Gravitons & GR: Does Curved Space-Time Exist?

[εllipse]

Would a graviton theory replace the curved geometry of general relativity? If gravitons are the cause of the gravitational force, does that mean space and time aren't really curved as Einstein thought?..

And maybe an unrelated question, is it better to think of curved space-time as a model that explains the world very well, rather than the way the world actually is?

 

[Haelfix]

No, the two theories are equivalent, at least in the linearized form and too first order. Its one of the great oddities/mysteries actually, a quantum theory of a spin 2 particle reproduces exactly (minus a tiny observationally negligable discrepancy) the same equations as General relativity. Its quite beautiful really, the overlap of the geometry within gauge fields and the spacetime geometry of Einstein.

However there are higher order quantum radiative corrections, and they are testable in principle (but not in practise), and there are rather deep conceptual questions within the field (like conformal anomalies, global topological problems, measurement problems, back reactions and the like) that have never really been understood. Supersymmetry was added to the mix in the eighties, and theorists almost thought they had the whole shebang solved, but then a rather unfortunate high order term proved that the theory was nonrenormalizable. That was then one of the principle motivations of string theory, as their formalism cured that divergence, otoh that theory has problems of its own, like excessive amounts of degrees of freedom and it gets really messy.

Its a strange situation, and quite amazing that one can get so close yet remain so far away.

 

[marcus]

And maybe an unrelated question, is it better to think of curved space-time as a model that explains the world very well, rather than the way the world actually is?

better to think of the world really being curved spacetime (the model used to talk about stellar collapse, observed kind of black holes, big bang, inflation scenarios, dark energy/accelerated expansion etc.)

I think it is is better to think of the "graviton" approximation as something that is typically useful in static almost linear situations. Haelfix indicates where the graviton approximation is applicable:

No, the two theories are equivalent, at least in the linearized form and to first order.

at least, that is, where there is not too great a concentration of matter, or too great fluctuation in curvature. Then you can approximate spacetime by a fixed 4D spacetime like e.g. minkowski flat (which is the spacetime associated with zero matter and zero curvature) and you superimpose on top of that a little ripple or perturbation. To FIRST ORDER that gives an approximation of the real thing, namely your basic curved spacetime.

you are approximating a dynamically curved thing by a static typically flat thing that you have added a little ripple or bump onto.

this is called a "perturbative" approach (take a fixed standardized typically flat thing and perturb it slightly, calculate first order effects)

conventional wisdom is that perturbative approach doesn't apply in regimes of high curvature and highly changeable geometry. It is a marvel (as Haelfix notes) that it works at all, even in the comparatively flat cases.

Would a graviton theory replace the curved geometry of general relativity? If gravitons are the cause of the gravitational force, does that mean space and time aren't really curved as Einstein thought?..

No, I don't think so. It is a marvel that the graviton picture works so well as a perturbative first-order approximation of reality.

Personally I doubt it is helpful to think of "gravitational force" bending lightrays as they pass by the sun by means of clouds of gravitons whizzing back and forth. It is more like those are just the geodesics in our curved spacetime. Nor does it help me to think of a black hole attracting another black hole by the two singularities sending clouds of gravitons whizzing back and forth between them---how do the gravitons escape from one to get to the other etc etc.

So I consider gravitons as marvelous wonderful approximations in the static typically flat cases they apply, and as elegant mathematical constructs. But in answer to your question they are not the fundamental " cause of gravitational force" and NO it does not mean that "space and time aren't really curved as Einstein thought?"

I'm following current research in non-perturbative quantum gravity, which has been making important progress recently. In the nonperturb. approaches to QG you do not use gravitons, you basically randomize the geometry of spacetime. so you have a quantum (curved) spacetime.
Non-perturbative quantum gravity typically means the kind of stuff at the Loops 05 conference
http://loops05.aei.mpg.de/
the graphics there can give some idea

if you do quantum gravity these days then most likely you think of spacetime as curved----only the curvature is uncertain as quantum things usually are.

 

[ohwilleke]

There is considerably dispute in the quantum gravity community over whether space-time itself is discrete, or whether only a quantum particle like a graviton should be used to model the universe.

Also, I'm inclined to think that FIRST ORDER and linearized is a bit of an understatement. Newtonian gravity gets you there.

 

[marcus]

There is considerably dispute in the quantum gravity community over whether space-time itself is discrete, or whether only a quantum particle like a graviton should be used to model the universe.
...

hi Ohwilleke!

let's focus on the nonperturbative QG community
this conference practically speaking defines that community
http://loops05.aei.mpg.de/

here is the list of invited speakers
http://loops05.aei.mpg.de/index_files/Programme.html

you will be teaching me something I'd like very much to know if you would please tell me which of these listed researchers thinks that a
graviton should be used to model the universe.

you may know something about one of these people that i don't, probably you do. Please give me an arxiv link to a paper by one of them that illustrates this.

when I read non-perturb QG papers I rarely if ever see mention of graviton (except as an approximation to variable geometry in almost flat or vacuum situations)

I also do not see spacetime modeled as a collection of PARTICLES. I see examples of discrete structure (discrete spectra of area and volume operators) but the structure is implemented with extended objects like networks or triangulations or spinfoams or manifolds. I don't see anything in the research that I would picture as made of point-particles or wavelets in some static medium.

You must have something "particular" in mind----some "particly" picture of spacetime in somebody's research article. I'd like to see it

For me it's a question of what are the right words and mental images to use in discussing current nonperturbative QG directions.

 

[marcus]

I am tryng to make sense of what you say here

There is considerably dispute in the quantum gravity community over whether ... a quantum particle like a graviton should be used to model the universe.
..

Well in Loop approach and the related spinfoam approach they use SPIN NETWORKS (a spin network extends throughout all space) and SPIN FOAMS (a spin foam is co-extensive with spacetime). These things serve to represent a state of the gravitational field, or a state of spacetime geometry. They have a certain combinatorial aspect because you can in principle describe them by listing data.

and using spin networks leads to area and volume operators with discrete spectra.

but I have never heard anyone describe a spin network or a spinfoam as a PARTICLE.

I never read where Ashtekar said that a spin network was "like a graviton".

That is just one sample approach, LQG, and one person. I am trying to see what you might mean.

let's focus on the nonperturbative QG community
this conference practically speaking defines that community
http://loops05.aei.mpg.de/

here is the list of invited speakers
http://loops05.aei.mpg.de/index_files/Programme.html
...

ohwilleke, if there is "considerable dispute" then does that include the non-perturbative QG crowd?

If you would, look down the list of representative QG people, the invited speakers, and tell me who would dispute with whom, about whether spacetime should be described as made of something "like gravitons" or some kind of point particle. You may be right! I don't know the work of everybody on the list, far from it.

 

[wolram]

There is considerably dispute in the quantum gravity community over whether space-time itself is discrete, or whether only a quantum particle like a graviton should be used to model the universe.

Marcus this may or may not be what Ohwileke is referring to.
http://en.wikipedia.org/wiki/Talk:Loop_quantum_gravity

 

[marcus]

There is considerably dispute in the quantum gravity community over whether space-time itself is discrete, or whether only a quantum particle like a graviton should be used to model the universe.

Marcus this may or may not be what Ohwileke is referring to.
http://en.wikipedia.org/wiki/Talk:Loop_quantum_gravity

Wow, there certainly has been a lot of brawling at Wiki over Lubos Motl (I thought generally questionable) criticisms of LQG! thanks for the link, wolram. I can't quite make sense of it but it seems like Lubos started a sort of pandemonium over there.

I actually don't know what "considerable dispute" ohwilleke means. there are lots of disputes that divide along lots of different lines. dispute between String and Non-String QG doesn't interest me so much.

I am more interested in hearing about some potential dispute among the Non-String QG people (I don't think the string approach has much future to it so what interests me is differences among the Non-String folk.)

 

[marcus]

Would a graviton theory replace the curved geometry of general relativity? If gravitons are the cause of the gravitational force, does that mean space and time aren't really curved as Einstein thought?..

And maybe an unrelated question, is it better to think of curved space-time as a model that explains the world very well, rather than the way the world actually is?

If this is a newb question then newb questions can be a big help. This question gets me to see something.

The thing that distinguishes the NonString QG people is that they treat GRAVITY AS GEOMETRY.
For people who do geometrical quantum gravity, gravity is not some force being propagated by some particle analogous to a photon or a virtual photon which is traveling through an inert static space.

If there were a static inert space and gravity were a force mediated by some particle, call it a graviton, then the job would be QUANTIZE THE PARTICLE.

But for people who do geometrical quantum gravity, the job is to QUANTIZE THE GEOMETRY, make a hilbert space that describes the uncertain shape of the universe, the imperfectly known shape of spacetime.
Because for them gravity is not a force mediated by a spray of little graviton-particles, for them gravity is geometry.

 

[marcus]

A couple of weeks ago selfAdjoint was saying how do we describe the non-string QG people.

It used to be we would just say "Loop Quantum Gravity" but that is no longer precise enough or general enough. there are a bunch different approaches.

We were trying different terms out, like B.I.QG (background independent QG) because for all of those people they don't start with a fixed background geometry. the model has a changing dynamic geometry.
And we also were saying QGATS (quantum gravity alternatives to string) because it is the non-string QG people we are trying to give a name to.

If you want to look at who those people are and what they call themselves you just go to the Loops 05 website
http://loops05.aei.mpg.de/

and you see that what they say right up front is
"...the annual international meeting on non-perturbative/background independent quantum gravity takes place..."

they don't mean just loop LQG, or spinfoams, or causal sets, or CDT, or Laurent Freidel's stuff, or Martin Reuter stuff, or whatever.

BUT THE NEWB WHO ASKS THE QUESTION PROBABLY DOES NOT HAVE A CLEAR IDEA OF " non-perturbative/background independent quantum gravity".

It suddenly dawns on me that what we should have been saying all along was GEOMETRIC quantum gravity.

That is, the approaches to QG that take seriously the idea that gravity is geometry and you are not going to get a satisfactory theory of spacetime by fixing on a static inert spacetime and letting graviton ripplets ripple across it.

Geometric QG (whatever the several approaches) always assumes that it is not going to work to quantize the particles or the superimposed ripples. Geometric QG takes for granted that you have to quantize the shape of spacetime itself.

Now this is maybe something which says it in ordinary language, and communicates, the way
"non-perturbative/background independent" does often not communicate.

 

[ohwilleke]

I am using the word quantum gravity is a broad sense. Any theory which includes a graviton is a quantum gravity theory. LQG is also a quantum gravity theory.

If you look at the big picture, long term history, people coming from the astro-physics direction (black holes and what not), have tended to look for quantatizing the geometry approaches, while people coming from the particle physics direct have tended to look for a spin-2 graviton.

My point is that there is not consensus on whether geometry or a particle is the right way to go in the quantum gravity world.

Note, incidentally, that one could have a discrete Minkowksiesque 4D space time, while still having GR gravitational effects mediated via a graviton.

 

[marcus]

Would a graviton theory replace the curved geometry of general relativity? If gravitons are the cause of the gravitational force, does that mean space and time aren't really curved as Einstein thought?..

And maybe an unrelated question, is it better to think of curved space-time as a model that explains the world very well, rather than the way the world actually is?

OK ellipse, now maybe I can reply.
I won't say which is better. Some people prefer to think of a fixed spacetime with gravitons going around mediating gravity force. Some people prefer to say "gravity is geometry". Each of them thinks or talks as if their idea is more fundamental.

I can tell you my personal opinion is that Einstein geometrical model of gravity is more precise, makes more accurate predictions, more widely applicable---so I think of it as more fundamental, more like what the world really is like. But people can have whatever opinion.

Now when it comes to quantizing, which is the premier world-class quest in theoretical physics today, there are two groups:

there are the STRING people who like the graviton picture of something moving in a fixed geometry, typically flat background but can be special curved cases too, and

there are the GEOMETRIC quantum gravity people.

The latter (maybe we can call them GQG) are especially fun to watch, in my view, because they have made extensive progress in just the past couple of years. So their conference this year is apt to be rather lively. they are going to have to discover whether they fight it out between competing GQG approaches or whether some will merge.

keep tabs on this site
the full program for October is supposed to be posted sometime this month
http://loops05.aei.mpg.de/

 

[wolram]

BY Marcus.
I am more interested in hearing about some potential dispute among the Non-String QG people (I don't think the string approach has much future to it so what interests me is differences among the Non-String folk.)
So its polyester for Marcus? and meat and two veg for me

 

[marcus]

and meat and two veg for me

just thinking about meat and two veg reminds me I haven't had lunch yet.
will be back in a while.

maybe i will start a thread about this "geometrical QG" descriptor.

I'm looking for a way of saying in clear ordinary language what that cluster of research lines is
that they call "nonperturbative background independent approaches to QG"

 

[wolram]

just thinking about meat and two veg reminds me I haven't had lunch yet.
will be back in a while.

maybe i will start a thread about this "geometrical QG" descriptor.

I'm looking for a way of saying in clear ordinary language what that cluster of research lines is
that they call "nonperturbative background independent approaches to QG"

Marcus, please come up with something interesting and not to Mathy,to give
some sort of incentive, for us plebs

 

[εllipse]

Thanks, everyone, that was very helpful. I am currently reading The Elegant Universe and thought ST was the best approach to unification so far (it sure gets a lot of publicity!), so I'm glad there are other things being actively pursued. And as far as math goes, I've just started working through MTW, so the only QM stuff I know is pop-sci. This GQG (I like that name, btw ) approach sounds like something I'd like better than ST, since I really love GR and the geometric approach. Are there any pop-sci books out on some of these other, non-string approaches?

Also, is it possible that the other forces could be thought of in terms of geometry, rather than particles.. kind of like Kaluza-Klein, I suppose?

 

[marcus]

popular-written sources on geometric QG approaches

for LQG:
Carlo Rovelli "Loop Quantum Gravity" article in physics world (online, will get link)
http://cgpg.gravity.psu.edu/people/Ashtekar/articles/rovelli03.pdf
Lee Smolin "Atoms of Space and Time" in SciAm January 2004 (not online, must find in library)
Lee Smolin "Three Roads to Quantum Gravity" book (not online, I haven't read but am told it is a good general audience book)
other semi-popular online articles at Abhay Ashtekar's website
http://cgpg.gravity.psu.edu/people/Ashtekar/articles.html


for Renate Loll's CDT:
here is a short notice by science writer Adrian Cho from the American Physical Society's "Focus" newsmagazine
http://focus.aps.org/story/v14/st13


there are some links in the nearby thread "Quantum Graffiti"
https://www.physicsforums.com/showthread.php?p=588926#post588926
I translated an article that was in a popular Dutch periodical, into English, and posted my rough translation in the "Quantum Graffiti" thread.

CDT (causal dynamical triangulations) is comparatively new, invented in 1998, involves having a computer assemble half a million or so triangle-like 4D building blocks in thousands of different random formations with quantum weighting. Add up all the ways spacetime can be, all possible geometries, to get a quantum weighted sum. From this, or from a statistical sampling, measure "average" geometric properties, it's a Feynmanesque path integral (add up all the ways how spacetime can get from this to that spatial configuration). CDT only last year reached the point where it could generate normal-looking 4D spacetime that at large scale on the average obey Einstein equation of GR. Microscopic behavior (at Planck scale) is not classical. It is new enough that there is not much popular writing about it!


we will keep our eyes out for more wide audience stuff on any type "GQG". there are several other approaches I didnt mention.

 

[εllipse]

Thanks, also, another question, just for clarification. Is it true that, in string theory, the graviton is the reason for gravity, rather than space-time curvature? Obviously, string theory does allow for space-time curvature, because the excess dimensions have to be curved in order to be so small, right? So we're not just talking about extending flat Minkowskian geometry to include extra dimensions.. But as far as gravity is concerned is it all due to the graviton, the same way the standard model attributes EM to the photon? That sounds kind of strange, because wasn't GR's approach to gravity the reason physicists began thinking of the possibility of space-time being curved? But if gravity, as far as string theory is concerned, is no longer due to space-time curvature, then the reason space-time can be curved has been replaced. It must now be curved so that string theory can include extra dimensions..

Do any of these "GQG" theories include excess dimensions?

 

[selfAdjoint]

Thanks, also, another question, just for clarification. Is it true that, in string theory, the graviton is the reason for gravity, rather than space-time curvature? Obviously, string theory does allow for space-time curvature, because the excess dimensions have to be curved in order to be so small, right? So we're not just talking about extending flat Minkowskian geometry to include extra dimensions.. But as far as gravity is concerned is it all due to the graviton, the same way the standard model attributes EM to the photon? That sounds kind of strange, because wasn't GR's approach to gravity the reason physicists began thinking of the possibility of space-time being curved? But if gravity, as far as string theory is concerned, is no longer due to space-time curvature, then the reason space-time can be curved has been replaced. It must now be curved so that string theory can include extra dimensions..

Do any of these "GQG" theories include excess dimensions?

The difference between (most) string theory and GR is not flat versus curved, but background dependent versus background independent. In GR, spacetime is a dynamic component of the physics, there is no "stage" upon which the physics takes place, rather the scenery transforms and is transformed by the action. By contrast string theory has strings and branes moving around in a fixed geometry. The compact dimensions are curved, yes, but they are assumed to have constant geometry - torus, Calabi-Yau manifold, or whatever, and just act as a background for the physics in the foreground.

 

[Ratzinger]

Thanks Marcus for these posts here. Always have been confused by my pop-sci books that first explain gravity as different from all other forces but then some chapters later they talk about the yet-to-discover gravitons that transmit gravity just like the bosons of all the other forces. That never made sense to me.

 

[marcus]

hello Ratzinger, I am glad to hear you found some of the posts here useful.
this morning I am in a muddle so you may find the following post makes things worse!

hi selfAdjoint, hope you will stay around in this thread especially if ellipse keeps on asking interesting questions. This next one is easy though.

...Do any of these "GQG" theories include excess dimensions?

The non-string QG approaches I'm familiar with (LQG, spin foam, CDT) do not NEED extra dimensions to work. But they can be formulated in different spacetime dimension besides 4D. The basic definitions of LQG, for example, can be stated with spacetime dimension equal to any N greater than 1 (as is done e.g. in Rovelli's "Primer of LQG"). You can build the various QG models in any dimensions you want, but for obvious reasons the case people are most interested in is 4D.

So you mostly see QG papers where the dimension is assumed to be the usual 4D (no "rolled up" stuff) or else it is a toy model of lower dimension 2D or 3D being used as part of a process of developing ideas and techniques.

Loll and co-workers Ambjorn and Jurkiewicz spent the first few years 1998-2002 with CDT practicing on 2D and 3D, before trying 4D. Developing intuition, methods, concepts.

Although it isn't usual, both Lee Smolin and Laurent Freidel have, at times, written papers incorporating 5D. I am unfortunately rather vague about that. Haven't studied those papers enough to say what they are about.

My answer to your question is to the extent of my limited knowledge of the "GQG" (the QG alternatives to string, treating gravity as geometry rather than as a force carried by some particle in a fixed space), none of the alternative QG approaches are bound to some special dimensionality.

they can all be formulated in dimension N

and the most interesting case is to take N = 4.

======================

as if that weren't wordy enough, keep in mind that when you formulate these QG approaches with N not equal 4, you may get different results in different dimensions. Hendryk Pfeiffer has a paper bearing on this.
http://arxiv.org/gr-qc/0404088 [Broken]
It is called "Quantum general relativity and the classification of smooth manifolds".
A quantum gravity theory may look similar or analogous in some other dimension but give different results.

Loll and Westra just showed that they can do the CDT path integral in 2D in a way that sums over possible spacetime topologies as well as over geometries possible for a fixed topology.
http://arxiv.org/hep-th/0507012 [Broken]
it is not obvious how they will extend this to 3D and then to 4D. For instance a sum might be finite in 2D and then blow up and turn out to be infinite when you try to calculate the analogous thing in 3D.
Including different topologies (in the weighted average, or "sum over histories"), instead of just including different spacetime geometries for one fixed spacetime topology, is considered a hard problem.

 

[marcus]

there is a monster lurking in the shadows here, which is the question of what is meant by "dimension"

with LQG it's all simple because if you decide to work in 4D then you establish a 4D differentiable manifold at the outset.
it has coordinate patches with four smooth coordinate functions
it looks like the traditional spacetime continuum we always think of
it just doesn't have any METRIC to start with, it has no GEOMETRY, no distances or angles or areas.

a differentiable manifold without a metric is like something limp and shapeless. defining a metric that gives it geometry is like putting starch in it, so it can have some shape.

So in LQG you start with something shapeless, but at least it has the basic differentiable structure that we expect and need to do 4D calculus. Actually in LQG it is even a little nicer---they typically start with a spacetime that is a limp R x S³, with space identified as a limp 3-sphere S³.

you start with a lot of conventional continuum structure, but no background geometry, and the game is to quantize the geometry that you put on the thing.

BUT IT DOESNT HAVE TO BE THAT SIMPLE
and in Loll CDT you don't start with a differentiable manifold, for example if you decide to work in 4D you might start with a topological space like
R x S³ without any differentiable structure

Loll and friends discovered how to divide that up into half a million essentially identical 4-simplexes and then to SCRAMBLE THE TRIANGULATION so you keep doing moves on it like shuffling a deck of cards and you eventually get RANDOM triangulations each of which is weighted with a quantum amplitude that comes from the Einstein equation (or Regge form of Einstein-Hilbert action)
or to put it without namedropping you get random triangulations weighted according to how nice they look by Gen Rel standards. And you add all these things up. And then you SEE what the macroscopic dimension is
IN THE LIMIT AS the simplexes get smaller and the number "half a million" gets larger.

They are able to do this computationally, with a computer simulation of spacetime, and also to some extent analytically.

the somewhat counterintuitive thing is that if you do this wrong you can get a quantum spacetime in the limit which HAS THE WRONG DIMENSION!
Like the 4-simplexes can call pile up on each other like sardines and all be in contact with each other and in the limit the dimension is infinite.

Or they can all feather out like frost fingers in the icebox or wax-drippings or stalactites in a cave and not be in enough contact with each other and in the limit have some awful macroscopic dimension like 1.9 or 2.5.
You measure the dimension (either in the whole spacetime or in a spatial slice) by seeing how volume depends on radius, in the limit, which basically means how much contact the simplex building blocks have with each other on the average in the limit.
You go out a step or two and you see how much the volume increased. this means counting blocks. in 3D it is supposed to increase like the 3rd power.

This is an experimental quantum spacetime. It is not built IN some surrounding familiar 3D or 4D space which forces everything to be glued together in a sensible way. It is not built imbedded in a surrounding. It IS spacetime. And so (momentarily thinking in a lower dimension 2D case, for convenience) there is no limit to how many equilateral triangles you can glue together around a given point. you think you can only glue 6 equilateral triangles around a point, but no, you can glue as many as you want. Or as few, at least within reason---you might decide never to have more than 3 surrounding a point, or 4.

So the simplicial gravity people had to wrestle with it just to get 2D! They were putting together equilateral triangles (essentially, not always quite equilateral) to get an experimental quantum 2D spacetime and the result kept not being macroscopically 2D!

A bunch of people actually worked on this in the 1990s. then Loll and Ambjorn saw how to do it in 1998. that was the beginning of CDT. But then they had to wrestle to get it to work in 3D! Finally in 2004 they got it working in 4D.

In this approach the macroscopic DIMENSION IS NOT A GIVEN. You start with a microscopic dynamical principle of how spacetime organizes itself at Planck scale, and the macroscopic dimension and other macro features have to EMERGE dynamically when you run the model.
================

so when we talk about LQG and CDT in the same breath, both being "geometric" quantum gravities, both being background independent non-perturbative quantum gravities, already there is a possible confusion when you say dimension.

because in LQG, dimension is the familiar old take-for-granted idea that you expect and yes LQG works in every dimension but the interesting one is four.

while in CDT you don't take the macroscopic dimension for granted it is a dynamical variable, just like other features of spacetime, that has to emerge. you have to set up microscopic workings of spacetime that, in effect, explain why the overall largescale effect looks like 4D to us.

 

[εllipse]

Wow, I didn't realize how complicated that question would be! Thanks a bunch, marcus and selfAdjoint. I think I've got a general idea, although some of that was hard for me to follow. I can't think of any more questions on the topic, but I'd like to re-ask something I asked earlier but wasn't addressed due to being in the midst of other questions..

Is it possible that the other forces could be thought of in terms of geometry, rather than particles.. kind of like Kaluza-Klein, I suppose?

 

[marcus]

Is it possible that the other forces could be thought of in terms of geometry, rather than particles.. kind of like Kaluza-Klein, I suppose?

particles "thought of in terms of geometry..."
but geometry of WHAT? what is your mathematical playdough that you are molding the shape in?
is it a classical differentiable manifold, or something else?
current work in quantum gravity is groping for a new model of
the continuum----a continuum that is not a differentiable manifold.

up till now essentially every kind of continuum in physics, virtually every surface and hypersurface and phase space and "world sheet" and "brane" and kaluza-klein and einstein-spacetime has been a differentiable manifold----or a lattice approximation of one.

selfAdjoint may know of some important counterexamples, I dont.

The Diff. Manif. has been the paradigm of the continuum in physics, and usually it's even more structured: usually it's not only a differentiable manifold but one with a smooth metric as well. It has been a knee-jerk reflex to immediately picture anything geometrical in those terms. What do you define any geometry on or in or of? Why, in or on or of some differentiable manifold.

Hopefully, as I say, it would be more structured than a generic Diff. Manif.-----like having a smooth metric already defined on it, and maybe even flat! If you were really lucky it would simply be flat Euclidean or Minkowski space----they are differentiable manifolds, but special extra nice ones.

Lie groups are other examples, fiber bundles, they are all special kinds of differentiable manifolds, with extra structure---this is how, for 150 years, we have treated the idea of a continuum. This is where geometry lives (as a mathematician customarily thinks of it)

But in the arduous effort to quantize Gen Rel, a new model of the continuum has been emerging recently that is not a differentiable manifold.

A differentiable manifold has the same dimension at all scales, small and large. This new model of continuum can be 4D at large scale and 2D at small scale and in between 2 and 4D at intermediate scale.
it can be foamy or fractally at very small scale, and maybe have wormholes, but look perfectly nice and conventional at large scale.

Reconstructing the Universe
http://arxiv.org/abs/hep-th/0505154

Ellipse you are asking this question at a time when, as I see it, a fundamental change is taking place. So if you ask "might particles be represented by something in this new continuum with its smallscale foam, and possible topological tangles, could a particle be some kind of traveling defect in the smallscale structure?" then I would have to say I don't know because this new version of the continuum is very new to me and so far very few people have studied its possibilities.

but on the other hand if you ask "might particles be represented by classical geometric accessories and flourishes in a conventional differentiable manifold, then I would say that I doubt it very much because I do not think that EVEN EMPTY SPACETIME can be adequately represented with the old model of a continuum!

So neither way can I give you a satisfactory answer.

On the other hand, someone still holding true to the faith that string theory will eventually connect up to nature CAN give you a satisfactory answer. It may be a wrong answer but at least you get told something. Strings worldsheets and branes ARE conventional oldfashion differentiable manifolds and they LIVE in surrounding space which is also a conventional oldfashion differentiable manifold. String is very much an Old Continuum theory. And a believer in that can say YES particles can be represented by geometry because isn't a string simply a geometrical object? You think it is made of vibrating energy? What baloney. You have to be kidding, it is not "energy", it is vibrating geometry that is all a string is, it is a bit of oldfashion continuum geometry pure and simple. And representing particles is what it likes to do best!

You can believe him but I don't, because string has had some awful shocks in the past 3 years and has kind of come to a standstill. This is how it looks to an outsider, at any rate. Research publication declining, people getting out or changing their focus from fundamentals over to cosmology and astrophysics. Scholarly citations of string papers declining. Confusion about the 10¹⁰⁰ vacua, or possibly the infinite number of vacua, the "Landscape", despair of finding the right vacuum or groundstate that describes physics in the nature we know. Uncertainty about what to predict, when it comes to specifics, as regards upcoming experiments.

So you can get answers, but they may not be worth much.

what I am doing myself is, instead of trying to forecast how they will represent particles on the new spacetime, I am focusing on the new picture of spacetime that is emerging. when that is finished then it will be time to see about the particles and fields etc to build on it.

 

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Would a graviton theory replace the curved geometry of general relativity? If gravitons are the cause of the gravitational force, does that mean space and time aren't really curved as Einstein thought?..

And maybe an unrelated question, is it better to think of curved space-time as a model that explains the world very well, rather than the way the world actually is?

Depending on the results of Model, one can define a number of Relative consequences of Graviton evolution.

For instance, are Gravitational 'Signals'...particles or waves?..if the Graviton is a particle (signal), does the model allow for Dark-Matter to be corresponding wave (signal)?

Is there a Gravitational Particle-Wave Duality?

There are certain models of course where the linear transformations in certain compact Dimensions take on complex dynamics, for instance there are functions whereby MASS>ENERGY can have a partial-function during transformation, simplistically transfoming from one to product to another: MASS>particle/\wave>ENERGY becomes observer reliant at source, but tranforms into a non-observer product(hidden variable, or Dimensionally excluded) at the End-point of tranference, you cannot know for sure that you are at the 'source or end' as a dynamic observer?

So thus, the Graviton at one location would be defined as a Particle of gigantic proportions, and quite literally could be envisiged as a extended 'Particle-Halo' around a Galaxy for instance. And from another perspective, the Graviton can be compacted to the Quantum scale of 'Charge-Halo' for fundamental Quantum Field Theory.

 

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More on Gravitons

How does string theory account for gravitational time dilation? Would getting hit by a gravition cause your clock to slow down?