How to recover time paradoxes in canonical theories of gravity?

MTd2

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?

mfb

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

Which paradox?

MTd2

Like killing my own grandfather. Weird solutions with closed time like curves.

mfb

I like Novikov's self-consistency 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 time-like curves.

MTd2

Basically, I want to know what happens to the wrong solutions within LQG, for example. If possible, I'd like to see how to recover them. Is it necessary to just get the classical limit and then you have the wrong solutions or do they appear in the full theory?

tom.stoer

It is correct that the full solution-space 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 non-trivial 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.

Demystifier

Let me first restrict myself to canonical formulation of CLASSICAL gravity. It can be formulated in terms of local Hamiltonian DENSITY, which does not require GLOBAL foliation of spacetime into 3-hypersurfaces. A local decomposition of spacetime into space and time is sufficient. In this way, one can recover solutions with closed time-like curves with a canonical approach.

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 time-like 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/gr-qc/9210011

tom.stoer

Really?

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]

Finbar

Why does a global foliation of space-time not allow for closed time-like curves? Surely one can just identify two space-like 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 time-like curves is probably not physical. So picking the topology to be

M = R * Ʃ

seems reasonable.

Demystifier

Yes.

Demystifier

Good points.

marcus

More good points.

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

MTd2

To ask in principle if you can use a hypersurface to slide with a patch of space faster than light, but without causality problems. That is, we could find a loophole physics to implement tachyons without causality problems.

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

marcus

Thanks! I understand the significance better now.

sshai45

@marcus,

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

marcus

Well, in the Tomita treatment, time is a parameter of change rather than a pseudo-spatial dimension. So "spacetime" is simply a mathematical construct that does not correspond to nature. I suppose there are various versions of time including observer times experienced by various observers, and also including the observer-independent Tomita time that is simply the real number parameter of the Tomita flow.

T-flow is a one-parameter 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 T-flow thread.