
#1
Dec2012, 10:42 AM

P: 1,925

Canonical theories of gravities imply a foliation of spacetime into 3 hypersurfaces, each one labeled by time and related by lapse functions. It happens that this makes time unlike space, which is bad, since in GR, time is another coordinate. It also makes closed time like surfaces to classically disappear.
How do we recover the paradoxes? 



#2
Dec2012, 05:21 PM

Mentor
P: 10,830

Even in GR, the time coordinate is different from the others  it has a different sign in the metric.
Which paradox? 



#3
Dec2012, 05:34 PM

P: 1,925





#4
Dec2012, 05:43 PM

Mentor
P: 10,830

How to recover time paradoxes in canonical theories of gravity?
I like Novikov's selfconsistency principle (in particular together with quantum mechanics, but it works in classical mechanics, too), but different branches of wave functions might work as well.
GR does not imply that there have to be closed timelike curves. 



#5
Dec2012, 05:55 PM

P: 1,925





#6
Dec2012, 06:33 PM

Sci Advisor
P: 5,307

It is correct that the full solutionspace of GR contains closed timelike curves (CTCs); in canonical GR or QG one uses an M³ * R foliation i.e. assumes global hyperbolicity in order to define the theory; doing that one loses CTCs b/c they violate global hyperbolicity.
Having no fuly developed conanical theory of QG it is hard to say what will happen. Especially in LQG the Hamiltonian is neither a time evolution operator (b/c H ~ 0 due to diff. inv.) nor is a quantized and consistent H known. So we can only speculate that something like CTCs will never exist. They seem to exist in other approaches like fully covariant approaches based on the path integral, but even there it's difficult b/c 1) we do not know how to (consistently) define the path integral (e.g. a measure on the space of metrics); only certain truncations are known; I guess that fixing diff inv is nontrivial and that there are something like Gribov copies etc. 2) even if we could do that we don't know how to extract CTCs (or other geodesics) b/c the PI is a PI on the space of geometries not on the space of paths within one geometry; so there is no curve in the PI. Perhaps CTCs vanish even in the PI approach b/c they have zero measure. 



#7
Dec2112, 04:17 AM

Sci Advisor
P: 4,491

Now how to generalize it to QUANTUM canonical gravity? It can also be formulated in terms of local Hamiltonian density (operator). However, unlike classical canonical gravity, quantum canonical gravity has the problem of time*. Depending on how exactly you resolve that problem (see the reviews below), you may or may not recover closed timelike curves in canonical quantum gravity. *For reviews, see K. Kuchar, www.phys.lsu.edu/faculty/pullin/kvk.pdf C. J. Isham, http://arxiv.org/abs/grqc/9210011 



#8
Dec2112, 04:45 AM

Sci Advisor
P: 5,307

Does that mean that you prefer [tex]h(x) ~ \sim 0[/tex] instead of [tex]H[N] = \int_{\Sigma^3} N(x)\,h(x) ~ \sim 0[/tex] 



#9
Dec2112, 07:52 AM

P: 343

Why does a global foliation of spacetime not allow for closed timelike curves? Surely one can just identify two spacelike hypersurfaces at times t=0 and t=T so its periodic in time.
Isn't the point more about the topology of spacetime? Classical general relativity is not a dynamical theory of spacetime topology. One must fix the topology of spacetime and then solve the Einstein equations on that manifold, with particular boundary conditions, to find physically meaningful spacetime geometries. Of coarse mathematically one can consider all solutions to GR on all possible topologies. But in physics causality is important. My QM teacher in undergraduate once wrote the following equation Physics = Equations + boundary conditions Unless you use both physically meaningful equations of motion and boundary conditions you're not doing physics anymore. So I would surgest that picking boundary conditions or spacetime topologies which imply closed timelike curves is probably not physical. So picking the topology to be M = R * Ʃ seems reasonable. 



#12
Dec2112, 12:21 PM

Astronomy
Sci Advisor
PF Gold
P: 22,803

So one can ask, what's the point of this thread? Why should one want to "recover time paradoxes"? 



#13
Dec2212, 07:39 AM

P: 1,925

Or, to implement in general relativity, off shell integration without violating causality in principle. 



#14
Dec2212, 09:20 AM

Astronomy
Sci Advisor
PF Gold
P: 22,803





#15
Dec2712, 02:22 AM

P: 55

@marcus,
So do you have any comment on this issue? Especially in light of the stuff discussed on the "does time exist" thread. 



#16
Dec2712, 10:25 AM

Astronomy
Sci Advisor
PF Gold
P: 22,803

Tflow is a oneparameter group of transformations defined on the algebra of all possible observations, denoted M. I would think that, by definition, M does not contain impossible observations. But I am not an expert in these matters, I just find them interesting. So I've collected links to research articles that you can read in post#2 of the Tflow thread. 


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