liometopum said:Can the Planck relation, and the Heisenberg and the time-energy uncertainty principles be derived, or produced, from the equations of General Relativity?
Nugatory said: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.
DaleSpam said:I cannot think of anything in GR which is similar to a QM operator and commutator.
Primordial said: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?
liometopum said:Can the Planck relation, and the Heisenberg and the time-energy uncertainty principles be derived, or produced, from the equations of General Relativity?
liometopum said: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?
Planck's constant doesn't appear in the Einstein field equations.
in classical mechanics, the commutator of two observables is always zero, whereas the Poisson bracket is not...
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.
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.
liometopum said:Can the Planck relation, and the Heisenberg and the time-energy uncertainty principles be derived, or produced, from the equations of General Relativity?
The Planck relation, first proposed by Max Planck in 1900, describes the relationship between energy and frequency of a photon. In the context of General Relativity (GR), it refers to the relation between the energy of a particle (such as a photon) and its corresponding frequency in curved spacetime.
In GR, spacetime is curved by the presence of massive objects. This curvature affects the path of particles traveling through spacetime, and thus also affects their energy and frequency. Through the equations of GR, it can be shown that the energy of a particle is directly proportional to its frequency, resulting in the Planck relation.
The uncertainty principles, first proposed by Werner Heisenberg, refer to the fundamental limit on our ability to simultaneously measure certain pairs of physical quantities, such as position and momentum. In the context of GR, the uncertainty principles are related to the uncertainty in the measurement of spacetime curvature. This is because the curvature of spacetime is affected by the energy and momentum of particles, and the uncertainty principles limit our ability to know these quantities precisely.
Yes, GR does predict the existence of the Planck scale. This is the scale at which the effects of quantum mechanics become significant in curved spacetime. At this scale, the curvature of spacetime becomes so extreme that the classical equations of GR break down and must be replaced by a theory that combines both GR and quantum mechanics.
Yes, there have been several experimental confirmations of the Planck relation and uncertainty principles in the context of GR. One example is the observation of gravitational lensing, where the path of light is bent by the curvature of spacetime, providing evidence for the Planck relation. Additionally, measurements of the uncertainty in the position and momentum of particles in curved spacetime have also been made, supporting the uncertainty principles in GR.