Einstein’s “Clock Hypothesis” and Mössbauer Experiments in a Rotating System

2.1 The Generalised Relativity byFriedman et al.

Insight into the theory by Friedman and his collaborators can be found in [12], [15]. The unorthodox generalisation of special relativity by Friedman et al. is essentially based on the negation of the CH and the postulation of a maximal acceleration a_m. It is important to note that this upper bound is not derived a priori but is merely contingent upon the a posteriori inputting of an arbitrary magnitude by hand based on empirical measurements. In fact, the true measured value is the ratio a/a_m, where a is the acceleration achieved in a given experiment. It is known that, under laboratory conditions, extreme values of a for macroscopic objects can be reached in rotor experiments where the centrifugal acceleration a=ω2r can delineate large numbers (10⁶–10⁷) g for modern rotor systems, with ω being the angular rotation frequency, r the rotor radius, and g the acceleration of free fall on the surface of the earth. Therefore, Mössbauer experiments in rotating systems, which combine the feasibility of obtaining large ratios of ω2r/am with an exceptionally high sensitivity of the Mössbauer effect to the relative energy shifts of resonant lines, represent a promising tool in the search for effects violating the CH under laboratory conditions.

Hence, if one is to give credence to the approach by Friedman et al., it is possible to “reverse-engineer” the value of the actual energy shift between emission and absorption lines in a rotating system as expressed by the inequality (3) by simply relying on a maximal acceleration a_m with a value much lower than its original estimation in (1).

In order to evaluate the appropriate value of a_m, which could seemingly explain the inequality (3), we can refer to the transformation between two accelerated frames. In the generalised relativity by Friedman et al., it has the form of a usual Lorentz transformation with the replacement of the ratio v/c by a/a_m [12]. One can show that, in this case, the relationship between the energy of an emitted radiation E₀ (for a source fixed on the rotational axis) and the energy of the absorbed radiation E (for an absorber located at the rotor edge) is given by the relation [12]

Therefore, the relative energy shift between the resonant lines of the source and the absorber takes the form

where u = ωr.

Comparing (2) and (6), one can straightforwardly determine the coefficient k as the function of a maximal acceleration a_m:

Trying to estimate the value of maximal acceleration a_m on the basis of (7), Friedman et al. originally used the result of the experiment by Kündig [4], which we have re-analysed and corrected in [3], claiming that it is much more reliable than the results of modern experiments [7], [8], [9], yielding (4) and (5). As we have shown in [13], the argument by Friedman et al. against the results (4) and (5) rests on their fallacious assumption about the influence of the linear Doppler effect on the shape of the measured resonant line in a rotating system. However, one can easily demonstrate that, in the case where both the source and the absorber are fixed on a rotor, the linear Doppler effect totally vanishes [13]. At the same time, we do not look more closely at this problem at the present stage, since the difference between the rectified datum of the experiment by Kündig [4] upon our re-analysis in [3] (k = 0.596 ± 0.05) and the results (4) and (5) does not matter so much in the evaluation of the postulated maximal acceleration a_m. In particular, substituting in (7) the radial coordinate of the resonant absorber in the experiment by Kündig [4] (r = 9.3 cm), we obtain at k = 0.596

One can see that the value (8) is much smaller than the initial estimation of a plausible maximal acceleration as given in (1).

In order to verify this result, recently Friedman et al. carried out two Mössbauer experiments in a rotating system, where they applied resonant synchrotron radiation focused as a narrow beam on a spinning disc [10], [11].

Such experiments, where a source of resonant radiation rests in the laboratory, have the principal disadvantage in comparison with the traditional scheme (where a point-like source of resonant radiation and the resonant absorber are both fixed to a spinning disc) because of the emergence of a linear Doppler effect between the immovable synchrotron source and orbiting absorber, which leads to substantial broadening of the measured resonant line. This, in turn, results in a proportional decrease of the measurement sensitivity with respect to the energy shifts of such a line.

Besides, it is obvious that, for a synchrotron source, the rotor vibrations as a whole distort the measurement results, whereas in the traditional measurement scheme, where both a compact source and a resonant absorber are fixed on a rotor, only relative vibrations between them should be taken into account – which are anyway much smaller than the absolute rotor vibrations.

As a result, the width of the resonant line of a rotating absorber, when measured with a resonant synchrotron beam, exceeds the proper width of the line tens of times at the tangential velocities of around 100 m/s [10], [11]. In contrast, in traditional experiments with a compact resonant source fixed at the centre of the rotor along with a co-rotating resonant absorber at the rotor’s rim, the measured width of the resonant line remains comparable to its proper width even at near-sonic tangential velocities of 300 m/s because of the absence of the linear Doppler effect.

These facts explain the failure of the authors of the experiments in [10], [11] in measuring the relative energy shift between emitted and absorbed resonant lines as a function of the tangential velocity of the absorber, so much so that they could not even estimate the coefficient k in (2). Under these conditions, Friedman et al. focused their efforts on another method for the determination of possible influences of acceleration on the relative energy shift between emitted and received resonant radiation: namely, they compared the Mössbauer spectra of the resonant absorber at different angular positions [designated in [10], [11] as positions (a) and (b)], where the projection of centrifugal acceleration on the axis of the synchrotron beam has the same values but opposite signs. Hence, according to (6), the relative energy shift of the resonant lines between these positions – as tagged in [10], [11] under the designation of the “alignment shift” (AS) – is determined by the equation

An advantage of this method is the independence of AS on the finite distance between the axis of the synchrotron beam and the rotational axis [10], [11], so that it might seem, at first glance, that AS should be fully defined by the supposed maximal acceleration according to (9).

As is usually done in Mössbauer spectroscopy, the relative energy shift between emission and absorption lines (in our case, the relative alignment shift) can be expressed via the conditional relative velocity between the source and the absorber, corresponding to the same linear Doppler shift, i.e.

with

The convenience of definition (10) is the fact that u_AS can be directly obtained from the measured Mössbauer spectra in positions (a) and (b) plotted in the velocity scale.

In the last experiment by Friedman et al. [11], which differs from their earlier experiment [10] by some methodological improvements, the authors actually observed a quadratic dependence of u_AS on the angular frequency ω, as predicted by (9), up to 200 rev/s. In particular, at the rotor radius of R = 5.0 cm, which was used in the experiments in [10], [11], and the rotational frequency of 200 rev/s, the estimated value of AS came out to be u_AS = 0.41 mm/s, with the relative measurement uncertainty being less than 10 %. Concurrently, Friedman et al. noticed that the contribution of any effects related to the non-random character of rotor vibrations had a much smaller value. Thus, Friedman and his collaborators concluded that the observed AS betokens the violation of the CH as a result of the existence of a maximal acceleration in nature, which can be estimated from (9) as

Hence, at ω = 2π × 200 s⁻¹, R = 0.05 m, and via the measured value u_AS = 0.41 mm/s, we obtain [11]

The value in (12) is two orders of magnitude smaller than the maximal acceleration (8) derived earlier by Friedman et al., which we have rectified based on the experiment by Kündig [3].

One should mention another estimation of this alleged maximal acceleration via the Mössbauer effect, which is shifted all the way up to at least

as obtained in [28] in the temperature-shift measurements with Mössbauer spectroscopy of ⁶⁷Zn, which is characterised by the extremely narrow width of its resonant lines.

Nevertheless, in the opinion of Friedman et al., the three different results (8), (12), and (13) still do not create any doubts as to the correctness of their methodological approach, but rather indicate that a_m is not a universal quantity, as was also contended in [2], and, according to these authors, can be much smaller than that originally postulated in (1).

At the same time, there appears strictly no way to attribute any physical meaning to such conflicting values for a supposed accelerational upper bound in experiments using the same tool – Mössbauer spectroscopy – under similar conditions and rotor parameters.

What is more, in our recent paper [29] we have shown that the latest result by Friedman et al. (12) is far from convincing, and can be explained by the fluctuation δr of the position of the rotational axis, which is always present for any real rotor system. In particular, as is estimated in [29], the value of u_AS at the level 0.41 mm/s (which is used in the derivation of (12)) is observed at δr ≃ 0.3 μm, which cannot be well controlled in the experiments [10], [11].

Next, we emphasise that the estimations of possible values of maximal acceleration by Friedman et al. [(8) and (12)] are anyway much smaller than the limited values of maximal acceleration evaluated via the experiments on accelerators of charged particles. First of all, we refer to the experiment [26] aimed at measuring the lifetime for negative and positive muons. In this experiment, the muons with the Lorentz factor 29.3 rotated on a circular orbit with the diameter of 14 m in the presence of a magnetic field of suitable magnitude, which was orthogonal to the rotating plane. In this case, the centripetal acceleration of muons due to the magnetic Lorentz force was about a = 1.3 × 10¹⁶ m/s². Further, taking into account that the relative measurement uncertainly of the proper lifetime of muons was about 2 × 10⁻⁴, one can estimate a possible maximal acceleration (which could affect the measured lifetime of muons beyond the indicated uncertainty):

We add that the analysis of the muon experiments on accelerators performed in [27] gives even stronger limitation with respect to the influence of acceleration on the particle’s lifetime: with a centripetal acceleration of muon near 10¹⁸ m/s², the estimated deviation from the behaviour of an ideal clock does not exceed 10⁻²⁵.

This and other results presented above create serious doubts about the validity of the generalised version of relativistic kinematics by Friedman et al., with the value of maximal acceleration of the order 10¹⁷–10¹⁹ m/s². On the contrary, the accelerator experiments with positive and negative muons, orbiting in the presence of a suitable magnetic field, rather indicate the independence of the clock rate on its acceleration, as in effect originally assumed by Einstein.

At the same time, as we have already mentioned and will detail below, we anticipate that for bound systems of particles (where composing of such systems requires delivering a finite work to its constituents), there appears one more mechanism of violation of CH, which exhibits itself in the Mössbauer rotor experiments as the extra energy shift between emission and absorption resonant lines on a rotating disc. Here, it is important to stress that the accelerated charged particles, orbiting in the presence of the magnetic field, do not represent bound particles in the meaning defined above, because the magnetic Lorentz force does not constitute any work. Therefore, the indicated way of possible violation of CH for mechanically bound systems (like resonant source and resonant absorber fixed on a spinning disc) cannot be realised in the experiments where charged particles are accelerated in a magnetic field.

A consistent way to describe this mechanism of violation of CH is given within the YARK gravitation theory, given in the following.

2.2 Mössbauer Rotor Experiments from the Point of View of YARK Theory

An introductory framework of YARK theory of gravity has been drawn, e.g. in [16], [17], [18], [19], [20], [21], [22]. Here, we present some principal points appertaining to our approach.

The main motivating factor for the development of YARK theory was to achieve a symbiosis between gravitation and quantum mechanics, as well as the elimination of any difficulties with respect to the definition of gravitational energy. One can realise that the necessary condition for meeting the aforementioned criteria is the denial of a purely metric approach as had been applied in GTR and in extended theories of gravity.

Needless to say, it is known that purely dynamic theories of gravity (where the notion of “force” is defined independently of the space-time metric; like, e.g. in post-Newtonian theories) are incompatible with available experimental facts.

These difficulties are overcome in YARK theory, which can be classified neither as a purely metric theory nor as a purely dynamic theory: it rather successfully combines the properties of both of them. In particular, in YARK, the force exerted on a test particle in the presence of gravity represents a non-vanishing entry in any frame of observation (just like in dynamic theories); while, at the same time, the origin of this force is explained via the variation of the metric of four-space, which makes YARK similar to metric theories in its many applications.

The combination of dynamic and metric properties is achieved via the principal postulate of YARK theory, which defines the overall energy of an object moving in the presence of gravity as [16], [17], [18]

Here, m₀ is the rest mass of the object in the absence of gravity, γ is its Lorentz factor, and E_B is the static binding energy of the object at the given location (i.e. this is the work delivered to the object – assumed initially at rest at the given location – in order to carry it quasi-statically to an infinity away).

In fact, (14) states that the rest mass of the object is not of a constant magnitude but is rather altered by a gravitational environment by the value E_B/c².

Further, because of the close quantum mechanical relationship between the quantities “mass”, “energy”, “frequency”, “time”, and “size”, the variation of the rest mass of a test particle by the static binding energy affects the time rate for the particle and induces a corresponding transformation of spatial intervals in the presence of gravity [16], [17], [18]. Thus, the gravitational field alters the metric of space-time, so that the expression for the space-time interval in spherical coordinates and in the radially symmetric case reads as

where the factor α=GM/rc2, with G being the gravitational constant and r the distance from the centre of the immobile host body M to the point of observation. Here, r is the radial coordinate, θ is the polar coordinate, and φ is the azimuthal coordinate, and all of them are defined with respect to a distant observer located outside of gravity.

Further, the action of a particle in YARK theory, defined in a usual way as S=−m0c∫ds, yields the following Lagrangian in the metric (15) [19], [20], [21], [22]:

where γ0=(1−v02/c2)−1/2, with v₀ = dl/dτ being the velocity of the particle measured by a local observer, dl the path element, and τ the proper time. Hence, the force F acting on the particle, its momentum p, and its energy E are defined by the corresponding relationships:

Comparing now (19) with the known expression for the energy of the test particle in GTR [30]

we see that the terms describing the effect of gravity in (19) and (20) coincide with each other to the accuracy c⁻³; i.e. 1−2α≈1−α≈e−α. Therefore, GTR and YARK do converge in the limit of a weak gravitational field, and both provide a successful explanation of the classical astrophysical observations (e.g. gravitational redshift, precession of the perihelion of Mercury [31], [32], gravitational lensing [33], and Shapiro delay [34]).

In addition, YARK theory also achieved considerable successes in explaining modern observations where the weak relativistic limit is abandoned (e.g. derivation of the alternating sign for the accelerated expansion of the Universe without the need to involve a notion of “dark energy”, presentation of the Hubble constant in an analytical form, and elimination of the “information paradox” for black holes of the YARK type [18], [19]).

In the available publications about YARK theory, we still did not consider the topical problem of rotation of galaxies and its connection to the existence of dark matter, where significant modifications of GTR are possible (see, e.g. [35], [36]). This will be done in a separate contribution, but now we address the most recent experimental results in space-time physics, which, among other things, can also be considered as crucial tests of YARK theory. In this regard, it is important to emphasise that YARK theory remains the only alternative to GTR, which provides an adequate account of the GW150914 and GW151226 signals announced by LIGO beyond the hypothesis about gravitational waves [22]. In addition, the recent indications with respect to the practically null bending of high-energy γ-quanta under Earth’s gravity [37] find a successful explanation in the framework of YARK theory, too [21]. Last, but not least, the significant achievement of YARK theory remains its quantitative agreement with the experimental results (3) and (4) obtained in modern Mössbauer rotor experiments.

A possible divergence between the predictions of GTR and YARK theory at a measurable level in this kind of experiments stems from the fact that, for an accelerated motion of particles in the absence of gravity, one cannot separate (19) and (20) into particular contributions describing the effects of inertial and non-inertial motion, so that for such a motion, the specific terms distinguishing GTR and YARK from each other might emerge already in the order (u/c), where u is a typical velocity characterising the motion of a system in question. For example, under rotational motion, the factor α entering into (17)–(19) is defined as α=acentrifugalr/c2=u2/c2. The substitution of this value into (19) and (20) reveals their disparity already in the order (u/c)², which corresponds to the accuracy in the presentation of (2). Hence, the coefficient k calculated in this latter equation can be different in GTR and in YARK even though both theories agree well with respect to numerous astrophysical observations dealing with commonly observed gravitational effects.

We emphasise the differing predictions made by GTR and YARK theory with respect to the coefficient k in (2) as indicative of how the respective geometries framed by both theories in a rotating disc should also be different.

The spatial geometry of the rotating disc in GTR is well known (see, e.g. [30]), and it yields k = 1/2 in (2), which, for a laboratory observer, is explained by the second-order Doppler effect and leaves no room for the violation of the clock hypothesis by Einstein.

The spatial geometry of the rotating disc in YARK theory is defined to the accuracy c⁻² by the equation [13]

at the time moment when the tangential velocity u in the considered point of the disc r is collinear to the x-axis.

The physical meaning of (21a–c) has been disclosed in [13]. Because of centrifugal acceleration, the spatial intervals are stretched homogeneously in the x, y, and z directions by the conformal factor (1−u2/2c2), in YARK theory. However, the Lorentz contraction of scale along the tangential velocity (directed along the x-axis at the considered time moment) exactly counterbalances the stretching effect along the x-direction, so that the x-coordinate remains unchanged within the adopted accuracy of calculation, c⁻². We notice that this eliminates the Ehrenfest paradox. Similarly, the increase of the mass of the rotating particle by the Lorentz factor is counterbalanced by the decrease of this mass by (1−u2/2c2) times because of the acceleration effect, so that m = m₀ up to the adopted accuracy of the calculations, c⁻² [13].

At this stage, it is important to emphasise the full compatibility of YARK theory with quantum mechanics the way we had substantiated it in [16], [17], [18], [38]. As pointed out in [13], this circumstance is profoundly crucial in the analysis of the Mössbauer effect where a resonant nucleus is to be considered analogous to a quantum particle localised in a crystal cell. On that account, the energy levels of such a nucleus should be defined via the Dirac equation in the spatial metric (21a–c), where the crystal cell can be modelled as a three-dimensional potential hole with infinite walls. As had been pointed out in [13], when we calculate the difference of the energy levels between a co-rotating resonant nuclear source and a resonant absorber characterised by different radial coordinates, it is sufficient to replace the Dirac equation with the Schrödinger equation, alongside the replacement of the variables according to (21a–c).

Thus, presenting a resonant nucleus in a crystal cell as a particle in a box with sizes a, b, and d along the respective coordinate axes, we obtain the solution of the Schrödinger equation in the form

where n_x, n_y, and n_z are the corresponding principal quantum numbers [13].

Therefore, for a resonant absorber on the rotor rim, and for a resonant source on a rotating axis (where u = 0), we obtain

So, taking for simplicity a = b = d (which is actually the case for the majority of resonant absorbers used in the known Mössbauer rotor experiments), we derive the relative energy shift between emission and absorption resonant lines as

Hence, (25) yields

Thus, the prediction of YARK theory (26) perfectly agrees with the recent measurement data (4) and (5), and is also in conformity with the Kündig [4] and Champeney et al. [5] outcome (3) the way we corrected them in [3].

One may add that the coefficient k in (26) can be presented as the sum k=12+16, where the term 1/2 stands for the usual transverse Doppler effect and the term 1/6 emerges as a result of the effect of acceleration as seen by a laboratory observer. Thus, (26) exhibits the violation of the CH as the specific effect for mechanically bound systems (in our case, resonant source and resonant absorber fixed on a rotating disc) predicted and explained by YARK theory, beyond the hypothesis about a possible existence of a maximal acceleration in nature.