Is space-time discrete or continuum?

[micromass]

I suppose you could have more than one metric defined on the same manifold. That sounds like a somewhat arbitrary thing to do. As you say, there could be a metric assigned for analytic purposes and a spacetime metric. But they both result in a number given two points on the manifold. Is a metric still a metric if it results in the same number for various paris of points? Or does that contradict the definition of a metric?

Sure, you can have various pairs of points which all lie in the same distance of each other. For example, we always have $d(p,p)=0$, so a point always lies within distances $0$ of itself. And this is true for any point.

It's true that a manifold admits many metrics. There is no metric that is the most natural. So yes, assigning a certain metric to the manifold is rather arbitrary.

If you're dealing with a Riemannian manifold however, then you can always find a metric that is most natural.

 

[WannabeNewton]

friend, a space-time metric is completely different from an analytic metric. A space-time metric allows you to compute inner products between vectors in each tangent space to the manifold at a given point. An analytic metric allows you to find the distance between different points on the manifold with respect to that metric. The input for a space-time metric is a specific point on the manifold and two vectors in the tangent space to the manifold at said point; the result is the inner product of these two vectors. The input for an analytic metric are two points on the manifold and the result is the distance between these two points as defined by that metric.

In the case of connected Riemannian manifolds, one can use the Riemannian metric to define a metric in the analysis sense: http://en.wikipedia.org/wiki/Riemannian_manifold#Riemannian_manifolds_as_metric_spaces_2 but note that this is for Riemannian and not pseudo-Riemannian manifolds.

 

[friend]

friend, a space-time metric is completely different from an analytic metric. A space-time metric allows you to compute inner products between vectors in each tangent space to the manifold at a given point. An analytic metric allows you to find the distance between different points on the manifold with respect to that metric. The input for a space-time metric is a specific point on the manifold and two vectors in the tangent space to the manifold at said point; the result is the inner product of these two vectors. The input for an analytic metric are two points on the manifold and the result is the distance between these two points as defined by that metric.

In the case of connected Riemannian manifolds, one can use the Riemannian metric to define a metric in the analysis sense: http://en.wikipedia.org/wiki/Riemannian_manifold#Riemannian_manifolds_as_metric_spaces_2 but note that this is for Riemannian and not pseudo-Riemannian manifolds.

Can you give examples of a degenerate metric, whether on the manifold or on the tangent space of it? What examples are there of a metric giving the same distance or the same inner product for basically an infinite number of nearby points or vectors, whatever close means with a degenerate metric?

 

[tom.stoer]

Suppose you have two light-like, orthogonal vectors, i.e. <x,x> = <y,y> = <x,y> = 0; then you also have <λx,λx> = <μy,μy> = <λx,μy> = 0 for arbitrary constants λ,μ. The inner product is defined in terms of coordinates <x,x> = x_ax^a = g_abx^ax^b

 

[WannabeNewton]

A metric tensor is non-degenerate by definition. The difference between a Riemannian metric and a pseudo-Riemannian one is that the latter is not positive definite. This leads to things like non-zero vectors having vanishing "norm" (such vectors are called null vectors/lightlike vectors).

 

[micromass]

A metric tensor is non-degenerate by definition. The difference between a Riemannian metric and a pseudo-Riemannian one is that the latter is not positive definite. This leads to things like non-zero vectors having vanishing "norm" (such vectors are called null vectors/lightlike vectors).

And although I risk to sound repetitive, I want to add that:

- A metric tensor (synonym: Riemannian metric) is defined on the tangent spaces. It is an inner product on the tangent space $T_pM$ that varies smoothly from tangent space to tangent space

- A distance function (also called a metric in analysis) is defined on the manifold itself. It is not an inner product. It simply is a function giving you the distance between two points.

The difference between these two concepts is crucial. Please try to understand it well. Not understanding this has lead to a lot of confusions in the past.

 

[friend]

Suppose you have two light-like, orthogonal vectors, i.e. <x,x> = <y,y> = <x,y> = 0; then you also have <λx,λx> = <μy,μy> = <λx,μy> = 0 for arbitrary constants λ,μ. The inner product is defined in terms of coordinates <x,x> = x_ax^a = g_abx^ax^b

OK, it seems all vectors on the light-cone have zero norm. This is not the same as a discrete spectrum, where zero in one possible values along with other possible values.

 

[tom.stoer]

There is absolutely no relation between the pseudo-norm ||x||² = <x,x> has absolutely nothing to do with eigenvalues.

 

[friend]

There is absolutely no relation between the pseudo-norm ||x||² = <x,x> has absolutely nothing to do with eigenvalues.

If the metric is quantized, then the inner product is quantized, right?

 

[tom.stoer]

You still confuse quantized and discrete

Quantized means that you are using quantum mechanics.

In QM:
- momentum is always quantized and sometimes it's discrete
- angular momentum is always quantized and discrete

In QG approaches
- the metric (or some other structure related to a manifold) is always quantized
- in LQG the manifold is replaced by a discrete structure during quantization
- in LQG length, area and volume will probably have discrete eigenvalues; this is not clear b/c the operators with discrete spectrum are no observables; other operators which are observables are not known afaik
- in other approaches with quantized metric but continuous manifold I do not know what happens to these eigenvalues; in these approaches the manifold is not replaced by a discrete structure
- in CDT the manifold is replaced by a discrete structure before quantization; length, area and volume are discrete b/c of discretization (trivially) w/ and w/o quantization
- whether the spectrum of physical observables remains discrete after the discretization of CDT is removed is unclear to me.

So even you are by precise regarding the question not all details are known; QG is work in progress.

 

[ryan albery]

If space-time has discreet values, or intervals- there's no way we have to measure those intervals. Every measurement we conduct (even weighing something depends on the relative passing of time) is based, fundamentally, upon time.

 

[tom.stoer]

If space-time has discreet values, or intervals- there's no way we have to measure those intervals. Every measurement we conduct (even weighing something depends on the relative passing of time) is based, fundamentally, upon time.

Perhaps I should add that given a quantum mechanical observable O does not imply that we know how to construct a measurement device for O. It simply means that O represents a quantity measurable in principle; how to measure it in practice cannot be derived from O.

 

[friend]

Perhaps I should add that given a quantum mechanical observable O does not imply that we know how to construct a measurement device for O. It simply means that O represents a quantity measurable in principle; how to measure it in practice cannot be derived from O.

Perhaps it would be instructive to tell us how in principle one would measure a discrete metric in quantum gravity. Or for that matter what it means and how we would measure, even in principle, ANY quantum nature of quantum gravity. I find myself not really understanding what that's supposed to mean.

 

[tom.stoer]

I do not see any direct way to measure space-time discreteness. But there are indirect methods, namely to measure effects induced by discreteness, especially violation or deformation of local Lorentz invariance, i.e. a corrections to E² = p² + m². For light propagation this means that speed of light propagation could become frequency-dependent. Experiments have ruled out these corrections up to a certain order.

 

[DimReg]

I do not see any direct way to measure space-time discreteness. But there are indirect methods, namely to measure effects induced by discreteness, especially violation or deformation of local Lorentz invariance, i.e. a corrections to E² = p² + m². For light propagation this means that speed of light propagation could become frequency-dependent. Experiments have ruled out these corrections up to a certain order.

I imagine you know this already, but it's worth pointing out for general audiences that not all theories with discreteness predict local lorentz invariance violation. As far as I know, LQG these days is believed to be local lorentz invariant.

 

[tom.stoer]

I imagine you know this already, but it's worth pointing out for general audiences that not all theories with discreteness predict local lorentz invariance violation. As far as I know, LQG these days is believed to be local lorentz invariant.

Yes, I agree, this is an important remark.

One must not confuse discreteness with a kind of fixed lattice structure or something like that. The main differences are that
1) spacetime discreteness may allow for dynamical creation and annihilation of "spacetime atoms"
2) spacetime becomes subject to "superpositions of spacetime states" in quantum gravity

This means that spacetime discreteness does not necessarily violate the quantum version of the continuous classical symmetries.

 

[friend]

Yes, I agree, this is an important remark.

One must not confuse discreteness with a kind of fixed lattice structure or something like that. The main differences are that
1) spacetime discreteness may allow for dynamical creation and annihilation of "spacetime atoms"
2) spacetime becomes subject to "superpositions of spacetime states" in quantum gravity

This means that spacetime discreteness does not necessarily violate the quantum version of the continuous classical symmetries.

Thank you Tom. Your efforts are appreciated.

If energy is quantized and mass of particles are quantized, then it stands to reason that curvature calculated from GR for that energy is quantized, at least in the rest frame of those particles.

 

[tom.stoer]

But energy is usually NOT quantized; action is quantized, and E=nhf is quantized in terms of number of photons n; but frequency is NOT quantized in general, only for specific systems and emission / absorption processes.

 

[ryan albery]

Interesting to contemplate the presence of big G in the equation of Newton's gravity (along with Einstein's) as being analogous to Planck's constant with the quantization of 'things'.

 

[friend]

How dependent is quantum gravity research on the assumption that Newton's constant, G, and Planck's constant h remains constant as the region of interest approaches the Planck scale? And what proof is there that these values don't change with very small scales?

 

[tom.stoer]

Regarding G (and Λ) expectation is that they become scale-dependent.

Asymptotic safety is an approach which tries to quantize gravity based on the assumption of smooth spacetime plus UV completeness using renormalization group theory http://www.percacci.it/roberto/physics/as/faq.html