Does GR produce the Planck relation and uncertainty principles?

1. Jan 18, 2013

liometopum

Can the Planck relation, and the Heisenberg and the time-energy uncertainty principles be derived, or produced, from the equations of General Relativity?

2. Jan 18, 2013

Staff: Mentor

No. The equations of GR are continuous and permit arbitrarily small differences of space and time, so don't even hint at any quantum mechanical principles.

3. Jan 18, 2013

liometopum

Thanks Nugatory!

4. Jan 18, 2013

TrickyDicky

I can't see how that matters, Schrodinger equation is also continuous as a differential equation. I'm not even sure QM is more related to discrete spacetime as opposed to continuous spacetime as you imply.

5. Jan 18, 2013

Staff: Mentor

TrickyDicky is correct.

Nugatory's "no" is correct, but the reason is not quite right. The uncertainty principle falls out of the commutation relationship between operators on the wavefunction, not some discretization of space and time. I cannot think of anything in GR which is similar to a QM operator and commutator.

6. Jan 18, 2013

Primordial

Nugatory : Consider this, gravity descrides the relative position of the photonsphere and photons which occupy this sphere are quantized packets of energy relative to the black hole. Right?

7. Jan 18, 2013

Staff: Mentor

Viewing observables as operators on a space of state functions is, AFAIK, not limited to QM; you can do the same thing in classical mechanics (including GR).

The Poisson bracket is the classical analogue of the QM commutator:

http://en.wikipedia.org/wiki/Poisson_bracket

http://en.wikipedia.org/wiki/Canonical_commutation_relation

As the first Wikipedia page notes, Poisson brackets are used in the Hamiltonian formulation of classical mechanics, of which the ADM formalism in GR is an example.

However, as the second page notes, Poisson brackets are not *identical* to commutators: in classical mechanics, the commutator of two observables is always zero, whereas the Poisson bracket is not.

That is the key difference between classical mechanics and QM: in QM you can have observables that don't commute. Which, as you note, gives rise to the uncertainty principle.

8. Jan 18, 2013

Staff: Mentor

Thanks PeterDonis, that was useful.

I was vaguely aware of Poisson brackets, but have never actually used them.

9. Jan 19, 2013

Staff: Mentor

GR describes the curvature of space, and that curvature in turn determines the paths of anything in that region, including light.

I'm not sure where you're going with your question... there's nothing in the description of curved space that requires, suggests, or even hints at a requirement that light be quantized.

10. Jan 19, 2013

liometopum

Hi Nugatory:

That was perceptive!

Stephen Hawking derived a specific energy for black hole radiation, a quantum of energy associated with a BH mass. Didn't he use GR to do it? Isn't Hawking radiation quantized?

11. Jan 19, 2013

bcrowell

Staff Emeritus
No. Planck's constant doesn't appear in the Einstein field equations. Therefore it's not possible to derive anything quantum-mechanical from them.

Hawking radiation doesn't come from just GR. It comes from GR plus other physics.

12. Jan 19, 2013

Naty1

yes.

So the quantum mechanics viewpoint would be to note that Max Planck discovered physical action at small scales takes place in discrete steps, not continuous ones. Action at the sub atomic scale is quantized.

So, for example, the wave function of an electron in free space can take on continuous values, but when in an orbital is constrained to discrete values...quantized energy levels.

PeterDonis:
Never saw that...another perspective; thanks....

13. Jan 19, 2013

Naty1

TrickyDicky posts:

I'm sure no math whiz, but that just seems an overstatement.

Wikipedia says:

http://en.wikipedia.org/wiki/Schrödinger_equation#Quantization

That was my rather vague recollection, but why is it so?
Haven't studied such in a LOOOOOONG time.....

14. Jan 19, 2013

stevendaryl

Staff Emeritus
Here's something that is a heuristic argument that quantum effects cannot come from GR alone. GR has two adjustable parameters: G and c. Using those two parameters, there is no way to get something that has the dimensions of h-bar, the fundamental constant of quantum mechanics.