Comment on " Analyses of Mössbauer experiments in a rotating system : Proper and improper approaches

The paper again shows the correctness of the remarkable result of Prof. C.Corda concerning the Mössbauer rotor experiment as new proof of general relativity, which has been awarded by the Gravity Research Foundation. The paper also shows various very elementary mistakes, misunderstandings and flaws by the self-colled « YARK group », which is a group of fringe researchers who attempts to promote wrong science, in particular, against the relativity theory.


Introduction
In paper [1] published recently A. L. Kholmetskii et al.argued that S.A.Podosenov et al. paper [2] wrong and cannot explain nowel Mössbauer experiment in a rotating system. However paper [1] contains a lot of principal mistakes: (i) first notice that [1] based on rejection the Einstein equivalence principle (EEP) (see [3]) and misconception in basic notion of GRT. By using these misconception A. L. Kholmetskii et al.argued that: "the problem of the physical interpretation of the observed energy shift between emission and absorption lines in a rotating system under the framework of general relativity open".
Ofcourse after rejection the EEP under the framework of general relativity one explains nothing.
(ii) secondary note that paper [1] based on misleading in quantum optics.
In quantum optics we dealing with the probability density w t, r , r 3 such that w t, r , dt is the probability of the photon registration near point r , between instant t and t dt by a detector with angular size 1sr and located at the point r 3 .The probability density w t, r is not highly localized in free space except unphysical 1D case known from literature (see [4],5B 1.2).A single photons cannot be localized by using photodetector. Nevertheless in order to disprove S.A.Podosenov et al.
paper [2], A. L. Kholmetskii et al. in [1] naively argued that -quanta is a point particle and propagate along highly localized classical trajectory. But this is a missconcept. Remark 1. But more importantly, these authors deliberately mislead readers by reporting absolutely false information about the experimental results stated in the classical papers [5]- [9]. A. L.Kholmetskii et al. wrote (see [1] p.5): However, "it becomes obvious that Podosenov et al. [26] did not even realize the fact that Eq. (11) indicates a red shift of the frequency of the resonant radiation (i.e., R 0 ), whereas the equality k 2/3 in Eq. (4), obtained in the experiments [9][10][11][12] corresponds to the blue shift of the resonant radiation when R 0 . We add that the same blue shift of the frequency of the resonant -quanta has been obtained in all other Mössbauer rotor experiments [1][2][3][4][5][6] (see corresponding ref. [5]- [9] in this paper) in the configuration where the source of resonant radiation was located on the rotational axis, and the resonant absorber was mounted on the rotor rim". "In these experiments, we did not repeat the approach by Kündig, who based himself on a linear Doppler modulation of the energy of the emitted resonant -quanta, because some unaccounted-for systematic errors in the evaluation of the coefficient k in Eq. (4) do inevitably emerge (see, e.g. [10]). Thus, we did not try to repeat directly the measurement scheme by Kündig, but followed the experimental scheme used in [1][2][3][4]6], where the source of resonant radiation is rigidly fixed on the rotor axis".
(iii) Note that in this simplified version the sign of the energy shift is not measured (iv) In Kündig experiment the energy shift is measured sucussesefully. The equation (3) describes the Kündig's experimental data. For instance, if R a R e , the energy that a photon must have for being absorbed by the absorber is smaller than the energy of the photon emitted by the emitter. In this case, for restoring frame K attached to the accelerated absorber, the problem could be treated by the principle of equivalence and the general theory of relativity. The centrifugal force acting on the absorber is then interpreted as a gravitational force with the potential 1 2 R a 2 2 . 4 Thus, the observer in K will come to the conclusion that his clock is slowed down by the gravitational potential. The frequency v a measured in his frame of reference is given to a first approximation by a s 1 2 /c 2 1/2 s 1 /c 2 . 5 The fractional energy shift is Remark 5.We would like to point out that the "minus" sign on the rhs of Eq. (3) corresponds to the red shift of the energy of the resonant radiation, where E a E s in accordance with true classical relativistic prediction is based on general relativity treatment [10]- [11].
2. Wrong theoretical descriptions of the Mössbauer experiment in a rotating system. Note that wrong prediction is given by Eq.(1) and named by A. L.Kholmetskii et al. the "classical" relativistic prediction is a classical mistake based on misunderstood what we really measured using rotating absorber of -quanta located on the rotor rim.
There exist a lot authors which naively treated Mössbauer experiment in a rotating system mistakenly using formula (4) for Doppler frequency (energy) shift it follows the frequency of absorbed radiation a reads as [12] a where 0 is the proper frequency of gamma-quanta, u A is the velocity of point A at the emission moment, and u B is the velocity of point B at the absorption moment, see Fig.0. From Eq.(4) finally one obtains (see [8]) There exists apparent contradiction between Eq.(9) and Eq.(6) from Kündig paper [9]. Remark 6. (i) Note that a "proof" of the Eq.(8) from T. Yarman et al. [12] wrong since this "proof" implicitly uses a postulate named in literature "Hypothesis of Locality".
(ii) Remind that the Hypothesis of Locality [13]- [15] is tacitly assumed that: any accelerated observer measures the same physical results as a standard inertial observer that has the same position and velocity at the time of measurement. For practical purposes, the hypothesis of locality replaces the accelerated observer by an infinite sequence of otherwise identical momentarily comoving inertial observers. Every inertial observer is endowed with a natural orthonormal tetrad frame in Minkowski spacetime. Therefore, the same holds for an accelerated observer by the hypothesis of locality.
(iii) A restricted hypothesis of locality is the so-called clock hypothesis, which is a hypothesis of locality only concerned about the measurement of time. This hypothesis implies that a standard clock in fact measures , d 1 2 t dt, along its path; is then the proper time along this accelerated path. According to most experiments, the hypothesis of locality seems to be true. No experiment has yet shown the hypothesis of locality to be violated (outside of radiation effects). Remark 7. Note that for the radiation effects in rotating frame the Hypothesis of Locality obviously wrong since Hypothesis of Locality contradicts with EEP, it follows from consideration below. Note that the energy of a particle of mass m at rest in a constant gravitational field is given by [10]- [11]: E mc 2 1 /c 2 where is the newtonian gravitational potential. If the particle is a nucleus in an exited state, one obtains where E is the energy difference between the two levels of the nuclear transition. Then, the energy difference between the two levels of the nuclear transition is modified by the gravitational potential by the multiplier 1 /c 2 . Thus the angular frequency of the nuclear transition is given by: 1 /c 2 , where being the transition frequency without gravitational field [10]- [11] . According to the weak equivalence principle, an acceleration field is locally indistinguishable from a gravitational one. Then, in a reference frame corotating with the rotor, the energy of a photon emitted by the source without recoil is given by: since 1/2 2 R s 2 is the pseudo -gravitational potential due to acceleration. Analogously, the energy of the photon that can be absorbed by the absorber is given by: Therefore E a E s 3.Non highly localizability of the probability density corresponding to one-photon state in 2D space dimension.
Remind that in quantum optics we dealing with the probability density w t, r , r 3 such that w t, r , dt is the probability of the photon registration near point r, between instant t and t dt by a detector with angular size 1sr and located at the point r 3 .The probability density w t, r is not highly localized in free space except unphysical 1D case known from literature (see [4],5B 1.2). Although below we will use the concept of the photon position vector r , we will keep in mind that in fact this is the position of the photon detector. We consider 2D space dimension wave packet with r 2 , but without los of generality. Let us consider a one-photon state of the form (see [4],complement 5B) The photodetection signal at time t and fixed point r r, , r r (see Fig.1) is given by [4]: where is angle r, X between axis X and vector r (see Fig.1), s is the sensitivity of detector and where We consider now a set of coefficients c l c l k l different from zero for values of k l distributed over some bounded region G k 0 of k-space of extent k x , k y about a value k 0 We thus obtain, at time t 0, a 2D wave packet localized in a volume of x-space with dimensions of the order of k x 1 , k y 1 . When the same set of coefficients c l c l k l , is substituted into (17), we thus obtain a photodetection signal (18) that differs from zero only within some bounded region G x of x-space of extent k x 1 , k y 1 . Remark 9.The volume of this region G x generally increases without limit as time goes by, and this in each space dimension. Therefore there is now well localized classical trajectories of such -quanta in physical 2D space dimension. The final result for inertial frame reads [16], (see Appendix A): where , 1 2 and 1 2 , (see Appendix C, Fig.2). It follows from Eq.(21) that under condition cos 1, with a probability P 1 the following constraint holds The final result for non inertial frame reads [16], (see Appendix D, Fig.3 where c r c r , r r, r 1 1 6 r c

.
It follows from Eq.(21) that under condition cos 1, with a probability P 1 the following constraint holds c r t RF r 0.

24
Remark 10.Note that the constraints (22) and (24) were obtained without any references to notion of the classical trajectories of -quanta [16]. By using proposed approach the fundamental C.Corda's result [22]- [24] can be recovered successfully by obvious way without any reference to unphysical notion of the classical trajectories of -quanta. Remark 11.(i) Note that in canonical literature, (see for example [4], [18], [19], [20], [21]) only unphysical specific forms of a one-photon state in one space dimension are considered. However such specific forms can be considered only as an simplification but rigorously, neither of these approximation is ever correct.This sometimes leads to misleading of the people and A. L.Kholmetskii et al. such of them.In contrast to the approach taken in Refs. [4], [18], [19], [20] we applayd a more realistic 2D picture [16].
(ii) Note that in paper [1] L.Kholmetskii et al. mistakenly argued that: "The constraint (8a) used in Ref. [26] implies that the resonant -quanta will propagate along the radial coordinate r of the rotating system, and hence, a laboratory observer would see the propagation of such -quanta along a curved path." (iii) This statement from L.Kholmetskii et al. [1] wrong and based on misconception meaning mentioned above, since such -quanta is well localized in k space and therefore is not well localized in x space except unphysical 1D space dimension. Thus a laboratory observer would see nothing since there is no any curved classical path mentioned in their paper [1] where , see Appendix D.Note that the interval-valued line element (25) corresponding to photodetection signal which propagate with a probability 1 in accordance with the interval-valued law is given by Eq.(C.9)-Eq.(C.10),see Appendix C.
The interval-valued transformation to a non inertial frame of reference t , r , , z rotating at the uniform angular rate with respect to the starting inertial frame (26)  The conservation law | g 00 | 1/2 E loc constant 30 valid for any time-independent interval-valued metric with g 0j 0 and for particles with both zero and non-zero rest mass. It describes how the locally measured energy of any particle or photon changes (is "red-shifted" or "blue-shifted") as it climbs out of or falls into a static gravitational field. For a particle of zero rest mass as photon, the locally measured energy E loc , and wavelength loc , are related by E loc / loc loc , where is Planck's constant. Consequently, the law of energy shift can be rewritten as Therefore, from Eq.(31),one gets where we use the proper time rather than the wavelength and where 10 is the delay of the emitted radiation, 11 is the delay of the received radiation, 1 Rc 1 , R is the radial coordinate of the absorber (see Fig.2-3) and v R , where is the tangential velocity of the absorber.In a gravitational field, the rate d of the proper time is related to the rate dt of the coordinate time by Using now again Eq. (26), we get where the equality follows from the issue that in the laboratory frame photodetection signal propagate with a probability 1, in accordance with the following interval-valued law r c t , 2 , 1 , 36 see Appendix C, Eq.(C.9).Hence, Eq.(33) becomes Note that the Eq. (15) is well approximated by Therefore the second contribution of order 2 /c 2 to the variation of proper time reads Note that r c 1 is the radial distance between the source and the detector. Then, one gets the Corda's desynchronization term

Conclusion
By using proposed approach based on point-free Lorentzian geometry [16], the fundamental C.Corda result [22]- [24] recovered successfully by obvious way without any reference to unphysical notion of the classical trajectories of -quanta. In additional note that YARK group papers [30]- [33] wrong and mast be rejected since in contrast with Kündig [9] YARK group did not measure the sign of the energy shift between emission and absorption lines but attributed this sign by own ubnormal meaning based on wrong Eq.(1).
Appendix. A.Two-dimensional wave packet and corresponding conditional photodetection probability density function. The photodetection signal at time t and fixed point r r, , r r (see Fig.2) is given by [2]: where is angle r, X between axis X and vector r (see Fig.1), s is the sensitivity of detector and where (ii) We thus obtain, at time t 0, a 2D wave packet localized in a volume of x-space with dimensions of the order of k x 1 , k y 1 . When the same set of coefficients c l c l k l , is substituted into (A.1), we thus obtain a photodetection signal (A.4) that differs from zero only within some bounded region G x of x-space of extent k x 1 , k y 1 .
(iii)The volume of this region generally increases without limit as time goes by, and this in each space dimension, there are specific form of a one-photon state in 1D space dimension for which the spreading effect does not occur [3].This kind of wave packet is not physically realistic, in the sense that it extends infinitely in the plane perpendicular to n.
(iv) However there are specific forms for which the spreading effect does not occur dramatically in 2D and 3D space dimension. An nontrivial example is the 2D wave packet we are about to discuss.
Consider now the case in which the wavevectors k l associated with the non-zero coefficients c l are all parallel to the same unit vector n (see Fig.2 where , (see Fig.2). Let us consider the case where all the modes have the same polarization . The coefficients c then depend only on the frequency , and a wave packet can be formed by considering a distribution peaking at some 0 , described by where g , l 0 is a function centred on 0 and having a typical half-width that is small compared with 0 . The function (A.11) will then be proportional to the Fourier transform g of g , yielding a wave packet with width of the order of 1/ . To carry out the calculation explicitly, the sum in (A.11) is replaced by an integral, introducing the one-dimensional mode density deduced from (A.10): The final result reads E r, t The photodetection probability density (A.5) reads w r, t 1 , g t r cos r, n c 2 .

A. 17
Spontaneous emission by a single atom in an excited state gives a one-photon wave packet. For this case one obtains the coefficients Note that the emitted light spectrum is described by a Lorentzian line centred at 0 , with width at half-maximum: We now write E r, t in the form (A.15). The Fourier transform of where H is the Heaviside step function, equal to 0 for 0 and 1 for 0. The final result reads where , (see Fig.2). Remark A.2.(i) Note that the probability density w t, r cannot be considered as the wave function of the photon, whose squared modulus, suitably normalized, gives the probability density for the presence of the photon, measured by a photodetector (see [3],5.6.).
(ii) It should not be thought that there is a position operator r for the photon corresponding to measurements by a photodetector. Therefore the probability density w t, r cannot be considered as the probability density of finding a photon exactly at point r 2 , but rather as an average probability density over some small area which cannot be smaller than where t t, t , 2 , 1 0. 5 2 , 1 .
Appendix. B.Quantum measurement on inertial relativistic frame of reference.Point-free Minkowski geometry.
In this appendix we introduce point-free Minkowski geometry [16], [25]- [26], related to relativistic quantum measurement on inertial relativistic frame of reference.
If we are to suppose that a quantum particle at a definite position x x 1 , x 2 , x 3 3 at instant t 0, T is to be assigned a state vector |t, x , and if further we are to suppose that the possible positions x i , i 1, 2, 3 are continuous over the range , and that the associated states are complete, then we are lead to requiring that any state | t of the particle at instant t 0, T must be expressible as with the states |t, x by -function normalised, i.e. x, t|t , x 3 x x t t . However well known that the notion of preparing a particle in a state |t, x does not even make any physical sense.The resolution of this impasse involves recognizing that the measurement of the position of a particle is, in practice, only ever done to within the accuracy, x x 1 , x 2 , x 3 say, of the measuring apparatus.In other words, rather than measuring the precise position of a particle, what is measured is its position as lying somewhere in a range x i 1 2 x i , x i 1 2 x i , i 1, 2, 3 Therefore if the particle is in some state | t , we can recognize that the probability P | t t, x, t, x of getting a result x with an accuracy of x between instants t t and t t. will be given by Remark B.1. Note that only under condition (B.3) the notion of position of a quantum particle at instant t holds in well approximation relevant to classical sense, i.e. as a definite point x 3 . 2.We assume now that there exists continuous vector-function x t : 0, T 3 such that for all t 0, T the following estimate is satisfied where c 1 , c 3 1, c 2 , c 4 1, are positive constants suth that c 2 x 1, c 4 t 1 and x 1, t 1. Remark B.2.Note that only under condition (B.4) the notion of trajectory of a quantum particle holds in well approximation relevant to classical sense, i.e. as continuous vector-function x t : 0, T 3 . 3.We assume now that at point x 3 the following estimates are satisfied P | t t, x, t, x 1 c 1 exp c 2 x 1,