by the gauge field with a charge dependent gravitational coupling. "

 

[Bob_for_short]

...It has some problems like predicting that black holes cannot exist, because there is a fundamental fixed Minkowski flat.

It is not so, the RTG predicts heavy black hole-like objects but it is free from non-physical singularities.

Logunov's formulation seems kind of arbitrary and unnatural...

It preserves the energy-momentum conservation and treats the gravitational field as physical rather than as the curvature of the true space-time. In this respect it is advantageous and quite natural.

The idea is say call the Minkowski metric g and the additional field h. So the total is g+h. Then the artificial assumption is that the other fields all know about g+h, and live on the full g+h geometry.

It is implementation of Einstein's equivalence principle for matter.

But the gravitational field h only knows about Minkowski geometry g and lives on the partial geometry g. Why should the gravitational field be treated differently?

Academician A. Logunov has given his motivations in many articles. As to me, I prefer working in the Minkowsky space-time where I have the energy-momentum conservation laws and the gravitational fields (static and waves) are as physical as the electromagnetic and other fields. There is nothing especially artificial in RTG. On the contrary, it is a return to good physics.

The problem of his approach is in revealing some crucial drawbacks of GR, it is an "anti-Einstein's" approach. Many are scared to follow it - they do not want to be excluded from researching. It is a serious matter.

There is a fresher reference: http://arxiv.org/abs/gr-qc/0210005

 

[Bob_for_short]

...And if you do this then you are permanently stuck with this clumsy stiff Minkowski space and cannot do things like black holes and big bangs and expansion and all the other really curved stuff. Like wearing a wooden shirt.

Do not tell me that the Riemann space-time is a custom-made suit. GR is even more non-linear and difficult than RTG.

Big Bang is an extrapolation backwards in time, not an experimental fact.

Black holes and singularities are a serious problem in GR, not good (physically acceptable) solutions.

It seems physicists started to believe too much in their weird results (black holes) and patches like bare particles, vacuum polarizations, appeared in late practice to somehow make senseless actions meaningful.

 

[Bob_for_short]

...The idea is say call the Minkowski metric g and the additional field h. So the total is g+h. Then the artificial assumption is that the other fields all know about g+h, and live on the full g+h geometry. But the gravitational field h only knows about Minkowski geometry g and lives on the partial geometry g.

It is not exactly so - the gravitational filed "knows" about both: g and h, but they do not enter as a sum (g+h) in the equations. Thus the Minkwsky space-time is in fact separated in the theory (RTG) from the effective Riemann geometry felt by matter.

 

[atyy]

Well, maybe Logunov had a good idea nonetheless - do you think these are in the same spirit?

http://arxiv.org/abs/hep-th/0009230
External Fields as Intrinsic Geometry
John Madore, Stefan Schraml, Peter Schupp, Julius Wess
"There is an interesting dichotomy between a space-time metric considered as external field in a flat background and the same considered as an intrinsic part of the geometry of space-time. We shall describe and compare two other external fields which can be absorbed into an appropriate redefinition of the geometry, this time a noncommutative one."

http://arxiv.org/abs/hep-th/0212262
Noncommutative Field Theories and Gravity
Victor O. Rivelles
"We show that after the Seiberg-Witten map is performed the action for noncommutative field theories can be regarded as a coupling to a field dependent gravitational background. ... This shows that noncommutative field theories can be seen as ordinary theories in a gravitational background produced by the gauge field with a charge dependent gravitational coupling. "

Following papers which cited Rivelles I came across:
http://arxiv.org/abs/0903.1015
Matrix Models, Emergent Gravity, and Gauge Theory
Harold Steinacker

The Steinacker paper is commented on by - in the same breath as quantum graphity:
http://arxiv.org/abs/0909.3834
Analogue Models for Emergent Gravity
Stefano Liberati, Florian Girelli, Lorenzo Sindoni
"The second option is probably the most viable, conceptually appealing, but most demanding in terms of new concepts to be introduced. If no reference is made to a background Minkowski spacetime, but rather the graviton emerges in the same limit in which the manifold emerges, then there is no obvious conflict with the Weinberg-Witten theorem. Simply, what is called the gauge symmetry in terms of fields living of spacetime is the manifestation of an underlying symmetry acting on the fundamental degrees of freedom in the limit when they are reorganized in terms of a spacetime manifold and fields (gauge fields and gravitons in particular). There are already two examples of this possibility, namely matrix models and quantum graphity models. In both cases, the very notion of spacetime manifold is immaterial for the foundations of the theory. The manifold and the metric are derived concepts, obtained in precise dynamical regimes of the theory. The interested reader can find additional comments and references in [10, 40]."

Maybe Marcus will like this one - it's got a bounce!
http://arxiv.org/abs/0903.0986
Cosmological solutions of emergent noncommutative gravity
Daniela Klammer, Harold Steinacker
Ooops, it's in 10D

 

[Haelfix]

Oye, there is no problems with causality or locality by splitting the metric up in terms of a weak field, provided that (wait for it), the field is weak! Just such a procedure is discussed in most textbooks, for instance MTW or some other GR book.

Pure relativists use this technique all the time, for instance when they might be interested in gravitational waves like at LIGO or when they are analyzing binary pulsars or other thorny time dependant solutions.

The problem is not that the approximation is invalid or breaks the 'spirit' of GR (whatever that means), instead the problem is that you can't always use it. This is completely analogous to situations in certain field theories where you can't always use a background field method and you have to get creative in order to make progress (here reffering to gauge fields, as opposed to a metric tensor).

Fortunately its nearly always the case that you can pick some frame or some physical context (perhaps sufficiently far from where there is violent geometry), that still captures the essential physics for most questions one might be concerned with. So for instance, if we are interested in black hole information loss, we know that our calculational tools clearly break down near singularities. So what do you do, you simply ask a question about horizons instead, for instance at a point right after the formation of the bh, and then compare it to one right after its evaporated. The complicated mess that takes place at the singularity is kind of bypassed that way (even if it contributes in some nontrivial way tothe final answer), but we still have a well defined problem that we can now calculate (and lo and behold, we end up with Hawking theorem and an information loss paradox).

 

[atyy]

Oye, there is no problems with causality or locality by splitting the metric up in terms of a weak field, provided that (wait for it), the field is weak! Just such a procedure is discussed in most textbooks, for instance MTW or some other GR book.

I've read that one can do g=minkowski+perturbation exactly as long as global harmonic coordinates exist (Eq 62, 63 http://relativity.livingreviews.org/Articles/lrr-2006-3/ ). Is it the case that such coordinates don't exist in strong fields? But it seems even the Kerr solution can be expressed in harmonic coordinates (http://arxiv.org/abs/gr-qc/0503018), so are such coordinates merely unknown , or provably non-existent for interesting physical cases?

 

[marcus]

I've read that one can do g=minkowski+perturbation exactly as long as global harmonic coordinates exist (Eq 62, 63 http://relativity.livingreviews.org/Articles/lrr-2006-3/ ).

I think you realized that the issue raised in the O.P. doesn't have to do with the weak field approximation (which Haelfix was talking about). It has to do with whether the gravitational field can be treated exactly and generically as g + h where g is minkowski.

So your reference to Will's review, section 4.3 (equations 62 and 63) would have been relevant if such an exact treatment were generically available. The key word is exact. Unfortunately it doesn't look like it is.

Perhaps this simple example will suffice, try to write the Schwarzschild solution as g+h where g is Minkowski.

Interesting reference though. Business about harmonic coordinates, how generally available? I don't know. If you read further, just past equation 64, it looks like what Will is talking about is a an approximation in any case. Just that one part of the approximation scheme (equation 62) is exact.

 

[atyy]

Perhaps this simple example will suffice, try to write the Schwarzschild solution as g+h where g is Minkowski.

What about the Kerr-Schild form given at http://en.wikipedia.org/wiki/Kerr–Newman_metric? The Schwarzschild metric is the Kerr-Newman metric with neither charge nor angular momentum.

 

[tom.stoer]

If you could use a QG path integral g = g° + h is simply a change of coordinates. Instead of Dg you can use Dh. That's OK.

The problem arises as soon as you start to use g° in order to "define" something, e.g.: what is a spacelike slice? what is the distance bewtween x and y? what is the lightcone at a certain point? In that case g° may or may not do the job. You need such definitions to make use of path integrals, commutation relations etc. I think the problem in classical GR may or may not exist - it certainly arises during the quantization.

In many cases the split g = g° + h is associated with a weak field limit for h, or, as it comes to path integrals, with perturbartion theory in h. This certainly the wrong approach:

1) we know that gravity is not renormalizable (at least not in the standard way); so doing perturbation theory is always difficult and requires great care.

2) we now from non-abelian gauge theories that gauge fixing in the path integral formalism will in many cases lead to so-called gribov ambiguities, i.e. ill-defined measures in the path integral. The fadeev-popov determinant becomes singular in these cases, which means that the gauge slice itself is ill-defined. It is interesting that especially Lorentz-covariant gauges (e.g. the Lorentz-gauge) suffer from these ambiguities, whereas e.g. the axial gauge (which breaks this explicit covariance) is well-defined. Unfortunately using non-Lorentz-covariant gauges is cumbersome as you have to check for covariance step by step (it is not lost but hidden: you have to check for the correct operator algebra, if it becomes anomalous, if physical amplitudes respect Lorentz-covariant whereas propagators do not.)
In QG one would have to check if the harmonic gauge is a global gauge slice or if it leads to similar ambiguities as well.

3) in QCD one knows that perturbation theory (weak field limit) already failes classically as e.g. the instanton configurations are of order 1/g²; so there is no perturbation expansion and no weak field limit which coveres both the vacuum and the one-instanton sector. In addition the restriction to one superselection sector (theta vacuum) is simply wrong. This need not be the case in QG, but it's a warning to everybody who thinks that we live in one superselection sector (e.g. according to topology) and can safely neglect the others. In order to justify this one must construct a globally well-behaved theory and derive if the superselection rule survives quantization.

 

[atyy]

Here's an example in quantum gravity where g=background+perturbation may be useful. I believe de Donder gauge is the same as harmonic coordinates:

http://arxiv.org/abs/0805.2909
Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation
Alessandro Codello, Roberto Percacci, Christoph Rahmede
"We decompose the metric into gμν = g(B)μν +hμν where g(B)μν is a background. We will refer to the field hμν as the graviton, even though it is not assumed to be a small perturbation. ... In this section we will restrict ourselves to the de Donder gauge"

 

[Fra]

You're right about the inital characterisation, realist views beeing a patten of facts. The problem of course, is that in an inference model all facts are infered. And there is no foolproof inference, therefore there are not unquestionable facts.

In my view, symmetry is an inferred regularities among evidences. But this renders the symmetry relative to the inference system.

IF you consider a very massive context, which is the case if we do particle physics experiments, the inferred symmetry of the small system can be inferred with high confidence levels, and represented by a massive structure.

In what you call a measurement theory, on the other hand, physics not only describes actual (measured) facts, but also structured systems of possible fact – I'm thinking of the quantum wave-function, describing the physical situation in which a certain type of measurement can be made, but where the outcome is not yet given.

If the quantum-type theory is fundamental, then symmetries have a natural explanation in terms of sequential levels of physical possibility-structure... i.e. where some aspects of physical structure become determinable before others.

There is sometimes a polarisation that either you take GR seriously, or you take QM seriously, leading to two different programs.

My view is neither. I see a measurement theory as fundamental BUT I am suggesting that we need to revise quantum mechanics. QM is currently way too simple IMO. This is part of what I called reconstructing an intrinisic measurement theory and also "probability theory" seems as an extension to logic. If you know how Jaynes introduces probability from pondering about quantification of degrees of beleifs and rational expectations, then I suggest a similar reconstruction but still in a different way which will incorporate quantum logic.

If you come from the path integral approach, what I am seeking is a deepering understanding of the meaning of path integral approach, where the path integral is seen as an inference machinery from with an action follows, but the point is that if you require the inference machiner to have a physical basis, the path integral is constrained in complexity, and thus it's forced to evolve to persist.

If I remember you like to keep QM as it is, pretty much like rovelli wants. The problem is that while you might think that you keep the mathematical structure of QM, the original beauty that begged for a quest for a measurement theory is abused IMO. QM is pretty much well tested for particle physics - ie very SMALL subsystems, probed with respect to a MASSIVE context. If we are talking about spacetimes and cosmological scale things, this type of abstraction is inappropriate IMO. IT has to do with the basis for statistics inference and probability theory too. In a massive context, the statistics about a small sysstem can be collected and stored. Picture the opposite, a small systrem trying retina statistics ov an overly massive environment. It doesn't work unless you don't mind information densities blowing up to infinity and similar things.

/Fredrik

 

[tom.stoer]

After eq. (32) a ghost action is specified. The operator between the ghost fields C-bar and C will certainly have zero eigenvalues for some background metric g. These zero eigenvalues correspond to singular gauges. Therefore this theory suffers from the same type of problems as explained above.

Note: in many cases these problems do not show up. In QCD one can use the (singular) Lorentz gauge for deep inelastic scattering and will never run into trouble. But this is only a certain sector of the theory where due to asymptotic freedom g << 1 and A = 0 is valid for perturbation expansion; reason is that all gauge fields stay "far away" from the Gribov horizon.

If one is interested in asymptotic safe gravity "g << 1" is no longer reasonable. All coupling constants could potentially cross Gribov horizons which have to be identified first. If one does not specify the background field one has to investigate all different sectors (Gribov domains) and check how they can be patched together. This analysis is missing.

I searched gr-qc for "Gribov". There is one paper mentioning this problem:
http://arxiv.org/abs/quant-ph/9611026
Title: Coherent State Approach to Time Reparameterization Invariant Systems M. C. Ashworth
(Submitted on 14 Nov 1996)
Abstract: For many years coherent states have been a useful tool for understanding fundamental questions in quantum mechanics. Recently, there has been work on developing a consistent way of including constraints into the phase space path integral that naturally arises in coherent state quantization. This new approach has many advantages over other approaches, including the lack of any Gribov problems, the independence of gauge fixing, and the ability to handle second-class constraints without any ambiguous determinants. In this paper, I use this new approach to study some examples of time reparameterization invariant systems, which are of special interest in the field of quantum gravity.

 

[Bob_for_short]

I've read that one can do g=minkowski+perturbation exactly as long as global harmonic coordinates exist ... Is it the case that such coordinates don't exist in strong fields?

In the frame of GR, splitting the metric tensor is artificial, - it is just a variable change, if you like, and nothing new can be obtained in this way. In particular, one arrives at the conclusion that such a splitting makes no sense.

Such a splitting should be done by construction - the real space-time is that of Minkowsky's and the gravitational filed is physical (RTG). The latter construction is preferable and intuitively attractive since it radically simplifies the theory. In particular, the harmonic coordinates in RTG are obtained as solutions of the field equations rather than as due to a coordinate choice, amongst many others, as in GR.

These are two different approaches and each should be checked for consistency. It is not correct to ask the "splitting" question in the frame of GR, obtain a negative answer, and extrapolate it to other theories because the negative answer belongs only to GR.

A. Logunov has clarified these questions and I find his work quite convincing (http://arxiv.org/abs/gr-qc/0210005).

 

[Haelfix]

If g is minkowski, than you definitely cannot reach say a FRW solution with a cc for instance. The perturbation 'h' would require infinite energy to reach the solution. Hence a different superselection sector.

No one ever does that though. There is no theory where the background metric is always fixed to be flat to begin with, in general its arbitrary (and actually has to be arbitrary at first) and you insert the appropriate metric for whatever physical situation is at hand..

In an analagous manner QCD's vacuum is a good example of the problem that you run into though with trying to do such a backround field procedure. It is an example of a vacuum that contains a great deal of nonperturbative modes (condensates, solitonic objects, possibly extended configurations and the like) and we only really understand the false vacuum structure of the theory. So the actual global <0|A|0> state is unknown and waiting to be solved for USD 1 million.

 

[tom.stoer]

If g is minkowski, than you definitely cannot reach say a FRW solution with a cc for instance. The perturbation 'h' would require infinite energy to reach the solution. Hence a different superselection sector.

Is there a classification of gravitational superselection sectors? I do not only mean a list but in addition a way to check how they could be possibly "connected".

In QCD you must take into account the different theta-vacua. Maybe in GR you can restrict to FRW and neglect flat space - I don't know. In general this depends on the applicability of the selection rules after quantization, e.g. if energy, (topological) charges etc. are conserved. In case of energy it depends in addition on the fact if tunneling is allowed or not. In addition it is by no means clear how to define energy in the GR context.

.

 

[Bob_for_short]

There is no theory where the background metric is always fixed to be flat to begin with, in general its arbitrary (and actually has to be arbitrary at first) and you insert the appropriate metric for whatever physical situation is at hand.

Wrong, the RTG and SR including electromagnetism are examples of the contrary.

If g is minkowski, than you definitely cannot reach say a FRW solution with a cc for instance.

Not all "results" of one theory should be reproduced with another. It is the experimental data that should be described.

 

[Bob_for_short]

By the way, in a plane Minkowsky world, where the forces are physical and the background is always the same, we can compare particle/photon trajectories in presence/absence of a heavy body and speak of gravitational effects safely.

If the gravity is completely geometrized (GR), presence and absence of a heavy body change the geometry. Is it correct to compare distances, angles, red shifts, etc., obtained in two different geometries? I think, each geometry has its own life. The only way to compare the effects is to have an interaction-independent background, like in SR and RGT. Otherwise such a comparison is meaningless.

 

[Haelfix]

Bob, there is no problem whatsoever with doing electrodynamics around say a Schwarzschild metric. Solving it might be hard, but there is nothing priviledged about flat space a priori.

As for classifying superselection sectors, that's an old story that was done a long time ago and you'd have to dust off some old GR tomes on the classification of spacetimes (done in the 60s I believe). I am by no means an expert on that, but I seem to recall that there are a bunch of different methods (some algebraic, others focus on conformal killing vectors and ADM mass etc). Trying to connect two spacetimes by wondering say about how much radiation you have to bring in from infinity is also a hard story, and far more recent. That involves topics like global stability theorems, the positive energy theorem and the like.

 

[Bob_for_short]

...but there is nothing priviledged about flat space a priori.

How about the energy-momentum conservation laws? The energy-momentum is additive in particles. It permits to distinguish them. It is a very important conceptual feature in physics - to name and define separate things.

 

[Haelfix]

Typically there is no obvious notion of a particle in curved spacetime to begin with. In GR, the energy momentum conservation is a highly local object, there is no unique global measure of such a concept.