Great Robert!!

I have the English version of those books at last...

for the response I have to think about it..

Very nice question, Robert.

Thanks for the question up votes.  I appreciate also the one attempted answer, but it really just puts the question off onto Fresnel's principle.

My request to anyone giving defending Huygens/Fresnel across curvilinear coordinate transformations is to point out a flaw in the diagram.  It is in the diagram where my understanding (right or wrong) lies.

Please do not overlook my request to point out other discussions in the literature.  Also, is there a presentation of how Fresnel principle works in various coordinate systems? 

It seems to me that in an "arbitrary" set of coordinates there is no compelling reason for the least time path to be a straight line.  But the Huygens principle is based on straight line expansion of the wavefront from each point.  In Schwarzschild (i.e. not local) coordinates, light is not just slowed at different points, it is bent.  Light in a refractive medium is not bent in the medium, only at boundaries.  This is tricky.  Light is always bent in gravity, not just slowed.  The bending is in addition to slowing.

BTW, in private email Klaus said something about the bending being distributed differently in the transformed coordinates.  That seems like a possible explanation.  But he did not give any references.  Maybe eventually he will post something here.

Does anyone know how Clifford's theory is connected to GRT? 

Actually, there is no "scale" on the diagram.  The vertical distance is arbitrary.  So I think the diagram is the same no matter how it is stretched.

If we go to Fermat's principle of "least time paths" then we see the need to calculate actual light travel times, and make the wavefront a set of points with identical travel times.  In Euclidean coordinates, such as the C & R of the distant observer in a Schwarzschild metric, light does not move in straight lines. 

The figure below attempts to draw the diagram taking equal time paths into account.  The red wavefront at the bottom is the modification to account for equal time paths, and it coincides with the "double bending" as you can see.  But the light path to the midpoint cannot be straight.  It is bent.  Therefore it does not get as far as a "straight" line at the nominal transverse coordinate velocity.  I will bet that when I do a detailed calculation, it gets just as far as the radial lines, making the Huygens analysis valid based on the radial coordinate velocity (which of course gives the right answer, but we've learned to be leery of coincidences).  Any takers on that bet?

   I have just looked up Eddington's explanation in "Space, Time and Gravitation". It is a hand-waving one, but it is good for general understanding. Something that puts me off---and which probably reveals my lack of familiarity with the usage of relativists---is that he gives the solar mass M in kilometers (this is probably the Schwarzschild radius). In the Appendix, Note 9, he gives the elementary formula M = r xv squared, which can be written as M = (beta times c)squared times r.  This gives kilometers only if you take beta times c (= v) as dimensionless; this doesn't make sense to me.

   My apologies if I don't answer the original question. It is not my intention to replace it by another one.

Thank you, Charles. So mine was not a stupid comment after all...

I was told there is book in German language by John Wheeler in which he pays tribute to Clifford. I never saw it, i blushingly admit.

@Prof. Rossler: do you mean "Gravitation" book by Misner, Thorne and Wheeler? If yes, that book is still available online. Ok i will check. Best wishes

Robert, 

"Light in a refractive medium is not bent in the medium, only at boundaries.  This is tricky.  Light is always bent in gravity, not just slowed.  The bending is in addition to slowing"

According to theory of waveguides for EM it is always an Impedance jump that makes the EM wave to be contained in a classical refractive medium. The impedance jump necessary in order to contain the beam is finite, metallic wave guide  or fiber optics (bouncing waves) (as in figure)

a) A particular impedance jump allows the wave guide to contain the beam in fiber optics, preventing the classical refraction where a dispersion of a part of the energy in the other medium occurs, only  reflection is allowed in the transversal direction of the wave guide.

b) In order to prevent a reflection backward along the direction of the wave guide, an impedance adaptation has to be done at the exit.

c) Part of the energy is anyway lost due to imperfections and temperature no wave guide is lossless, unless recurring to very sophisticated spatiotemporal solitons which exploits non linearity of the medium and imperfections.

Space-time fabric is "the wave guide" for EM waves, 

Refraction and  reflection  phenomena are interwined since a continuous change in the impedance of the medium occurs in presence of gravitation, lossless and no need of impedance adaptation..

Hard to say that bending is in addition to slowing...

Stefano your email comment about the onions was interesting.  I'll quote it here:

"Speaking about VLS, the time dilation and length contraction are automatically a consequence of it. Along the Earth geoid, equipotential surfaces, the "general speed of light " can be considered "constant. Space-time around a massive body is like a onion the more you go down the more frequent and thinner is the space inbetween surfaces…the more the speed of light varies…"

"Light passing through the close gravitational field of a massive body, crosses the onion's skins and is deflected accordingly..."

"The time like part and the space like part play equivalent roles in the light deflection, since the null geodesic Equation which is the one regulating the trajectory of the beam by definition… means that space-like and time like parts are equal…"

My comment about bending and slowing is novel and I wouldn't expect anyone to accept it right away (if ever).  But if you take the onion model of slowing, i.e. discrete layers, you'll find that is half enough.  The conventional wisdom is that one adds spatial curvature for the other half.  But it is much simpler to see the other half as simple falling, by equivalence, which also produces the right answer, exactly the right answer in fact even for sublight speeds.  The derivation is in my 2011 paper on inertia.

"But if you take the onion model of slowing, i.e. discrete layers, you'll find that is half enough."

Why half??

It just is.  Maybe you can decompose it into a "reason."  Why is Newtonian bending half enough?  Those are philosophical questions.  But it means you can take any two out of the three typical explanations of bending and add them to get the correct answer:

The last one I cannot calculate, but I infer it to be so because relativists say that the "total" bending is obtained by taking it into account.

See section VII of the 2011 paper.  It is only one page.  Linked below.

I guess I should add that I'm assuming the onion skin approach does not affect the total bending.  It is just a differential model of a time dilation gradient.  But the onion skin allows one to ask if anything is happening to the light between layers where it is bent?  I think it is plausible to assume it is still falling, that the falling has nothing to do with the Huygens bending, just from thinking about the onion skin model.  Once again, Stefano, you have handed me something quite interesting.  :)

Article Isotropy, equivalence and the laws of inertia

Thanks Robert for the appreciations...

I think I got  it....this is half the real ONION...

The light deflection is as Einstein calculated, all derived as factor 2 (1 time+1space) from the variation of the speed of light according to the  distribution of the masses...

The main fact is that the gravitational potential alone is not enough to account for the influence of the masses in the space-time. 

The PHI is  a scalar, is a local approximation of a TENSOR.

What is really needed here with huge distances (tera meters and big masses, Sun) is the real tensor from which the approximated Newton scalar potential is derived.

The onion with the PHI accounts only for the time dilation, but the extended-onion due to  the tensor influence would be enough. With the tensor we have the space-like influence back in the calculations....

Here we notice the difference of the WEP (NEP or the original EP ) experimentally verified in the Etvos experiment (not the Einstein extended EP or E-WEP) which is able to account only for the time part, and the GRT predictions of space-time distorsion for significant space distances.

It is very important to stress the fact that it hasn't anything to do with the EEP or SEP or the E-WEP, Einstein extension of the WEP, which was only an outcome of Einstein's mind to get rid of...and if they have any possible impact they have it local, mean infinitesimal.

This help us understand also the Mercury advance of the Perihelion...which cannot be explained without the SPACE distortion, the correction of the WEP and Special relativity together are not enough as already said by hundreds of authours including Schiff.. A further demonstration that WEP and SR can't make any GRT.

The only thing equivalence doesn't account for is the variation of gravity from place to place (i.e. that it is really NOT "constant" acceleration).  If there were anything else it didn't account for, it would be invalid,.  Then GR would be invalid  This is a much easier position to defend now than it was a few months ago (and has been for the last century), because an accounting of curvature has been given based on equivalence.  In the final (submitted) version I gave two different methods, which pretty much nails it.

I no longer believe it is even necessary to assume the E-WEP as you have been calling it.  the WEP and energy conservation should do it.  But it is much more convenient to assume the E-WEP, which I take as "measurements in a gravitational field which do not depend on the variation of acceleration from place to place must yield the same result as in an equivalent setup in an accelerated elevator far from gravitational fields.."

This removes the inept and vague restriction of locality, replacing it with the real quantity which equivalence does not describe, the variation in magnitude and direction of the acceleration vector.

"the WEP and energy conservation should do it."

I really don't think so.. and any extension of WEP regarding the only acceleration, which Einstein did in 1991 is totally inadequate. Such consideration can survive (if they can) only at infinitesimal distances (locality).

It doesn't actually matter.  It does not even matter if the EP is valid or not.  All that matters is that the math and derivations are consistent.  That if one accepts the EP it should either:

• A. lead to a clear contradiction (no one has really claimed this)
• B. leads to a theory of gravity in which the EP is completely valid and explains everything it should explain.

The case B has been ambivalent for 100 years without anyone invoking clearly case A.  All I have done is add (considerable) weight to case B.  I have shown that accepting equivalence leads to a gravitational theory in which one must choose an additional relation for the variation of acceleration from place to place, which determines the exact curvature coefficients in the metric.  But the presence of the coefficients and even their general magnitudes, and that one has to be the reciprocal of the other, is required by equivalence.  If you choose other coefficients, I can show it leads to case A.

Since these coefficients are all that is required to calculate full light bending (in fact without the further determination of the variation from place to place they are quite close enough using either the potential or acceleration laws of Newton), then equivalence explains full light bending.  Even if one cannot draw it with the simple falling diagrams.

QED

That if one accepts the EP it should either: A. lead to a clear contradiction (no one has really claimed this)

a) Clear refutation of E-WEP there have been (Synge, Fock, Ohanian) and today it is not anymore accepted at finite distances, the SEP is  acceptable in an infinitesimal volume.

b) In the GP-A report there is a big mistake in the attempt to comply the EEP...but somebody maybe didn't even think to  jeopardize their carreeer in 1976-1979 when the equivalence between accelerated frame and gravitational field was still generally accepted.

c) I can see that your proposal is based on the correction of the Newton's relation...the correction of the NEP or the original well tested EP, which is reasonable..

TOTH and MOFFAT did it successfully with a quite complicated exponential law in order to explain the whole galactic gravitational behaviour, being able to get rid of the dark matter.

It is a very generalised scalar potential giving birth to acceleration of masses which can account for the motion of the convetional matter.

So it is reasonable that the  law, whose modification you proposed, can account for the lensing in the weak field limit, why not.

Charles,

not only in General Relativity, in general it cannot be applied for finite distances. And I'm talking about the SEP not the original EP tested in the Etvos experiment .I'm talking about the unhapply extended thought " we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system."

Please look up the SEP. As Robert says, it has never been refuted.

Who said the opposite?

But not at the distance of meters. WEP and UFF have been tested with sufficient accuracy instead even at longer distances.

 it is just a silly mathematical point with no bearing on practical physics

Ask him then...

Atomic clocks nowadays can falsify the equivalence of accelrated frames and gravitation just with a programmed robot, revealing the presence of a gravitational field or not. So at least the equivalence is invalidated over .1 meter which is the actual capabilities of atomic clocks. And the better and the smaller atomic clocks the lower will be the inequivalence limit.

Charles,

You have also misunderstood the ability of atomic clocks to measure gravitational fields at short distances,

which is precisely equivalent to their ability to measure uniformly accelerating frames at  similar distances..

I don't really think so. If you sustain that the atomic clocks measure accelerated frames at similar distances you make me laugh.

There are too many comments above with vague claims for me to follow and either agree or refute.  But I'll pick on one point and ask a question.

1. The EP with regard to equivalence of gravity and acceleration is applicable only to constant acceleration.  This is not restricted in space per se.  It's just that you can't find to gravitational field which is constant over a volume of space.  So it is not testable over an extent of space.  However, as a postulate, it's good over arbitrary spatial extent.  But you cannot use it easily because you have to account for field gradients.  In other words, the EP as used by Einstein has always been a mathematical principle more than a physical one.  Everyone always knew that by detecting spatial variations in acceleration due to gravity, one could detect a real gravitational field.  No one ever disputed this either.

(With regard to the falling-rate EP, you can only test this for similarly moving objects, i.e. you can't compare a mass to a light ray because they won't stay in the same place ... but the falling-rate EP has no spatial limitation otherwise.  Objects will fall at the same rate in a vacuum in a real gravitational field, irrespective of the field gradient)

2. If Stefano thinks he can detect being in a gravitational field without doing so by means of detecting the spatial variation in acceleration (either magnitude or angle) that would be interesting.  But I doubt it so strongly that I won't read some long paper.  It is not even worth considering to that degree.  I would read a short argument to assess whether any particular idea had merit.  This would completely undermine a lot of modern physics, and as the saying goes, strong claims require strong evidence.

Robert,

"The EP with regard to equivalence of gravity and acceleration is applicable only to constant acceleration,  you cannot use it easily because you have to account for field gradients"

exactly...how can it be used as an approximation of a real gravitational field??

However, as a postulate, it's good over arbitrary spatial extent

You mean it is given for granted..............

As an euristic tool it seems to work fine in finiding similar general phenomenologies like predicting a frequency shift of a photon...but the detailed analisys of what happens in the two configurations accelerated and gravitational, reveals totally different quality of the phenomena.

The doppler shift of the first order implies that some energy is provided to the mass-photon ensamble by the kinetics of the masses, rising (or lowering) the energy of the absorbed photons.

The gravitational Redshift doesn't imply any energy exchange between the photons, during their free path, and the gravitational field or the masses at stake which are static. The photon is not chnaged in its initial energy, it is the gravitational potential difference which allows different energy absorption for twin atoms so that the frequency "perceived" is different.

It is the old story of the gravitational potential energy of photons.  Einstein had put it forward in his 1911 paper, Feynman  too followed him in 1960's in one of his famous lectures, allowing the photon to be massive and not only him unfortunately fell initially in this trap.

According to GRT and QM the photon cannot have a mass, when energy becomes mass, by getting absorbed, it is not a photon anymore. (gravitation photons clocks attached). No gravitational potential energy of photons is allowed .  This is proved also with the help of the GP-A experiment results. The only explanation possible is the different absorption capabilities of twin atoms at different gravitational potentials.

"With regard to the falling-rate EP, you can only test this for similarly moving objects, i.e. you can't compare a mass to a light ray because they won't stay in the same place"

You can compare, according to the predictions of the principle, the system before and after absorption and see if energy conservation is satisfied, in the reference system of the "planet" where the free fall process occurs. If that is complied then no reason for objection, if not, any principle has to be discarded.

Everyone always knew that by detecting spatial variations in acceleration due to gravity, one could detect a real gravitational field.

Obvious.

If Stefano thinks he can detect being in a gravitational field without doing so by means of detecting the spatial variation in acceleration ....

if we are talking about a real gravitational field, it is enough as I already told you, two sufficiently sharp atomic clocks. They don't delay  in the same accelerated frame, as also pointed out by Victor Toth in one thread, but they do it certainly in a real gravitational field. Unless we affirm that an ideal omogeneous gravitational field doesn't allow clocks to delay.. and photons to shift either.  

This would invalidate completely even the eurstic effectiveness of the Einsten application of the WEP in his 1911 paper, stating that Einstein's results were totally fortuitous ( going from bad to worse....).

Einstein tried to extrapolate something which in "principle" was arbitrary though intuitive ..

He made something strictly local and belonging to single masses with their relevant center of masses as stated in the EP, become quasi-local let's say in a finite region.

He was reasonably worried only about  forces both in the accelerated and free falling frames..

Regarding specifically the EP,  F=ma --> mgg =miai  the EP or WEP, was  experimentally verified in the Etvos ... The WEP can be also reformulated according to Galilean free fall and becomes the UFF, conceptually is equivalent, but always regards the relation one to many between a massive object and his center of mass and single, poinlike small masses,  free falling.

The NEP regards planetary motion, m*(MG/r2) = ma(r)i , very well tested as well.  Under a certain accuracy it is falsified by the Mercury perihelion, GRT is a much better predictor.

Einsten's WEP in 1907 was arbitrary.: ...mgg = miai --> g = ai    numerically holds but it is strictly local.. thinking of extending it to a finite region of space, though small,   by giving the same properties to what is  in between masses has drawbacks.

The phenomenologies regarding forces inside the acc and real grav systems are the same with good approximation, but effects of higher order, like time dilation are not..

Article Gravitation, photons, clocks

Article A Falsification of the extension of the Equivalence Principle

Article About Time dilation in accelerated frames

It is necessary to answer only one of Stefano's questions (points) to show that he and I are talking about unrelated topics.  I think it is a matter not so much of disagreement as disinterest - by either of us in the other's topic.

Stefano asks: "how can it [equivalence] be used as an approximation of a real gravitational field?"

No one has ever suggested such a thing.  Except over infinitesimally small regions of space, equivalence does not model gravity at all.  Equivalence is not an approximation of gravity.

Rather, equivalence provides theoretical insight about the effects of gravity on time and space.  Stefano has previously stated he has no interest in theoretical constructs, so the conversation with Stefano must end here. 

For others (i.e. if your name is not Stefano, meaning I do not expect Stefano to make take any interest in or make any reply to anything below this line, since it is entirely theoretical), the theoretical use of the EP as a postulate is a matter of convenience.  One could just as well use some other postulate, and there has been a suspicion since about 1960 that one could start only with special relativity, the conservation of energy, and the universality of gravity.  But for some people, obviously not Stefano, the EP provides a shorter path.  To use this path, one must analyze first in a hypothetical situation of gravity with constant acceleration, determine the relation of the relativistic effects to gravity, and then as a second step make it real by choosing a law of gravity, or deriving it from the Einstein field equation.  If one doesn't care for theoretical arguments to achieve accurate practical theories, of course they will not like this procedure.  It is pointless to criticize it on the basis that equivalence represents something that is not claimed in the procedure.

   Apparently, Robert, you are one of the people for whom "equivalence provides a shorter path" to "theoretical insights about  the effects of gravity on time and space" ? 

Raul, not that there is anything "short" about a path that was 100 years in the taking.  But since Schiff's conjecture about all the effects of equivalence being implied by energy conservation is not formally proven, then by that measure I suppose it is short-er than that, yes.

To clarify what I mean by the 100 years ... of course Einstein loudly lists equivalence as a postulate, but his "derivation" is sufficiently obscure that people have been arguing ever since over what role exactly equivalence plays.  I was referring to my recent analysis which achieves static gravity via the EP and a separate determination of a law of acceleration.  Many similar proposals have been made and rebutted.  We'll see if mine gets published and if so will anyone rebut it.  I used a shotgun approach, so they'd have to rebut at least two different approaches.

I'd like to point out that Stefano contributed one of the approaches.  He may not care for equivalence, but he is instinctively good at pointing things out whether he likes it or admits it or not.  :D

Robert

"To clarify what I mean by the 100 years ... of course Einstein loudly lists equivalence as a postulate, but his "derivation" is sufficiently obscure that people have been arguing ever since over what role exactly equivalence plays"

   miai= mgag results true   in the ETVOS torsion balance Equilibrium experiment, it is a comparison between static forces  ..

such equation is  strictly punctual in the center of the masses of the material objects.

If I simplify the masses there, I lose information.

The equation with only accelerations has to be punctual too, strictly local in order to be derived from the equivalence of the forces.

IF then I consider it valid from infinitesimal to finite, saying that it can predict phenomena at "human scale" like the gravitational Redshift, it is another  arbitrary assumption.

The question I ask is, why such predictions of EEP actually work for light??

Is it because light does not feel gravitation as a force and any acceleration at all and moves along geodesics?

Very interesting sentence of EDDINGTON, pag. 68 of SPACE TIME AND GRAVITATION:

"We have said that no experiments have been able to detect a difference between a gravitational field and an artificial field of force such as the

centrifugal force. This is not quite the same thing as saying that it has

been proved that there is no diifference. It is well to be explicit when a

positive generalisation is made from negative experimental evidence. The

generalisation which it is proposed to adopt is known as the Principle of

Equivalence."

The equivalence beteween a centrifugal field and gravitational field...this is reasonable

Charles,

I recognise that light does not feel acceleration and follows geodesics being massless.. I don't recognise at all that real objects begining from fermions which being massive disturb heavily the metric of the space time locally can ever follow a geodesic.

Stefano

Charles,

for sure GRT does not account for quantum particles, but I simply said massive objects beginning from the smallest mass we know which is the fermion. In any case, the perturbation in the space-time which locally a fermion creates is very significant locally it is a singularity of the EM field and its mass density is very remarkable. Every object we know is made of fermions and the fact that  objects do not significanly perturb the space time in their nearest proximity is not physically acceptable.

Stefano

Localised to the extent that it respects  the indetermination principle of Heisemberg regarding position and momentum.

It is not said though. Only measuring disturbs the actual location/momentum of an electron, if there is no photon or other particles to disturb its position or momentum, the Heisemberg Principle should not apply. Nothing forbids that its location is fixed at a certain time, though impossible to be known.

Dear Robert,

"No one has ever suggested such a thing.  Except over infinitesimally small regions of space, equivalence does not model gravity at all.  Equivalence is not an approximation of gravity"

Great to hear such thing, I fully agree.

Equivalence in terms of mapping accelerated to gravitational configuration in finite situations, seems to be a useful predictive tool in absence of spatial curvature (finite but not long distances) it is interesting to understand what is behind it which makes it work.. some surprise may rise..

We are close to being able to do without the gravity-acceleration version of equivalence, which you (i.e. Stefano) have called Einstein-WEP.  If we use only falling rate equivalence (Galilean), another way of saying gravity's effect is universally the same, then it should not be limited to small extents.  The only thing E-WEP was used for historically anyhow was to get time dilation, which is available via energy conservation and Planck's law.

Robert,

"If we use only falling rate equivalence (Galilean), another way of saying gravity's effect is universally the same, then it should not be limited to small extents"

the UFF (universality of free fall) for non charged and non rotating bodies holds to a very good approximation. The point to clarify is orbiting objects are considered a particular case of free falling ones... do you think that in such cases (of orbiting objects) we can apply the concept of inertiality??

Very clever, Stefano.  Yes but there is a catch.  Free fall is a function of location, and unless the orbiting objects are bosons (therefore capable of occupying the same location at the same time) we cannot actually compare the free fall rates of two objects. 

However, if you do the experiment at different times with two objects, not changing anything else, their free fall paths (i.e. orbits) should be identical, don't you think?

if you do the experiment at different times with two objects, not changing anything else, their free fall paths (i.e. orbits) should be identical, don't you think?

sure, but this is not very related to inertiality but to repeat same conditions which should give same results...

The objects are different.  Only the initial position and velocity are the same.  I thought that's what you meant, would they follow the same path.  Otherwise I didn't understand your comment.  : )

Sure Robert,

the equivalence principle in free fall. Yes it is surprisingly valid also for orbiting laboratories at low orbits in which the gravitational field almost cancels out. . the Einstein's sentence: the gravitational field cancels out in the proximity of a (free falling) orbiting object has to be true ...  and this makes me quite unconfortable              

Stefano:

   If we agree that an  orbiting lab is also freely falling, it is natural to suppose that it does so because of some external gravitational field...so we cannot say that the latter cancels out. 

We (humans instinctively) have two ways of determining lack of action:

• Did I feel a force?
• Did I see a change?

The two can become strangely linked as for example when sitting on a train or plane with little other reference than the limited view through a window of another such vehicle.  When the eye detects relative motion, usually the body will at first feel it.  It is only after some investigation and looking around that the false feeling of motion fades away.  An inertial measurement unit will of course not be fooled, but neither can it think about physics and post on RG.

If you and I are in spacesuits at the same orbital radius, we will cross paths (or bump into each other) twice on each orbit.  Our eyes will tell us we are accelerated, but our IMU's will dispute that (discounting the bump).

So I argue it is just as natural to say that if there is no apparent relative motion nothing is acting on an object.  That would include me sitting in my chair, though you (generic you, I have no one in particular in mind) may legitimately consider gravity to be pulling me down and/or earth pushing me up. 

The trouble with earth pushing me up is that there is no result of the pushing.  This does not jibe with our idea of pushing, really.  But if I walk off the edge of a cliff there is a strongly noticed result of gravity pulling.  All of a sudden everything moves and my stomach flutters exactly like when the train moves.  Or when the other train moves.  (The similarity of the two makes ordinary relativity much easier to accept than gravitational relativity.)

Mathematically, a universal acceleration operator in Euclidean space is indistinguishable from a particular spacetime geometry which can be computed as corresponding to the operator.

I don't know if this has been formally proved, but I think I can prove it.  The gist is, any universal acceleration operator can be replaced by a particular spacetime geometry, and any spacetime geometry so produced can be again replaced by Euclidean geometry and a universal operator.

There remains a question as to whether there exist spacetime geometries which are not derivable as above, and are physically realizable.  In that case, they could never be replaced by the Euclidean space and the universal operator.  These can exist as theoretical solutions to the field equations in several gravitational theories, including GR, but none have been observed.  Black holes can indeed be replaced by the E-space and the operator.  It is only spacetimes that have an alternate path out of a closed box, without going through the surface of closure, that cannot be replaced by Euclidean space and an operator. (i.e. wormholes)

This, by the way, answers another question I posted about the necessity of curved coordinates.  I now feel it is immaterial and superfluous to prefer one description over the other.  They should both be taught on the same footing.

It is quite interesting the description of Fock in the link.

https://archive.org/stream/TheTheoryOfSpaceTimeGravitation/Fock-TheTheoryOfSpaceTimeGravitation#page/n407/mode/2up

Please read the paper, may be helpful for you.  A new approach to the correction of

Galilean transformation.

There has been a comment recently in the Journal of Physical Mathematics about light bending in General Relativity.

It appears that when the internal structure of photons is clearly understood, there is no need to resort to GR spacetime curvature to explain the correct light bending angle.

http://www.omicsonline.com/open-access/on-de-broglies-doubleparticle-photon-hypothesis-2090-0902-1000153.php?aid=70373

Andre, your paper looks extremely interesting.  I was not even aware of this aspect of de Broglie's work.  I will study it next week. 

Glad you like it Robert.

If you really get into the 3-spaces geometry and the possible nature of kinetic energy, you are in for many more surprises.

Andre, I can't be sure what you mean exactly, but generally that is what I've been working on for 10 years now.  I posted a link to your paper on another question thread that has seen some discussion lately, here https://www.researchgate.net/post/Is_it_possible_to_derive_the_constant_uniform_velocity_of_light_the_Lorentz_transform_without_starting_from_the_principle_of_relativity