Albert Einstein departed from constant light speed in General Relativity in his 1921 lectures at Princeton. The reference is given in his equation 107 where L is the ratio between local light velocity in a gravity field divided the usual constant light speed in flat space.
https://einsteinpapers.press.princeton.edu/vol7-trans/369
Researchers including Peter Bergmann argued that the direction of light would change but not the magnitude. Other researchers including Max Born argued that Einstein intended the magnitude of light speed to change in a gravity field. The answer is found in Einstein's following equation for alpha where L occurs in the denominator as well as in the differential, showing that Einstein's velocity of light is a scalar, the local speed of light in a gravity field relative to the cosmic standard. Local light speed slows down as gravity increases, and approaches zero at an event horizon, explaining why nothing escapes.
As analysis continues the widely accepted equations of energy and momentum make an incomplete set when local light speed is not constant. The set can be completed by Einstein's equation 107, giving exact calculations for well defined cases.
Results show that when local light speed changes, the local measurement of mass also changes. Electric and magnetic properties of vacuum space also change.
Planck h is the only parameter of space time that doesn't change when Einstein's equation 107 is applied in energy and momentum equations over all examples of General Relativity.
In exchange of energy between physical objects and vacuum space where E=hf for quanta and dE/df = n*h for large collections of quanta, the equation 107 gives an answer (n=1) which is general relativity. In the hierarchy of Plancks at extreme high energy density (n>1) h becomes variable. Other cases of extreme low energy density (n=0) represent the local end of time and collapse of a false vacuum.
In terms of Nobel Prize winner Roger Penrose the values of n may be used to represent his U unitary evolution in space time, and his R state reductions may be expressed by the change from one n to another.
Parameter n is a quantum number and h represents the quantized spin angular momentum of localized stress energy curvature in vacuum.
Implication of this representation is that vacuum space time has a quantum state that doesn't change over all of General Relativity. It doesn't explain why h remains constant when other local properties of vacuum change.
Why Is Planck h Constant In General Relativity?
It isn't. General relativity is a classical description. The fact that Lorentz transformations aren't global symmetries (since spacetime is curved-that's how it's possible to understand that the speed of light isn't globally defined in curved spacetime) doesn't have anything to do with Planck's constant, in the guise of its defining the scale of quantum fluctuations.
E = h f is just an expression of dimensional analysis-it doesn't mean, by itself, either that the prefactor to h f must be an integer, nor does it imply that the fluctuations of energy about this value, in general relativity, are due to quantum effects. The fact that energy is quantized in the technical sense of the term-that it's possible to resolve individual photons-is beyond the domain of general relativity, where the energy-momentum tensor on the RHS of Einstein's equations, first of all, doesn't know what are the fields that produce it, for it to appear there and, second, is a classical quantity, that makes sense when it's not possible to resolve the individual quanta of the fields that produce it.
In addition: In special relativity energy depends on the reference frame and in general relativity, also. In special relativity the vacuum state is uniquely defined, in general relativity it isn't, precisely because invariance under time translations is a local, not a global, symmetry in curved spacetime.
Thank you Stam Nicolis for your informed comment.
About energy I agree that energy is not conserved in relativistic transformations because not all observers will notice a sequence of events in the same order, also for other reasons.
Integer multiplier of Planck h is not proven but is supported by other researchers in topics of Squeezed Quantum States, Scale Relativity, and Hierarchy of Plancks. Something new might be discovered in this topic.
General Relativity as you pointed out is a classical representation. Albert Einstein described it as a second order approximation suitable for most purposes, but subject to rapid improvement by other researchers. He truncated high order terms to preserve covariance with approximations in his equations that follow equation 91, and continued with additional approximations in equations 92, 93, and 94 to get the field equations. Einstein made additional approximations in equations 98, 102, 103a, and 106. GR is not exact in classical representation, and not quantized as you pointed out.
Planck h is constant in GR, but not necessarily constant in an exact calculation or in scientific principles.
Some researchers have calculated RHS from Lagrange Density, giving different answers flatter space than conventionally obtained from stress energy. One example is FLRW where dark energy is invented to flatten space in compensation for a mistake in curvature that Lagrange Density corrects.
The present question is asking if GR assumptions can be challenged in cases that have not been experimentally confirmed. I'm interested in the high energy density cases where second order curvature is not sufficient, and quantum effects become significant.
Thanks again for your comments.
Once more: GR is a classical theory and, within that framework Planck's constant is just a quantity that has the dimensions of angular momentum. It doesn't make sense to ascribe to it any particular significance for GR. In most applications of GR to rotating systems there are other quantities that provide a more appropriate scale for angular momentum than Planck's constant.
``High density'' of matter doesn't mean that quantum effects can be described by introducing terms that are higher powers of the curvature.The description remains classical, since the equations of motion are being solved. Nor does it mean that, if the density is high enough, such higher order terms are either necessary or sufficient.
There's a huge literature on this subject and none of these modifications change, qualitatively what's, already, described by the Einstein-Hilbert action.
GR is exact, as a classical theory, in the sense that it is complete: The equations of motion do admit solutions that can be proved to be unique and this includes black hole spacetimes.
There's been huge progress since Einstein's time, so it's not useful reading his lectures in order to understand the subject.
The FLRW solution is one solution, that's useful for describing certain effects in cosmology. It doesn't represent any contradiction.
Regarding challenging the assumptions of GR, the only assumption is invariance under diffeomorphisms. That wasn't understood immediately, but it is now. The reason it doesn't make sense to challenge that is that the answer is known: The mathematics is inconsistent if invariance under diffeomorphisms is broken.
It is possible to extend GR and the most general way to do so is, also, known, it's supergravity. Which includes the most general set of fields, beyond the metric, that can determine the spacetime geometry, in a mathematically consistent way. Which realization of supergravity is appropriate (there are many) is an open question.
Supergravity is, still, a classical theory, it should be stressed.
agree with Stam. h has no part to play in GR.
When photons zing past the sun and are deviated, at least in initial GR approximation, the speed of light is still c all along the path.
A ratio GM/(d cc) is dimensionless , d is the radius of sun, M is its mass.
It is small everywhere in our local solar system, can be used as a measure
of the strength of gravity in general. The angular deviation is 4 times this ratio in GR. In this calculation, photon energy just drops out, no dispersion as in refraction.
Energy conservation becomes lost in GR, at least in any simple form.
Thankyou both Stam Nicolis and Juan Weisz
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I am working outside the GR constraints, but not on super symmetry or super gravity. Those fields are well enough represented by specialists and their thirty or so theories.
In consideration of deep space transport at high speed, especially in extension of the concept Polarizable Vacuum, it became necessary to predict how and where kinetic energy is transported, a topic neglected in science. The conclusion using equivalence is that kinetic energy is carried in a local stress energy field in and around a fast moving vehicle as Einstein recommended in 1949. Lagrange Density treats the kinetic energy as a source of reverse curvature opposing the curvature of gravity when LD occurs on the RHS in field equations.
An implication is that the physical vehicle exchanges energy with the local stress energy field during acceleration and deceleration in quanta described by h and f, although GR does not say this. It is well understood that GR must coexist with quantum effects, both of which may be valid at the same time in many cases.
The 30 famous supergravities were not helpful for answering questions of physical limits at high speed. Help came from an old classical theory TGD that uses Hierarchy of Plancks with n and h, but is still classical and agrees with GR in all cases of GR that have been verified, differing in extreme energy densities where space is folded into higher order curves than GR second order.
TGD allows that 4D space can become folded into layers somewhat like an accordion by sufficiently high energy density, which is kinetic energy in my high speed cases, allowing a worm hole to tunnel from one layer to an other. Some researchers try to do this with GR in pictures, but not showing the math that is higher than second order curve. In classical GR a worm hole takes a longer path and more time than transit in the space time continuum. Also 4D is not enough dimensions to produce a worm hole, so the use of GR there is in the topic of super gravity without saying why so much curvature has occurred. I use kinetic energy to produce the folded layers in front of the high speed vehicle.
Answering your comment, I realize that much work has been done to extent GR, but in ways that have not removed the approximations I listed in the 1922 publication. About exactness I feel that approximations and discarded terms do not produce an exact result, although it may be precise in many cases. Extrapolation is not wise when part of the math is left out. Higher order terms are not the reason for using h. Exchange of energy with the local vacuum is the reason for using h. Other descriptions of angular momentums are available in physics, but h is the one that describes a physical object giving energy to vacuum or receiving such energy from vacuum.
It was necessary to use h for bringing GR equation 107 into conventional energy and momentum solutions where local c varies with acceleration. Although h does not appear in the final result the parameter n does appear and relates to the Penrose state reduction R. In classical GR n is equal to one and can be left out. Someone else might find a way to reach the result with out using h and not containing parameter n, but without n the result will have infinities, suggesting the math is wrong. Penrose state reduction R prevents infinities in the math, and Penrose U unitary evolution represents the intervals between successive R processes.
The result brings velocity terms into Polarizable Vacuum theory along with frame dragging, the lack of which was the main objection to Polarizable Vacuum. Also there emerges a theory of high speed transport in deep space free of infinities, based on well established work of others, and making predictions that may be tested, plus giving explanations of Big Bang cause and effects.
You are correct that it isn't GR, or supergravity, but it is internally consistent, extension of accepted science, in agreement with verified GR, and predicting what machines to build and what tests to do with them.
Thanks again for your comments which are always helpful.
The answer to this question depends on why you think h is ever constant. If you consider quantum mechanics in terms of wave particle duality, h is the discrete unit of action that is generated between wave crest antinodes. In my opinion, that is a consequence of the Euler relationship for complex numbers, and the fact that action causes the phase change in the wave. The fact that angular momentum has the same dimensions as action is merely a cause for confusion. and something that complicates QM for those that think it is that angular momentum is quantised. Since formal GR does not involve quantum effects or include wave particle duality, h is not present in it, so you cannot say whether or not it is constant. However, to decide you must first properly represent action, and not simply in terms of the observer.
I would like to add that it is misleading to talk of the local speed of light being different from c and even going to zero at the Schwarzschild radius. That speed should rather be called a coordinate speed, because it is coordinate dependent. If the Schwarzschild metric is described by Kruskal-Szekeres coordinates instead, the (coordinate) speed of light will not be zero at the event horizon.
If the term "local speed of light" is reserved to mean "what a local observer will measure as the speed of light" (in terms of standard rulers and standard clocks), then the local speed of light will remain c. It is therefore as constant as ℏ...
Here is a short calculation for the radial local speed of light in the Schwarzschild metric to prove my point:
We have ds2 = (1-2M/r) c2dt2 - (1-2M/r)-1 dr2 = 0, giving for the radial coordinate speed of light dr/dt = (1-2M/r) c. The length of a coordinate stationary standard ruler corresponding to a coordinate interval dr is: dl=(1-2M/r)-1/2 dr, the proper time interval of a coordinate stationary observer dτ=(1-2M/r)1/2 dt, so the local speed of light in radial direction for such an observer is
dl/dτ=(1-2M/r)-1/2 dr/((1-2M/r)1/2 dt)=(1-2M/r)-1 dr/dt = c. Since the result is independent of r, it holds at the event horizon by continuity.
A similar calculation for azimuthal or polar direction of the light shows that you get c in all cases (and independent of the choice of coordinates).
Dear Jerry Decker ,
The "Local speed of light" being the speed of light measured by an observer locally (Local Lorenz In-variance), is indisputably c from RADAR experiments in absence of gravitation. On the other hand the speed of light measured non-locally in presence of gravitation does not give the same results.
It is clear that in the Shapiro Delay experiment the speed of light during its RADAR path Earth-Venus-Earth was not constant c at all.
Although its path is not straight as calculated from GR, that light-bending involves a length variation of a few meters (astonishingly!!), negligible in comparison to the delay of 200 microseconds which undergoes light in its return path corresponding to 60KM (at the speed of light). The Shapiro time delay can be justified only if the RADAR front travelled at a lower speed than c. We have only one clock, same as the only clock used to measure the speed of light within a platform without any influence of gravitation.
The RADAR whose beam approached the Sun two times, showed clearly that the speed of its microwaves along its trajectory was not c.
IT is variance with the same RADAR whose trajectory did not approach the SUN but joined Planets with no Sun in the middle, whose result would be c.
Preprint SHAPIRO TIME-DELAY, Curved 4D space-time or Variable speed of light
K. Kassner thank you for reply.
Einstein set ds to zero to calculate light speed in a gravity field one paragraph before equation 107. You have a special case of nonrotating central force which also calculates light speed in the same way.
ds2 = (1-2MG/rc2) c2dt2 - (1-2MG/rc2)-1 dr2 = 0
Then dr/dt is locally measured light speed which means ds is zero.
dr / dt = c (1-2MG/rc2)
In Einstein terms L is used.
L = dr/cdt = (1-2MG/rc2)
With Equivalence and Lagrangian the kinetic energy is added.
L = (1-2MG/rc2 + v2/c2)
And when far from a gravity source the second term becomes small.
L = (1 + v2/c2)
Local light speed faster than c is measured by a traveler who ages less than the non accelerated frame.
This last equation agrees with result of Max Born obtained obtained in much the same way page 357 in his Dover edition..
ξ/τ = ( -g44/g11)1/2
"Therefore the velocity of light is much greater for some directions of the light ray than its ordinary value c, and other bodies can also attain much greater velocities.
Also page 285 of 1922 edition, has the same paragraph. https://archive.org/details/einsteinstheoryo00born/page/n7/mode/2up
Born and Einstein discussed the Born book in letters before it was published.
The question was about h, but the local variation in c is one cause for asking.
I am copying my answer from a similar thread. "The Planck constant h is constant, but it is not a constant in the usual sense - it is a unit of action required to make a Schrödinger wave complete a period. If it is not constant, then the Schrödinger equation is false. So if you have evidence whether or not that equation is violated, there is your answer."
Note h is not an energy, a length, or anything else - it is a unit of action.
Ian Miller Planck h when multiplied by fractions and integers describes the spin angular momentum of subatomic particles. Researchers just mention the fractions and integers presuming that others will know that they are multiplied by h.
Planck h is a property of space time and measure of angular momentum either potential or real.
There is no problem in Schrödinger equation because energy hν is conserved, such that if h becomes twice the usual quantity possibly triggered by some extreme energy density then ν becomes half the previous value. In this way stress energy density can be contained in higher concentrations with out causing space time to become degenerate.
Roger Penrose wrote about U process and R process, where in this case R is the state reduction by which h can become an integer multiple of the standard value. A different U process evolves before and after the R process.
In this representation state reduction does not change Schrödinger equation, even if the state reduction occurs during a wave cycle.
Of more interest is the quantum mechanical matrix equation, the canonical commutation relation of Max Born, suggesting that a change occurs in the matrices for location and/or momentum when h is incremented or decremented.
pq - qp = hI/2πi
Considering that momentum must be conserved in a state reduction, the location probability matrix must undergo state reduction when h does.
Also affected are the probability density functions which provide the logical next step in this discussion. Once again the momentum must be conserved, causing the probabilities of location to undergo state reduction, considering that the state reduction may occur during a measurement.
Jerry Decker
My answer was that h is the action required to complete a period for the wave component because action determines the phase. The general relationship is ψ = Aexp(2πiS/h) and a period occurs when S/h = 1. Half quanta arise when it takes two cycles to generate a period, which is generally recognised for the spin. The reason is, there are no nodes in the wave function, so you need two cycles, one to generate a crest and one a trough. Failure to do that means no wave.
Ian Miller
Thank you for reply.
The ψ function is a square root of probability for finding the related particle in a given location. In my previous comment I pointed out that the probability of location must change during state reduction to conserve energy and momentum. The ψ wave will complete a cycle, but with different geometries.
Notice that action S has units of angular momentum both in classical and in quantum use. In classical systems with many possibilities the one that occurs has a smallest possible S. In quantum use the location with highest S is the most likely place to find the related particle.
The wave equation can also be written in momentum space, which Paul Dirac referred to as the Heisenberg representation.
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