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TrickyDicky
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GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.
There are indications, are there not?, that geometry is also discrete when probed at small enough scale. What are these indications? You certainly know some. We can't be sure yet, but there are reasons to suspect geometric discreteness---what are some of them?TrickyDicky said:GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.
Sten Odenwald seems to think so - http://www.astronomycafe.net/qadir/BackTo286.htmlmarcus said:I wonder if this old (1983) paper of Rafael Sorkin is relevant.
http://www.perimeterinstitute.ca/personal/rsorkin/some.papers/31.padova.entropy.pdf
He argues for geometric discreteness simply based on the finiteness of bh entropy
Thank you for introducing me to this wonderful paper. I said space is continuous by definition because I don't know what the stuff in between would be called if space is discrete.marcus said:For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.
http://www.signalscience.net/files/Regge.pdf
That goes back to 1960 when Regge was at Princeton. His 1960 paper was called
General Relativity Without Coordinates
It is a famous paper that introduced a famous approach (Regge calculus) that succeeded in doing Einstein General Rel without using coordinates.
That link gives a fax of the original paper.
Since then there have been many ways developed to treat spacetime using discrete entities, like simplices, cell-complexes, or graphs, rather than using the usual continuum model (the differential manifold).
Just to take one example, there is so called "CDT" (causal dynamical triangulations) as developed by Renate Loll and Jan Ambjorn, with others.
I wouldn't say that a clear winner has emerged in the competition for "best discrete model of spacetime geometry" but it cannot be said that it is
"continuous by definition." : ^)
Causal Sets, as developed by Rafael Sorkin and Fay Dowker, with others. If anyone wants they can do a search by author names at arxiv.org and find a large collection of paper that appeared over the past 10 years, and some earlier.
But it seems to me that ideal measurements measure what IS. If what IS is not continuous then it would not measure as continuous. a simple example would be making measurements of the distance between a series of fence posts that are 10 feet apart. It just doesn't make sense to talk about a measurement of 23.78... feet in that realm.TrickyDicky said:Ideal measurements are obviously discrete ...
TrickyDicky said:GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.
marcus said:Sorkin's approach to geometric discretness preserves Lorentz Invariance
marcus said:For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.
marcus said:For the sake of general awareness of what we're talking about maybe we should clearly acknowledge that neither "space" nor "space-time" are necessarily continuous by definition and they certainly do not necessarily involve coordinates.
http://www.signalscience.net/files/Regge.pdf
That goes back to 1960 when Regge was at Princeton. His 1960 paper was called
General Relativity Without Coordinates
It is a famous paper that introduced a famous approach (Regge calculus) that succeeded in doing Einstein General Rel without using coordinates.
That link gives a fax of the original paper.
Since then there have been many ways developed to treat spacetime using discrete entities, like simplices, cell-complexes, or graphs, rather than using the usual continuum model (the differential manifold).
Just to take one example, there is so called "CDT" (causal dynamical triangulations) as developed by Renate Loll and Jan Ambjorn, with others.
I wouldn't say that a clear winner has emerged in the competition for "best discrete model of spacetime geometry" but it cannot be said that it is
"continuous by definition." : ^)
Causal Sets, as developed by Rafael Sorkin and Fay Dowker, with others. If anyone wants they can do a search by author names at arxiv.org and find a large collection of paper that appeared over the past 10 years, and some earlier.
Wow, so to you GR is quantizable and discrete. I think this indicates your admitting not being well versed in GR is somewhat an ironic understatement.Torbjorn_L said:I don't think that is correct.
GR describes spacetime as smooth, but you can insert singularities like particles from quantum field theories in semiclassical physics or perhaps idealized black holes (I'm not well versed in GR).
GR is known to be an effective, quantizable theory...
These are important points, to the first I would reply that universe is not necessarily the same as spacetime, for instance if we make the distinction between spacetime as something different from its content.bahamagreen said:Is it meaningful to ask if these two questions are the same?
1] Is the universe continuous?
2] Is spacetime continuous?
And... continuity of what, exactly?
Is it meaningful to distinguish asking about the continuity of what comprizes the universe or spacetime, or asking about the continuity of the "physical laws" (in the universe or spacetime)?
Since in reality there are no ideal measurents by definition we find that physical measurents must introduce some sort of scaling and coarse graining that deviates from what otherwise would be a scale independent process, this is usually dealt with when studying the renormalization group.phinds said:But it seems to me that ideal measurements measure what IS. If what IS is not continuous then it would not measure as continuous. a simple example would be making measurements of the distance between a series of fence posts that are 10 feet apart. It just doesn't make sense to talk about a measurement of 23.78... feet in that realm.
td21 said:I don't know what the stuff in between would be called if space is discrete.
No argument that there is a point below which we cannot make (and likely never will be able to make) actual measurements but I don't get how that relates to whether or not what we are measuring is continuous, based on theory. If I have a meter stick that will only measure down to a millimeter, that does NOT constrain my height to be an integer multiple of millimeters.TrickyDicky said:Since in reality there are no ideal measurements by definition we find that physical measurents must introduce some sort of scaling and coarse graining that deviates from what otherwise would be a scale independent process, this is usually dealt with when studying the renormalization group.
George, I have a different perspective. I think of QG as representing GEOMETRY. A theory would not necessarily contain a set that represents the "points of space-time". A theory should have a representation of quantum states of geometry, and perhaps transition or boundary amplitudes. But it might not have a set which are imagined to be the points of space-time. So "countable" might not refer to any thing.George Jones said:Presumably, "discrete" means that, a set, spacetime is countable.
TrickyDicky said:GR models of the universe describe it as a continuum, a smooth manifold, on the other hand the universe contains matter and matter is considered discrete.
marcus said:There are indications, are there not?, that geometry is also discrete when probed at small enough scale. What are these indications? You certainly know some. We can't be sure yet, but there are reasons to suspect geometric discreteness---what are some of them?
Jacobson's 1995 derivation of GR as a thermodynamic equation of state?
So what might be the geometric molecules of which GR is the EoS?
Finite entropy beyond bh horizon?
Continuum GR cannot be entirely right since it fails at extreme density?
marcus said:Note that Sorkin's approach to geometric discretness preserves Lorentz Invariance.
This is one of the first things proved in the Perimeter lectures on Causal Sets, that Dowker and Sorkin presented a year or two back.
Same with LQG, there is that paper by Rovelli and Speziale proving Lorentz invariance.
So we don't want to get geometric discreteness MIXED UP with the old idea of "graininess" which somehow leads to higher energy gamma arriving EARLIER or as other people imagined LATER than other light. That old idea never seems to go away, does it? : ^)
There is it cropping up in that "Astronomy Cafe" piece by Sten Odenwald! The old L.I.V. idea ("lorentz invariance violation")
We have to learn to make a sharp distinction.
George Jones said:Presumably, "discrete" means that, a set, spacetime is countable.
George Jones said:Presumably, "discrete" means that, a set, spacetime is countable.
Of course, I'm not equating discrete measurements in that sense to a discrete universe. I'm bringing in the theme about scale independence that I find perhaps more specific mathematically than the wider concepts of discrete and continuous.phinds said:No argument that there is a point below which we cannot make (and likely never will be able to make) actual measurements but I don't get how that relates to whether or not what we are measuring is continuous, based on theory. If I have a meter stick that will only measure down to a millimeter, that does NOT constrain my height to be an integer multiple of millimeters.
That seems like something interesting to try. Maybe if someone comes in with a new QG theory we could ask themTrickyDicky said:Of course, I'm not equating discrete measurements in that sense to a discrete universe. I'm bringing in the theme about scale independence that I find perhaps more specific mathematically than the wider concepts of discrete and continuous.
I believe that discrete models are by definition not scale invariant, thus all the examples you cite, but I'm not so sure the opposite is true.marcus said:That seems like something interesting to try. Maybe if someone comes in with ;a new QG theory we could ask them
"Is there a minimum geometric quantity in your theory?"
Like, for example, "Is there a minimum measurable area?" If there is, then it would seem that their theory is NOT scale invariant.
I remember reading a CDT paper by Renate Loll some years back in which she proved that the simplices of CDT had to be at least a certain size. I forget how she proved it, I'm sorry to say. It wasn't put in at the start, the size arose somehow. That would indicate that CDT is not scale invariant. But there could be some disagreement about that.
What about AsymSafe QG? Do you know, or does anybody know, a reason why that would not be scale invariant? One would think, since it is such a close imitation of GR, that it would be scale invariant. But the AsymSafe people claim that it experiences spontaneous dimensional reduction at very small scale. The space-time dimension dwindles down from 4 to around 2.
I wish I knew more about this. It seems like something interesting to consider.
With LQG it is straightforward. LQG has a length scale. It is clearly not scale invariant!
there is also a minimum positive area that one can measure.
What does it matter? I mean that seriously. Maybe "discrete" is a bad term, the word having been used so indiscriminately/inconsistently over the years that is has gotten dull.TrickyDicky said:I believe that discrete models are by definition not scale invariant, thus all the examples you cite, but I'm not so sure the opposite is true.
PeterDonis said:I think it's worth noting that the converse is not necessarily true: that is, points of spacetime being a countable set does not necessarily mean spacetime must be discrete. The rational numbers are a countable set, but they are also continuous (at least with the standard ordering on them).
Just say that when one interprets the GR equation symbols as calculus limits on the smaller scales; the equations are stochastic. In other words the classical GR equations are summaries/extreme cases of some stochastic equations where the independent variables happen to be random at a small scale.td21 said:Thank you for introducing me to this wonderful paper. I said space is continuous by definition because I don't know what the stuff in between would be called if space is discrete.
martinbn said:The rational numbers are not continuous because they are not complete
TrickyDicky said:Wow, so to you GR is quantizable and discrete.