GR Background Independence: Indeterminate and Non-local?

[RandallB]

REF: Smolin; Perimeter Institute;
-- “The case for background independence”

What does background independence mean. Are statistical probabilities needed to compare separated events. If the background that physics works on varies relative to distant locations in GR does that mean GR is indeterminate?

For example: Two particles starting out from the same place separate at relativistic speeds and approach separate black holes at escape speed, remain outside the horizon and use the curve in space-time to change direction by about 120 degrees putting them on courses that should bring them back together at a point some distance from their start. With mass speeds and locations of all elements well defined, GR should be able to calculate the exact initial trajectories needed to effect a collision.

However, background independence between the two, should not allow us to determine a correct relative relationship between the two reference frames associated with the two approaching particles. That is; by using an independent background GR to calculate the speed and location of the other particle, relative to the new local reference frame of one particle used as reference; we can only predict within a range of some statistical probability the exact speed and location of that other particle as it comes back into observable local proximity of our selected reference particle. Leaving us unable to reliably predict or even guide our local into a collision without making current local observations of the other approaching particle. Meaning an element of chaos or uncertainty within the theory is involved.

If Smolin is correct about an independent background GR, this directly implies that GR is also indeterminate. And in that sense this would mean GR in also classically non-local.
That is even GR does not require ‘Bell local causality’ and therefore is a theory that could be compatible with the experiment proofs of Aspect, and others, that so far show a viable theory would need to be non-local.

I’ve done the best I can to understand background independence and this seems a logical conclusion to me.
Does someone know; is this Smolin view of GR non-local?
Or if my logic above needs correcting, comments please.
BUT in layman’s classical terms, please.
NOT in some redefinition of “local” derived from some other theory.
I’d rather this thread not degenerate into another argument about how which non-local theory is correct and better than other non-local theories.
Those posts should start a new thread in the quantum forum anyway so redirect us to one there if it is important.
That way maybe this thread can remain focused on;
do some views of GR define it as Non-Local?

 

[JesseM]

However, background independence between the two, should not allow us to determine a correct relative relationship between the two reference frames associated with the two approaching particles. That is; by using an independent background GR to calculate the speed and location of the other particle, relative to the new local reference frame of one particle used as reference; we can only predict within a range of some statistical probability the exact speed and location of that other particle as it comes back into observable local proximity of our selected reference particle.

I don't follow your logic here. Why should background independence prevent us from having a single deterministic prediction about what the relative velocities of the particles will be when they pass each other (assuming we measure the relative velocities in a local inertial coordinate system)?

 

[RandallB]

I don't follow your logic here. Why should background independence prevent us from having a single deterministic prediction about what the relative velocities of the particles will be when they pass each other (assuming we measure the relative velocities in a local inertial coordinate system)?

Yes picking any single inertial coordinate system with well defined elements including speeds, locations, and vectors; using Classical Newtonian Physics or Special Relativistic physics by definition of those theories should allow us define a deterministic indication of exactly how they will meet.

But as I’m understanding background independent GR, this is not still true. And the problem here may well be my not getting what the real meaning of background independence is.

What I get from it, is that as the two particles experience significant GR effects, as they pass by significant masses imparting significant accelerations, they are in widely separated locations with independent backgrounds. With the significance of the independent backgrounds being that you do not have direct relative relationships between the backgrounds they are experiencing. Leaving them with no determinate way to relate to the motion of the other particle in a not well defined background. An element of uncertainty is introduced because the backgrounds are not consistent, nor do the vary in a reliably predictable way that would allow for a deterministic mathematical way make a clear prediction. Hence as I see Somlin view of GR here, it is that GR is indeterminate and likely even non-local.

Now I admit I find the issue of background independence hard to follow, So if I’m wrong about what this view of GR means – just what is the significance of background independence?
Is it nonsense or does it mean something.

 

[Hurkyl]

"Background independent" means that the theory wasn't told before-hand how space-time looks... instead the theory describes how space-time looks.

For example, when Newtonian physics, you start with the assumption that space-time is split into 3D Euclidean space and 1D Euclidean time. In SR, you essentially start with the assumption that space-time is a 4-D Minowski space-time. These are background dependent.

But you don't do that in GR -- the geometry of space-time is one of the variables in the theory. This is background independent.

 

[vanesch]

But you don't do that in GR -- the geometry of space-time is one of the variables in the theory. This is background independent.

Indeed - and I fail what this has to do with "non-locality".
GR to me is both background - independent (meaning, as Hurkyl said, that no pre-defined geometry is imposed - except maybe its signature :blush: ), and local (in that every relevant relationship can be expressed as a function of stuff that is only function of a spacetime point and a limited number of their derivatives and that we never need more than stuff defined over a small neighbourhood).

 

[Hurkyl]

And also local in the causal sense of SR -- the conditions at a point are completely determined by a slice of its past light-cone.

 

[RandallB]

Well maybe my analogy of what I thought GR Background Independence is just way of any sensible mark.
But, sorry guys these explanations are just way to trivial to consider them useful.

Smolin address it as “relational” and having implications in the ‘relational vs. absolute debate’. By the end of his 36 paper he is speculating on how relational non-local hidden variables may be useful in a bell valid solution.

Maybe someday I’ll come across something that makes the significance of background independence clear.

 

[Hurkyl]

The significance of background dependence is this -- if your theory requires you to tell it what the background looks like, then it is incapable of telling you how the background interacts with itself, how other stuff affects the background, et cetera.

General Relativity suggests that a theory of gravity needs to explain how the background interacts with itself, and how other stuff affects the background. If this suggestion is right, then a background dependent theory cannot be adequate.

 

[coalquay404]

Well maybe my analogy of what I thought GR Background Independence is just way of any sensible mark.
But, sorry guys these explanations are just way to trivial to consider them useful.

Smolin address it as “relational” and having implications in the ‘relational vs. absolute debate’. By the end of his 36 paper he is speculating on how relational non-local hidden variables may be useful in a bell valid solution.

Maybe someday I’ll come across something that makes the significance of background independence clear.

Smolin usually talks about relational spacetimes in the context of attempting to describe spacetime in a fundamentally Machian context. Over the past ten years there has been some interesting work done on this in the context of classical general relativity but it's still unclear whether or not it works even in that case. It's therefore probably a bit premature to start thinking about its relationship to quantum mechanics. That said, there are one or two really interesting results from the relational point of view, most important of which is probably the realisation that, even in a classical sense, spacetime isn't necessarily fundamental but rather is an emergent concept.

 

[JesseM]

Over the past ten years there has been some interesting work done on this in the context of classical general relativity but it's still unclear whether or not it works even in that case. It's therefore probably a bit premature to start thinking about its relationship to quantum mechanics. That said, there are one or two really interesting results from the relational point of view, most important of which is probably the realisation that, even in a classical sense, spacetime isn't necessarily fundamental but rather is an emergent concept.

Do you know any good online papers or books on the "interesting results" you describe? I doubt I'd understand much of it but I'd like to have some information on this for future reference. I remember Julian Barbour was trying to find a "relational" formulation of GR, did he ever come up with anything?

 

[coalquay404]

Well, perhaps the canonical example is this one, and references therein:

The Physical Gravitational Degrees of Freedom
Anderson, et al
gr-qc/0407104

The reason why this paper is interesting is that pretty much all of GR (including the Lichnerowicz-York equation!) is reproduced from Barbour's relational standpoint. I'm still uncertain about the variational method that they employ, but it does look like a promising approach, particularly since it demonstrates that spacetime is an emergent structure even in the classical regime.

The last time I spoke to Julian was in December so I'm unsure of the current level of progress on this. I might email him this week to see what his current thoughts are.

 

[Careful]

Smolin usually talks about relational spacetimes in the context of attempting to describe spacetime in a fundamentally Machian context. Over the past ten years there has been some interesting work done on this in the context of classical general relativity but it's still unclear whether or not it works even in that case. It's therefore probably a bit premature to start thinking about its relationship to quantum mechanics. That said, there are one or two really interesting results from the relational point of view, most important of which is probably the realisation that, even in a classical sense, spacetime isn't necessarily fundamental but rather is an emergent concept.

How, how, RandallB, you cannot impose any idea Smolin might have about the Mach principle or what a relational theory ought to be on GR ! First of all, the Mach principle is a philosophical notion and has no precise meaning as far as I am aware. Some interpret it that the inertia of a body depends on its relative position to neighbouring matter distributions in the Universe, this could lead to Brans Dicke theory. Although such theory violates the strong equivalence principle, it is clearly *local* in the sense that the equations of motion for all fields are second order hyperbolic.

Again, GR as it stands now is already violating the no conspiracy hypothesis in Bell's theorem. One should be very careful in adding extra ``memory capacity'' to elementary particles (fields determining the inertia of a particle for example) so that they synchronise better with the environment. This is what I called in the other thread superdeterminsm and it is as local as applepie is american.

Such appearantly non-local or precausal effects are known for 80 years in classical Maxwell theory (which we all agree upon to be local ), where the Lorentz-Dirac equation has solutions with preacceleration.

Now, I don't know about the work of Barbour at all, so let me ask some silly questions. If there is no time, you need to define speed with respect to a reference observable (say the moon) as d/dt O = {M, O} where O is the position of me (say), M is everything which is to be known about the moon and {,} is our Poisson bracket. O and M need to be observables, so they have to commute with the constraints H_i (dynamics) :
{H_i, O} = 0. I guess it is this what he is talking about ... (but RandallB - all this comes from a *spacetime* theory and the foliation is of no importance here). But such thing would not be a reworking of GR at all, that is simply describing evolution of stuff with respect to other stuff in the same theory - there is even no foliation needed for that (phase space is a covariant concept). Apart from the fact that it is quasi impossible to get a decent ``localized'' observable out of this procedure technically, I can do the same thing a spacetime view.
So what is this all about ? Are Barbour and friends looking for a different theory (Brans Dicke like) or are they trying to get a deeper insight into spacelike correlations between events (especially the neighboring solid stars) which might develop ``over time'' in standard GR?

Careful

 

[RandallB]

How, how, RandallB, you cannot impose any idea Smolin might have ...

I wasn't imposing anything on GR. I was asking a question to those that may know more than I on this issue ---- ? are you confusing me with someone else.

Again, GR as it stands now is already violating the no conspiracy hypothesis in Bell's theorem. ... This is what I called in the other thread superdeterminsm and it is as local as applepie is american.

Glad you brought it up again as I missed it the first time (I'm not sure where first was, but I'll look up Bell Theorem conspiracy hypothesis). I take it you do not agree with others that GR is local, as GR must be in some sense non-local in violating that no conspiracy hypothesis in Bell's theorem. Or is doing so somehow not non-local, just in violation of Bell?

Don't know where you used superdeterminsm before please direct me. As compared to apple pie you must mean it is NOT local as apples are NOT American (European import from long ago & I think they imported it as well.)

 

[Careful]

** I wasn't imposing anything on GR. I was asking a question to those that may know more than I on this issue ---- ? are you confusing me with someone else. **

Of course not, I remember you very well; how could I ever forget you ? :tongue2:

**
Glad you brought it up again as I missed it the first time (I'm not sure where first was, but I'll look up Bell Theorem conspiracy hypothesis). I take it you do not agree with others that GR is local, as GR must be in some sense non-local in violating that no conspiracy hypothesis in Bell's theorem. Or is doing so somehow not non-local, just in violation of Bell?
**

Of course GR is local, even Brans Dicke is local. The conspiracy issue has *nothing* to do with (non-)locality; it is just a measure of ``intelligence'' of subsystems to synchronise (just like we humans have group wavelike behavior although there is no magic action at a distance between us). You repeatedly make this mistake; perhaps study Bell chapter 12, and in particular the paper of Clauser, Horne and Shimony on this issue.


**Don't know where you used superdeterminsm before please direct me. As compared to apple pie you must mean it is NOT local as apples are NOT American (European import from long ago & I think they imported it as well.) **

Well, because we produce it, does not imply that we eat it. So, you might think about Californian orange juice.

Actually, my questions about Barbour were adressed to Coalquay404.

 

[coalquay404]

I'll give a detailed response about the relational viewpoint tomorrow. Unfortunately, we've just had rather a large World Cup party here in Cambridge and I've had too many drinks to be of much use

 

[George Jones]

I'll give a detailed response about the relational viewpoint tomorrow. Unfortunately, we've just had rather a large World Cup party here in Cambridge and I've had too many drinks to be of much use

If you've had a large party today and England hasn't played in two days, what's going to happen on Saturday if England beats Portugal?

 

[RandallB]

need a new word for local (or non-local)

Of course GR is local,... The conspiracy issue has nothing to do with (non-)locality; it is just a measure of ``intelligence'' of subsystems to synchronize (just like we humans have group wavelike behavior although there is no magic action at a distance between us). ... perhaps study Bell chapter 12,

GR is local which might imply it could never complete a BELL solution;
BUT GR is “violating the no conspiracy hypothesis in Bell's theorem” and in that there is enough ‘free’ ‘external’ variable (chpt 12) to give GR enough mathematical freedom (or whatever) to allow it to complete a Bell solution. Even though defined as “local” it can produce “magic action at a distance” as you put above.

Thus GR as a theory has power and flexibility greater than fundamental Special Relativity. That is where SR is not allowed to take advantage of other theories to gain some external freedom of some kind, and is restricted to a Classical 3D reality with time being only a Newtonian measure of change.

Under these conditions experiments have currently shown that a Classical, (even with SR) solution can not produce the required Unknown Variable for a valid Bell solution.

So for my vocabulary please:
=Any theory that cannot produce a Bell Solution as above is “A” (or “non-B”).
=And all theories that can define a valid Bell Solution is “non-A” (or “B”). These would include most all variations of QM whether defined as local or non-local.

I see the communications problem in my using the word 'local' for “A” and complaining that theories with whatever freedom needed, defined within them to allow for a Bell Solution, should not be using the word local.

BUT IF convention is that these extra freedom theories, GR included, are by convention to be called local, -- I guess I can do that.
BUT, I need a clear, accepted, and well recognized term for “A” or “B” that all will accept as differentiating between what I had called ‘local’ and cannot solve Bell; and all those that can solve Bell (non-local or not non-local as they define themselves). You know what I need, what is the replacement word for my use of ‘local’. Does science have such an accepted term for “A” or “B” since ‘local’ in "A" seems to be unacceptable?

If not I’ll just take to calling GR, BM, MWI, etc. non-local local theories. And those unable to produce variables for a Bell solution local local theories (e.g. Classical).

 

[Careful]

**GR is local which might imply it could never complete a BELL solution **

Bell = locality + CFD + no conspiracy + ...

**
BUT GR is “violating the no conspiracy hypothesis in Bell's theorem” and in that there is enough ‘free’ ‘external’ variable (chpt 12) to give GR enough mathematical freedom (or whatever) to allow it to complete a Bell solution. Even though defined as “local” it can produce “magic action at a distance” as you put above. **

Right, but that is also the case for matter fields in SR ; there is no need whatsoever to make the lightcone dynamical in order to achieve that; in contrast to what you say below (think about it !) :

**
Thus GR as a theory has power and flexibility greater than fundamental Special Relativity. That is where SR is not allowed to take advantage of other theories to gain some external freedom of some kind, and is restricted to a Classical 3D reality with time being only a Newtonian measure of change. **


Locality (for a fixed spacetime metric) is very easily defined as follows : any correlation between spacelike separated events can be traced back to a specific interaction mechanism in their common causal past. Actually, this still allows for the inequivalent notions of (a) Einstein locality and (b) local causality (I guess it was called). Einstein Locality states that both agents can have some indefinite memory and that the correlations come from interactions in the *entire* mutual past, while local causality restricts the memory to the most recent common past. QM in the standard orthodoxy is non-local in the sense that the probability of some *outcome* of a measurement by B is immediately infuenced by what happened at A. Bohmian mechanics is non local in the sense that orbits of particles depend upon the global foliation and the specific whereabouts of the other particles ``now''.

Careful

 

[Haelfix]

Ideally you want your theory to be both background dependant and background independant. For instance, let's say you have some manifestly background independant theory where the full metric tensor is varied in the action (eg GR).

For calculational reason, you then might want to split it into a background dependant part (a fixed metric) + perturbations around it. This then let's you expand around the classical solution and see for instance, the presence of gravitational waves (something that is completely hidden in the previous formalism). If you can't do that, it means you have no classical saddle point, and that's very bad, particularly for quantum mechanics where the whole point of quantization requires the presence of such an entity.

So you really want both parts to be present in your theory. At this time, only general relativity has it both ways.

 

[RandallB]

Ideally you want your theory to be both background dependant and background independant. For instance, let's say you have some manifestly background independant theory where the full metric tensor is varied in the action (eg GR).

For calculational reason, you then might want to split it into a background dependant part (a fixed metric) + perturbations around it. This then let's you expand around the classical solution and see for instance, the presence of gravitational waves (something that is completely hidden in the previous formalism). If you can't do that, it means you have no classical saddle point, and that's very bad, particularly for quantum mechanics where the whole point of quantization requires the presence of such an entity.

So you really want both parts to be present in your theory. At this time, only general relativity has it both ways.

Allow me to feedback what you said and you can tell me if I received it close to correct.

General Relativity is background independent, BUT unlike other theories is able to hold a portion of itself in a background dependent form in order to retain some fix on a ‘classical saddle point’.
Other theories are not able to do so as they are:
- exclusively background independent; or
- exclusively background dependent; or
- if they can create a formulation that is background independent it cannot be done so as to retain the needed background dependence in part at the same time to establish the entity described as a ‘classical saddle point’. (And might this be part of QM being “non-local”?)

Now if I’m tracking fairly well with this –
Does GR have a solid enough lock though this entity or ‘classical saddle point’ to build a complete solution to something like entanglement or double slits to be considered “causally local” ?
(I’ll stick with ‘causally local’ for now - I don't see a distinction vs. Einstein Local yet).
Or might the math involved, within the background independence part of GR Smolin et.al. are working on, be not quite sufficient to make that “causally local” claim yet?

 

[coalquay404]

Ideally you want your theory to be both background dependant and background independant. For instance, let's say you have some manifestly background independant theory where the full metric tensor is varied in the action (eg GR).

This doesn't make sense. Background-independence implies one simple thing: that the field equations of the theory are invariant under the action of the diffeomorphism group of the target space. Unsurprisingly, saying that we want a theory to be both background-independent and background-dependent is an oxymoron.

For calculational reason, you then might want to split it into a background dependant part (a fixed metric) + perturbations around it. This then let's you expand around the classical solution and see for instance, the presence of gravitational waves (something that is completely hidden in the previous formalism). If you can't do that, it means you have no classical saddle point, and that's very bad, particularly for quantum mechanics where the whole point of quantization requires the presence of such an entity.

So you really want both parts to be present in your theory. At this time, only general relativity has it both ways.

This is very, very wrong. Perturbation theory in GR is a level of abstraction above the diffeomorphism-invariance of the theory. The fundamental property of GR is that it is fully background-independent. Assuming that we can have a first-order perturbation of the metric of the form

$$g_{ij} \approx \eta_{ij} + h_{ij}$$

is a *physical* assumption designed to make calculations easier in the linearized regime, nothing else. In particular, these types of perturbation expansions of the metric do not alter the central diff-invariance in GR.

 

[Careful]

**
is a *physical* assumption designed to make calculations easier in the linearized regime, nothing else. In particular, these types of perturbation expansions of the metric do not alter the central diff-invariance in GR**


Well, provided you take into account the full expansion, and assuming that the latter makes sense. Relational mechanics ?

 

[coalquay404]

**
is a *physical* assumption designed to make calculations easier in the linearized regime, nothing else. In particular, these types of perturbation expansions of the metric do not alter the central diff-invariance in GR**


Well, provided you take into account the full expansion, and assuming that the latter makes sense. Relational mechanics ?

I'll try to get to the relational idea over the next couple of days. I wrote something on it last night but it's *way* too long to post here.

 

[Haelfix]

Sorry Coalquay I think you misunderstand what I am saying, not that I am disagreeing with you.

By definition, when you go into linearized perturbation theory, you are fixing a metric. This is a background dependant formulation of general relativity. What is lost is *manifest* background independance, not the full symmetry of active spacetime diffeomorphisms (it is hidden, but of course still there as it must), you could say that this choice spontaneously breaks this symmetry.

Assume for instance that we had invented linearized GR first, and did not have a good formulation for the full field equations. You would not necessarily know that GR is fully background independant a priori. You might worry about back reactions not from huv, but rather from Nuv in the perturbative expansion. Indeed, you would have to run consistency checks, to see if the form of the equations comes out the same with different choices of the fixed background metric. (It does!)

This is the situation string theory is in (often called background dependant), even though people are pretty convinced it *is* background independant in that the choice of the metric Guv is arbitrary, and the form of the observables comes out the same regardless of which choice is made.

Contrast that with LQG, which is manifestly background independant, but has no classical limit (it seems to have no classical saddle points).

Are we in agreement?

 

[vanesch]

BUT GR is “violating the no conspiracy hypothesis in Bell's theorem” and in that there is enough ‘free’ ‘external’ variable (chpt 12) to give GR enough mathematical freedom (or whatever) to allow it to complete a Bell solution. Even though defined as “local” it can produce “magic action at a distance” as you put above.

No. What Careful is pointing out is that, given that GR is a deterministic and local theory, the "choices" made at spacelike distances (of which dependence of the "other" probability is Bell's point), are not really choices, but are ALSO determined by the common past lightcone. In other words, in a truly deterministic theory, there are no choices.

What Bell does, is to show that one cannot obtain the quantum probabilities by assuming that the probability of outcome of event A (under "choice" of setting a, and hidden variables of the common past L), and the probability of outcome of event B (under the choice of setting b, and common hidden variables of the common past L):

P(A,B | a,b,L) = P(A | a,L) x P(B | b,L)

cannot hold in all cases, because from the above assumption follow the Bell inequalities which are violated in quantum theory.

Here, "a" stands for the "free choice of setting" of the analyser at side a, and "b" stands for the "free choice of setting" of the analyser at side b, and these "free choices" are supposed to be made arbitrarily, and independently, at spacelike distances.

But what Careful points out, is that in the frame of a deterministic theory, this is begging the question, because in a deterministic theory, free choice doesn't exist, and hence a and b are ALSO determined by stuff from the past ; in other words, by L. Now, if we recognize that a is a function of L, and b is a function of L, then Bell's claim becomes empty, because the entire trick was that a doesn't appear in P(B | b, L), and that b doesn't appear in P(A | a, L). But we can now simply write P(B | L) and P(A | L), and by giving L, a and b are also given, and hence it doesn't even make sense to say that P(A | a, L) "doesn't depend on b", because b is known when L is known.

Strictly speaking, this is correct, and Bell made some extra assumption, which Careful calls the "no conspiracy" assumption, which is that the "free choice of a or of b" doesn't really exist, and is of course determined by L, but that we assume somehow that a and b will be function of a part of L which is statistically independent fro the part of L that will determine the outcome at A. Although this is indeed a totally unfounded assumption, we nevertheless always make it when doing experimental work: we assume that things which "look rather independent" ARE statistically independent (whether there is free will or not). You assume that throwing dice over here is statistically independent from throwing dice over there, even though both are determined sensitively by common past initial conditions. This hypothesis is indeed unfounded, but without it, you cannot do any experimental work.

 

[Careful]

** You assume that throwing dice over here is statistically independent from throwing dice over there, even though both are determined sensitively by common past initial conditions. This hypothesis is indeed unfounded, but without it, you cannot do any experimental work. **

Well, Bell made still other assumptions (which I consider much more dangerous) and one can debate whether these ``loopholes'' are technical or fundamental. I disagree however with your statement in the following sense: in order to make a *theory*, one has to rely upon the absence of conspiracy in *some* experimental situations (otherwise one is guessing blindly). However, once these experiments have provided a theory T, which we consider to be *complete*, then one has to check whether conspiracy can happen or not within T (and conspiracy happens a lot in QM). It is not the problem for experiment whether conspiracy happens or not, it is however up to T to predict experimental outcome !

Careful