No prior geometry and QG

  • I
  • Thread starter mieral
  • Start date
  • #1
203
5

Main Question or Discussion Point

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!
 

Answers and Replies

  • #2
Haelfix
Science Advisor
1,950
212
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)
 
  • Like
Likes atyy
  • #3
203
5
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."
 
  • #4
Haelfix
Science Advisor
1,950
212
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.
 
  • Like
Likes atyy and mieral
  • #5
203
5
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?
 
  • #6
Haelfix
Science Advisor
1,950
212
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 Schwarschild 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.
 
  • Like
Likes atyy
  • #7
203
5
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?
 
  • #8
Haelfix
Science Advisor
1,950
212
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..
 
  • Like
Likes atyy
  • #9
203
5
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?
 
  • #10
atyy
Science Advisor
13,904
2,174
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.
 
  • #11
203
5
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 refered 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.
 
  • #12
haushofer
Science Advisor
Insights Author
2,319
696
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.
 
  • #13
haushofer
Science Advisor
Insights Author
2,319
696
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).
 
  • #14
203
5
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?
 
  • #15
203
5
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?
 
  • #16
PeterDonis
Mentor
Insights Author
2019 Award
30,053
9,241
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.
 
  • Like
Likes mieral
  • #17
203
5
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.
 
  • #18
588
43
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.
 
  • #19
PeterDonis
Mentor
Insights Author
2019 Award
30,053
9,241
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.
 
  • Like
Likes mieral
  • #20
203
5
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)

  • String-nets giving rise to gapless helicity ±2 excitations with no other gapless excitations[57]
  • String Theory
  • Loop Quantum Gravity
 
  • #21
PeterDonis
Mentor
Insights Author
2019 Award
30,053
9,241
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.
 
  • #22
julian
Gold Member
584
105
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
 
  • #23
julian
Gold Member
584
105
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 cant 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 dont 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!
 
  • #24
PeterDonis
Mentor
Insights Author
2019 Award
30,053
9,241
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.
 
  • #25
haushofer
Science Advisor
Insights Author
2,319
696

Related Threads on No prior geometry and QG

Replies
6
Views
3K
  • Last Post
Replies
5
Views
2K
Replies
27
Views
4K
  • Last Post
2
Replies
28
Views
5K
Replies
22
Views
5K
  • Last Post
2
Replies
29
Views
6K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
14
Views
4K
  • Last Post
Replies
2
Views
2K
Top