Does GR produce the Planck relation and uncertainty principles?

[liometopum]

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

 

[Nugatory]

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

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.

 

[liometopum]

Thanks Nugatory!

 

[TrickyDicky]

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.

I can't see how that matters, Schrödinger 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.

 

[Dale]

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.

 

[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?

 

[PeterDonis]

I cannot think of anything in GR which is similar to a QM operator and commutator.

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.

 

[Dale]

Thanks PeterDonis, that was useful.

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

 

[Nugatory]

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?

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.

 

[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?

 

[bcrowell]

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

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

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?

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

 

[Naty1]

Planck's constant doesn't appear in the Einstein field equations.

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:

in classical mechanics, the commutator of two observables is always zero, whereas the Poisson bracket is not...

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

 

[Naty1]

TrickyDicky posts:

I can't see how that matters, Schrödinger 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.

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

Wikipedia says:

The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur... One example is energy quantization: the energy of an electron in an atom is always one of the quantized energy levels, a fact discovered via atomic spectroscopy.

http://en.wikipedia.org/wiki/Schrödinger_equation#QuantizationThat was my rather vague recollection, but why is it so?
Haven't studied such in a LOOOOOONG time...

 

[stevendaryl]

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

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.