Very good question, Sadeem.  The point has been argued in the literature as to whether equivalence produces the bending.  I will cite pivotal papers that

• introduce the local-global terminology
• dispute that anything other than local calculations are necessary while maintaining (and there is a long history) that neither equivalence nor flat spacetime is sufficient
• and finally showing that indeed equivalence is sufficient to explain and calculate all bending

It is not freely downloadable so I'll quote sections:  "The present paper is writ ten in reaction to a false rumor that has a certain currency in the literature. This asserts that , since Einstein’ s equivalence principle is somewhat vague and heurist ic, none of it s conclusions can be fully trusted. In particular, its conclusion about light bending is held to be contradicted by Nordstrom’ s second theory which contains the equivalence principle and is in effect based on conformally flat spacetime: it is alleged that because of the latter there can be no light bending in that theory ... To clarify the situation, it is essential to make the distinction between two types of light bending, local and global."

In this paper, "local" bending results from the light keeping pace (or falling) with a reference mass in a rocket or in a small area of a gravitational field. 

"Global" bending is not calculated, but portrayed in words describing a figure which is not drawn: "There is ... a second contribution.  We consider the bent Newtonian light path [which oddly the authors have named the local bending] to be the central line of a narrow strip which we imagine to be cut out of the plane.  This strip we now glue onto what is know as Flamm's Paraboloid.  This is essentially an infinite plane with a circular funnel-shaped hole in the middle, somewhat like the wide end of a trumpet, and it represents the real geometry of the central plane of the sun's field in which the ray lies.  A little experimenting with such a curved strip will quickly convince the reader that the depression in the middle will impart an extra amount to the total deflection"

Surely this paper with first grade paper construction explanations only got past reviewers because of the stellar reputations of the authors.  But I digress...

Then note "The failure of the equivalence principle to fully account for the total deflection ... has been analyzed and discussed in this journal [Am. J. Phys.] by many authors from different points of view [11 references!]"

They calculate by three methods and conclude 1/3rd of the deflection rate is due to acceleration with respect to local inertial frames, and 2/3rd is due to spacetime curvature.  There is a parameter q which if zero represents no curvature which they say confirms the "local equivalence of accelerating frames in flat spacetime and the corresponding stationary frames in the gravitational field in the limit of zero curvature."

Their comments about q=1 are even more interesting, as they say "The agreement of calculations ... with q=1 verifies that the approximations made in the latter are valid in calculating a LOCAL quantity like the RATE of deflection [emphasis added]."   In other words, they disagree that light bending can be distinguished based on local and global.  On that I would agree with them, though the rest of their argument about "flatness" is suspicious in light of the publications we will encounter later.

It seems to me that Ferraro's paper represents the considered "opinion" of the "physics establishment," at least in the United States.  I could find no later works rebutting Ferraro.

That leaves only finding simpler ways to explain it (Ferraro's abstract paper has no figures), and convincing the stragglers that cutting and pasting lines on funnels does not refute the completeness of equivalence in describing bending.

I offer my own simpler calculation below, with figures.

Article The equivalence principle and the bending of light

Thank you Robert. nice answer. So the issue is still opened.

The principle strong of equivalence (EEP) is valid where there is no curvature (Einstein). Where there is curvature is impossible to eliminate it (Logunov) therefore EEP does not apply . And where there is no curvature EEP is false (author, see http://www.cirjap.com/ojs/index.php/jap/article/view/66N)

@Gomez, that link is not to a specific article, can you correct?

There are a lot of misunderstandings even among brilliant physicists about the equivalence principle, because it is not a statement about a particular spacetime.  Think of it is an emulation.  No SR reference frame is equivalent to curved spacetime in GR.  However, for a rocket which is accelerating, there is no direct SR analysis for its frame since it is not inertial. 

What I believe Einstein was saying was that if one analyzes from the rocket viewpoint - which can done by continuously adopting instantaneously co-moving reference frames - the effect will be as if there were curved spacetime.  But there is no one inertial frame in which equivalent trajectories will appear.

I am sorry. I correct for this address.

Thank you

http://www.alfonsoleonguillen.net/Critifal.pdf

Robert

I read the Ferraro's paper and my coclusion in conguction with your comments in other question which is your's, as follow's

To compare what you and Charles said about the Huygens' bending and equivalence principal with Ferraro’s paper can we conclude that half of bending is due to equivalence principal and the other half is due to Huyghens' principal which is described by the bending of light in 3-space in Ferraro's paper?

Gomez

Can we consider this as a special case where the equivalence principal is inapplicable in curved space time or it refers to a general one?

I think that the essence of gravitational atraction corresponds to the property of energy in any system, What I mean is that two systems atract each other with a force in proportion to their energy. Since photons have obviously energy, a photon gas is atracted by any other energy, like the energy MC^2 of the sun. Here I do not see the need to talk about equivalence principle or curved or not curved space: of course I am thinking in Newtonean gravitational atraction, and specially in the bending of star light when passing near the sun. In short, I believe that the bending of light is locally related to the gravitational field created by an energy Mc^2.

Fadhil

EEP does not apply to curved spacetimes.

Hmm.. who said the EEP "applies" to curved spacetimes?  No one that I know.  The EEP does not apply to a spacetime.  It applies to the relationship between an accelerated reference frame in flat spacetime, and a stationary reference frame in a gravitational field (pre-GR without curvature notion), holding by assumption that in a small volume an observer would find them indistinguishable to first order [i.e. ignoring 2nd order Taylor series terms that have to do with divergence of spherical gravity fields and non-uniformity of observations at different positions in an accelerating rocket]. 

Using this assumption, and some fancy mathematics, Einstein came up with the curved spacetime formulation.  Tests of equivalence are routinely considered to be tests, therefore, of GR on which it is based.  If a first order effect were found to be dis-similar in an accelerating setup, it would be bad news for GR.  However, two things:

Fortunately for GR, all first order effects, including the falling and bending of light, are detectable in the accelerated rocket (in thought experiments).  This actually was only settled recently, really in the Ferraro paper. 

Our friend Charles Francis has been kind enough to write up these results, which were somewhat hard to ferret out of the maze of papers and opinions and ways of speaking about the problem, here:

http://rqgravity.net/Gravitation#BendingOfLight

http://rqgravity.net/Gravitation#BendingFromEEP

The problem that may arise if we assumed the bending of light is due to time only, is (from my point of view at least)  light has a limit velocity so logically it must have a limit of time if there is such limit. I wish I can prove it, but it's not easy at all. By all means if there is such time limit this means this idea will be violated. Robert and Charles, Do you agree?

A system accelerated, existing in a flat spacetime, applying the strong equivalence principle, we can describe it as a gravitational system at rest. A gravitational system, according to GR existing in a curved spacetime, can we describe it as an uniformly accelerated system, existing in a flat spacetime, by applying the strong principle of equivalence?

Does strong equivalence principle is result of relationships ....?

Sadeem,

As I understand, photons travelling at the speed of light have ZERO proper time (d tau) You know the "Robertson Walker Lemaitre Friedmann" metrric,

           (c d (tau)^2  =  (c dt)^2   -  R^2 (dr)^2

for photos tau is constant, and the photon equation becomes

             c dt = R (dr)

For a non expanding universe R = constant and the speed of light c is = dr/dt where r is coordinate space and t coodinate time.

So, I do not see how we can talk of time only bending light.

)

Dear Antonio Alfonso-Faus

I totally agree with you. That's what I'm trying to say to Charles and Robert.

The equivalence principle states that matter is affected by gravity only through the coupling of its energy-momentum tensor to the metric-which is the only field that determines spacetime properties. If one uses as energy-momentum tensor that of the electromagnetic field, one finds the result for the bending of light. If spacetime required additional fields (e.g. Brans-Dicke scalars or fields from supergravity) then the result would differ and this is one way one can imagine detecting their presence.

In my view, time is controlled by inertia, and whatever interaction determines inertia is more likely to be akin to entanglement than to QFT.  While QFT interactions are limited to lightspeed, it is well known that entanglement interactions have no regard for relativistic limits.  Think of inertia as position entanglement of multiple masses. 

After so many reactions I just refer to footnote 9 of my paper: "Einstein's Zurich Notebook and his Journey to GR", arXiv:1106.0900.

As a friend of the late Juergen Ehlers I recall in this connection a lively discussion on the issue with the two authors in Munich, long before they published their paper.

I think footnote 10 on p. 14 of http://arxiv.org/pdf/1106.0900v1 is what is meant, correct?

All falling and bending present in a gravitational field has a counterpart in an accelerated reference frame.  Some of it isn't so obvious and it took us a week or two on another thread to ferret out the conclusions finally reached by physicists after 100 years of papers on the subject.  See the links above provided by me to Charles' writeup.  (page 1  of this thread, bottom).  It really isn't feasible to repeat the discussion fresh with every newcomer.  Suggest newcomers read all of this thread and Charles' links as a minimum before commenting. 

The problem that you may face if you assumed only time affecting bending of light is ignoring a real physical parameter which is spacial curvature. This may lead to conflict in the case of non static gravitational potential. Because you can't generalize your theory there.

Remark to Sadeem: The Einstein equivalence principle implies for light rays the geodesic law. See, e.g. my book "General Relativity", Springer-Verlag, 2013, especially Sects. 2.3 and 2.4, p.19--

Dear Charles Francis:

"According to the equivalence principle, at each point on the path of light, the geometry is locally equivalent to a frame with uniform acceleration a relative to an inertial frame".

Is there at a  point curvature different to cero?

Dear All,

I have followed this discussion with great interest, particularly since the Einstein law for light deviation in a gravitational field has been poorly investigated and confusingly explained. I am sorry for repeating some material but since the thread has changed coordinates here we go.

The problems emerge when the argument of the Equivalence Principle is applied to photons. Here the most flagrant violation notably sticks out, i.e. that the bending of a light ray in an accelerated box will be half as large as the bending in a box at rest in a gravitational field.

The standard explanation, often given, is that the “second half” of the bending comes from the amount of space curvature, notwithstanding the fact that this space curvature should already be a consequence of the (strong) Equivalence Principle.

To me the simplest way out of this conundrum is to formulate the general physical problem as a secular like operator equation in terms of energy and momenta (cf. the Klein-Gordon or Dirac equations), adjoined with the conjugate operators in terms of time and position. The interconnected form, of the space-time- and matter-momentum formulation, therefore, in principle includes the specific tensor traits of gravitational interactions, see my RG page for more details

The physical interplay between the conjugate partners automatically takes care of the space-time “dependence”. This imparts the the Einstein’s law of general gravity all in commensuration with the strong equivalence principle.

The formulation in its simplicity yields Einstein relativity (note however that the “physical” interpretation becomes quite different allowing both quantum operator and classical variable representations) and thus gives a non-contradictory theory for the presently discussed light deflection in a gravitational field as well as the perihelion motion of the planet Mercury (see also the Kepler article from my RG page).

Regarding the Ferraro paper:

Here Ferraro is using an approximation to the Schwarzschild solution (Ø(r) small). He appears to have switched the components of the static metric, but that does’nt seem to influence his end result.

Second the strategy used amounts to estimating the bendings as separate contributions from the spatial and temporal parts of the metric and then comparing the results. It does not become clear in this portrait, why the force felt by a non-zero rest mass particle would be different from light as both "particles" are subject to the same gravitational field.

In my own articles I derive the Schwarzschild gauge, corresponding to an exact solution of Einstein’s equations, see my RG page. From this follows the light bending, perihelion of Mercury (for details see my) etc. as direct consequences of the formulation. It does not make sense to separate the various contributions from the temporal and the spatial parts.

Antonio Alfonso-Faus: You said “that two systems attract each other with a force in proportion to their energy”. This is unfortunately not true.

Of other hand if local is understood as a infinitisimal lapse of a curved spacetime and “the equivalence principle, if understood as the possibility of excluding the gravitational field in an infinitesimal region, is not correct since there is no way in which we can exclude the curvature of space (if it is nonzero) by selecting an appropriate reference frame, even with in a given accuracy". Logunov A and Mestvirishvili M. (1989), The Relativistic Theory of Gravitation. Mir Publisher Moscow,15-16.

Dear Charles Francis:

Is Logunov bad ?

@Charles, is there a type in your immediately preceding post ... two occurrences of "clocks do not keep time" ... one of them should be something else?

@Alfonso Gomez, the posts between you and Charles are a bit hard to follow as to what each of you are saying.  Are you questioning whether acceleration in a rocket is equivalent to remaining stationary in a gravitational field, if the rocket is also in a gravitational field?  (i.e. the equivalence experiment is done with background curvature)  Or are you saying that an accelerating rocket in a flat background cannot produce or explain some effect that happens with stationary objects in a gravitational field?  Notice that I only said "produce or explain some effect," not that spacetime in the equivalence setup was actually curved.  To an inertial observer it will not be, but to the accelerated observer, various effects combine to give equivalent result.  It is not that spacetime structure is the same, but that effects are equivalent.  The continuous change of reference frame by the acceleration mimics curved spacetime, I think is a good way to think of it.

@Erkki J. Brändas, would you care to recommend just one of your many papers for an introduction to the ideas you mentioned in your post?  The least difficult one, presumably.  Thanks.

Instead of many words, I recommend to perform the following two calculations for light rays in the Earth field, say.

1. Determine the ray orbits for the Rindler wedge. (I recall that this is obtained by a coordinate transformation from the Minkowski metric.) The simplest way to do this is to look at the geodesics of the corresponding Fermat metric, which agrees with the metric of hyperbolic 3-space. Up to higher orders one obtains Einstein's result of 1907 (in Bern), which he obtained on the basis if the equivalence principle,

2. Do the same for the almost Newtonian approximation for the metric of the homogeneous Earth field. At first sight the result seems to differ by a factor 2. But this is only apparent, as can be seen with a simple coordinate transformation that has to be performed in order to compare the two results physically. (Do not misinterpret coordinates as physical lengths.)

Erkki, your comment:

"Antonio Alfonso-Faus: You said “that two systems attract each other with a force in proportion to their energy”. This is unfortunately not true."

In my last paper here the gravitational constant G is proportional to c^4, regardless of any time variation of either G or c. This is so for a constant lambda.

Then the gravitational atraction between two masses M and m, (Newton), is proportional to the product GMm  that is proportional (to Mc^2).(mc^2), the product of their relativistic energies. I think that this is fortunate true and explains why photon gasses atract each other: a force proportional to the product of energies, not matter.

Dear Robert and Francis,

Thanks for your interest. Although I am a chemical physicist my scientific advisor was a theoretical physicist and I did learn early about the “problem” at least from the interpretative point of view. I will look at the papers suggested by Francis.

In the mean time I will send to both of you two very simple texts privately, since they are still under copyright, i.e. Erkki J. Brändas: Arrows of Time and Fundamental Symmetries in Chemical Physics. Int. J. Quant. Chem. 112, 173-184 (2013), and E. J. Brändas: The Relativistic Kepler Problem and Gödel’s Paradox. in Quantum Systemsin Chemistry and Physics. Progress in Methods and Applications., eds. K. Nishikawa, J. Maruani, et al., Springer Verlag, Vol. 26, 3-22 (2012).

The trivial argument regarding the Equivalence Principle as applied to photons, viz. that the bending of a light ray in an accelerated box should be half as large as the bending in a box at rest in a gravitational field, should have a simple answer without any references to additional bendings and/or (not exclusive “or”) elevators working in prior fields.

Best

erkki

Instead of many words, I recommend to perform the following two calculations for light rays in the Earth field, say.

1. Determine the ray orbits for the Rindler wedge. (I recall that this is obtained by a coordinate transformation from the Minkowski metric.) The simplest way to do this is to look at the geodesics of the corresponding Fermat metric, which agrees with the metric of hyperbolic 3-space. Up to higher orders one obtains Einstein's result of 1907 (in Bern), which he obtained on the basis if the equivalence principle,

2. Do the same for the almost Newtonian approximation for the metric of the homogeneous Earth field. At first sight the result seems to differ by a factor 2. But this is only apparent, as can be seen with a simple coordinate transformation that has to be performed in order to compare the two results physically. (Do not misinterpret coordinates as physical lengths.)

I am a bit confused regarding which thread to answer so here is my answer again:

Dear Robert and Francis,

Thanks for your interest. Although I am a chemical physicist my scientific advisor was a theoretical physicist and I did learn early about the “problem” at least from the interpretative point of view. I will look at the papers suggested by Francis.

In the mean time I will send to both of you two very simple texts privately, since they are still under copyright, i.e. Erkki J. Brändas: Arrows of Time and Fundamental Symmetries in Chemical Physics. Int. J. Quant. Chem. 112, 173-184 (2013), and E. J. Brändas: The Relativistic Kepler Problem and Gödel’s Paradox. in Quantum Systemsin Chemistry and Physics. Progress in Methods and Applications., eds. K. Nishikawa, J. Maruani, et al., Springer Verlag, Vol. 26, 3-22 (2012).

The trivial argument regarding the Equivalence Principle as applied to photons, viz. that the bending of a light ray in an accelerated box should be half as large as the bending in a box at rest in a gravitational field, should have a simple answer without any references to additional bendings and/or (not exclusive “or”) elevators working in prior fields.

Best

erkki

Instead of many words, I recommend to perform the following two calculations for light rays in the Earth field, say.

1. Determine the ray orbits for the Rindler wedge. (I recall that this is obtained by a coordinate transformation from the Minkowski metric.) The simplest way to do this is to look at the geodesics of the corresponding Fermat metric, which agrees with the metric of hyperbolic 3-space. Up to higher orders one obtains Einstein's result of 1907 (in Bern), which he obtained on the basis if the equivalence principle,

2. Do the same for the almost Newtonian approximation for the metric of the homogeneous Earth field. At first sight the result seems to differ by a factor 2. But this is only apparent, as can be seen with a simple coordinate transformation that has to be performed in order to compare the two results physically. (Do not misinterpret coordinates as physical lengths.)

Instead of many words, I recommend to perform the following two calculations for light rays in the Earth field, say.

1. Determine the ray orbits for the Rindler wedge. (I recall that this is obtained by a coordinate transformation from the Minkowski metric.) The simplest way to do this is to look at the geodesics of the corresponding Fermat metric, which agrees with the metric of hyperbolic 3-space. Up to higher orders one obtains Einstein's result of 1907 (in Bern), which he obtained on the basis if the equivalence principle,

2. Do the same for the almost Newtonian approximation for the metric of the homogeneous Earth field. At first sight the result seems to differ by a factor 2. But this is only apparent, as can be seen with a simple coordinate transformation that has to be performed in order to compare the two results physically. (Do not misinterpret coordinates as physical lengths.)

Dear Charles thank you

Dear Robert:

Gravitation is locally equivalent to acceleration. This is the strong principle of equivalence (EEP). I know two notions of local:

- any point of a curved spacetime and the variant of a tangent spacetime at one point (point by event) of a curved spacetime. The application of EEP to a curved spacetime to some people seems like a joke because in both cases there are no curvature (one point or in a tangent plane spacetime). Of other hand, in a point is not possible to make experiments, for example, his rocket is at Cape Canaveral and you can not put the rocket in uniform accelerated movement because its trajectory will occupy many points.

- in one infinitesimal region where the curvature has no effect on the experiments. But, to Logunov is not possible eliminate the curvature, i.e. the effect of the curvature in the experiments. For example, his rocket is in uniform accelerated movement, under action of the gravity of the Earth, within of an infinitesimal region, much as you want. In any Cape Canaveral control station it  will trace trajectories of bodies initially en rest, inside rocket, that will cut at the center of masses of the Earth.

Dear Antonio,

The problem arises because in relativity theory the force law, the momentum law and the energy law are not compatible. In a sense this is one reason to entail tensorial descriptions.

Dear Erkki:

Locally, at the lab system, there is total compatibility: tensorial descriptions must obey locally the special relativity theory. Otherwise it would be a WRONG THEORY. The zero covariant divergence of the tensorial Einstein field equations, div. T_mu nu = 0   ensures the conservation equations. Thank goodness¡

Antonio,

The force-, momentum- and the energy laws are incompatible even in the special theory.

Erkki:

Certainly I do not think so. First, the second law of Newton establishes that the rate of change of momentum dp/dt  (p the momentum of a particle, t time) is equal to the force (if any)  applied to this particle. Zero force, then constant momentum p. This is a relativistic relation, confirmed in special relativity theory. And as far as energy is concerned, there is a very well known relativistic equation, well confirmed too, that states E^2 = (pc)^2 + (E_o)^2 , E the total relativistic energy E = mc^2, E_o the rest energy equal to m_0 c^2, m_0 the rest mass .

Now just a bit f humor: your statement, "The force-, momentum- and the energy laws are incompatible even in the special theory." may be true in some other galaxy, if different from our Milky Way.

Antonio,

Here we go:

Energy law, case of central force: E=mc^2 (1–µ/r);   µ=GM/ c^2  [G=gravitational constant, µ=gravitational radius of the mass M].

One obtains first from the momentum law: d(mc^2)=v•dp=dr•(dp/dt) by the use of the relation m=m_0/(√1-v^2/c^2).

However the force law imparts dr•(dp/dt)=dr•f  from which one obtains by using the expression for E above that dE=dr•f (1–µ/r) for all dr. Except for a component perpendicular to dr one gets a modification of the force: f=nf(mM/r^2)( 1–µ/r)^(-1) with n=-r/r.

It thus appears that we get an adjustment of the force law by the extra factor above. This altered law breaks down at r= µ, which happens to be one half of the Schwarzschild value.

I am still in the Milky Way! Where are you?

Erkki,

It seems to me that your maths are not right, or at least not very clear.

Let us see: Energy law, case of central force E = mc^2 - GMm/ r

This is just a computation of the total energy E: total relativistic mc^2  plus gravitational potential energy  - GMm/r  . This is a well known energy law. When r is very large E = m_0 c^2    and for r small enough to have E = 0  then you get a limit for r, that is the gravitational radius of M,  r_g = GM/c^2  i.e.  one half of the schwarzschild value as you have seen.

Now, the force law is jus the atractive force on m as  - GMm/r^2 . at distance r between M and m. The work done by this field is then   -GMm/r^2 . dr   for a radially moving particle m approaching M. If you integrate the equation correctly, m is varying with r, then you have no discrepancy.

Dear Professor Alfonso-Faus,

Looking at your publications at the RG page as well as your experience and competence in the theory of general relativity, I am surprised that we have this discussion. Somehow we must have entered the wrong track and miscommunicated.

All I am trying to convey is the triviality that the three laws quoted in my previous posting are consistent with each other, or in other words compatible in the classical formulation of Newton’s law.

The force law, introducing the momentum, and writing the force law in the momentum form is consistent with the energy E as a sum of kinetic and potential energies, and with the important conclusion that E is a constant of motion. Hence dE=0 gives you back the Newtonian gravitational force, which demonstrates the compatibility of a so-called privileged system. Therefore one can e.g. use all three forms in calculating the orbits of the particle (in the gravitational field associated with the large mass M), the classical Kepler problem.

As discussed in my previous posting this is not true in the relativistic case. Hence starting with the force law, as Einstein did, would not give the same result as if one had started with the energy law.

My simple demonstration, which I believe is well-known (I can give references if you want), imparts in particular that if one “take the same route” as we did above for the classical case, one ends up with an extra factor, (1–µ/r)^(-1), as a modification of the force law.

It is interesting to test the various forms in a calculation of the perihelion movement of the planet Mercury, see my RG page.

Best erkki

Dear Professor Brändas:

Thank you very much for your kind reply. Yes, I agree with you that there has been a misunderstanding from my side: I was thinking in classical terms, may be semi-classical. I was not familiar with the different result you have pointed out for the force law. Thank you again.

Best wishes,

Antonio

That modification of the force law and various other modifications, some little known like acceleration, are derived very simply in my 2011 paper, linked below.

Article Isotropy, equivalence and the laws of inertia

Robert, the Ferraro paper which claims "The post-Newtonian parameter does not enter the deflection angle" is certainly not the "establishment opinion". If this notion makes sense, then the establishment is presented by Will.  He writes

"It is interesting to note that the classic derivations of the deflection that use only the corpuscular theory of light, by Cavendish in 1784 and von Soldner in 1803 [61], or that use only the principle of equivalence, by Einstein in 1911, yield only the “1/2” part of the coefficient in front of the expression in Eq. (16). But the result of these calculations is the deflection of light relative to local straight lines, as established for example by rigid rods; however, because

of space curvature around the Sun, determined by the PPN parameter γ, local straight lines are bent relative to asymptotic straight lines far from the Sun by just enough to yield the remaining factor “γ/2”. The first factor “1/2” holds in any metric theory, the second “γ/2” varies from theory to theory. Thus, calculations that purport to derive the full deflection using the equivalence principle alone are incorrect."

http://arxiv.org/abs/1409.7871v1

Dear Ilja,

I do not agree with your interpretation. For instance in a critical examinations of the Principle of Equivalence, a physicist will measure that the bending of light in an accelerating box will only be half as large as the bending in a box at rest in a gravitational field. The common answer, often given by relativists, you included, is that the “second half” of the bending comes from the amount of space curvature, notwithstanding the fact that this space curvature should already be accounted for as a consequence of the Equivalence Principle.

I have calculated in detail both the light deflection and the perihelion motion of planet Mercury, see my RG page, directly from the Equivalence Principle and the results clearly shows the difference in behaviour between zero- and non-zero restmass particles all in agreement with Einstein's laws.

Erkki, the argument has a quite strong base. The point is that there are different metric theories of gravity, and these different theories give different angles. This is the whole point of the PPN formalism to compare different such theories. 

But all metric theories of gravity share with GR the property that the matter Lagrangian is covariant, the matter fields interact only with the metric, and in the same way as in GR.  So, in all these theories, the Einstein Equivalence Principle holds. 

Ilja, there is an implication in your comment that light bending cannot be determined from equivalence.  But when I studied the comment closely, I could find nothing to support that.  So what are you saying?

All acceptable metric theories must give the same result in the solar system, because they all must match the measurements there.  End of that discussion.

Any calculation of anything from "equivalence" must first choose how to match an actual gravitational law, as equivalence does not have either a 1/r potential law or a 1/r^2 acceleration law built in to it.  These other laws would vary according to which metric theory was chosen.  It makes no discernible difference in the solar system, curvature is not great enough, but in other cases you simply have to extract whatever the law is that a particular metric theory is implying, and using equivalence then you can calculate, well, pretty much anything, though it may be tedious.  The reason is that all metric theories, kind of by definition, uphold equivalence.  And therefore any deviation from motion predicted from the theory would also deviate from the motion predicted by equivalence combined with the chosen (or implied) law.

Robert, what I'm saying is that the Einstein Equivalence Principle (EEP) is fulfilled in all metric theories of gravity.  (The Strong Equivalence Principle not, it is stronger.) So, what can be derived from the EEP does not go beyond the properties shared by all metric theories of gravity.  

Among the metric theories of gravity will be, of course, acceptable ones (which agree with observations) as well as unacceptable ones (which disagree). But acceptable or not is irrelevant if the question is derived from EEP or not. If there are two metric theories of gravity, thus, two theories where the EEP holds, and they predict different values for light bending, then light bending cannot be derived from EEP alone.  Point.  That at least one of the two - because it predicts a wrong value for light bending - has to be false is completely irrelevant if the question is if the light bending can be derived from EEP or not.  In this case, even a wrong theory can give a counterexample. 

Robert and Ilja,

To avoid confusion let me state the following. The WEP (weak equivalence principle a la Newton) does not hold comparing the deviation of light- and non-zero restmass particles in a strong gravitational field like the sun.

EEP introduce abstract metrics that should be adapted according to the problem under discussion.

The SEP (strong equivalence principles), i.e. "there is no observable distinction between the local effects of gravity and acceleration" predicts as far as we now, no paradox or counter examples.

Finally the problem of a particle (with zero or nonzero testmass) outside a massive object (could be  black hole) yields the Schwarzschild gauge. In this fundamental case EEP and SEP gives the same result while violating WEP.

Erkki, I continue to insist that, following Will, there is no "result" given by the EEP.  The EEP gives metric theories, but there are different metric theories with different results. 

I do not think that we disagree either, but I also insist that my comment is correct.

It appears Ilja and I agree on all substantive points, but not how to talk about it.  Therefore I assume we will continue to argue bitterly because that is what GR people do!  Finally, I feel like one of the insiders.

On the other hand Erkki, whose views I normallly find insightful, appears to have missed the boat altogether.

In summary: 

No one, ever, as far as I know (I've read dozens of papers on the subject) ever made such a claim as to say one could derive gravity from ONLY the EEP without any law of radial dependence.  Some authors do not make a big deal of it.  It is so obvious to them, they just switch the potential from height in constant acceleration with the potential law of gravity with only a handwave.  Schiff for example. 

This puts them on the wrong track if agreement with GR is intended, because GR preserves the inverse square acceleration law, and both cannot be preserved.  Klaus has the only treatment of this short of full GR derivation that I've seen and he promised to co-author a paper with me.  If someone is interested I'll find his post and give a link to it.

Robert,

As far as I can see I agree with your points 1-4. Where did I "miss the boat" according to your view?

Erkki, I got that impression from: "The WEP (weak equivalence principle a la Newton) does not hold comparing the deviation of light- and non-zero restmass particles in a strong gravitational field like the sun."  Perhaps you did not mean what it sounded like?

Dear Robert,

Maybe the wording was unfortunate, but if I use the "universality of free fall" there occurs a paradox when a physicist will measure the bending of light in an accelerating box, since the latter will only be half as large as the bending in a box at rest in a gravitational field. The answer that the “second half” of the bending comes from the amount of space curvature sounds inconsistent to me, since the space curvature should already be accounted for as a consequence of the "universality".

Erkki, the bending of light measured in the box is correct.  The assumed reference point is wrong.  I have temporarily withdrawn the paper where I explain this because of an unrelated snafu, but Klaus and I are working on a rewrite and I'll try to remember to send it to you.  Maybe you can review it for us?

Maybe this is just wording, but I would say that WEP is wrong but SEP and EEP is correct. Of course I could try to review your "snafu" if I could be able to follow all the details.

To me the WEP is about the weight of things, as much as the falling rate.  It's usually checked on a torsion balance, not by actually measuring falling rate.  Since matter is at the very least composed partly of energy (and probably entirely), the WEP does apply to energy.  If it did not apply to photons, then we should have an inequivalence in weight owing to the ratio of EM field mass in an object as compared to total mass.

I thought just like you until I became fond of Asif's theory of the electron.  I realized that I would have to accept that the WEP applied to photons if any kind of theory similar to Asif's had a chance.  Photons have to have the correct weight.  Without the same falling rate as other particles, they cannot have the correct weight.

Robert, You say "To me the WEP is about the weight of things, as much as the falling rate" and "Photons have to have the correct weight.  Without the same falling rate as other particles, they cannot have the correct weight.

Are you saying that the mass should be twice that of "m" according to E=mc^2? 

Erkki, since this has been endlessly discussed in another thread, how about just waiting for our paper.  I will briefly summarize what we found, but I do not want to get into defending it.

Everything has to have the same universal falling rate.  So-called double-bending does not imply anything about the falling rate, certainly not twice the acceleration.  Such would violate local equivalence.  All "mass" relativistic and otherwise must also have the same inertia.  Thus falling rate and "weight" are inextricably linked.

To compare bending in equivalence with bending in a gravitational field is apples and oranges.  What is the zero bending angle in an equivalence setup?

It is not "straight across" because that requires infinite velocity.  In other words, one cannot actually talk about zero bending angle in equivalence. 

If we presume some sort of Newtonian reference, some kind of particle not slowed by gravity (which cannot exist except in hour heads), then it still would not cross the equivalence frame instantly, and would stay at the same height as any un-accelerated object.  Unlike light, our hypothetical Newtonian ray would not slow down, and thus it would get across the equivalence frame (elevator, rocket) a little quicker, and therefore exit a little higher.  But its falling rate is not changed by this.

Charles, I would mostly agree, with the exception that SEP is IMHO wrong. Because it is incompatible with quantization.  

Ilja,

SEP is not supposed to be compatible with quantization.

The SEP (strong equivalence principles), says that "there is no observable distinction between the local effects of gravity and acceleration" and it predicts as far as we now, no paradox or counter examples.

String theory or branes or similar yields quantization of gravitational waves, but I do not think that SEP should quantize "acceleration" in order to give Einstein's GR-laws or to produce black-hole-type objects. SEP is simply a "classical" relativistic principle.

Erkki, principles are principles, they are general properties of whole classes of theories, not invented to support particular theories. Say, energy conservation was invented at a time when quantum theory was not known but is also compatible and used in modern quantum theories. And, of course, as it is used in QT it is based on a quite different formulation than used at the time it was invented. 

Similarly for the SEP. It was invented for GR, but many people hope that it will survive in QG too.  Of course in a different formulation, not using words like "acceleration", but with the same essence.  This essence is named today "background independence".  

And it is my thesis that this background independence is impossible in quantum theory, see http://arxiv.org/abs/0909.1408 for a justification of this claim.

Charles, we have already seen that your theory refuses to predict results even for an elementary two particle quantum gravity experiment which can be without problem considered by Schrödinger theory with Newtonian potential. Do the computations for this elementary experiment and we will see if your theory is background-free. 

Charles, the experiment is a simple one: You  have a superpositional state, like in a double slit experiment.  You have a test particle which gravitationally interacts with the particle in the superpositional state. The question is if this interaction is sufficient to destroy the superposition (so that we will not see, after this interaction, a typical interference picture, or not - then we will see it.

This is a simple, well-defined experiment, and the outcome should be predicted by any theory of quantum gravity.  Newtonian gravity makes a prediction.  Your theory refuses to make a prediction.  Any remarks about equivalence of a Newtonian potential with whatever acceleration are completely irrelevant, the point remains that we have an experiment, Newtonian quantum gravity trivially computes a prediction for the outcome, but we have yet to see a background-indpenedent theory which allows to compute a prediction for this triviality. At least your theory, judging from your reaction, predicts nothing. 

Charles Francis,

I have taken a look at your "Teleparallel" paper. I simply do not believe that your scaling relations between locally Minkowski coordinates is correct. For instance, if you are going to determine the redshift for photons in a strong gravitational field, the flat Minkowski tangent space will not suffice, since it corresponds to a locally Lorentz invariant picture.

As Ilya has pointed out the correct background behaviour most be respected.

It is in fact surprising that taking energy-momentum and its conjugates time-space into consideration in a consistent fashion one gets agreement with Einstein's laws and SEP.

@Brändas:

"Maybe the wording was unfortunate, but if I use the "universality of free fall" there occurs a paradox when a physicist will measure the bending of light in an accelerating box, since the latter will only be half as large as the bending in a box at rest in a gravitational field."

This is wrong. In a uniform gravitational field, and local means the approximation of uniformity to be acceptable, the bending will be the same as in an accelerated system in flat space having the same acceleration.

Have a look at my post on: 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?_tpcectx=qa_overview_following&_trid=545b5ee1cf57d7854d8b468f_

discussing this issue in detail. The post will show up on top, as it is the most popular post of that thread so far.

Regarding your prior statement concerning the strength of the gravitational field of the sun, I would object that this is a pretty weak gravitational fied in terms of the theory. The light ray considered in the calculation of deflection never gets closer to the center of the sun than its surface, and there the relevant quantitiy Φ/c2 determining whether gravitational effects are strong or weak is on the order of 10-5 or smaller, so the field is really very very weak and all calculations can be done perturbatively.

Charles, there is such a theory as "Newtonian quantum gravity", this is simply the name I have used for standard multi-particle Schrödinger theory with Newtonian potential.  Once large velocities play no role in the experiment, the non-relativistic theory is completely sufficient to handle this thought experiment.  

If you think there is no such thing as a gravitational interaction, try to jump out of the window. If in a particular interpretation (the spacetime interpretation) of a particular classical theory (GR) the gravitational interaction is named differently, this is completely irrelevant for the comparison of the description of this simple thought experiment in different theories (which anyway have to be necessarily quantum, given the experiment). 

Charles, you may have a point that Schrödinger theory with Newtonian potential does not handle gravity as a quantum field with own degrees of freedom. But the classical Newtonian theory does not do this too, but is also named a theory of gravity, and I don't think that naming the quantization of a classical theory of gravity quantum gravity is wrong.

Whatever, names are not important.  The point is that an acceptable theory of quantum gravity has to be able to make a prediction of the outcome of this experiment.  And also that this is completely unproblematic in Schrödinger theory with Newtonian potential, name it as you like, I continue to name it Newtonian quantum gravity. 

And, no, it makes sense to consider approximations. This is necessary in particular in situations where we simply do not have a better theory.  Of course, in this case we have a better theory, my own described in http://arxiv.org/abs/gr-qc/0205035 which does not have a problem with the computation too, because it has a background.  But quantum GR does not exist, and so it cannot be used to improve the prediction of the outcome of this though experiment. 

Or, it the case if it exists in your work, provide here the rules how to improve the non-relativistic approximation which can be easily computed with Schrödinger theory with Newtonian potential. (Or the error made by the relativistic computation made on the basis of my theory, whatever you like.)

Charles, I have described the experiment here already several times, which is possible because it is that simle. 

You create a superpositional state of one sufficiently heavy particle.  Something sufficiently close to psi(x) sim delta(x-x_l) + delta(x-x_r).

Then you use a test particle which interacts gravitationally (or however you name this) with this superpositional state.  In Newtonian gravity (or however you name it) this interaction would be described by an initial state (delta(x-x_l) + delta(x-x_r)) phi_0(y) which, interacting via the Schrödinger equation with H sim p_y^2/2m + GMm/|x-y| (ignoring the H for the heavy particle which holds it at delta(x-x_l) + delta(x-x_r)) leads to the final state

delta(x-x_l)phi_l(y) + delta(x-x_r)phi_r(y). 

Then, finally, we make a measurement of the state of the heavy particle. The result depends on the scalar product .  If it is 1, then there was no measurement, the final experiment shows the undistorted interference pattern.  If it is 0, the position is measured, no interference pattern will be visible.  Other results are possible too and lead to corresponding results for the final measurement.  So, the scalar product is something observable in this Newtonian variant of quantum gravity (or however you name it): The result of the final measurement depends on this scalar product.

Now I'm interested in how to compute GR corrections for this simple experiment.

So, word games (if gravity should be named gravity or Newtonian quantum gravity should be named Schrödinger theory with Newtonian potential or so) we have had.

That to read the first two quite simple pages of the introduction of the paper (the remaining part is relevant only if you want a proof of a theorem that it is impossible to create a background-free theory, not for understanding the experiment itself) is, I think, also only a quite cheap excuse. Whatever, now everybody can see how cheap it is.

And, instead of waiting for the next excuse, I will explain what is the problem if you want to compute relativistic corrections for  .  

Once you have a heavy source particle and a light test particle, it would be quite natural to compute the wave functions phi_l and phi_r on corresponding metrics g_mn^l resp. g_mn^l, which describe the metric created by the environment together with the heavy particle localized in x_r resp. x_r.   After this, you have two nice pairs (g_mn^l, phi_l) and (g_mn^r, phi_r), but you have no connection between the metrics, and no prescription how to compute the scalar product .

You can use one system of coordinates on (g_mn^l(y), phi_l(y)), and one on (g_mn^r(y), phi_r(y)).   And then simply compute int phi^*_l(y) phi_r(y) dy.  But, once you have chosen the coordinates for (g_mn^l(y), phi_l(y)), how to choose the "same" coordinates on (g_mn^r(y), phi_r(y))?  If you simply use other coordinates, you can change the scalar product in an arbitrary way. 

@Kassner

Many thanks for your answer and reference to a more detailed answer at another thread. I am sorry that I missed that one. I will answer you here before looking at the answers at Robert’s question regarding “Is it possible to …..”

I agree with what you say there except the statement “So particles and light do not fall at different accelerations" and here is why:

If you look at the relativistic Kepler problem (yielding the perihelion motion of planet Mercury) and look at the hyperbolic case (in spherical coordinates), one obtains for small angles  ϕ =φ-π/2 that the deviation for a particle with non-zero rest mass passing a sphere with large mass M is given by 2ϕ =(2μ/R)(c/υ0 )2, with μ being the gravitational radius, R=(α1+β)-1 ,  α1≈α(1-3αμ)-1, α=f M/A2 where A is the area velocity, f the gravitational constant and υ0 the velocity at r –›∞. Note that β>α1 is coupled to the energy to correspond to the hyperbolic case.

For a photon with the velocity υ0 =c the result becomes 2ϕ =(4μ/R), which clearly shows that zero restmass particles have f–›2f.

Loosely speaking the deviation comparing a photon with a neutrino (with small mass and velocity close to c) would be close to a factor 2 if measurable. Whatever approximations used in the relativistic calculation for comparing deviations in various cases the qualitative part will always relate back to the mnemonic rule “f–›2f ”.

The actual calculation is available on my RG page or I can send it to you.

Charles, When I say that "the background behaviour must be respected" I also include the appearance of the Schwarzchild singularity at r= 2 μ, where  μ equals the gravitational radius.

Charles,

If you look at the the deflection of light by the sun, one can approximate the sun as spherically symmetric object and then use the Schwarzschild gauge. This metric displays the appropriate background behaviour for Mercury's perihelion motion and Einstein's law of light deviation. It becomes singular at  r= 2 μ.

Charles, sufficiently heavy particles are, of course, also well-described by quantum theory, because things like decoherence are also part of quantum theory. (Ok, the Copenhagen interpretation may have a problem with it, but dBB certainly does not have a problem with this.) 

But, let's look again at the question.  I have proposed an experiment, given the formulas to compute the observable effect, which depends on  .  I have explained two extremal cases: =1, the gravitational interaction with the test particle changes nothing, because it is too weak.  And =0, the superposition is destroyed, because the position is measured by the gravitational interaction.  I have asked how to compute relativistic corrections. 

Your answer, roughly speaking:  "If the masses are too heavy, we have something similar to Schrödinger's cat. Therefore your experiment is void." 

Sorry, but this is simply the case where =0 - there is too much interaction with the environment which destroys the superposition. Known as decoherence. 

So, another variant of the problem: Compute the mass where we switch from usual particle masses where gravity is negligible to the "too heavy" mass where Schrödinger's cat is applicable. Not with handwaving about "Schrödinger's cat", but with formulas for relativistic corrections for .

"A measurement of a single particle does not show an interference pattern in any variant of quantum theory."

Of course, the interference pattern will be visible only if the experiment is repeated sufficiently often.  But the interference pattern in a double slit is, clearly, the prediction for the probability distribution of single particle theory. There is, in every instance of a double slit experiment, only one particle going through the slits (at least such experiments have been done), and the probability where it appears is defined by the interference pattern for each particle taken alone.  The interference pattern is not a consequence of some interactions between different particles.

I hope you know this. Else, it would not make much sense to continue to talk with you about quantum theory. 

Charles, it becomes funny. 

First, you have tried one extremal value, a la the mass is to heavy, so no superposition, but Schrödingers cat, which corresponds to the case = 0. 

Now you try the other extremal case, gravity is much too weak, one would have to wait millions of years to obtain an effect. Which corresponds to the case = 1, or a mass which is too small.   

These two extremal cases are trivial, nice that we agree about this (at least I hope so).  But the question about what would be the critical mass, or other critical parameters of this experiment (like distance between the slits and the time of interaction) remains open. Newtonian quantum gravity (or how you name it) has clear prescriptions how to compute it, namely by computing .  And I continue my request, give the formulas how to compute relativistic corrections. 

And, please note, it is not the question about the order of magnitude of the masses, time scales, or distances, and the particular way to prepare the superpositional state is irrelevant, because this is a thought experiment, not a proposal for a real one. The point of such a thought experiment is to check the consistency of proposed theories: They have to be able to compute the predictions for these thought experiments too.  If not, then, of course, to consider quantum gravity at all is "unscientific nonsense" or so. Ok, nobody is obliged to consider quantum gravity as something worth to be considered - but you have claimed to have something like such a theory. 

Your name-calling against dBB interpretation is irrelevant, decoherence vs. collapse is irrelevant here too, given that what counts is if an interference pattern would be observable or not, and not what you think about interpretations, the minimal interpretation is completely sufficient here. 

Charles,

Yes I use the the word gauge here, since the Schwarzschild metric plays a particular role in my derivation – normally the Schwarzschild metric is part of the gravitational gauge group – this is why. Hope you have no problems with it. Sincce the latter contains matter it has the proper background behavior of curved space.

Can explain what you mean?

Thanks Charles for slowing me down!

My focus, like you and Ilja, is to find ways to make general relativity, GR, commensurate with quantum mechanics, QM. Therefore my concern starts with a search for what is the fundamental difference(s) between GR and QM, looking at the simple model of a large spherical non-rotating mass (or black hole entity).

One key difference is that in QM energy-momentum is conjugate to space time and If you want to keep this (primary) structure one can show that it leads uniquely to the Schwarzschild metric, SM. If you take this result to the classical world you easily connect with Einsteins equations and the property of background independence.

However, my conclusion, based on this is that commensuration with QM is only possible in the SM, and that it leads to a blackhole singularity as a "real physical" object, rather than curiosity of the SM, and which can be further investigated.

If you and Ilja have serious objections, please do not hesitate ...

The only clear and well-defined way to treat locally light bending in a mathematical intrinsic way is in terms of geodesic curvature. In all cases I know (and have looked at) the result agrees in linear approximation, as expected, with the one Einstein's in 1907 in Bern. Globally, the light bending depends , of course, on the particular theory. There is, for instance no such light bending in the Einstein-Fokker theory, but locally one obtains the same result as in GR (on the linearized level). I could provide details of the mathematics, which is quite instructive. For instance, in the Einstein-Fokker theory, whose Fermat metric (that determines the null geodesics) is euclidian, the geodesic curvature of the straight lines with respect to the spacial metric of the EinsteinFokker 4-metric (that is conformally flat) has a non-vanishing geodesic curvature, that leads to the same local light curvature as in GR, and also to the one of the preliminary static metric of Einstein in 1912 (in Prag), which gives globally the wrong answer (by the famous factor 2).

So all this confirms what was expected.