mössbauer experiments in a rotating system

Mössbauer experiments in a rotating system: Recent errors and novel interpretation T. Yarman1, A.L. Kholmetskii2, and M. Arik3 1

Okan University, Istanbul, Turkey & Savronik, Eskisehir, Turkey Department of Physics, Belarus State University, Minsk, Belarus, tel. +375 17 2095482, fax +375 17 2095445, e-mail: [email protected] 3 Bogazici University, Istanbul, Turkey 2

Abstract. We consider the energy shift between emission and absorption lines in the Mössbauer experiments in a rotating system in the view of a recently reported extra component of such a shift, in addition to the usual relativistic time dilation effect, expressed by the inequality k>0.5 in the expression for the relative energy shift ∆E/E=-ku2/c2 (u is the orbital velocity, and c the light velocity in vacuum). We consider the recent attempts of re-interpretation of the Mössbauer rotor experiments, showing their incorrectness, and analyze the proposed explanations of the physical origin of this extra energy shift. This way we stress that the energy of nuclei located in crystal cells rotating at the edge of the rotor, should be determined via relativistic quantum mechanics, involving the special geometry of a rotating disc, displayed by YARK (Yarman-Arik-Kolmetskii Gravitational Approach). We show that the standard theory yields k=1/2; applying YARK, we obtain k=2/3 in a perfect agreement with experimental data.

1. Introduction It is known that the first (and major) series of Mössbauer experiments in a rotating system had been carried out soon after the discovery of the Mössbauer effect at the early 60th (see, e.g. [16]). In these experiments, an absorber orbited around a source of resonant radiation (or viceversa). The goal was to verify the relativistic dilation of time for a moving resonant absorber (source), which induces the relative energy shift between emission and absorption lines at the value (1) ∆E/E=±ku2/c2, where k=0.5 according to the Special Theory of Relativity (STR), u is the tangential velocity of absorber, c is the light velocity in vacuum; the sign “+” corresponds to the case, where a source orbits around an absorber, and the sign “-” corresponds to the reverse case, where an absorber orbits around the given source (see Fig.1, where the latter configuration is shown). For subsound u≈300 m/s, the value of ∆E/E has the order of magnitude 10-12, which can be reliably measured with Iron-57 Mössbauer Spectroscopy, providing the relative energy resolution of the resonant γ-quanta of about 10-14. All of the authors of the mentioned papers reported the value of k=0.5 with the accuracy about 1 %, confirming thus the general relativistic prediction. Later the usual special relativistic dilation of time had been confirmed with much better precision (108 …10-9) in the experiments on ion beams [7, 8], and this achievement seems to have deprived physicists of further interest in repetition of the Mössbauer experiments in the rotating systems. New wave of interest to the Mössbauer experiments in a rotating system emerged after publication of the paper [9] (stimulated by the predictions made by Yarman et al [10], specified below in section 4), where serious errors in the data processing in the available experiments of 1960’s were revealed. First of all, we imply here the experiment by Kündig [1], which was much more informative and reliable than the other experiments of the early 1960’s [2-6]. He was the only one who applied the first order Doppler modulation of the energy of γ-quanta on a rotor at each fixed rotation frequency ν, implementing an oscillating motion of the source along the radius of the rotor. By such a way, Kündig recorded the shape and the position of resonant line on the energy scale versus the rotation frequency and thus his results were practically insensitive to the presence of mechanical vibrations in the rotor system, which, due to the chaotic nature of

2 vibrations, broaden the resonant line, but do not affect its position on the energy scale. In contrast, other authors [2-6] measured only the count-rate of detected γ-quanta at each fixed ν, and their results were not protected from the distortions induced by the vibrations. However, analyzing the paper by Kündig [1], the authors of ref. [9] found a number of serious computational errors committed by Kündig, and made their own re-estimation of the coefficient k in eq. (1), based on the raw data of [1]. As a result, they obtained k=0.596±0.006, (2) which drastically deviates from the relativistic prediction and many times (the order of magnitude and more) exceeds the estimated uncertainty [9]. What is more, they have shown that the experiment by Champeney et al [6] distinguished by numerous experimental data with various resonant absorbers, is well fitted into k>0.5, too. In particular, the re-estimation with regards to Champeney et al, unlike what these latter authors had originally reported, yields k=0.61±0.02 [9] 1. Concerning other experiments mentioned above [2-5], the absence of raw data in these publications does not allow deriving any independent estimation of the coefficient k in eq. (1). Based on these results we conjectured that in rotating systems, the energy shift between emission and absorption resonant lines is induced not only via the standard time dilation (which is measured alone in the experiments with ion beams [7, 8] dealing with an inertial motion), but also via some additional effect, which induces an excess of ∆E/E in comparison with the standard relativistic prediction. We now firmly believe this additional effect is induced by the sole acceleration, which, according to the original statement by Einstein, had been assumed to be very negligible, if non-existant (the so called clock hypothesis, assuming the independence of clock rate on its acceleration [11]). Thence, for further clarification of the situation, new measurements of the coefficient k were required. The first modern Mössbauer experiment in a rotating system had been carried out in 2008 in Minsk [12], which had confirmed the presence of the extra-energy shift between a resonant source (located on the rotational axis) and a resonance absorber on the rotor rim, yielding the value of k=0.68±0.03. Later, this result had been corrected to the value [13] k=0.66±0.03, (3) due to temperature effects in Mössbauer measurements, when the data with respect to unexpectedly low Debye temperature of one of resonant absorbers used in experiment [12], had been reported in 2009 [14]. Recently, one more experiment on this subject became necessary and has been implemented in Istanbul University [15], which is characterized by the essential improvement of the parameters of the rotor system in comparison with the experiment [13]. The result obtained k=0.69±0.02, (4) agrees with the result (2) within the measurement uncertainty. We stress that the modern Mössbauer rotor experiments [12, 15] are based on a novel methodological approach in comparison with the older experiments [1-6], which allows eliminating the influence of vibrations in the rotor system on the estimated value of k, without the first order Doppler modulation of the energy of γ-quanta, applied in the Kündig experiment [1]. The main reason in the favour of this novel approach is the presence of some instrumental factors in the experiment [1], affecting the measured energy shifts of resonant lines, which hardly could be reliably estimated. One of such instrumental factors most likely related to the non-controlled variation of the parameters of oscillating piezo-element, driving the source in the Kündig experiment, versus the rotational frequency. In ref. [1], Kündig mentioned that the piezoelectric constant of his transducer does not practically vary due to the pressure effect up to 1

We would like to stress that the measurement uncertainly in eq. (2), as well as in the result of Champeney et al experiment, has a purely statistical origin, and still does not include possible systematic errors. This statement is obvious with respect to the experiment by Champeney, where no correction to the vibrations of the rotor system had been made. A possible origin of systematic error in the Kündig result (2) is explained below. 2

3 the maximal rotation frequency 35000 rpm. However, he appears not to have taken into account the decrease of resonant frequency of piezo-element with the increase of rotational frequency of the rotor, caused by considerable variation of the effective weight of this piezo-element. For example, at the angular rotation frequency ω≈3660 s-1 (which corresponds to 35000 rpm), the centrifugal acceleration even at the small distance r=1 mm from the rotational axis (which is larger than the size of piezo-element), is equal to about 13.4x103 m/s, and exceeds three orders of magnitude the acceleration of “free fall” on Erath. Since in the calibration measurements of work [1], the piezo-element operated at the frequency adjusted to its resonance, the decrease of the resonant frequency in a rotating system definitely leads to an underestimation of the energy shift between emission and absorption lines of a resonant source and rotating absorber, and does reduce, though a little bit, but still, the estimated value of k in eq. (2). We have thus good reason to believe that just the presence of the related instrumental error can explain just a little reduction of the value of k yielded by the experiment [1] (eq. (2)) in comparison with the recent results (3), (4). At the same time, it seems practically impossible to estimate exactly the rather minor influence of this factor on the measured coefficient k. Thus, we discarded the alternative of a direct repetition of the experiment by Kündig and, like in the experiments [2-6]. Thereby, we decided to measure the count-rate of resonant γquanta by detector at different rotational frequencies ν [12, 15] according to the configuration of Fig. 1. However, in contrast to the experiments [2-6], the level of vibrations in the rotor system was also evaluated via the measurement of absorption curves (i.e., the dependence of detector’s count-rate on ν) for two different resonant absorbers, whose resonant lines are shifted on the energy scale with respect to each other approximately by their linewidth. The idea of this method is based on the fact mentioned above that the chaotic vibrations do broaden the resonant line, but do not affect its area and position on the energy scale. Hence, an equal broadening of shifted lines of these absorbers, caused by vibrations, should induce quite different variation of the detector’s countrate with the change of rotational frequency. Therefore, implementing the joint processing of absorption curves obtained with both resonant absorbers, we could separate the variation of detector’s countrate, caused by the energy shift (1), from the distortions of countrate due to vibrations, and to eliminate the influence of such vibrations on the measured value of k. The algorithm of data processing for the realization of this method has been described in ref. [12]. It is important to emphasize that this algorithm allows obtaining an “unbiased estimation” of k, where the contribution of all possible instrumental factors can be explicitly accounted for. The publication of results (2)-(4) in a series of the papers mentioned above [9, 12, 13, 15] induced further discussions, which can be conditionally divided into two branches: 1 – to put in doubt the general methodology of Mössbauer measurements in a rotating system [16, 17]; 2 – to provide the explanation of the strong inequality k>0.5 [18-21], recognizing the correctness of the methodology of the alleged experiments [1-6, 12, 15]. In section 2 we consider the available attempts to re-interpret the measurement results in Mössbauer rotor experiments and show that all of them are erroneous. In section 3 we analyze the available explanations of the inequality k>0.5 proposed up to date, and show that they cannot be adopted as satisfactory. In section 4 we demonstrate that the experimental data (3), (4) find their consistent explanation in the framework of the gravitational approach by Yarman, developed further and presently abbreviated as YARK (Yarman-Arik-Kholmetskii) theory to designate the collaborative work of the co-authors. In particular, we show that the geometry of a rotating disc occurs substantially different in YARK theory and in general theory of relativity (GTR). Therefore, the measurement of a relative energy shift between the resonant lines of a source and absorber, which is sensitive to the geometry of a rotating disc, is capable to distinguish the predictions of these theories at the experimental level. As is known, GTR predicts k=0.5 in eq. (1). In section 4 we show that in the framework of YARK, k=2/3, and this prediction exactly coincides with the results (3) and (4). Finally, we conclude in section 5. 3

4 2. The attempts of re-interpreting of Mössbauer experiments in a rotating system Few years ago the papers by Friedman & Nowik [16] and Zanchini [17] had been published, where the authors doubted the common interpretation of Mössbauer rotor experiments, aimed to measure the relative energy shift between emission and absorption resonant lines in the order (u2/c2). As the outcome, they suggest to consider the major part of these experiments to be erroneous. We start our analysis with the paper by Friedman & Nowik [16], where the authors exposed their hypothesis on the possible existence of a universal maximal acceleration in Nature (see sub-section 3.1) and claimed that among the Mössbauer rotor experiments, performed up to date, only the experiment by Kündig [1], as re-analyzed in [9], is correct, whereas all other experiments on this subject are erroneous. In order to substantiate this assertion, Friedman & Nowik carry out their own calculation of the relative energy shift between emission and absorption lines for the configuration, where a resonant absorber rotates, while a source of resonant radiation centered on the rotational axis is at rest in a laboratory frame. For this configuration, the strong aberration effect does emerge, which leads to the substantial broadening of resonant line as the function of finite sizes of source, absorber and the divergence of gamma-beam. Based on this result, Friedman & Nowik claim that the broadening of resonant line, observed directly in the experiment by Kündig, takes place due to this aberration effect, but not due to mechanical vibrations in the rotor system, on the contrary to what Kündig assumed. Furthermore, Friedman & Nowik assert that all other Mössbauer experiments in rotating systems, both the old and modern [2-6, 12] are erroneous due to the missing aberration effect calculated by Friedman & Nowik. However, the calculations by Friedman & Nowik had been carried out for the configuration, which was not realized in the known experiments [1-6, 12, 15], where both the source and absorber are rigidly fixed on a rotor, and even if the source is located on the rotational axis, it is also involved into rotational motion. In other words, in a real configuration, realized in all of the mentioned experiments [1-6, 12, 15], the source is at rest in the rotational frame, but not in the laboratory frame. Let us show that for this configuration, the aberration effect, calculated by Friedman & Nowik [16] and causing the component of relative energy shift proportional to (u/c), completely to disappear, and the entire energy shift between emission and absorption lines is proportional to the ratio u2/c2 (to the accuracy of calculations c-2), regardless of finite sizes of source, absorber, and the divergence of gamma-beam as well. In order to prove this statement, it is sufficient to show that for two arbitrary points A and B on the rotor surface (see Fig. 2, where A stands for the point-like source, and B for the pointlike absorber), the relative energy shift between emission and absorption lines does not contain the linear terms of order (u/c). For a laboratory observer, the frequency of an emitted γ-quanta is equal to [22]

ν em =

ν 0 1 − u A2 c 2

,  n ⋅ uA   1 − c   where ν0 in the proper frequency of γ-quanta, uA is the velocity of point A at the emission moment, and n is the unit vector along the direction of propagation of γ-quanta, emitted from the point A towards the point B. Correspondingly, the frequency of absorbed radiation reads as

4

5 n ⋅ uB   n ⋅ uB  2 2   ν 0 1 − u A c 1 − c  c    , (5) ν ab = = 2 n ⋅ uA  2 2 1 − uB c 2 1 − u B c 1 −  c   where uB is the velocity of point B at the absorption moment. In order to calculate the frequency (5), we designate rA, ϑA the radial and angular coordinates of the point A at the moment of emission of γ-quantum, and rB, ϑB the radial and angular coordinates of the point B at the moment of absorption of γ-quantum, correspondingly, see Fig. 2. With these designations, we have the following components: (rAB )y rB sin ϑB − rA sin ϑA (r ) r cosϑB − rA cosϑA nx = AB x = B , ny = , (6a-b) = rAB rAB rAB rAB (7a-d) u Bx = ωrB sin ϑB , u By = −ωrB cos ϑB , u Ax = ωrA sin ϑ A , u Ay = −ωrA cos ϑ A , where rAB is the distance between the point A at the emittance time moment and point B and absorption time moment. Hence, substituting eqs. (6) and (7) into eq. (5), we derive:  n ⋅ uB  ν 1 − u 2 c 2 1 − nxu Bx + n y u By  ν 0 1 − u A 2 c 2 1 −  0 A   c c   =  = ν ab = ⋅ + n u n u n u     2 2 A y Ay 1 − u B c 2 1 −   1 − u B c 2 1 − x Ax c   c  

ν em 1 −



ν 0 1 − u A 2 c 2 1 − 

(rB cosϑB − rA cosϑA )ωrB sin ϑB − (rB sin ϑB − rA sin ϑA )ωrB cosϑB  rAB c

 =

 (r cos ϑB − rA cos ϑ A )ωrA sin ϑ A − (rB sin ϑB − rA sin ϑ A )ωrA cos ϑ A  2  1 − u B c 2 1 − B rAB c    ωr r sin (ϑ A − ϑB )   ν 0 1 − u A 2 c 2 1 − A B 2 2 rAB c   = ν 0 1− uA c . (8) 2 2 ωrB rA sin (ϑ A − ϑB )  2 2 1 − u c B  1 − u B c 1 − rAB c   Thus, we see that the terms of nominator and denominator, which contain the linear components in (u/c), mutually cancel each other, so that the frequency (energy) shift is determined by the second order Doppler shift (or time dilation effect) alone, if we do not include the extra energy shift, discussed in the present paper. From the physical viewpoint, the absence of linear Doppler shift proportional to (u/c) in eq. (8) is explained by the fact that the distance between the emitter A and receiver B remains constant during a rotation. Since the equation (8) was derived for two arbitrary points A and B on a rotor surface, it also remains in force for spatially extended source and absorber, and do not depend on the divergence of gamma-beam. The only point is that for such spatially extended source centered on the rotational axis, the tangential velocities uA at the edge of the source and at its center differ from each other, which can cause a broadening of the emitting resonance line. However, for a source of resonant radiation, sufficiently compact, and bearing typical configurations of Mössbauer rotor experiments, this effect is quite negligible, as numerically estimated by Kündig in ref. [1]. Thus for the practical purpose we can well put uA=0 for a compact source, so that the relative frequency (or energy) shift becomes: 2 ∆E v0 −ν ab 1 u = = 1− ≈ − B2 , 2 ν0 E 2c 1 − u c2 B

when the extra energy shift is not included. 5

6 This result shows a conclusive irrelevance of the analysis by Friedman & Nowik [16] implemented for the case, where the source rests in the laboratory frame. For real configurations of all of the Mössbauer experiments in a rotating system [1-6, 12, 15], where both the source and the absorber are rigidly fixed on a rotor, eq. (1) remains valid, and confirms the correctness of the methodological approach of these experiments. One more attempt to put in doubt the correctness of the known Mössbauer experiments in rotating systems was done by Zanchini in ref. [17]. The essence of his criticism can be expressed via three claims: - In the Mössbauer rotor experiments, when the source of radiation is located on the rotational axis and the absorber is on the rotor rim, the energy shift is blue, whereas all of the authors of these experiment assumed a red shift. - The authors of these experiments missed the contribution of the first order Doppler effect to the energy shift between emission and absorption resonance lines, so that the up to now available interpretation of these experiments is erroneous. - With such a re-interpretation of Mössbauer rotor experiments, Zanchini refers to a work [23], which, in fact, puts in doubt the physicality of time dilation effect in STR. The fallacy of these claims is commented in details in ref. [24]. For convenience, here we briefly reproduce some of these comments. In particular, with respect to the first claim by Zanchini, we notice that the authors of Mössbauer rotor experiments never stated that the frequency shift of resonant radiation is red, when an absorber orbits around a source. In contrast, the experimental results presented in refs. [1-6, 13, 15] clearly show that for the orbiting absorber, the energy shift is blue (i.e., the position of resonant line of a source is shifted at right upon the frequency/energy scale with the increase of rotational frequency). One should only mention ref. [2], where the authors actually entitled their paper as “Measurement of the red shift in an accelerated system using the Mössbauer effect”. This title is mistakable indeed, because the authors of this paper reported, in fact, on the blue energy shift of resonant radiation in a rotating system. Perhaps, such an error in the title of the paper [2] can be explained by its coupling with one more paper of the same research group [25] published on the adjacent pages of the same Journal, where they dealt with the measurement of the frequency red shift of gamma-radiation in a gravitational field. Further, considering the second claim (the missing of the first order Doppler effect contribution), Zanchini suggests to consider a number of problems, dealing with the emittance of light by a source S and its detection by the observer O (being at rest with respect to S), or by the observer O’ (moving with respect to S). Further he analyzes these problems either in the frame O, or in the frame O’, in order to derive the corresponding expressions for the Doppler shift. This approach is undoubtedly vital in the framework of the STR, dealing with an inertial motion. However, it cannot be extended to the analysis of Mössbauer experiments in rotating systems (on the contrary to what Zanchini did), where the absorber of resonant radiation (O’) rotates, and thus is represented by a non-inertial frame. Therefore, in this kind of experiments the frames O and O’ are not equivalent to each other, and due to this reason the analysis of Zanchini cannot be extended to the case of resting source and rotating absorber (as viewed in a laboratory frame). In particular, one can easily understand that for the laboratory observer, the tangential velocity of rotating absorber is always orthogonal to the line joining S (at the emittance moment) and O’ (at the receiving moment), and no linear Doppler effect (which bothers Zanchini) emerges. Thus, only the time dilation effect is responsible for the frequency/energy shift of resonant lines, as the equation (8) obtained above fully confirms. Concurrently this equation also invalidates the third claim by Zanchini (the doubts in reality of the time dilation effect); for more details see ref. [24].

6

7

3. Proposed explanations for the extra energy shift in Mössbauer rotor experiments 3.1. Generalization of special relativity by Friedman et al An attempt to explain the inequality k>0.5 was done by Friedman et al on the basis of their generalization of the STR (see [18-20]) with the negation of the clock hypothesis by Einstein. Thereby the authors postulated the presence of an universal maximal acceleration am in Nature and proposed a modification of space-time transformation between uniformly accelerated frames, which is reduced to the usual relativistic transformation in the limit am→∞. In particular, with respect to the second order Doppler effect in rotating systems, they derived an expression (in the case, where the source of radiation is located on the rotational axis) [20]: −1 2

 Rω 2  R 2ω 2  1 − 2  E0 , E = 1 + a c  m   where E0, E are the energies of emitted and absorbed radiation, correspondingly, R is the radial coordinate of the absorber, and ω is the angular rotational frequency. Hence the relative energy shift between emission and absorbtion lines can be written as ∆E E0 − E Rω 2 R 2ω 2 u 2  1 c2  . = ≈− − = − 2  + (9) E Eo am 2c 2 c  2 Ram  Thus, comparing eqs. (1) and (9), we find that in the extended relativity of Friedman et al, the coefficient (10) k = 1 2 + c 2 Ram . The authors also pointed out that the Mössbauer experiments in rotating systems represent the convenient tool for a test of their hypothesis, due to two reasons: - A high sensitivity of the Mössbauer effect to the relative energy shift of resonant lines. - A large centrifugal acceleration in these experiments (up to 107 m/s2), which is directed along the line joining the resonant source and the absorber. Thanks to these features, the Mössbauer rotor experiments become much more sensitive to the assumed existence of maximal acceleration am, than, for example, the experiments in particle physics with any kinds of accelerators. In particular, Friedman et al conjectured that the inequality k>0.5 can be explained by their eq. (10). Taking the result of Kündig experiment (2), being reanalyzed in ref. [9] as the most reliable, they estimated the maximal acceleration as [20] am≈1019 m/s2. (11) This is indeed a huge acceleration from a practical point of view, which exceeds by many orders of magnitude the typical acceleration of particles in accelerators. At the same time, the hypothesis about a maximal acceleration implies the existence of a fundamental time unit tfundamental = c am , which for the estimated value (11) yields tfundamental≈3x10-11 s. (12) This time interval corresponds to an electromagnetic radiation with a wavelength of about 1 cm, and this appears to be too large, in order to be considered as the basis of any adopted fundamental time unit. Rather one can suppose that the fundamental time unit is determined by the relationship t fundamental = lP c ≈ 0.54 ⋅10 −43 s , (13) (Sasha, use the symbol “x” for multiplications.) -35 where lP≈1.616x10 m is the Planck length. In this case, the universal maximal acceleration, if it exists, is defined via the equation am = c 2 lP ≈ 5.5 × 1051 m / s 2 . Then eq. (10) reads as

k = 1 2 + lP R , and the deviation of k from 0.5 becomes non-observable in the Mössbauer rotor experiments due to the tiny ratio lP/R≈10-34. 7

8 Thus, the huge difference (about thirty orders of magnitude) between the fundamental time units, estimated via the corrected result of Kündig experiment (eq. (11)) and via the Planck length (eq. (13)) represents a strong argumentation against the hypothesis by Friedman et al with the involvement of the value of maximal acceleration (11). One more concrete argumentation against the hypothesis by Friedman et al is related to the observation that, according to eq. (10), the measured value of k should depend on the rotor radius R. Therefore, if eq. (10) along with the estimation (11) is correct, then in the experiment by Kündig [1] (where R=9.3 cm), Champeney et al [6] (R=4.2 cm), Kholmetskii et al [12] (R=30.5 cm), Kholmetskii et al [15] (R=16.1 cm), the values of k must be different (see Table, the last column). However, all of these experiments yield the comparable values of k, whose conservative average is about 0.64±0.02. Comparison of basic properties of Kündig experiment [1], Champeney et al. experiment [6], and Kholmetskii et al. experiments [12, 15] Parameter Experiment 1

Kündig Champeney et al Kholmetskii et al. [12] Kholmetskii et al. [15]

νmax, rev/s

umax, m/s

2

Rotor radius, cm 3

Measured value of k in eq. (1)

4

Centrifugal acceleration, in “g” 5

6

Estimated value of k according to Friedman (eq. (4)) 7

586 1400 120 260

9.3 4.2 30.5 16.1

340 370 230 260

1.3⋅105g 3.1⋅105g 1.8⋅104g 4.2⋅105

0.596±0.006* 0.61±0.02* 0.66**±0.03 0.69±0.02

0.60 0.72 0.53 0.79

*as re-estimated in ref. [9] **as specified in ref. [13] Our observation is, no doubt, strongly against the hypothesis by Friedman et al, at least with the value of maximal acceleration (11). Thus, even if a maximal acceleration exists, its numerical value should be much larger than (11). 3.2. Possible dependence of time rate of charged particle on an electric potential at its location One more attempt to explain eqs. (3), (4) was presented in the paper of our team [21], where we supposed a possible dependence of a time rate dτ of resonant nucleus with the mass m and electric charge Ze (Z being the atomic number) on the electric potential at its location, i.e. (14) dτ e = (1+ eϕ mc 2 )dt γ , where dt is the time interval, measured in empty space for a resting charged particle outside the electric field, and γ is the Lorentz factor, associated with the rotational motion of a nucleus. Applying eq. (14) to the Mössbauer experiments in rotating systems, one should notice the huge centrifugal acceleration of resonant absorber (up to 106 m/s), which is never achieved in any other kind of experiments with condensed matter. Therefore, the resonant nuclei of absorber experience the effect of the local electric field, counteracting the centrifugal force, with the appearance of corresponding local electric potential on the nuclei. Thus, the hypothesis (14) becomes testable in such experiments. The authors of [21] proposed to calculate the difference of effective electric potentials on resonant nuclei of source and absorber via the work done on each nucleus of absorber, in order to bring it from the rotational axis to the rotor rim. This way, using eq. (14), they derived the coefficient (15) k = 0.5(1 + Z A) , 57 where A is the atomic mass number. For the resonant isotope Fe (used in the experiments [1-6, 12, 15]), Z=26 and A=27. Hence, k=0.728, which qualitatively agrees with the results (3) and (4). 8

9 At the same time, the authors of [21] noticed that for a charged nucleus situated in a crystal cell, the effective electric potential is not a well-defined quantity. Rather, the way applied to the estimation of the electric potential ϕ in eq. (14) is well applicable to a freely rotating proton, whose centrifugal force is counteracted by a suitable external electric field. Therefore, the applicability of eq. (14) remains not fully convincing in the case we deal with resonant nuclei in a crystal lattice of a rotating absorber. 3.3. Attempt to explain the inequality k>0.5 in the framework of General Theory of Relativity (GTR) The most recent attempt to interpret the inequality k>0.5 has been achieved in ref. [26], where the author claims that the results (3), (4) fully confirm the GTR, if one takes into account an effect, missed (to the author’s opinion) in the previous analyses of Mössbauer rotor experiments: the synchronization of clocks of a resonant source (located on the axis of a rotating frame) with a clock of detector of γ-radiation (located outside the rotor system) for an observer attached to the spinning source. The author of [26] claims that this effect leads to the additional relative energy shift z2 = u 2 6c 2 , next to the relative energy shift due to the time dilation effect for an orbiting absorber z1 = u 2 2c 2 . Thus, the total energy shift measured in the Mössbauer rotor experiments should be defined as the sum of z1 and z2, which yields the coefficient k = 1 2 +1 6 = 2 3, thus (according to him) in a great agreement with the data (3), (4). However, this explanation is based on a total misunderstanding of the Mössbauer effect methodology and, in particular, the role of detector of γ-quanta in Mössbauer spectroscopy. As is well-known, in the transmission Mössbauer spectroscopy (which is just the case for Mössbauer rotor experiments), the detector of γ-quanta is used to measure the intensity of resonant radiation of a source, passing through an absorber, which varies with the change of a relative energy shift of resonant lines of the source and absorber. Having measured this intensity with a high enough statistical quality, we can determine the relative energy shift of resonant lines, if their shapes are known. Here we specially emphasize that the detector measures the intensity, but not the energy of γ-quanta. Rigorously speaking, in practice of Mössbauer spectroscopy, where the spectra of resonant sources usually contain several γ-lines, the detector of γ-quanta should be energysensitive to some extent, so to select the line of resonant radiation in the total spectrum emitted by the source. More specifically a source 57Co used in 57Fe Mössbauer spectroscopy, emits the lines 6.3 keV, 14.4 keV (resonant line) ≈ 123 keV and ≈ 137 keV [27]. In these conditions, the relative energy resolution 10…20 % with respect to the resonant line 14.4 keV occurs sufficient to select this line and to cut off other lines, increasing by such a way the effect/background ratio in Mössbauer measurements. For example, in the experiments [12, 15], the detector represented Ar-Xe proportional counter with the energy resolution about 15 % for γ-quanta 14.4 keV. This means that a measurable variation of intensity of its output pulses can happen at the relative variation of the energy of γ-quanta at the level 10-2…10-1, which is more than ten (!) orders of magnitude larger than the value z2 = u 2 6c 2 ≈ 10 −13 , derived in ref. [26]. Therefore, in all of the Mössbauer rotor experiments mentioned above [1-6, 12, 15], the detectors of γ-quanta are completely insensitive to the energy shifts ∼ 10-13, which makes the interpretation of the inequality k>0.5 in ref. [26] totally irrelevant. One can add that the variation of measurement time of intensity of output pulses of detector at the relative value 10-13, as predicted in [26], exceeds many orders of magnitude the relative statistical error in the determination of this intensity (≤ 1 %), and also is impractical. Thus, the actual effect, which is measured in the Mössbauer rotor experiments, is the intensity of γ-radiation of a spinning source, passing across a rotating absorber, as the function of 9

10 a relative shift of their resonant lines. Since the width of these lines has a typical value 10-12, their relative shift on the energy scale at different rotational frequencies induces a measurable variation of intensity of γ-radiation, passing across a resonant absorber, at the level of few percent. The corresponding variation of the intensity of output pulses of detector allows us to calculate the value of such energy shift, when the shapes of resonant lines of the source and the absorber are known. As the outcome, we conclude that in the Mössbauer rotor experiments, only the relative energy shift between the resonant lines of a source and absorber is measured, which yields z=0.5 (according to the designation of [26]). Therefore, the obtained experimental results (2)-(4) are in a drastic contradiction with this latter prediction indeed.

4. Interpretation of Mössbauer experiments in a rotating system in the framework of YARK gravitation theory YARK theory principally differs from GTR in its philosophical foundations, and its description can be found in refs. [28-34]. Here we reproduce the main points of this theory: 1. YARK theory is built directly on the basis of the law of energy conservation appertaining to gravitational interaction, which for a massive object moving in the presence of gravity is essentially based on the law of energy conservation, postulated in the form of (16) E = γm0 ∞ c 2 (1 − EB m0 ∞ c 2 ) , where m0 ∞ is the rest mass of a given object, measured at infinitely far away from everything else, γ is the Lorentz factor associated with the motion of the object at hand, and EB is the “static binding energy”, i.e. the energy one has to furnish to the object if it were at rest at the given location, to bring it quasi-statically to infinity. Eq. (12) tells us that the rest mass of the object m0 ∞ is altered by a gravitational environment by the value of the static binding energy EB (in the units c=1). In other words, gravitational energy is considered to be localized inside the gravitating object rather than being distributed across “a field of interaction” between masses. 2. The variation of rest mass of a test particle by a static binding energy, quantum mechanically, affects the time rate for the particle, which simultaneously induces a corresponding transformation of spatial intervals, as observed from the synchronous reference frame co-moving with the particle. However, no warping influence by the host massive body on space-time geometry is assumed [33]. As shown [28-34], this approach allows the elimination of ambiguities that would otherwise arise regarding the definition of “gravitational energy” with respect to an arbitrary reference frame, where it remains a non-vanishing quantity in all plausible frames of reference. This means, in particular, that the energy-momentum tensor in the presence of gravity remains to be a true tensor, not only at linear space-time transformations, but, unlike GTR, in any other admissible transformations between any frames of references. 3. We arrive at the equivalence of gravitational and inertial masses similar to the widely classically acknowledged conceptualization. This we do, however, in such a way as to derive the proper mass of the object with the resulting equation of motion dropped out – so that, the equation becomes free of the rest mass of the object under consideration. As is well known, this property can be called the weak equivalence principle. Therefore, particles with different rest masses do happen to acquire the same acceleration in a given gravitational field. Moreover, the equivalence of gravitational and inertial masses always allows us to choose a reference frame whereby the local geometry becomes pseudo-Euclidean by intrinsic design. Be that as it may, a particle in any such frame of reference undergoes to “experience” the presence of the gravitational medium simply due to the variance in its rest mass as compared to what it would have been in the absence of gravity. 4. In YARK, the laws of energy conservation, momentum conservation and angular momentum conservation in flat Minkowskian space-time are taken as fundamental. These are 10

11 carried only by particles in accordance with quantum field theory. The classical concept of field energy is assumed to be only a classical approximation. 5. In a synchronous reference frame attached to a test particle, resting in a point rm of gravitation medium, the space-time interval for the entire space acquires the form

(

ds 2 = c 2 g 00 dt0 + g11dx0 + g 22 dy0 + g11dz0 = 1 − EB (rm ) m0 ∞ c 2 2

2

2

2

) (c dt 2

2

2

0

)

− dl0 , (17) 2

where dl0 = dx0 + dy0 + dz0 . Eq. (17) manifests a Minkowskian-like metric, where the Christoffel symbols are equal to zero, and the diagonal metric coefficients {1, -1, -1, -1} are multiplied by the conforming factor 2

2

(1 − E (r ) m

2

2

)

2

c 2 ; with the latter explicitly depending on the spatial coordinate of the particle of concern, and implicitly depending on time for a moving particle. Here we specially emphasize that eqs. (17) describe the metric of space-time, as seen in a synchronous frame of a test particle. For a set of different particles, the conforming factors in eqs. (17) are different, too, so that these equations do not describe a real metric of space-time, and rather play an auxiliary role in the definition of the Lagrangian of particle in the presence of gravity (see item 7). 6. The real space-time in a gravitation field remains flat and instead of the geodesic postulate of GTR, the laws of energy and angular momentum conservation in Minkowskian space-time are regarded as fundamental. 7. The action for a test particle is defined in the common way: (18) S = −m0 ∞ c ∫ ds , B

m

0∞

which in the metric (17) yields the Lagrangian L = −m0 ∞ e −α (rm )c 2 1 − v0 c 2 , 2

(19)

where α (r ) = GM rc 2 in the radially-symmetric case. Here G is the gravitation constant, and M is the mass of the source of gravity. The corresponding energy of test particle in a gravitational field reads as (20) E = γm0 ∞ e −α c 2 , and the laws of energy and angular momentum conservation are sufficient to derive the motion of this particle. It is straightforward to see the similarity of equation (20) with the total relativistic energy obtained under the framework of GTR in the limit of a weak gravitational field [35]: (21) mγ c 2 = γm0 ∞ c 2 1 − 2α ≈ γm0 ∞ c 2 (1 − α ) ,

which coincides with eq. (20) γm0 ∞ c 2 e − α ≈ γm0 ∞ c 2 (1 − α ) for a weak gravitational field. It will be fair to mention that, Yarman, in fact, arrived to all of the results presented above along with just the law of energy conservation embodying though the mass and energy equivalence of the STR, thus without having to make usage of the classical Lagrangian formalism. Thus, both the YARK equation (20) and GTR equation (21) produce the same result up to order c-3, and naturally both yield Newton’s equation of motion in the non-relativistic limit. Therefore, YARK provides the successful explanation of cornerstone astrophysical observations (e.g., gravitational lensing, Shapiro delay, precession of the perihelion of Mercurial planets, gravitational red shift), considered usually as the experimental confirmation of GTR. 8. Finally, in the metric (17), the local velocity v of the peripheral object remains the same within the framework of YARK theory for resting observers, located in any spatial point. The same result holds for the speed of light c, which remains a universal constant according to YARK in empty space, as well as under gravitation, when assessed by either the local observer or the distant observer.

11

12 The obtained coincidence in the definition of energy of a massive object in a static (or stationary) gravitation fields in GTR (eq. (21)) and in YARK theory (eq. (20)) to the accuracy of calculation c-3 does not mean, however, that both theories straight yield identical predictions in the case of a weak gravitational field. In particular, it has been found in ref. [33] that the cosmological model of the Universe, constructed in YARK at the time epoch far enough from the Big Bang event (where the approximation of the weak gravitation field becomes applicable) allows, in particular, to eliminate the “dark energy puzzle”, to explain the alternating sign of the acceleration of the Universe expansion, to derive analytically the Hubble constant, and to propose ways for understanding other non-explained in GTR observations in the Universe [33]. Below we apply YARK theory to the calculation of the relative energy shift between resonant lines of a source and absorber in the Mössbauer rotor experiments, highlighting that in the framework of YARK, the validity of eq. (16) can be extended not only to the case of a real gravitation field, but to the case of force field in rotating systems, too, with a suitable redefinition of the static binding energy EB. At this stage one should recall the full compatibility of YARK gravitational theory with quantum mechanics, as substantiated in refs. [30-32, 34]. This circumstance becomes important in the analysis of Mössbauer experiments, where a resonant nucleus should be considered as a quantum particle, localized in a crystal cell (potential hole). The behaviour of such nucleus is described via the corresponding Relativistic Dirac Equation (see, e.g. [36]), and in the case of rotational motion of a resonant absorber, this equation should be transformed from a laboratory frame to a set of Lorentz frames, co-moving with the resonant absorber at different time moments. We omit here the straightforward derivation of the energy of rotating resonant nucleus via the Dirac equation, applied to a set of its co-moving Lorentz frames, which as expected leads to E = E0 γ .

(

)

−1 / 2

where E0 is the energy of resting nuclei of a resonant source, and γ = 1 − u 2 c 2 . Then, the decrease of overall energy by γ for a rotating nucleus signifies a proportional decrease by γ times of the distances between its energy levels. Hence, the relative energy shift in the Mössbauer rotor experiments is equal to (E − E0 ) E0 ≈ − u 2 2c 2 , to the accuracy of calculations c-2, achievable in modern measurements, which then yields k=0.5. However, considering the nuclei in a rotating system in the framework of YARK theory and addressing to eq. (16), we can determine the static binding energy needed to bring quasistatically a massive particle from one spatial point r0 to another spatial point r, both belonging to the rotating system, and, in the limiting case, we can determine the static binding energy needed to bring quasi-statically the given particle from the edge of the rotating system to the center of rotation. For further progress, it is natural to adopt that the rest mass m0 ∞ of a point-like particle located on the rotational axis (r0=0) is not altered by a rotation in comparison with its value outside the rotating frame. Therefore, in the absence of gravity, the geometry of space-time for an observer, located on the rotational axis, remains Minkowskian. Next, considering an arbitrary point r, we have to determine the alteration of the geometry in its vicinity due to rotation with respect to Minkowskian geometry on the rotational axis. For this purpose we calculate the static binding energy EB of the object with the rest mass m0 ∞ , as the work done on the object, in order to bring it quasi-statically from the point r to the 0

rotational axis. Hence, E B = ∫ m0 ∞ω 2 rdr = m0 ∞ω 2 r 2 2 = m0 ∞ u 2 2 (where ω is the angular r

rotational frequency), and EB m0 ∞ c 2 = u 2 2c 2 .

(22) 12

13 Thus, introducing space-time elements dt0, dx0, dy0, dz0 on the rotational axis, and dt, dx, dy, dz in the spatial point r, and using the metric (17), we obtain the following relationships for an observer on the rotational axis in the point r: dt = dt0 (1 − u 2 2c 2 ), dx = dx0 (1 − u 2 2c 2 ) , dy = dy0 (1 − u 2 2c 2 ) , dz = dz0 (1 − u 2 2c 2 ) , (23a-b) which differ, in general, from the corresponding relationships in rotating systems, derived in GTR (see, e.g. [35]). It is that in YARK, the size of a rotating object stretches in all directions; this is yielded by quantum mechanics owing to the rest mass decrease of the rotating object bound to the centrifugal force field. (Whereas, in GTR, sizes are changed along the tangential displacement only, and the alleged change in return is a contraction.) We have further to be recall that in YARK the overall relativistic energy of a rotating object remains the same as its plain rest energy (it bears prior to rotation). In other words, in YARK, bringing an object to a rotation means, pumping out of it, rest mass, and this, as much as the binding its develops to the centrifugal force field, that comes into play. The overall relativistic mass of it though, remains the same. All of these conjoint occurrences will be taken care of, below. Anyway, we have to determine the corresponding temporal and spatial intervals dtL, dxL, dyL, dzL in the point r, as seen by a laboratory observer, located outside the rotating system. For this observer, the particle in the point r has the tangential velocity u = ω × r , and we suppose that at the considered time moment, this velocity is parallel to the axis x. Hence, involving the known relativistic effects of time dilation and contraction of a moving scale along the direction of motion, we obtain: (24a-b) dt L = dt γ , dx L = dx γ , dy L = dy , dz L = dz . -2 Combining eqs. (23), (24), and adopting the accuracy of calculations c (where 1 γ ≈ 1 − u 2 2c 2 ), we obtain with regards to the moving point r: dt L = dt0 1 − u 2 c 2 , dxL = dx0 , dy L = dy0 1 − u 2 2c 2 , dz L = dz0 1 − u 2 2c 2 . (25a-d) In other words, after all, we have stretching of the sizes of the rotating object in y and z directions, as explained right above; the time dilation occurs conjointly; the stretching along the x direction though is compensated by the usual Lorentz contraction. We have to stress that for a laboratory observer, the total mass of an orbiting particle remains exactly equal to its proper rest mass to the adopted accuracy of calculations. Indeed, for an observer on the rotational axis, the rest mass of the particle in the point r is equal to m0 = m0 ∞ (1 − u 2 2c 2 ) due to eqs. (16) and (22). Hence, for a laboratory observer we have: (26) mL = γm0 ∞ (1 − u 2 2c 2 ) ≈ m0 ∞ . It is worth to remind that the relationships (25a-d), obtained earlier in refs. [10, 28] for a rotating system, suggested to conjecture that the coefficient k in eq. (1) should be equal to unity according to eq. (25a), and this prediction motivated the re-analysis of Kündig’s experiment [1], and subsequent implementation of the experiments [12] and [15]. In this respect, the recently obtained experimental results looked puzzling: on the one hand, we get indeed k>0.5. On the other hand, the coefficient k turns out to be less than unity (see eqs. (3), (4)), in a little disturbing divergence with eq. (25a). However, we should recall the actual shift of energy levels of an orbiting resonant nucleus, must rigorously be determined via the appropriate quantum mechanical analysis, framed by YARK, and not kust the classical metric relationship between the time rates of an orbiting and resting particles. This circumstance was not too essential in the framework of standard approach, where, as we have mentioned above, the classical time dilation effect, and the Dirac equation for a resonant nucleus, confined in a rotating crystal cell, yield the identical result k=0.5. However, as we will

(

)

(

)

(

)

13

14 see below, this is not the case what YARK theory induces, and thus, equation (25a), describing the change of time rate for a rotating particle with respect to a laboratory time, does not mean as yet that the same relationship is directly extended to the energy levels of quantum system, though anyway eq. (25a) should be involved into the calculation of these levels along with other relationships (25b-d), (26). Thus, we have to address to the Dirac equation of the nucleus, located in a crystal cell, modifying the variables of this equation according to eqs. (25), (26) in the case, where the crystal (resonant absorber) rotates. To the accuracy of calculations c-2, sufficient for analysis of experimental data, we can simplify further calculations, replacing Dirac Equation by Schrödinger Equation via taking into account the obvious relationship (27) EDE = ESE + O(u 2 c 2 ), where EDE, ESE stand for the energy levels, representing the respective solutions of Dirac Equation and Schrödinger Equation. Next we use the fact that the relationship (27) is valid for both the resonant nuclei of a source (located on the rotational axis) and absorber (located on the rotor rim). Hence, the replacement of variables according to eqs. (25), (26) in the Dirac equation does not affect the terms of the order c-2 and higher in eq. (27), when the calculations are carried out to the accuracy c-2. Therefore, when we determine the difference of the energy level of resonant nuclei ∆E of a source and absorber (which is just the subject of measurement), the term O u 2 c 2 disappears, and ∆EDE ≈ ∆ESE . Thus, it is enough to determine the change of energy levels of resonant nuclei on the rotational axis and on the rotor rim via Schrödinger equation, describing the motion of particle in a small box, where the replacement of variables (25), (26) is applied. For resonant nuclei of a source, fixed on the rotational axis, the time-independent Schrödinger equation has the standard form (see, e.g. [37]): ∂ 2ψ (r ) ∂ 2ψ (r ) ∂ 2ψ (r ) 8π 2 m0 E (28) + + + ψ (r ) = 0 , ∂x 2 ∂y 2 ∂z 2 h2 where ψ (r ) is the wave function, m0 is the rest mass of the nucleus, E is the energy, and h is the Planck constant. For infinite walls of the potential hole (the box with the sizes a, b, d along the respective coordinate axes), eq. (28) defines the energy levels as follows [37]: n y2 h 2 nx2 h 2 nz2 h 2 , (29) + + E= 8m0 a 2 8m0 b 2 8m0 d 2 where nx, ny, and nz are principal quantum numbers associated with the given directions. Further, writing Schrödinger equation for resonant nuclei in a rotating absorber, we have to implement the substitution of variables in eq. (28) according to the relationships (25) and (26). Hence we obtain: 8π 2 m0 E ∂ 2ψ (r ) ∂ 2ψ (r ) ∂ 2ψ (r ) (30) + + + ψ (r ) = 0 . ∂x 2 ∂y 2 1 − u 2 c 2 ∂z 2 1 − u 2 c 2 h2 It will be useful to recall briefly the following facts and occurrences used in eq. (30): - The source at the center and the absorber at the rim, are co-rotating. - The mass of the nuclei of absorber is reduced as much as the static binding energy it develops along with the centrifugal force, with respect to the co-rotating observer at the center, but at the same time, for a laboratory observer it is increased by the Lorentz factor, associated with the rotational motion of absorber. Ultimatley, the overall relativistic mass of the absorber becomes the same (see eq. (26)). - The size of the absorber due to the i) stretching because of the centrifugal binding (eq. (23b), ii) to the conjoint contraction of the same amount along the direction of motion for a laboratory observer, remains untouched (see eq. (25b)).

(

)

(

)

(

)

14

15 - The sizes of the absorber in directions perpendicular to the displacement, are stretched because of the centrifugal binding, ii) and they should remain at the latter sizes through the rotation, as seen by a laboratory observer (see eqs. (25c-d)). One can see that the solution of eq. (30) with respect to the energy levels differs from the solution of eq. (28) by the replacements b' → b 1 − u 2 2c 2 , d ' → d 1 − u 2 2c 2 . Hence,

(

(

)

)

(

(

)

)

n h 1− u c nh n h 1− u c . (31) + + 2 2 8m0 a 8m0b 8m0 d 2 Adopting the equality a=b=d, which is reasonable for crystals used in Mössbauer rotor experiments (characterized by simple Mössbauer spectra), we obtain  2u 2  n2h2  u2 u 2  3n 2 h 2  2u 2  1 + 1 − 2 + 1 − 2  = 1 −  = E (source )1 − 2  . E (absorber ) = 8m0 a 2  c c  8m0 a 2  3c 2   3c  Therefore, the relative energy difference is equal to E (absorber ) − E (source ) 2u 2 (32) =− 2 . E (source ) 3c Since the relative shift of the proper energy levels of resonant nuclei of absorber and source is proportional to the relative shift of their total energy (32), then eq. (32) directly determines the coefficient k in eq. (1), i.e. k=2/3. (33) The result (33) obtained in YARK theory, faultlessly coincides with the recent measurement data (3), (4), just as much with the Kündig data [1] corrected in ref. [9]. E (absorber ) =

2 x

2

2 y

2

2

2

2 z

2

2

2

5. Conclusion First of all, we would like to emphasize that the extra energy shift (i.e. the excess of the ratio ∆E/E over the classical relativistic prediction based on the time dilation effect), originally revealed through the re-analysis of old Mössbauer experiments in a rotating system [9] and further confirmed in the modern experiment of our research group [12, 15], actually represents a rather unexpected effect, which thus strongly required its physical explanation. Up to the moment, three possible explanations had been advanced. The first one (in the chronological order) is based on the hypothesis by Friedman et al [16, 18-20], which implies the negation of the clock hypothesis by Einstein, and the existence of a universal maximal acceleration am. In the framework of this hypothesis Friedman et al developed the extended version of relativistic dymanics, which yields eq. (9) for the relative energy shift between emitted and received radiation in a rotating system, when a source of radiation is located on the rotational axis. However, according to eq. (9), the measured ratio ∆E/E should depend on the radial coordinate R of absorber, which is not supported by the experiments. In addition, the estimated value of a maximal acceleration (11) fitted into the result of Kündig experiment, as re-analyzed in [9], gives the fundamental time interval t fundamental = c a m ≈ 3x10 −11 s , which seems too high, in order to be a fundamental time unit. These observations practically invalidate the hypothesis by Friedman et al, at least at the estimated value (11) for the maximal acceleration. One way out is to assume the value of am to be much larger than (11), thereby the second term in rhs of eq. (10) becomes negligible. Then once again, the entire hypothesis by Friedman et al about the existence of such maximal acceleration falls to be inapplicable toward the explanation of the observed extra energy shift in the Mössbauer rotor experiments. The second possible explanation of physical origin of the extra energy between emission and absorption lines in rotating systems had been advanced by our research group with the involvement of the hypothesis about the dependence of time rate of charged particles in an electric potential at its location [21], described by eq. (14). It is important to notice that the Mössbauer rotor experiments are characterized by the huge centrifugal acceleration of resonant 15

16 absorber (up to 106 m2/s), which is never achieved in any other kind of experiments with condensed matter. Therefore, the resonant nuclei of absorber experience the effect of the local electric field, counteracting the centrifugal force, with the appearance of corresponding local electric potential on the nuclei, and the hypothesis (14) becomes testable. This way, using eq. (14), the authors obtained the coefficient k=0.728 in eq. (1) for the resonant isotope 57Fe, which qualitatively aggress with the experimental data (3), (4). At the same time, as noticed in [21], for a charged nucleus situated in a crystal cell, the effective electric potential is not a well-defined quantity, and the applicability of eq. (14) remains not fully convincing to the Mössbauer experiments in a rotating system. One more attempt to explain the results (2)-(4) has been recently done in ref. [26], where the author (Corda) pointed out the difference in the synchronization of clocks of a spinning source and detector of gamma-quanta, as seen by an observer attached to the source. The author calculated the related component of the energy shift of resonant radiation, which in his opinion should be added to the energy shift component due to the time dilation effect between the source (located on the rotational axis) and absorber (located on the rotor rim). As a result, Corda obtained k=2/3 in eq. (1), which seems in a good agreement with the data (2)-(4). However, summing up two components of the relative energy shift mentioned above, the author of [26] tacitly assumed the equal sensitivity of resonant absorber and non-resonant detector to the energy shifts of γ-radiation at the relative value about 10-13, which is obviously totally inadmissible for typical detectors of Mössbauer spectroscopy, having a relative energy resolution at the level 10-2…10-1. Thus, the conclusion of [26], stating that the result k=2/3 represents a new confirmation of GTR, is utterly erroneous. All the more that, neither Einstein nor Kündig (who conducted the very first Mossbauer measurements on the subject), aimed to state and measure the alleged coefficient. Thence the actually measured value in Mössbauer rotor experiments of the relative energy shift between the lines of the resonant source and the resonant absorber, leading very clearly to the inequality k>0.5 is, in a strong contradiction with GTR. Finally, in section 4 we addressed to YARK gravitational theory, extended to rotational systems. We further stressed that the energy of a resonant nucleus in a crystal cell should be determined via the Dirac equation with taking into account the relativistic effects for an orbiting absorber. This way we explicitly determined the geometry of a rotating disc in the framework of YARK theory and found the relationships (25) and (26), being valid in a laboratory frame, where the measurements are carried out, along with a co-rotating source and absorber. We have shown that the replacement of variables according to eqs. (25), (26) in quantum mechanical description of resonant nuclei yields the coefficient k=2/3, in a perfect agreement with the results (3), (4) obtained in modern Mössbauer rotor experiments. Acknowledgment Special tanks are due to Assoc. Prof. Dr. Ozan Yarman, who provided very many precious hours of great discussions and skilfull efforts, which considerably helped to finalize the present manuscript. References 1. W. Kündig, Phys. Rev. 129, 2371 (1963). 2. H.J. Hay et al., Phys. Rev. Lett. 4, 165 (1960). 3. H.J. Hay, in: Proc. of Second Conference on the Mössbauer effect, ed. by A. Schoen and D.M.T Compton (Wiley, New York, 1963) p. 225. 4. T.E. Granshaw and H J. Hay, in: Proc. Int. School of Physics, “Enrico Fermi” (Academic Press, New York, 1963) p. 220. 5. D.C. Champeney and P.B. Moon, Proc. Phys. Soc. 77, 350 (1961). 6. D.C. Champeney, G.R. Isaak and A.M. Khan, Proc. Phys. Soc. 85, 83 (1965). 16

17 7. R.W. McGowan et al., Phys. Rev. Lett. 70, 251 (1993). 8. I. Bailey et al., Nature 268, 301 (1977). 9. A.L. Kholmetskii, T. Yarman and O.V. Missevitch, Phys. Scr. 77, 035302 (2008). 10. T. Yarman, V.B. Rozanov and M. Arik, in: Proc. Int. Conf. Physical Interpretation of Relativity Theory (Moscow Bauman State University, Moscow, 2007), p. 187-197. 11. A. Einstein, The Meaning of Relativity (Princeton University Press, Princeton, 1953). 12. A.L. Kholmetskii, T. Yarman, O.V. Missevitch and B.I. Rogozev, Phys. Scr. 79, 065007 (2009). 13. A.L. Kholmetskii, T. Yarman and O.V. Missevitch, Int. J. Phys. Sci. 6, 84 (2011). 14. F.G. Vagizov, E.K. Sadykov and O.K. Kocharovskaya, in: XI Int. Conf. Mössbauer Spectroscopy and its Applications. Book of Abstracts (Ekaterinburg, 2009) p 184 (in Russian). [15] A.L. Kholmetskii, T. Yarman, M. Arik and O.V. Missevitch, AIP Conf. Proc. 1648, 510011 (2015); “Mössbauer experiments in a rotating system and their physical interpretations”, Can. J. Phys., in press. 16. Y. Friedman and I. Nowik, Phys. Scr. 85, 065702 (2012). 17. E. Zanchini, Phys. Scr. 86, 015004 (2012). 18. Y. Friedman and M. Semon, Phys. Rev. E 72, 026603 (2005). 19. Y. Friedman, Ann. Phys. 523, 408 (2011). 20. Y. Friedman and Yu. Gofman, Phys. Scr. 82, 015004 (2010). 21. A.L. Kholmetskii, T. Yarman and O.V. Missevitch, Eur. Phys. J. Plus 128, 42 (2013). 22. C. Møller, The Theory of Relativity (Clarendon Press, Oxford, 1973). 23. J. Essen, Nature 202, 787 (1964). 24. A.L. Kholmetskii, O.V. Missevitch and T. Yarman, Phys. Scr. 89, 067003 (2014). 25. T.E. Cranshaw, J.P. Schiffer and A.B. Whitehead, Phys. Rev. Lett. 4, 163 (1960). 26. Ch. Corda, Ann. Phys. 355, 360 (2015). 27. Chemical Applications of Mössbauer spectroscopy/ ed. V.I. Goldanskii and R.H. Herber (Academic Press, New York, 1968). 28. T. Yarman, Foun. Phys. Lett. 19, 675 (2006). 29. T. Yarman, Fond. de Broglie 29, 459 (2004). 30. T. Yarman, Int. J. Phys. Sci. 5, 2679 (2010). 31. T. Yarman, Int. J. Phys. Sci. 6, 2117 (2011). 32. T. Yarman, The Quantum Mechanical Framework Behind the End Results of the General Theory of Relativity: Matter Is Built on a Matter Architecture (Nova Publishers, New York, 2010). 33. T. Yarman and A.L. Kholmetskii, Eur. Phys. J. Plus 128, 8 (2013). 34. T. Yarman, A.L. Kholmetskii, M. Arik and O. Yarman, Phys. Essays 27, 558 (2014). 35. L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, 2nd ed. (Pergamon Press, New York, 1962). 36. J.C. Davis Jr., Advanced Physical Chemistry (Chapter 10) (The Ronald Press Company, New York, 1965). 37. W.J. Moore, Physical Chemistry (Chapter 12) (Prentice Hall Publication, New York, 1955).

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18

Rotor chamber

ν

Source

Output window

Rotor Detector

Absorber Collimator

Fig. 1. General scheme of Mössbauer experiments in rotating systems. A source of resonant radiation is located on the rotational axis; an absorber is located on the rotor rim, while a detector of γ-quanta is placed outside the rotor system, and it counts γ-quanta at the time moments (just shown in the figure), when source, absorber and detector are aligned in a straight line.

18

19

B

γ

rB

ω

rAB

ϑB

rA

n uB

A

ϑA uA

Fig. 2. Diagram of calculation of the Doppler effect in a rotating system between a point-like emitter (located in the point A at the emittance moment) and point-like receiver (located in the point B at the receiving moment).

19