Is GR about fixed curved background or dynamical?

[mieral]

1. Is GR about fixed curved background or dynamical...When Einstein first proposed GR.. did he mean it to have fixed curved background or dynamical?

2. As I understand it. At present. We treat GR as fixed curved background.. so that when we do QFT in curved spacetime, we fix the stress-energy tensor everywhere so that we have a well-defined solution to the EFE, and that fixes the spacetime geometry everywhere. But is this how it should be done.

3. What would happen if you don't fix the stress-energy tensor everywhere but it is dynamical.. can't you do any QFT this way. Is one of the purpose of String theory to make the string Planck size so that you don't have to deal with this problem of dynamical stress-energy tensor.. or do Strings able to somehow still have well defined solution to the EFE in spite of the stress-energy tensor not fixed everywhere.. how does it do that?

4. What other quantum gravity theories able to have well defined solution to the EFE in spite of the stress-energy tensor not fixed everywhere.. how does that particular QG theory do that?

5. And is this problem about dynamic background and still having well defined soluton the EFE exactly the purpose of quantum gravity?

I'm not sure of most sentences above.. so please emphasize whether the answer is yes or no to the 5 questions. thanks!

 

[Haelfix]

What is your background here? The premise you are making is incorrect and is likely based off pop science treatments. Have you studied GR and QFT?

Forget about quantum gravity for a second and ask the following question. BLackholes and Hawking radiation are studied using the semi classical theory. If the theory was nondynamical, then why would people talk about black hole evaporation? If the theory was truly nondynamical, nothing would ever change.. I encourage you to try to square that with what's really being done (which involves a subtle approximation scheme)

 

[mieral]

What is your background here? The premise you are making is incorrect and is likely based off pop science treatments. Have you studied GR and QFT?

Forget about quantum gravity for a second and ask the following question. BLackholes and Hawking radiation are studied using the semi classical theory. If the theory was nondynamical, then why would people talk about black hole evaporation? If the theory was truly nondynamical, nothing would ever change.. I encourage you to try to square that with what's really being done (which involves a subtle approximation scheme)

I saw this message by Peterdonis which made me asked the questions in this thread.

atyy: "Well, the formalism of particles on a fixed curved background is only an approximation. We do expect the motion of all particles to modify the 4D spacetime. So Smolin is right to emphasize background independence."

Peterdonis: "Yes, but if we go beyond that approximation we are going beyond GR, and I was answering your question about what we get when we apply GR. GR has a fixed curved background--in the sense that when we do QFT in curved spacetime, we fix the stress-energy tensor everywhere so that we have a well-defined solution to the EFE, and that fixes the spacetime geometry everywhere. The stress-energy tensor we use can have "back reaction" terms in it which take into account the energy in the quantum fields, but it can only do so in an averaged sense. That's why this approach is only an approximation."

I created this thread to see others input of the above. Is it really true GR has a fixed curved background and Rovelli saying it should not be the case and it should be dynamical in the sense of no prior geometry or what atyy described thus:

"Curved spacetime alone does not mean background independence. The crucial idea of background independence is that if particles move in a different way, then spacetime curvature is different, ie. each pattern of spacetime curvature corresponds to one pattern of particle motion. In a curved fixed spacetime (ie. no background independence), each pattern of spacetime curvature can correspond to more than one pattern of particle motion.
In curved fixed spacetime, there is no coupling between the energy of the particles that move and the curvature of spacetime.
In contrast, Einstein's equation says that the energy of all particles couples to spacetime curvature."

 

[Haelfix]

Peters statement is roughly correct (actually there are small propagating fluctuations that are fully dynamical, as well as large diffeomorphisms on the boundary), but you have neglected the second part of his statement. Namely that the backreaction terms are exactly what puts the dynamics back into the full space time.

A bad analogy is that It's a little bit like trying to recreate a movie from the stills. Each picture is a fixed entity, but when you put them all together you can piece together the full dynamical thing. Of course you introduce approximation artifacts (aliasing), and you miss high frequency information, but a similar story works here.

The real problem is not that the approximation is bad or inconsistent, it's merely incomplete.

 

[mieral]

Peters statement is roughly correct (actually there are small propagating fluctuations that are fully dynamical, as well as large diffeomorphisms on the boundary), but you have neglected the second part of his statement. Namely that the backreaction terms are exactly what puts the dynamics back into the full space time.

A bad analogy is that It's a little bit like trying to recreate a movie from the stills. Each picture is a fixed entity, but when you put them all together you can piece together the full dynamical thing. Of course you introduce approximation artifacts (aliasing), and you miss high frequency information, but a similar story works here.

The real problem is not that the approximation is bad or inconsistent, it's merely incomplete.

Does this only occur in the Planck scale.. is this the essence of the need for quantum gravity to handle and make it exact instead of approximate? They say quantum gravity is needed to understand the singularity or inside the black hole. So if they can create equations that can make particles couple to spacetime curvature, then the equation can automatically make it exact and solve what goes on in the Planck scale? If yes. What are the other goals of quantum gravity?

 

[Haelfix]

So defining what quantum gravity actually is, is a difficult problem. Worse it's hard to even define the problem that quantum gravity is supposed to solve. Different approaches really come at it from wildly different angles and don't necessarily answer the same thing.

Roughly speaking we would like to quantize the gravitational field described by Einsteins equations (or something close to Einsteins equation), or show that such a thing doesn't exist and is only an approximation to something more fundamental.

The semiclassical treatment corresponds to solving the quantization problem in a very strange part of the parameter space. In our example for a Schwarzschild black hole it amounts to solving the problem in the case where Newtons constant is sent to zero, the mass of the hole goes to infinity such that the Schwarcshild radius stays fixed. So we use this known solution, to try to guess the behavior of the full theory (which we call a particular solution from the full quantum gravity theory). What that full theory will tell us, is of course unknown, but it is expected that we will learn about the fate of mathematically pathological objects in the classical theory, like singularities.

 

[mieral]

In the Physics World article "Loop Quantum Gravity" by Carlo Rovelli, he mentioned that: "General relativity is not about physics on curved spacetimes, asymptotic space–times, or connections between theories defined over different backgrounds. It is the discovery that there is no background; no space–time.The challenge for the physicists of the 21st century is to complete the scientific revolution that was started by general relativity and quantum theory. For this we must understand quantum field theory in the absence of a background space–time. Loop quantum is the most resolute attempt to address this problem."

I'd like to know if there are other quantum gravity approach that is not loop quantum gravity where the following is fulfilled too: "You can have a background with test objects in GR, but once we do physics with objects that play an integral role in 'shaping' spacetime there is no longer a physics on a background but the physics and the background are the same thing."

It is very elegant.. but are we stuck with Loop quantum gravity? For those who like the idea but dislike LQG. What other QM approaches have the same elegant ideas above?

 

[Haelfix]

Yes, so background independence, no prior geometry, things like that. They are also difficult to define mathematically, and differ somewhat between approaches, especially in so far as quantum gravity is concerned. I wanted to separate their notion from dynamical gravity, bc strictly speaking they are very different things. Dynamical gravity is a physical property of the classical gravitational field and is a necessary requirement for all approaches, background independance is usually more of an aesthetic requirement on the form that a theory can take. For instance you can write the classical theory of GR in such a way that makes it manifestly background dependant and in another equivalent form that makes it manifestly background indépendant. It is very difficult to use the property as a theory sieve however and that's where most of the pop sci accounts veer of a ledge, and where your questions go wrong.

My advice is to learn the theories you want to learn, and stick to the physical and mathematical predictions that the theory outputs first and foremost, and you can worry about the aesthetics and what it all means later..

 

[mieral]

Yes, so background independence, no prior geometry, things like that. They are also difficult to define mathematically, and differ somewhat between approaches, especially in so far as quantum gravity is concerned. I wanted to separate their notion from dynamical gravity, bc strictly speaking they are very different things. Dynamical gravity is a physical property of the classical gravitational field and is a necessary requirement for all approaches, background independance is usually more of an aesthetic requirement on the form that a theory can take. For instance you can write the classical theory of GR in such a way that makes it manifestly background dependant and in another equivalent form that makes it manifestly background indépendant. It is very difficult to use the property as a theory sieve however and that's where most of the pop sci accounts veer of a ledge, and where your questions go wrong.

My advice is to learn the theories you want to learn, and stick to the physical and mathematical predictions that the theory outputs first and foremost, and you can worry about the aesthetics and what it all means later..

Do you think the word background independence or no prior geometry must be reserved only for region in the Planck scale (area of quantum gravity).. because conventionally.. according to atyy:

"Classical GR is background independent. This is a traditional way of saying that GR has no prior geometry. In special relativity there is a prior geometry of flat spacetime. It is prior geometry because no matter how much matter you put on it, the spacetime is still flat. In GR, you cannot specify your geometry first then put matter as you wish, because matter curves spacetime. Nor can you put matter first, because there is no meaning to "where" without spacetime. So you must put matter and geometry on at the same time, so the geometry is not prior to the matter. This is the sense in which GR has no prior geometry."

So when they mention about background independence in quantum gravity circle.. do they mean in the Planck scale or in the context of atyy classical GR?

 

[atyy]

I think you can forget about background independent. It's not a very useful distinction, and different people mean different things when they say it. Classical gravity can be formulated without a background, and with a background.

 

[mieral]

I think you can forget about background independent. It's not a very useful distinction, and different people mean different things when they say it. Classical gravity can be formulated without a background, and with a background.

In another thread.. you contradicted yourself by stating:
"Curved spacetime alone does not mean background independence. The crucial idea of background independence is that if particles move in a different way, then spacetime curvature is different, ie. each pattern of spacetime curvature corresponds to one pattern of particle motion. In a curved fixed spacetime (ie. no background independence), each pattern of spacetime curvature can correspond to more than one pattern of particle motion.
In curved fixed spacetime, there is no coupling between the energy of the particles that move and the curvature of spacetime.
In contrast, Einstein's equation says that the energy of all particles couples to spacetime curvature."

How do you reconcile your above statement that "Classical GR is background independent". You meant quantum gravity people referred to Planck scale no prior geometry when they talked about "background independent" that has different meaning to the classical GR's case that you described above? please say yes or no to make more clear the distinction and if I understood you right.

 

[haushofer]

Some people have the tendency to regard the Einstein equations as some sort of "static background producing machine", on which we then consider particles and fields to live on. Background independency states that if the fields/particles evolve, the background evolves with it; they are coupled. This is already at the classical level; no QG required.

 

[haushofer]

So when they mention about background independence in quantum gravity circle.. do they mean in the Planck scale or in the context of atyy classical GR?

Already classical. The problem with the quantum case however is that in quantum field theories symmetries restrict the correlators. Theories which are general covariant due to background independency necessarily have correlation functions which are constant all over spacetime (see e.g. Zee's GR book).

 

[mieral]

Some people have the tendency to regard the Einstein equations as some sort of "static background producing machine", on which we then consider particles and fields to live on. Background independency states that if the fields/particles evolve, the background evolves with it; they are coupled. This is already at the classical level; no QG required.

Please comment what you understand by "fixed curved background".. I thought the opposite was "dynamic curved background" but Haelfix said it wasn't. So what should we call the opposite of "fixed curved background"... "unfixed curved background"?

Also remember what Peterdonis said that "GR has a fixed curved background--in the sense that when we do QFT in curved spacetime, we fix the stress-energy tensor everywhere so that we have a well-defined solution to the EFE, and that fixes the spacetime geometry everywhere."

What would happen if we don't fix the stress-energy tensor everywhere? How do you make QFT that won't fix the spacetime geometry everywhere?

 

[mieral]

Yes, so background independence, no prior geometry, things like that. They are also difficult to define mathematically, and differ somewhat between approaches, especially in so far as quantum gravity is concerned. I wanted to separate their notion from dynamical gravity, bc strictly speaking they are very different things. Dynamical gravity is a physical property of the classical gravitational field and is a necessary requirement for all approaches, background independance is usually more of an aesthetic requirement on the form that a theory can take. For instance you can write the classical theory of GR in such a way that makes it manifestly background dependant and in another equivalent form that makes it manifestly background indépendant. It is very difficult to use the property as a theory sieve however and that's where most of the pop sci accounts veer of a ledge, and where your questions go wrong.

My advice is to learn the theories you want to learn, and stick to the physical and mathematical predictions that the theory outputs first and foremost, and you can worry about the aesthetics and what it all means later..

How can you write the classical theory of GR in such a way that makes it manifestly background dependant and in another equivalent form that makes it manifestly background indépendant?

What mathematical concepts does it fall under or is involved? Is it diffeomorphisim invarance, general covariance? what?

 

[PeterDonis]

We treat GR as fixed curved background.. so that when we do QFT in curved spacetime

You're mixing up two different things here.

When we are doing classical GR, we solve the Einstein Field Equation to find out what the spacetime geometry is. We don't assume a fixed background.

When we are doing semi-classical QFT in curved spacetime on a fixed background, we take some classical solution of the Einstein Field Equation (which we found by the above method), and use it as a fixed background on which to do QFT.

we fix the stress-energy tensor everywhere

Yes, but we don't have to do that just once. We can make multiple tries. If the quantum fields we find have significant stress-energy of their own, their stress-energy (more precisely, the expectation value of the stress-energy operator corresponding to the fields) might not be consistent with the stress-energy tensor we fixed when we started out. But we can keep on trying different possibilities until we find a self-consistent solution for both at the same time--a set of quantum fields on a curved spacetime which is also a solution of the Einstein Field Equation when the expectation value of the stress-energy operator for those fields is used as the stress-energy tensor. (This is called taking "back reaction" into account, as I described in what you quoted from me earlier in this thread. But, as I noted there, it is only an approximation, because we are using the expectation value of the stress-energy operator, which is only a kind of average.)

Is it really true GR has a fixed curved background

No. See above. Don't confuse classical GR with semi-classical QFT in curved spacetime.

 

[mieral]

You're mixing up two different things here.

When we are doing classical GR, we solve the Einstein Field Equation to find out what the spacetime geometry is. We don't assume a fixed background.

When we are doing semi-classical QFT in curved spacetime on a fixed background, we take some classical solution of the Einstein Field Equation (which we found by the above method), and use it as a fixed background on which to do QFT.

Why do we have to use fixed background on which to do QFT.. why not unfixed background to do QFT? And what is the standard word for "unfixed background"?

Yes, but we don't have to do that just once. We can make multiple tries. If the quantum fields we find have significant stress-energy of their own, their stress-energy (more precisely, the expectation value of the stress-energy operator corresponding to the fields) might not be consistent with the stress-energy tensor we fixed when we started out. But we can keep on trying different possibilities until we find a self-consistent solution for both at the same time--a set of quantum fields on a curved spacetime which is also a solution of the Einstein Field Equation when the expectation value of the stress-energy operator for those fields is used as the stress-energy tensor. (This is called taking "back reaction" into account, as I described in what you quoted from me earlier in this thread. But, as I noted there, it is only an approximation, because we are using the expectation value of the stress-energy operator, which is only a kind of average.)
No. See above. Don't confuse classical GR with semi-classical QFT in curved spacetime.

 

[RockyMarciano]

How can you write the classical theory of GR in such a way that makes it manifestly background dependant and in another equivalent form that makes it manifestly background indépendant?

By letting the concept be mathematically ill-defined enough to allow contradiction in the term(that is being both A and not A, dependent and independent at once) without affecting the rest of the math

What mathematical concepts does it fall under or is involved? Is it diffeomorphisim invarance, general covariance? what?

It doesn't clearly fall under a clear mathematical concept because of the above, but it is loosely related to the ones you mention(although general covariance is not a well defined mathematical notion either).
The physical problem comes with the requirement of independence of coordinates for any plausible physical theory, that in GR's case is associated to background independence, but since you can express GR both as background dependent and independent, this has caused a certain amount of eyebrow raising over the years. But everything is fine of course.

 

[PeterDonis]

Why do we have to use fixed background on which to do QFT.. why not unfixed background to do QFT?

Because with the current tools we have to do QFT, you have to know the background spacetime (and it has to be locally Lorentz invariant) in order to construct the theory at all. In other words, we do not have a version of QFT (that I'm aware of) in which we can dynamically solve for the QFT and the background spacetime at once. The best we can do is what I described before, where if we come up with a QFT whose expectation value of the stress-energy tensor doesn't match the fixed background spacetime geometry via the Einstein Field Equation, we go back and try again.

what is the standard word for "unfixed background"?

I don't know if there is one.

 

[mieral]

Because with the current tools we have to do QFT, you have to know the background spacetime (and it has to be locally Lorentz invariant) in order to construct the theory at all. In other words, we do not have a version of QFT (that I'm aware of) in which we can dynamically solve for the QFT and the background spacetime at once. The best we can do is what I described before, where if we come up with a QFT whose expectation value of the stress-energy tensor doesn't match the fixed background spacetime geometry via the Einstein Field Equation, we go back and try again.
I don't know if there is one.

If someday we develop a QFT in which we can dynamically solve for the QFT and the background spacetime at once. Is it automatically called Quantum Gravity even though it doesn't quantize the gravitational field described by Einsteins equations?

And for others as well. Which of the following quantum gravity approaches try to dynamically solve for the QFT and the background spacetime at once? (list taken from Wikipedia entry on quantum gravity)

• Asymptotic safety in quantum gravity

• Euclidean quantum gravity

• Causal dynamical triangulation[47]

• Causal fermion systems,[48][49][50][51][52][53]

• Causal sets[54]

• Covariant Feynman path integral approach

• Group field theory[55]

• Wheeler–DeWitt equation

• Geometrodynamics

• Hořava–Lifshitz gravity

• MacDowell–Mansouri action

• Path-integral based models of quantum cosmology[56]

• Regge calculus

• Scale relativity

• Shape Dynamics

• String-nets giving rise to gapless helicity ±2 excitations with no other gapless excitations[57]

• Superfluid vacuum theory a.k.a. theory of BEC vacuum

• Supergravity

• Twistor theory[58]

• Canonical quantum gravity

• E8 Theory

• Quantum holonomy theory[59]

• String Theory

• Loop Quantum Gravity

 

[PeterDonis]

If someday we develop a QFT in which we can dynamically solve for the QFT and the background spacetime at once. Is it automatically called Quantum Gravity even though it doesn't quantize the gravitational field described by Einsteins equations?

I have no idea. That's question about words, not physics.

Also, rather than try to extend the semi-classical approach in this way (with quantum fields but a classical background spacetime), all of the quantum gravity approaches I'm aware of are trying to quantize spacetime--or at least to build a quantum theory of something whose classical limit looks like spacetime, i.e., like the geometric structure related to stress-energy that is described by Einstein's Equations.

Which of the following quantum gravity approaches try to dynamically solve for the QFT and the background spacetime at once?

None of them, as far as I know. See above.

 

[julian]

Once you accept/treat spacetime geometry as dynamical, you have to accept that a fundamental symmetry is diffeomorphism invariance - this implies that a solution of Einstein's equations is not a single curved spacetime but an equivalence class of spacetimes related to each other through diffeomorphisms. This symmetry expressly forbids a priori individuation of the points of a spacetime manifold as spatio-temporal events - background-independence. If you think this BI is "aesthetic requirement" - then try exctracting physical meaninful physical predictions given this symmetry, this is a rather non-trivial task!

Already classical. The problem with the quantum case however is that in quantum field theories symmetries restrict the correlators. Theories which are general covariant due to background independency necessarily have correlation functions which are constant all over spacetime (see e.g. Zee's GR book).

See the paper (an early paper in a series of papers) "Particle scattering in loop quantum gravity" by Rovelli at el

https://arxiv.org/pdf/gr-qc/0502036.pdf

They say

"A well-known difficulty of background independent quantum field theory is given by the fact that if we assume (1) to be well-defined with general-covariant measure and action, then then-point function is easily shown to be constant in spacetime (see for instance [3]). This is the difficulty we address here."

The basic idea is

"Consider a diffeomorphism invariant theory including the gravitational field. Assume that the equations above hold, in some appropriate sense. The field ##\phi## represents the gravitational field, as well as any eventual matter field, and we assume action and measure to be diffeomorphism invariant. Two important facts follow [6]. First, because of diffeomorphism invariance the boundary propagator ##W[\phi,\Sigma]## is independent from (local deformations of) the surface ##\Sigma##. Thus in gravity the left hand side of (3) reads ##W [\phi]##. Second, the geometry of the boundary surface ##\Sigma## is not determined by a background geometry (there isn’t any), but rather by the boundary gravitational field ##\phi## itself."

A generally covariant definition of ##n-##point functions can then be based on the idea that the distance between physical points–arguments of the ##n-##point function is determined by the state of the gravitational field on the boundary of the spacetime region considered. The claim is this way correlation functins can be formulated in a fully background–independent manner

 

[julian]

I'd like to know if there are other quantum gravity approach that is not loop quantum gravity where the following is fulfilled too: "You can have a background with test objects in GR, but once we do physics with objects that play an integral role in 'shaping' spacetime there is no longer a physics on a background but the physics and the background are the same thing."

It is very elegant.. but are we stuck with Loop quantum gravity? For those who like the idea but dislike LQG. What other QM approaches have the same elegant ideas above?

See the Newton Lecture 2010 given by Witten:



In this talk he argues that it is impossible for local scalar field that depends on a spacetime point ##x## to be gauge invariant under diffeomorphisms (well, except the trivial case of a field which is constant over all of spacetime - similar to the argument that says if the theory is diffeomorphism invaraint then the correlation functions must be constant over all of spacetime). He says:

"Now in the context of gravity there can't be a gauge invariant local field ##\phi## of ##x## where ##\phi## is the field and ##x## is the spacetime point ... The reason is that ##x## itself isn't gauge invariant, Einstein's gauge symmetry - the principle of general covariance - invaraince of the theory under diffeomorphisms of spacetime - the gauge symmetry exactly acts on ##x## and therefore it is impossible for a local field ##\phi## that depends on ##x## to be a gauge invariant concept in General relativity..."

"...So a theory of quantum gravity is actually not going to have local fields that are functions of spacetime, as we have in other branches of physics ... A theory with gauge-invariant local fields cannot describe quantum gravity."

I think possible Witten is talking about Dirac observables? Anyway, so the good thing about strings, if I understand what Witten is alluding to, is that you may be able to have gauge-invariant NON-local fields as strings are extended objects instead of point particles. This is intriguing but I don't know how well developed the idea is.

However, Rovelli and general relativists tend to interpret general realtivity as a relational theory, where Dirac observables are relegated. To formalise things Rovelli introduces two different notions of obseravble:

Partial observable: a physical quantity to which we can associate a (measuring) procedure leading to a number.

Complete observable: a quantity whose value can be predicted by the theory (in classical theory); or whose probability distribution can be predicted by the theory (in quantum theory).

(A complete observable gives the correlation between partial observables, and is actually a one-parameter family of Dirac observables).

So it is possible for a local scalar field to have physical revelance - for example, in classical and quantum cosmology people often use a scalar field as a clock variable with respect to which other measurable quantities evolve. This is a relational view.

By the way, mieral, I'm a fan of LQG!

 

[PeterDonis]

this implies that a solution of Einstein's equations is not a single curved spacetime but an equivalence class of spacetimes related to each other through diffeomorphisms

You are misstating this somewhat. A "single curved spacetime" is a geometric object, which is characterized by its geometric invariants. There will be an equivalence class of descriptions of this single curved spacetime in different coordinate charts, which we can think of as mathematical solutions of Einstein's equations expressed in these different coordinate charts; and these descriptions will be related to each other through diffeomorphisms. But all of those descriptions will have the same geometric invariants; that's how we know they are all describing the same single curved spacetime.

This symmetry expressly forbids a priori individuation of the points of a spacetime manifold as spatio-temporal events - background-independence

No, it doesn't. It just means you have to individuate the events by geometric invariants, not by their coordinates.

 

[haushofer]

No, it doesn't. It just means you have to individuate the events by geometric invariants, not by their coordinates.

I'd say it does; that's the whole point of Einstein's hole-argument. See e.g.

http://www.rug.nl/research/portal/e...ed(fb063f36-42dc-4529-a070-9c801238689a).html

page 25 onward. (I guess you know this, so I'm probably misunderstanding something, but this is how i formulated it in my thesis ;) )

 

[RockyMarciano]

You are misstating this somewhat. A "single curved spacetime" is a geometric object, which is characterized by its geometric invariants. There will be an equivalence class of descriptions of this single curved spacetime in different coordinate charts, which we can think of as mathematical solutions of Einstein's equations expressed in these different coordinate charts; and these descriptions will be related to each other through diffeomorphisms. But all of those descriptions will have the same geometric invariants; that's how we know they are all describing the same single curved spacetime.

No, it doesn't. It just means you have to individuate the events by geometric invariants, not by their coordinates.

It seems to me it is not clear here what the invariants should be, or are you suggesting that geometric invariants are compatible with diffeomorphism invariants in a theory with dynamic(changing) metric fields. If the metrics(and their derived curvatures) are considered geometric how can they be invariant if they are changing in the theory? Mathematically either diffeomorphism invariance or geometric invariance must go, no?

Background independence in a dynamic gravitational theory implies EFE solutions are no geometric invariants to depend on in the background, while background dependence is the view you comment above with a "single curved spacetime" as a geometric invariant object that simply can be described by different coordinates related by diffeomorphisms that are not invariant between all possible curved spacetimes(so no general diffeomorphism invariance in this interpretation). Are both views really compatible in your opinion? It doesn't seem so by your quoted answers, could you clarify?

 

[PeterDonis]

are you suggesting that geometric invariants are compatible with diffeomorphism invariants in a theory with dynamic(changing) metric fields

The word "changing" is misleading here. Spacetime is a 4-dimensional geometric object; it doesn't change. Each event in this 4-dimensional geometric object is individuated by the values of geometric invariants at that event; those values don't change. "Change" here just means that we look at the values of invariants at different events and see that they are different. It doesn't mean the geometry itself is "changing".

So the answer to your question is "yes", but in order to understand why, you have to understand the issue with the word "changing" that I just described.

 

[PeterDonis]

that's the whole point of Einstein's hole-argument.

I think the issue here is over the word "manifold". If you restrict that word to just mean the mathematical model (not the geometric object being modeled), then yes, the hole argument means you can't individuate points in the manifold by geometric invariants, because you can always apply a diffeomorphism that "moves" a given geometric invariant to a different point in the manifold. But that doesn't "move" the geometric object itself.

To see the distinction, consider: the surface of the Earth is a geometric object. There are points on it individuated by invariants, such as Big Ben in London (call this point A) and the Empire State Building in New York (call this point B). Now suppose I have two charts of the Earth, one using stereographic projection about the North Pole and one using a Mercator projection centered on the equator. These two maps can be thought of as two different "manifolds" related by a diffeomorphism, and considered that way, points A and B are different points in the two manifolds (roughly, because they have different coordinates in the two charts). This is my understanding of what Einstein was trying to say with his hole argument. But points A and B are still the same points on the Earth itself--the geometric object that the charts are modeling. I don't see that the hole argument refutes that point, which is the point I have been making.

In your thesis, you say that points can only be physically interpreted after one introduces a metric; this is basically the same thing I'm saying. "Geometric invariants" are derived from the metric, so if you don't have a metric, you don't have any geometric invariants.

 

[RockyMarciano]

The word "changing" is misleading here. Spacetime is a 4-dimensional geometric object; it doesn't change. Each event in this 4-dimensional geometric object is individuated by the values of geometric invariants at that event; those values don't change. "Change" here just means that we look at the values of invariants at different events and see that they are different. It doesn't mean the geometry itself is "changing".

So the answer to your question is "yes", but in order to understand why, you have to understand the issue with the word "changing" that I just described.

I would say the word changing or dynamic metric field is clear enough and doesn't refer to the change you are describing. Basically because this concept of being dynamic is used to differentiate GR from SR(and virtually most other theory), and SR spacetime(Minkowski's) is certainly also considered a 4-dimensional geometric object that doesn't change in the sense you mean, so how would this sense of change distinguish GR from SR(or other theories)?

No, I would say that when talking about dynamic gravitational fields one must think about what really changes in GR metrics with respect to SR, which is the curvature, different sources configurations give different curvatures from different metrics that are solutions of the EFE, this is what leads to a background independence from any particular metric. This is what actually is different from between GR and other theories, that the sources determine not only fields on a spacetime but spacetime itself, and in this sense the spacetime is dynamic in a new sense.

 

[julian]

You are misstating this somewhat. A "single curved spacetime" is a geometric object, which is characterized by its geometric invariants. There will be an equivalence class of descriptions of this single curved spacetime in different coordinate charts, which we can think of as mathematical solutions of Einstein's equations expressed in these different coordinate charts; and these descriptions will be related to each other through diffeomorphisms. But all of those descriptions will have the same geometric invariants; that's how we know they are all describing the same single curved spacetime.
No, it doesn't. It just means you have to individuate the events by geometric invariants, not by their coordinates.

Just to check, are you referring to mere coordinate transformations? I ask as it is a common misunderstanding that, in the context of GR, diffeomorphisms mean mere coordinate transformations; no, you are meant to understand diffeomorphisms as a mathematician defines them. For example transforming a doughnut-shaped manifold into its coffe-cup-shaped copy. Diffeomorphisms are the true gauge symmetries of GR. Such transformations are more radical than mere coordinate transformations. They are distiguished from mere coodinate transformations in that they change the functional form of the metric tensor function while remaining in the same coordinate system. Therefore, these other spacetimes within the equivalence class under diffeomorphisms are truly goemetrically distinct.

 

[julian]

In post 22 I meant to write "...then the ##n##-point function is easily shown to be constant in spacetime..."

 

[RockyMarciano]

I think the issue here is over the word "manifold". If you restrict that word to just mean the mathematical model (not the geometric object being modeled), then yes, the hole argument means you can't individuate points in the manifold by geometric invariants, because you can always apply a diffeomorphism that "moves" a given geometric invariant to a different point in the manifold. But that doesn't "move" the geometric object itself.

To see the distinction, consider: the surface of the Earth is a geometric object.

But how is a spherical surface an analogy to GR manifolds? In what sense is a sphere dynamical in the way GR metrics are?

 

[julian]

For those unaware of Einstein's hole argument, let me sketch it.

First of all let me get something out of the way. Say you have a metric tensor function that solves Einstein's equations in ##x-##coordinates, let us denote it ##g_{ab} (x)##. Given another coordinate system, denote it the ##y-##coordinates, there should exist a metric tensor function in the ##y-##coordinate system that imposes the same geometry that ##g_{ab} (x)## imposes in the ##x-##coordinates. We want this to also be a solution of the field equations. This is guaranteed if write the field equations in tensor form. Standard textbooks tell you about this.

However, standard textbooks often miss what about what I am about to explain (Einstein didn't miss this!). O.K. say we have the vacuum field equations in the ##x-##coordinates:

##
R_{ab} (x) = 0 \qquad Eq (1)
##

this is an horrendous differential equation where the independent variable is ##x##. Let us consider another coordinate system, call them the ##y-##coordinates. Einstein required that the laws of physics take the same form in all coordinate systems. Therefore the vacuum field equations in the ##y-##coordinates coordinates should be given by exactly the same differential equation but now the independent variable is ##y##:

##
R_{ab} (y) = 0 \qquad Eq (2)
##

So as soon as we find a metric tensor function, denote it ##g_{ab} (x)##, that solves the field equations in the ##x-##coordinates, simply write down exactly the same function but replace ##x## by ##y## and this will solve the field equations in the ##y-##coordinate system! Denote this new metric tensor function ##\tilde{g}_{ab} (y)##. Now, because it has the same functional form as ##g_{ab} (x)## but belongs to a different coordinate system, it imposes a different spacetime geometry! This may come as a shock to some people, but it is correct.

Now comes the problem - roughly the Hole argument. What if the ##x## and ##y-##coordinates coincide at first but differ after ##t=0##. You will have two geometrically distinct solutions after ##t=0## but which have the same initial boundary conditions at ##t=0##! EEk! The conclusion is that GR does not uniquely predict the spacetime geometry after ##t=0##! Einstein initially recoiled from this and dropped the principle of general covariance only to return to it. The resolution was to realize that we have a gauge transformation, and we have to understand what is then physically meaningfull given this gauge symmetry.

So what was this transformation taking ##g_{ab} (x)## to ##\tilde{g}_{ab} (y)##? Well, first observe that as they have the same functional form they satisfy:

##
g_{ab} (x^1 = u_1,x^2 =u_2, x^3=u_3, x^4=u_4) = \tilde{g}_{ab} (y^1 = u_1,y^2 =u_2, y^3=u_3, y^4=u_4)
##

where ##u_1,u_2,u_3,u_4## take values with the region of overlap between the two coordinate charts. If you think about this relation, you will realize that these two solutions are related by taking the metric tensor function ##g_{ab} (x)## and actively dragging it over the manifold while keeping the coordinate lines “attached” (see the figure I have provided - the value of ##\tilde{g}_{ab}## at ##P## coincides with the value of ##g_{ab}## at ##P_0##). I won't go into the details of it, but this corresponds precisely to a diffeomorphism as a mathematician would define it.

You can read more about the Hole argument and what is physically meanifull in Rovelli's book, a draft version available at

http://www.cpt.univ-mrs.fr/~rovelli/book.pdf

p.s. if you do a coordinate transformation on ##\tilde{g}_{ab} (y)## going from the ##y-##coordinate system to ##x-##coordinate system, the resulting metric tensor function in the ##x-##coordinates will have a different functional form to ##g_{ab} (x)##, as I alluded to in my previous post.

 

[PeterDonis]

how is a spherical surface an analogy to GR manifolds?

It's a geometric object.

In what sense is a sphere dynamical in the way GR metrics are?

A sphere is an unchanging 2-dimensional geometry. Spacetime is an unchanging 4-dimensional geometry. The only difference between them is that the sphere has a positive definite metric and spacetime does not.

It is true that we aren't considering the sphere as being produced by solving any field equations, whereas spacetime is. But that difference has nothing to do with how we individuate events in the manifold by geometric invariants. See below.

I would say that when talking about dynamic gravitational fields one must think about what really changes in GR metrics with respect to SR, which is the curvature, different sources configurations give different curvatures from different metrics that are solutions of the EFE

This is all true, but, as above, it is irrelevant to the question of how we individuate events in a given 4-dimensional spacetime geometry. In order to have a given manifold, you must already have solved the EFE, so all the "dynamic" stuff you are talking about is already done. Once you have a given geometry, individuating points in it has nothing to do with how you derived it from a field equation. You just look at the invariants derived from the metric that you have already found. You don't change the metric in the course of doing that. So any talk about "changing" or "dynamic" metrics is irrelevant to that particular question, of how points in the geometry are to be individuated, which is the question at issue in the hole argument.

 

[PeterDonis]

you are meant to understand diffeomorphisms as a mathematician defines them. For example transforming a doughnut-shaped manifold into its coffe-cup-shaped copy

But doing that changes the geometric invariants, and therefore changes the physics (if we are talking about applying such an operation to a spacetime geometry). The doughnut and the coffee cup are different geometries. Similarly, two spacetimes which are related by a diffeomorphism of the kind you describe (an "active diffeomorphism", as opposed to a "passive" one) are different geometries. And different geometries in GR means different physical predictions, so there is no point in asking which points in the different geometries are "the same".