Einstein's equivalence principle doesn't apply to light

Introduction: ____________ Excerpts from Wikipedia: The Einstein equivalence principle : "In the physics of relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. They are related to the Copernican idea that the laws of physics should be the same everywhere in the universe, to the equivalence of gravitational and inertial mass, and also to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force experienced by an observer in a non-inertial (accelerated) frame of reference." According to Einstein, "An observer in a windowless room cannot distinguish between being on the surface of the Earth, and being in a spaceship in deep space accelerating at 1g." Thought experiment: __________________ Einstein proposed an experiment involving two elevators (Strobel's Astronomy Notes): "One at rest on the ground on the Earth and another, far out in space away from any planet, moon, or star, accelerating upward with an acceleration equal to that of one Earth gravity (9.8 meters/second2). (Modern readers can substitute ``rocket ship'' for Einstein's elevator.) If a ball is dropped in the elevator at rest on the Earth, it will accelerate toward the floor with an acceleration of 9.8 meters/second2. A ball released in the upward accelerating elevator far out in space will also accelerate toward the floor at 9.8 meters/second2. The two elevator experiments get the same result!" Cf. also the appendix below.

The results are different for a light pulse.

Let's remember: "To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a "being" (what kind of a being is immaterial to us) begins pulling at this with a constant force. The chest together with the observer then begin to move "upwards" with a uniformly accelerated motion." The "accelerating" elevator ___________________________ After a light pulse is emitted from the top of the accelerating elevator, the floor of elevator moves upward, hence the pulse will hit the floor after a time t < H/c, where H is the distance between the emitter and the receiver. After such time t, the light pulse will be at a distance d1 = ct from the top, and the floor will have travelled upward a distance d2= 1/2 gt^2. From d1 + d2 = H, or ct + 1/2 gt^2 = H, one gets t = [-c + sqrt(c^2 + 2gH)] / g = [(c/g) * (sqrt(1 + 2gH/c^2) - 1)], which is the time at which the pulse meets the floor. The corresponding upwards velocity of the floor is then v = gt = c * (sqrt(1 + 2gH/c^2) -1) As sqrt(c^2 + 2gH) is approximately equal to 1 + gH/c^2 + (gH/c^2)^2/2, v =~ c * (gH/c^2 + (gH/c^2)^2/2) From the formula v/c - 1, the observed blue shift is gH/c^2 + (gH/c^2)^2/2. The elevator is at rest on the ground _____________________________________ A light of frequency Nu(e) is emitted at a distance H from the surface of the Earth of mass M and radius R. At the surface of the Earth, the measured frequency of the light is Nu(o). From GR: ------- Nu(o)/Nu(e) = sqrt((1-2Phi(o)/c^2)/(1-2Phi(e)/c^2) =~ (1-Phi(o)/c^2)/(1-Phi(e)/c^2) =~ (1-Phi(o)/c^2)*(1+Phi(e)/c^2) =~ 1-Phi(o)/c^2+Phi(e)/c^2 As Phi(o) = -GM/Rc^2 and Phi(e) = -GM/(R+H)c^2, one gets Nu(o)/Nu(e) = 1 + GM/Rc^2 - GM/(R+H)c^2 = 1 + (GM/c^2)*(1/R - 1/(R+H)) = 1 + (GM/c^2)*(H/(R*(R+H)) The corresponding shift is thus (GM/c^2)*(H/(R*(R+H)). From the potential energy: ------------------------- The acceleration of gravity at the surface of the Earth is GM/R^2, where G is the gravitational constant. At the distance H, it is GM/(R+H)^2, hence the mean acceleration is g = GM/R(R+H). Let the energy of a photon emitted at the distance H from the surface of the Earth be hNu(e) and the energy of the photon measured at the surface be hNu(o), where h is the Plank's constant. Let m = hNu(e)/c^2 be the pseudo mass of the emitted photon. Its potential energy at the distance H is thus Ep = mgH, where g = GM/(R*(R+H)). When it reaches the surface of the body, the total energy of the photon is hNu(o) = hNu(e) + Ep = hNu(e) + hNu(e)/c^2 * g * H Hence, Nu(o) = Nu(e) (1 + gH/c^2) Nu(o)/Nu(e) = 1 + gH/c^2 Shift = gH/c^2. As g = GM/(R*(R+H)), the shift formula can be written Shift = (GM/c^2) * (H/(R*(R+H)) This shift is identical to the one obtained with GR. Therefore, one can safely use the difference of potential energy of the photon to calculate the shift when the gravitational potential is constant, i.e. when g = GM/R^2. It is important to note that in the GR formula, the gravitational potential in not constant, whereas in Einstein thought experiment, the chest (the elevator) moves upwards with a uniformly accelerated motion. Calculation of the shift from the difference of potential energy of the photon in a constant gravitational potential. __________________________________________________________ The potential energy of the photon at the distance H is then Ep = mgH, where g = GM/R^2. When it reaches the surface of the body, the total energy of the photon is hNu(o) = hNu(e) + Ep = hNu(e) + hNu(e)/c^2 * g * H Hence, Nu(o) = Nu(e) (1 + gH/c^2) Nu(o)/Nu(e) = 1 + gH/c^2 Shift = gH/c^2, which can also be written Shift = (GM/c^2) * H/R^2, against (GM/c^2)*(H/(R*(R+H)) for the GR shift. One could however consider that, as H is much smaller than R in the Einstein thought experiment, R+H =~ R. Then the GR shift is close to (GM/c^2) * H/R^2. But the correction factor (R+H/R) is far from negligible, and masks the fact that, as the GR formula is based on a gravitational potential which is a function of H, it cannot be used. So, one is left with two different shifts: gH/c^2 + (gH/c^2)^2/2, observed in the "accelerating" elevator, and gH/c^2, observed in the elevator "at rest" on the ground. Such difference, small but theoretically important, falsifies the Einstein equivalence principle (EEP), one of the basis of GRT. Conclusion: __________ Einstein tried to show with his thought experiment, that a *constant* gravitational field is equivalent to uniform acceleration. But such equivalence exists only for massive bodies, not for light, as the frequency shift due to a constant gravitational field is slightly different from the shift due to a corresponding uniform acceleration. The shift difference between the "accelerating" elevator and the elevator at rest on the ground is entirely due to the difference of length that the light pulse has to travel between the emitter and the receiver, i.e. H for the elevator "at rest", and H' = (c^2/g) * (sqrt(1 + 2gH/c^2) - 1) for the "accelerating" elevator. Such difference is not taken into account by General Relativity Theory. Even if it small, it has such an important theoretical implication, that GRT should be amended. APPENDIX ________ I. Albert Einstein (1879-1955). Relativity: The Special and General Theory. 1920. XX. The Equality of Inertial and Gravitational Mass as an Argument for the General Postulate of Relativity (http://www.bartleby.com/173/20.html) "We imagine a large portion of empty space, so far removed from stars and other appreciable masses that we have before us approximately the conditions required by the fundamental law of Galilei. It is then possible to choose a Galileian reference-body for this part of space (world), relative to which points at rest remain at rest and points in motion continue permanently in uniform rectilinear motion. As reference-body let us imagine a spacious chest resembling a room with an observer inside who is equipped with apparatus. Gravitation naturally does not exist for this observer. He must fasten himself with strings to the floor, otherwise the slightest impact against the floor will cause him to rise slowly towards the ceiling of the room. To the middle of the lid of the chest is fixed externally a hook with rope attached, and now a "being" (what kind of a being is immaterial to us) begins pulling at this with a constant force. The chest together with the observer then begin to move "upwards" with a uniformly accelerated motion. In course of time their velocity will reach unheard-of values -- provided that we are viewing all this from another reference-body which is not being pulled with a rope. But how does the man in the chest regard the process? The acceleration of the chest will be transmitted to him by the reaction of the floor of the chest. He must therefore take up this pressure by means of his legs if he does not wish to be laid out full length on the floor. He is then standing in the chest in exactly the same way as anyone stands in a room of a house on our earth. If he releases a body which he previously had in his hand, the acceleration of the chest will no longer be transmitted to this body, and for this reason the body will approach the floor of the chest with an accelerated relative motion. The observer will further convince himself that the acceleration of the body towards the floor of the chest is always of the same magnitude, whatever kind of body he may happen to use for the experiment. Relying on his knowledge of the gravitational field (as it was discussed in the preceding section), the man in the chest will thus come to the conclusion that he and the chest are in a gravitational field which is constant with regard to time. [...] We have thus good grounds for extending the principle of relativity to include bodies of reference which are accelerated with respect to each other, and as a result we have gained a powerful argument for a generalised postulate of relativity. We must note carefully that the possibility of this mode of interpretation rests on the fundamental property of the gravitational field of giving all bodies the same acceleration, or, what comes to the same thing, on the law of the equality of inertial and gravitational mass. [...] Guided by this example, we see that our extension of the principle of relativity implies the necessity of the law of the equality of inertial and gravitational mass. Thus we have obtained a physical interpretation of this law." II. The Confrontation between General Relativity and Experiment, by Clifford M. Will April 14, 2009 Marcel Luttgens