Spacetime

[thylacine]

[FortunaMaximus]

Chewy.

[51cards]

Is there any way to check expirimentally? If no then I'll just ask the IPU. Bless her holy hooves.

[MoreGentilythanU]

Those are not good words to describe spacetime. Please refer to books on the subject known as GR (general relativity) this should clarify for u maybe ( not really u won't be able to understand it because u need a Phd or to at least be hella good at math srry)

CHEERS!

[FortunaMaximus]

It's still chewy.

JEERS!

[thylacine]

[ QUOTE ]
Those are not good words to describe spacetime. Please refer to books on the subject known as GR (general relativity) this should clarify for u maybe ( not really u won't be able to understand it because u need a Phd or to at least be hella good at math srry)

CHEERS!

[/ QUOTE ]

LOLROFLMAO

Firstly, these surveys are for entertainment purposes only. I know perfectly well the limitations of these words. Secondly, I know about GR and QFT and I know what the various new theories are. I have a better understanding of mathematics than at least 99.9999% of people.

[govman6767]

[ QUOTE ]
[ QUOTE ]
Those are not good words to describe spacetime. Please refer to books on the subject known as GR (general relativity) this should clarify for u maybe ( not really u won't be able to understand it because u need a Phd or to at least be hella good at math srry)

CHEERS!

[/ QUOTE ]

LOLROFLMAO

Firstly, these surveys are for entertainment purposes only. I know perfectly well the limitations of these words. Secondly, I know about GR and QFT and I know what the various new theories are. I have a better understanding of mathematics than at least 99.9999% of people.

[/ QUOTE ]

MoreGentlythanyou just got served !!!!!!!!!

[bigpooch]

Okay, I know you are being tactful, but no need for
exaggeration! Don't you know there are super-nits on SMP ?

[FortunaMaximus]

[ QUOTE ]
[ QUOTE ]
Those are not good words to describe spacetime. Please refer to books on the subject known as GR (general relativity) this should clarify for u maybe ( not really u won't be able to understand it because u need a Phd or to at least be hella good at math srry)

CHEERS!

[/ QUOTE ]

LOLROFLMAO

Firstly, these surveys are for entertainment purposes only. I know perfectly well the limitations of these words. Secondly, I know about GR and QFT and I know what the various new theories are. I have a better understanding of mathematics than at least 99.9999% of people.

[/ QUOTE ]

Hyperbole much?

[malorum]

Whichever model you choose to use for the purpose in question.

The word "is" is somewhat ill-defined.

[ QUOTE ]
"What we observe is not nature itself, but nature exposed to our method of questioning...."

Werner Heisenberg

[/ QUOTE ]

[thylacine]

[ QUOTE ]
Okay, I know you are being tactful, but no need for
exaggeration! Don't you know there are super-nits on SMP ?

[/ QUOTE ]

[ QUOTE ]
Hyperbole much?

[/ QUOTE ]

You think I don't know the difference between 99.999%, 99.9999%, and 99.99999%?

[FortunaMaximus]

I'm sure you do. Your ego's spewing though. It's not all that classy. But if it makes you happy, go ahead.

[thylacine]

[ QUOTE ]
I'm sure you do. Your ego's spewing though. It's not all that classy. But if it makes you happy, go ahead.

[/ QUOTE ]

I was just letting "MoreGentilythanU" know how far off base he was. I was just stating the facts. "MoreGentilythanU" essentially said I was a clueless idiot, so I thought I would carefully quantify how wrong that characterization was. It's a pity it gets misperceived, but so be it. /images/graemlins/crazy.gif

BTW you, "FortunaMaximus", actually got the original post much better than "MoreGentilythanU" did.

[FortunaMaximus]

Why not. I mean, gum is an spacetime artifact. /images/graemlins/tongue.gif

Cheers.

[thylacine]

[ QUOTE ]
Why not. I mean, gum is an spacetime artifact. /images/graemlins/tongue.gif

Cheers.

[/ QUOTE ]

I don't disagree with you. Maybe you could have a survey with more options.

[FortunaMaximus]

Eh. I suppose. Then again, I'm fond of contrarian attitudes that still bring my point across.

Besides, I don't have to prove anything to anybody but myself. If others get helped because of that, well, that's a causative effect I'll never be ashamed of.

I've always gone out of my way to help others when I've felt in the correct frame of mind to do so. Costs nothing, really.

And for good reason. A net gain's a net gain, and the effects involved aren't -EV where I'm concerned.

[FortunaMaximus]

http://img115.imageshack.us/img115/1066/dsc0343720mounted20thylyr7.jpg

Cute-looking animal. Bet it's rather fast too.

[Speedlimits]

[ QUOTE ]
[ QUOTE ]
Those are not good words to describe spacetime. Please refer to books on the subject known as GR (general relativity) this should clarify for u maybe ( not really u won't be able to understand it because u need a Phd or to at least be hella good at math srry)

CHEERS!

[/ QUOTE ]

LOLROFLMAO

Firstly, these surveys are for entertainment purposes only. I know perfectly well the limitations of these words. Secondly, I know about GR and QFT and I know what the various new theories are. I have a better understanding of mathematics than at least 99.9999% of people.

[/ QUOTE ]

yeah right rofl. Why don't you have a phd then

[MoreGentilythanU]

I think my post was a tad harsh, I didn't mean to offend.
I have studied "spacetime" i.e. General Relativity to some extent.
The idea is that time is not so easy to define, and to do a rigourous mathematical definition of time we need to alter the standard (Newtonian) idea of a constant independent time parameter.
Of course that is just the main idea in Special Relativity. General Relativity links space, time and matter.
It shows that Gravitational forces are caused by the "interaction" (for lack of better word, I'm srry this is not too precise) between space and matter. Together with the special relativity this forms a very complete picture of spacetime and matter at normal energy levels and sizes.
Srry for the lecture, but I just wanted to state what is obvious before getting to the point. My main point is that if you want to construct some general description of spacetime it would generally take years of studying the finer points of the GR theory.
I think you obviously know a fair amount, probably more than me. However, if you want to describe spacetime I think you shouldn't trivialize it since it is deep and fascinating subject.


edit: I believe what I meant by the "interaction" of space and matter is that matter "curves" space in a well defined way. Distance is no rather simple and is a complex function known as the "metric" in GR.

[evank15]

Discrete as in quantized?

Graviton?

Am I missing something?

[Metric]

If you directly quantize the gravitational field via loop quantum gravity, you find that the operators corresponding to area and volume have a discrete spectrum. String theory, on the other hand, assumes a continuous manifold with local Lorentz symmetry on which to define the world-sheets of strings. So a case for the reasonableness of either possibility can be made.

[yukoncpa]

[ QUOTE ]
If you directly quantize the gravitational field via loop quantum gravity, you find that the operators corresponding to area and volume have a discrete spectrum. String theory, on the other hand, assumes a continuous manifold with local Lorentz symmetry on which to define the world-sheets of strings. So a case for the reasonableness of either possibility can be made.



[/ QUOTE ]
In general relativity, space time is continuous if I recall. So does string theory correspond with general relativity? I can envision a curvature of space ( sort of, although I don’t exactly know what space time really means in physical terms ). But how does sharing of virtual particles effect gravity? Is there any way this could be explained to a layman? If anyone can do it, I know you can Metric. I’m counting on you.

[thylacine]

[ QUOTE ]
If you directly quantize the gravitational field via loop quantum gravity, you find that the operators corresponding to area and volume have a discrete spectrum. String theory, on the other hand, assumes a continuous manifold with local Lorentz symmetry on which to define the world-sheets of strings. So a case for the reasonableness of either possibility can be made.

[/ QUOTE ]

When I put the survey up I was definitely associating String theory with a continuous spacetime and associating loop quantum gravity with a discrete spacetime. Actually I wanted to instead do a survey saying choose one from a list of purported theories of quantum gravity, but I couldn't be bothered figuring out the list (String theory, loop quantum gravity, and a bunch of other ones) so I did the above survey instead.

[Metric]

[ QUOTE ]

In general relativity, space time is continuous if I recall. So does string theory correspond with general relativity?

[/ QUOTE ]
At low energy/long distances, string theory reproduces GR, though at high energy/short distances there are corrections.

[ QUOTE ]
I can envision a curvature of space ( sort of, although I don’t exactly know what space time really means in physical terms ). But how does sharing of virtual particles effect gravity? Is there any way this could be explained to a layman? If anyone can do it, I know you can Metric. I’m counting on you.

[/ QUOTE ]
"Virtual particles" are a particle physics trick for implementing one of the main principles of quantum mechanics -- that everything that can possibly happen, contributes to the final outcome. So particle physicists sum over virtual particles... Gravity, though, is more than just particles running around on spacetime -- gravity IS spacetime. So if you're in a regime where you can do perturbation theory around a fixed "background" spacetime, you can use virtual gravitons, but if you really want to find out how things are built from the ground up, you've got to get rid of the background -- and thus the concept of virtual gravitons along with it. At least, this is the LQG approach -- a lot of string theorists may deny that this is necessary, but the LQG theorists will say that its the ultimate manifestation of the "background free" nature of physics -- build the theory non-perturbatively with no background at all.

[thylacine]

Metric says:

[ QUOTE ]

"Virtual particles" are a particle physics trick for implementing one of the main principles of quantum mechanics -- that everything that can possibly happen, contributes to the final outcome. So particle physicists sum over virtual particles... Gravity, though, is more than just particles running around on spacetime -- gravity IS spacetime. So if you're in a regime where you can do perturbation theory around a fixed "background" spacetime, you can use virtual gravitons, but if you really want to find out how things are built from the ground up, you've got to get rid of the background -- and thus the concept of virtual gravitons along with it. At least, this is the LQG approach -- a lot of string theorists may deny that this is necessary, but the LQG theorists will say that its the ultimate manifestation of the "background free" nature of physics -- build the theory non-perturbatively with no background at all.

[/ QUOTE ]

I am a mathematician, not a physicist, so I only vaguely understand these issues. But isn't it obvious that this (the bolface part of the quote) is true. My question is, how could it be that the string theorists don't see this fundamental objection to their entire enterprise. It just seems so obvious to me that string theory could not possibly lead to a fundamental theory (unless they address this objection). What are these people doing? Any insights Metric.

[Metric]

[ QUOTE ]
I am a mathematician, not a physicist, so I only vaguely understand these issues. But isn't it obvious that this (the bolface part of the quote) is true. My question is, how could it be that the string theorists don't see this fundamental objection to their entire enterprise. It just seems so obvious to me that string theory could not possibly lead to a fundamental theory (unless they address this objection). What are these people doing? Any insights Metric.

[/ QUOTE ]

This is a key part of the contention between string theorists and the LQG people. String theorists claim that it's enough to be able to go back and forth between (smooth) backgrounds -- and indeed, they can do this. So in a sense, the theory doesn't depend on which background is chosen -- you can do the calculation in background "A" or background "B" (your choice) and get a physical result. String theorists will argue that this is all that is needed to arrive at a "background independent" theory.

The point, though, is that all these backgrounds look the same locally -- they are all smooth manifolds with local Lorentz symmetry. This is a requirement of string theory, as currently formulated, thus LQG people tend to think this represents a kind of background.

LQG takes the view that Lorentz symmetry only needs to be emergent on large enough distance scales. At the Planck scale, there need not be any smooth Lorentz manifold geometry -- only discrete "atoms of spacetime." This is certainly a prediction of LQG, but the problem with LQG is getting from this level up to the macroscopic spacetime that we observe -- it's very computationally difficult. So this is where a lot of work in LQG is focused these days -- trying to get the correct large scale symmetries to show up in the relevant limit. Some string theorists think this is nuts -- that you should start with the manifold structure that we know and love, making connection to real-world observations much more likely. String theorists have their own problems, of course, but getting the right large-scale spacetime symmetries is not one of them, since it's assumed from the beginning.

[thylacine]

[ QUOTE ]
[ QUOTE ]
I am a mathematician, not a physicist, so I only vaguely understand these issues. But isn't it obvious that this (the bolface part of the quote) is true. My question is, how could it be that the string theorists don't see this fundamental objection to their entire enterprise. It just seems so obvious to me that string theory could not possibly lead to a fundamental theory (unless they address this objection). What are these people doing? Any insights Metric.

[/ QUOTE ]

This is a key part of the contention between string theorists and the LQG people. String theorists claim that it's enough to be able to go back and forth between (smooth) backgrounds -- and indeed, they can do this. So in a sense, the theory doesn't depend on which background is chosen -- you can do the calculation in background "A" or background "B" (your choice) and get a physical result. String theorists will argue that this is all that is needed to arrive at a "background independent" theory.

The point, though, is that all these backgrounds look the same locally -- they are all smooth manifolds with local Lorentz symmetry. This is a requirement of string theory, as currently formulated, thus LQG people tend to think this represents a kind of background.

LQG takes the view that Lorentz symmetry only needs to be emergent on large enough distance scales. At the Planck scale, there need not be any smooth Lorentz manifold geometry -- only discrete "atoms of spacetime." This is certainly a prediction of LQG, but the problem with LQG is getting from this level up to the macroscopic spacetime that we observe -- it's very computationally difficult. So this is where a lot of work in LQG is focused these days -- trying to get the correct large scale symmetries to show up in the relevant limit. Some string theorists think this is nuts -- that you should start with the manifold structure that we know and love, making connection to real-world observations much more likely. String theorists have their own problems, of course, but getting the right large-scale spacetime symmetries is not one of them, since it's assumed from the beginning.

[/ QUOTE ]

Thanks for the insight Metric.I am fascinated by this topic, but I don't have the physics background to deeply understand it. But my instinct is that there should be an utterly discrete model of physics, which means I wouldn't even allow any kind of continuum at a fundamental level, not even the complex numbers in Hilbert spaces.

By the way the LQG papers I tried to read (with shallow comprehension or less) still seemed to have some background manifold involved. Has there been any success in getting rid of this background manifold, as far as you know?

[Metric]

[ QUOTE ]
But my instinct is that there should be an utterly discrete model of physics, which means I wouldn't even allow any kind of continuum at a fundamental level, not even the complex numbers in Hilbert spaces.

[/ QUOTE ]
I very much sympathize with this idea -- I've seen it suggested before, but implemented in a very ad-hoc sort of way (simply discretizing the Bloch sphere, for example). It would be great if there were some kind of physical principle, perhaps phrased in information theoretic terms, that would univerally lead to something like this... But it's nearly untouched as far as I'm aware.

[ QUOTE ]
By the way the LQG papers I tried to read (with shallow comprehension or less) still seemed to have some background manifold involved. Has there been any success in getting rid of this background manifold, as far as you know?

[/ QUOTE ]
In LQG one starts with a manifold, and defines a connection "A" on it as a canonical variable. Passing to the quantum theory, one has L^2 functions of A (with respect to some special measure) that form the "states" of the theory. However, these alone are not the "physical" states of the theory -- we want the physical states to be invariant under arbitrary diffeomorphisms of the underlying manifold (this is the expression of the "background free" nature of LQG). After modding out by all diffeomorphisms, though, the underlying manifold has essentially no physical significance -- all physically relevant information is contained on abstract spin networks. So the manifold turns out to be just a mathematical tool with which to build the various state spaces -- it is not to be interpreted as a physical background structure that has to be specified in order for the predictions of the theory to make sense.

[MoreGentilythanU]

[ QUOTE ]
[ QUOTE ]
But my instinct is that there should be an utterly discrete model of physics, which means I wouldn't even allow any kind of continuum at a fundamental level, not even the complex numbers in Hilbert spaces.

[/ QUOTE ]
I very much sympathize with this idea -- I've seen it suggested before, but implemented in a very ad-hoc sort of way (simply discretizing the Bloch sphere, for example). It would be great if there were some kind of physical principle, perhaps phrased in information theoretic terms, that would univerally lead to something like this... But it's nearly untouched as far as I'm aware.

[ QUOTE ]
By the way the LQG papers I tried to read (with shallow comprehension or less) still seemed to have some background manifold involved. Has there been any success in getting rid of this background manifold, as far as you know?

[/ QUOTE ]
In LQG one starts with a manifold, and defines a connection "A" on it as a canonical variable. Passing to the quantum theory, one has L^2 functions of A (with respect to some special measure) that form the "states" of the theory. However, these alone are not the "physical" states of the theory -- we want the physical states to be invariant under arbitrary diffeomorphisms of the underlying manifold (this is the expression of the "background free" nature of LQG). After modding out by all diffeomorphisms, though, the underlying manifold has essentially no physical significance -- all physically relevant information is contained on abstract spin networks. So the manifold turns out to be just a mathematical tool with which to build the various state spaces -- it is not to be interpreted as a physical background structure that has to be specified in order for the predictions of the theory to make sense.

[/ QUOTE ]
holy crap I can't wait to get my Phd in this stuff, coolest discussion ever. Nh.

[thylacine]

[ QUOTE ]

In LQG one starts with a manifold, and defines a connection "A" on it as a canonical variable. Passing to the quantum theory, one has L^2 functions of A (with respect to some special measure) that form the "states" of the theory. However, these alone are not the "physical" states of the theory -- we want the physical states to be invariant under arbitrary diffeomorphisms of the underlying manifold (this is the expression of the "background free" nature of LQG). After modding out by all diffeomorphisms, though, the underlying manifold has essentially no physical significance -- all physically relevant information is contained on abstract spin networks. So the manifold turns out to be just a mathematical tool with which to build the various state spaces -- it is not to be interpreted as a physical background structure that has to be specified in order for the predictions of the theory to make sense.

[/ QUOTE ]

So I saw that you get a Hilbert space spanned (in some sense) by some kind of labelled knotted graphs (up to diffeomorphism) in a manifold. And I see how they could be described combinatorially in some way. But then if you stick in a surface, you Hilbert space seems to get bigger. For each knotted graph you had before, know you have many corresponding to all the different (up to whatever appropriate equivalence) ways the knotted graph can intersect the surface. Then you can stick more and more surface in, and get bigger and bigger Hilbert spaces, until you need every knotted graph (no longer up to diffeomorphism) on your original manifold.

This may be a bit vague, but hopefully you know the issue I'm getting at. So is there really an escape from the manifold in this formulation?

[Metric]

[ QUOTE ]
So I saw that you get a Hilbert space spanned (in some sense) by some kind of labelled knotted graphs (up to diffeomorphism) in a manifold. And I see how they could be described combinatorially in some way.

[/ QUOTE ]
Yes, these go by various names depending on how many of the constraints they solve. Cylindrical functions, spin networks (solve the Gauss constraint), s-knots also sometimes called "abstract spin networks" (solve the diffeomorphism constraint), etc.

[ QUOTE ]
But then if you stick in a surface, you Hilbert space seems to get bigger. For each knotted graph you had before, know you have many corresponding to all the different (up to whatever appropriate equivalence) ways the knotted graph can intersect the surface. Then you can stick more and more surface in, and get bigger and bigger Hilbert spaces, until you need every knotted graph (no longer up to diffeomorphism) on your original manifold.

[/ QUOTE ]
I am not sure what you're talking about here, but I am going to guess that you're talking about something like calculations of black hole entropy, which does involve a background spacetime and inserting a surface corresponding to the black hole horizon, and getting a Hilbert space of "surface geometry" from the spin networks. This is certainly an approximation -- the full quantum state of the spacetime isn't known, but putting in reasonable boundary conditions allows you to calculate the entropy anyway. In this case, the manifold on which you play (and surfaces you consider on it) is indeed supposed to be taken as physical spacetime -- a semiclassical approximation to the full quantum state. This is simply because it is too hard to write out the full quantum state -- this is somewhat analogous to QFT calculations of the EM field inside a conducting cavity. The full EM field, taking into account the atomic structure of the metal isn't explicitly calculated because that is too hard to write down -- instead, reasonable boundary conditions are written down, and the much simpler problem of just the fields in the cavity can be considered.

But this doesn't mean that the formalism in general depends fundamentally on a background spacetime manifold -- it does not. However, introducing one as an approximation can make certain calculations of interest many orders of magnitude easier/more practical to compute.

I am not exactly sure this is what you referring to, but hopefully it illustrates the point that you are worrying about...

[thylacine]

[ QUOTE ]
[ QUOTE ]
So I saw that you get a Hilbert space spanned (in some sense) by some kind of labelled knotted graphs (up to diffeomorphism) in a manifold. And I see how they could be described combinatorially in some way.

[/ QUOTE ]
Yes, these go by various names depending on how many of the constraints they solve. Cylindrical functions, spin networks (solve the Gauss constraint), s-knots also sometimes called "abstract spin networks" (solve the diffeomorphism constraint), etc.

[ QUOTE ]
But then if you stick in a surface, you Hilbert space seems to get bigger. For each knotted graph you had before, know you have many corresponding to all the different (up to whatever appropriate equivalence) ways the knotted graph can intersect the surface. Then you can stick more and more surface in, and get bigger and bigger Hilbert spaces, until you need every knotted graph (no longer up to diffeomorphism) on your original manifold.

[/ QUOTE ]
I am not sure what you're talking about here, but I am going to guess that you're talking about something like calculations of black hole entropy, which does involve a background spacetime and inserting a surface corresponding to the black hole horizon, and getting a Hilbert space of "surface geometry" from the spin networks. This is certainly an approximation -- the full quantum state of the spacetime isn't known, but putting in reasonable boundary conditions allows you to calculate the entropy anyway. In this case, the manifold on which you play (and surfaces you consider on it) is indeed supposed to be taken as physical spacetime -- a semiclassical approximation to the full quantum state. This is simply because it is too hard to write out the full quantum state -- this is somewhat analogous to QFT calculations of the EM field inside a conducting cavity. The full EM field, taking into account the atomic structure of the metal isn't explicitly calculated because that is too hard to write down -- instead, reasonable boundary conditions are written down, and the much simpler problem of just the fields in the cavity can be considered.

But this doesn't mean that the formalism in general depends fundamentally on a background spacetime manifold -- it does not. However, introducing one as an approximation can make certain calculations of interest many orders of magnitude easier/more practical to compute.

I am not exactly sure this is what you referring to, but hopefully it illustrates the point that you are worrying about...

[/ QUOTE ]

I got interested in LQG after reading the Lee Smolin article in Scientific American in about 2003/4, and I tried reading several things (about LQG as well as GR and QFT) a couple of years ago. I only have a patchy, vague understanding of it all, as you can probably tell. (I am a mathematician, not a physicist.) Anyway ....

with what I said above, one of the things I had in mind was the consideration of an area operator A. It seems that such an operator would be associated with a surface S. If x1 and x2 are equivalent spin networks (or something of that ilk) up to diffeomorphism (or something of that ilk), then presumably they are supposed to be the same element of the Hilbert space, so |x1>=|x2>. On the other hand I thought the area calculation would depend on how x1 and x2 intersect the surface S, giving different answers for A|x1> and A|x2>, so that the introduction of surface S meant that |x1> and |x2> would have had to have been considered distinct in the presence of S, even it they could have been safely identified in the absence of any surface. This is what I meant by needing a bigger Hilbert space. Then consider what happens when you consider all area operators.

A similar thing maybe happens when, for spin networks x1 and x3 (or something of that ilk) you want to define <x1|x3>. Maybe it would depend on how x1 and x3 are linked together giving rise to similar issues.

e.g. An unknot may intersect a surface in many ways. Two unknots may link together in many ways.

I know you can take linear combinations of everything in sight, but I don't think that causes any extra problem.

I can't tell if I am just talking complete nonsense, or if I am driving at some fundamental issue here (or something in between).

[Metric]

I see what you're saying now -- this is a rather subtle point. Let me quote from Carlo Rovelli's excellent text "Quantum Gravity."

"Recall that in classical GR we distinguish between a metric g and a geometry [g]. A geometry is an equivalence class of metrics under diffeomorphism. For instance, in three dimensions, the euclidean metric g_a_b = delta_a_b and a flat metric g'_a_b =/ delta_a_b are different metrics, but define the same geometry [g]=[g']. The notion of geometry is diffeomorphism invariant, while the notion of metric is not. On a given manifold with coordinates x, we can define a surface by S=(sigma^1, sigma^2) --> x^a(sigma^i). Then it makes sense to ask what is the area of S in a given metric g_a_b(x), but it makes no sense to ask what is the area of S in a given geometry, because the relative location of S and the geometry is not defined.
.
.
.
Now, in quantum gravity we find precisely the same situation. Above we have defined coordinate surfaces S and regions R, and their areas and volumes. Such coordinate surfaces and regions are not defined at the diffeomorphism-invariant level. However, we can nevertheless define surfaces and regions on the abstract quantum state |s> itself, and associate areas and volumes with them. A region R is simply a collection of nodes. Its boundary is an ensemble of links and defines a surface; we can say that this surface "cuts" these links. A moment of reflection will convince the reader that this is precisely the same situation in the classical theory."

Apparently this is a common source of confusion among people studying this stuff -- understanding what is and is not observable in diffeomorphism invariant physics is a very non-trivial thing.

[siegfriedandroy]

hasnt anyone seen back to the future, which discussed the space-time continuum?

[thylacine]

[ QUOTE ]
I see what you're saying now -- this is a rather subtle point. Let me quote from Carlo Rovelli's excellent text "Quantum Gravity."

"Recall that in classical GR we distinguish between a metric g and a geometry [g]. A geometry is an equivalence class of metrics under diffeomorphism. For instance, in three dimensions, the euclidean metric g_a_b = delta_a_b and a flat metric g'_a_b =/ delta_a_b are different metrics, but define the same geometry [g]=[g']. The notion of geometry is diffeomorphism invariant, while the notion of metric is not. On a given manifold with coordinates x, we can define a surface by S=(sigma^1, sigma^2) --> x^a(sigma^i). Then it makes sense to ask what is the area of S in a given metric g_a_b(x), but it makes no sense to ask what is the area of S in a given geometry, because the relative location of S and the geometry is not defined.
.
.
.
Now, in quantum gravity we find precisely the same situation. Above we have defined coordinate surfaces S and regions R, and their areas and volumes. Such coordinate surfaces and regions are not defined at the diffeomorphism-invariant level. However, we can nevertheless define surfaces and regions on the abstract quantum state |s> itself, and associate areas and volumes with them. A region R is simply a collection of nodes. Its boundary is an ensemble of links and defines a surface; we can say that this surface "cuts" these links. A moment of reflection will convince the reader that this is precisely the same situation in the classical theory."

Apparently this is a common source of confusion among people studying this stuff -- understanding what is and is not observable in diffeomorphism invariant physics is a very non-trivial thing.

[/ QUOTE ]


Thanks, that does clear up some confusion to some extent. At least I understood enough of what I had previously read to get confused. Where can I find `Carlo Rovelli's excellent text "Quantum Gravity."'? Is it a book, or a paper, or what? Thanks!

[Metric]

"Quantum Gravity" is a book available through amazon.com or wherever -- it is by far the best and most complete LQG text yet written. Rovelli used to have a preprint version in pdf format linked on his webpage, but I don't see it there anymore -- if you'd like the preprint version just pm me and I'll email you a copy.