Variation of rest mass scale in a gravitational field

TL;DR

The paper proposes that the rest mass scale varies with gravitational potential due to changes in the Compton wavelength, linking it to gravitational redshift and suggesting an experiment to test this variation.

Contribution

It introduces the idea that rest mass scale varies inversely with time scale in curved spacetime, connecting quantum scales with gravitational effects.

Findings

Rest mass scale varies with gravitational potential.

Proposes an experiment for MICROSCOPE satellite to test mass scale variation.

Links quantum scale variation to gravitational redshift effects.

Abstract

I argue that an angular momentum scale is necessary to explain energy-momentum propagation along a single null geodesic, the scale being known as Planck's constant $(h)$. If $h$ and $c$ (the velocity of light in vacuum), are considered to be given fundamental (constant) scales in local measurements, then the rest mass scale varies exactly inversely as the time scale varies in two different locally flat regions of a curved spacetime. As the time scale variation gives rise to gravitational redshift, the rest mass scale variation leads to a change in the Compton length scale associated with an elementary particle. I suggest an experiment for the MICROSCOPE satellite and report the expected outcome.

Equations

$$E_2=E_1\left(1+\frac{\Phi }{c^2}\right)$$

$${\nu }_2={\nu }_1\left(1+\frac{\Phi }{c^2}\right).$$

$$-E^2/c^2+|\vec{p}{|}^2=-m_0^2{c}^2$$

$$E^2/c^2=|\vec{p}{|}^2.$$

$$\frac{(x-ct)}{\lambda }=\frac{(x^{\prime }-c{t}^{\prime })}{{\lambda }^{\prime }}$$

$$E=\frac{hc}{\lambda }.$$

$$m_2=m_1\left(1+\frac{\Phi }{c^2}\right).$$

$$\Delta \lambda :={\lambda }_{\theta }-{\lambda }_0={\lambda }_e(1-\cos \theta ).$$

$$\Delta \lambda ={\lambda }_1\left(1-\cos {\theta }_1\right)={\lambda }_2\left(1-\cos {\theta }_2\right)$$

$$\frac{m_2}{m_1}=\frac{1-\cos {\theta }_2}{1-\cos {\theta }_1}=\left(1+\frac{\Phi }{c^2}\right)$$

$$⟹$$

$$\frac{\Phi }{c^2}\simeq r_0\left[\frac{1}{r_1+H}-\frac{1}{r_1}\right]=\left[\frac{1}{x+\Delta x}-\frac{1}{x}\right]\simeq -\frac{\Delta x}{x^2},$$

$$\frac{(\cos {\theta }_1-\cos {\theta }_2)}{(1-\cos {\theta }_1)}\simeq \delta \cot \frac{{\theta }_1}{2}.$$

$$\delta \simeq -\frac{\Delta x}{x^2}\tan \frac{{\theta }_1}{2}.$$

$${\delta }_{MICRO}\simeq -1.72\times 10^{-5}\tan \frac{{\theta }_E}{2}.$$

Full text

Introduction: Three most important physical dimensions that one perceives of in daily life are ‘mass’ $(M)$, ‘length’$(L)$ and ‘time’$(T)$, which are measured with predefined reference scales (or units) such as kilogram, metre and second, respectively [1, 2]. In special relativity, Einstein showed that the notion of length scale and time (or frequency) scale are not absolute concepts and vary in two frames with constant relative velocity (inertial) [10]. What is fundamental in special relativity is the velocity scale denoted by $c$, which is the ‘velocity’ of light in vacuum [10]. Another scale that is absolute, but not fundamental, in special relativity is the rest mass scale $m_{0}$ associated with a particle [3]. It is ‘absolute’ in the sense that it is invariant under Lorentz transformations i.e. $p^{I}p_{I}=-m_{0}^{2}c^{2}$ where $p^{I}$ is the four-momentum [18, 19]. It is not ‘fundamental’ because it is arbitrary. $m_{0}$ represents the rest mass scale of an elementary particle, which is rather understood as the Lorentz invariant mass scale associated with a field (e.g. see [38]).

Now, to address the effect of gravitation on light propagation, in ref. [12], just by assuming equivalence principle [13] and considering mass-energy conservation principle, Einstein concluded that the energy scale differs in two frames that differ by a gravitational potential due to a homogeneous field or equivalently in constant relative acceleration (non-inertial). However, while writing down the formula for gravitational redshift, Einstein simply associated ‘frequency’ with ‘energy’ and therefore, implicitly assumed the existence of a fundamental angular momentum scale that remains unaffected by gravity. The experimental verification of gravitational redshift by Pound and Rebka [21, 22] shows that this fundamental scale is none other than the Planck’s constant $(h)$. In the language of curved spacetime geometry, Einstein’s result is a manifestation of energy scale variation between two locally flat spacetime regions when the local observations are compared in a suitable coordinate system such that the Newtonian potential, with some approximation, implies the curvature [23]. However, the role of $h$ remains indispensable. For example, one can see section (6.3) of ref. [23], where the author ‘makes sense’ of the calculation by relating ‘frequency’ with ‘energy’ and requiring $h$ in the process (and hence invoking ‘quantum’ mechanical concept of ‘photon’ in the context of ‘classical’ gravity). I argue that a fundamental scale of angular momentum dimension, which is none other than $h$, is necessary to make sense of energy-momentum propagation along a single null-geodesic. It has, a priori, nothing to do with quantum mechanics and this is in sharp contrast with the presently accepted general understanding.

Now, considering $h$ and $c$ as fundamental scales, ‘kilogram’ is defined in terms of $\Delta\nu_{cs}$, which is the frequency scale associated with the transition between the hyperfine states of the unperturbed ground level of a Caesium (133) atom, in its rest frame [1]. I argue that if one considers $h$ and $c$ to be given, rather than determined, fundamental scales in local measurements, then the rest mass scale associated with an elementary particle varies exactly inversely as the time scale varies in two different locally flat regions of a curved spacetime. This is a hitherto unexplored, or rather ignored, facet of Einstein’s equivalence principle [27]. Nonetheless, the effect is quite clear from the definition of ‘kilogram’ in terms of $\Delta\nu_{cs}$. Since frequency scale variation suggests that $\Delta\nu_{cs}$ have different meanings at different locally flat regions of a curved spacetime [9], so does the notion of ‘kilogram’. Irrespective of this fact, I suggest a very doable experiment to directly test the rest mass scale variation, along with a predicted outcome for the MICROSCOPE satellite mission [25].

A review of Einstein’s thought process: With an aim to investigate the role of $h$ in the explanation of energy propagation along a null geodesic and to shed light on the mass scale variation due to gravity, I briefly mention the relevant results which were reported by Einstein in ref. [12]. Einstein considered the following thought process:

Two observers $O_{1}$ and $O_{2}$, relatively at rest with respect to each other, are equipped with a set of identical measuring instruments (implies identical scales for measurements). 2. 2.

Now, these observers, along with their corresponding set of instruments, sit at different potentials of a homogeneous gravitational field or equivalently have a relative non-zero acceleration (equivalence principle). 3. 3.

A certain amount of energy $E_{1}$, as measured by $O_{1}$ locally in its rest frame, is emitted, in the form of radiation, towards $O_{2}$. 4. 4.

$O_{2}$ receives this energy and performs a local measurement in its rest frame to yield the result $E_{2}$.

Einstein showed that, if energy conservation principle holds, then, approximately up to the first order in $\Phi/c^{2}$, $E_{1}$ and $E_{2}$ are related by the following equation

[TABLE]

where $\Phi$ is the potential difference between $O_{1}$ and $O_{2}$. Then, Einstein wrote the formula for gravitational redshift. To do that, Einstein simply assigned ‘frequency’ corresponding to radiation energy (and not ‘intensity’): “If the radiation emitted …… had the frequency….” (see the very beginning of section 3 of [12]). He arrived at the formula

[TABLE]

Einstein’s argument to interpret the frequency shift was based on the fact that the time scale (unit) for $O_{1}$ and that for $O_{2}$ do not remain identical due to the difference in gravitational potential.

Einstein’s untold assumption: It is interesting to note that Einstein did not use or mention, explicitly, the involvement of any fundamental scale of angular momentum dimension, that remains unaffected by gravity [24]. However, without this assumption it is impossible to pass on from eq.(1) to eq.(2). The experimental verification of Pound and Rebka [21, 22], indeed relied on methods, namely Mossbauer effect [33], that involve $h$. Therefore, Einstein’s untold assumption was that, besides $c$, $h$ is another fundamental scale that remains identical for $O_{1}$ and $O_{2}$ while performing local measurements. In what follows, I explain the necessity of this assumption.

An angular momentum scale and null geodesic: In special relativity, for a massive point[16] particle following timelike geodesic with four velocity $u^{\mu}$, one has $u^{\mu}u_{\mu}=-1$ and for four momentum $p^{\mu}=m_{0}u^{\mu}$ one has $p^{\mu}p_{\mu}=-m_{0}^{2}c^{2}$ which implies

[TABLE]

where $p^{0}=E/c$, $m_{0}$ is the rest mass scale uniquely characterizing the particle under consideration and $\vec{p}$ is the spatial momentum with respect to the observer. All these quantities are dimensionally well defined and the physical interpretation of the energy carried by the massive particle along the time-like geodesic is well posed in terms of rest mass energy (Lorentz invariant) and kinetic energy (observer dependent). The role of $m_{0}$ becomes justified while one checks that in the non-relativistic limit it plays the usual role in the kinetic energy term $\frac{1}{2}m_{0}v^{2}$ (see p.27 of ref. [18]).

To seek a similar understanding about the null geodesic is perfectly legitimate. However, for propagation along a null geodesic, there is no associated rest mass scale i.e. $p^{\mu}p_{\mu}=0$, which implies

[TABLE]

There is no expression for $E$ or $|\vec{p}|$ that can be cooked up from the Lorentz invariant energy-momentum tensor of electrodynamics. This is because, eq.(4) is a statement about propagation along a single ray at the velocity of light (with respect to any observer), which is a single null geodesic in the spacetime description. Whereas calculation of energy-momentum tensor involves an extended region of spacetime i.e. a bundle of rays or a congruence of geodesics are required and hence, the components of the energy-momentum tensor are flux densities [18, 39]. Further, added to all the above explanations, there is no non-relativistic limit of the null geodesic scenario from where one can draw any conclusion.

The only meaningful quantity that one can consider along a single null geodesic is the phase of the propagating light which is a Lorentz invariant quantity in special relativity [19]:

[TABLE]

where $(x,t)$ and $(x^{\prime},t^{\prime})$ are related by Lorentz transformations; $\lambda$ and $\lambda^{\prime}$ are the length scales associated with light propagation in the corresponding Lorentz frames. So, there is only one option to construct an expression for energy-momentum propagation along, or associated with, a null geodesic and that is to look into the scales involved in the expression of the phase. Doing so, one finds $c$ (Lorentz invariant) and $\lambda$ (observer dependent) are the only two such independent quantities. Therefore, it is necessary, on dimensional grounds, to introduce a fundamental angular momentum scale ($h$) and write

[TABLE]

Hence, one can conclude that association of $h$ is necessary to make sense of energy-momentum propagation along a null geodesic. Therefore, if one considers $h$ to be the manifestation of quantum physics (which is in accord with the present general understanding), then one encounters a classical-quantum dilemma in special relativity, or rather in the geometric description of spacetime itself.

Realizing mass scale variation with a modified thought process: To explain the effect of gravity on the rest mass scale of an elementary particle, I shall slightly modify and refine Einstein’s thought process. Since $h$ and $c$ are fundamental scales for an elementary particle, following de Broglie, I consider that its rest frame is associated with an energy scale $E_{0}$ and a frequency scale $(\nu_{0})$ by the following relation: $E_{0}=m_{0}c^{2}=h\nu_{0}$ [34, 35]. Then, the description goes as follows:

An observer $O_{1}$ considers $h$ and $c$ to be given scales and define all other scales in terms of those two. $O_{1}$ makes measurements with these scales. 2. 2.

$O_{1}$ studies the decay of a massive elementary particle in its rest frame, e.g. the decay of a neutral pion to two photons: $\pi^{0}\to\gamma+\gamma$, and measures the amount of released energy in the form of photons with the derived scales. 3. 3.

Let, $O_{1}$ measures $\alpha_{1}$ units of energy with the derived scale $E_{1}$, i.e. $\alpha_{1}E_{1}$ amount of energy is released in the form of radiation as measured by $O_{1}$. For $O_{1}$, it implies, $\alpha_{1}$ amount of mass in the scale $m_{1}=E_{1}/c^{2}$ is the rest mass of the pion, which follows from four momentum conservation. Another observer $O_{2}$ notes down the whole procedure while relatively at rest with $O_{1}$. 4. 4.

Then $O_{2}$ goes to a frame which differs by a gravitational potential from that of $O_{1}$. $O_{1}$ repeats the same experiment and sends the resulting photons from the pion decay to $O_{2}$. 5. 5.

$O_{2}$ receives the photons and measured the energy with the predefined scales and found that he did not receive $\alpha_{1}E_{1}$, but an amount $\alpha_{2}E_{1}$. 6. 6.

$O_{2}$ finds that the energy measurement yields $\alpha_{1}$ if the scale is redefined to be $E_{2}$ which is related to $E_{1}$ by the eq.(1). 7. 7.

$O_{2}$ concludes that the rest mass of the pion is $\alpha_{2}m_{1}$ or $\alpha_{1}m_{2}$ where

[TABLE]

This implies, if $O_{2}$ studies the decay of a pion by bringing it to relatively at rest with respect to him/her, then the rest mass will come out to be $\alpha_{1}$ if measured with the scale $m_{2}$, but not $m_{1}$.

Although it sufficed for the above thought process to consider only the kinematics (the four momentum conservation), the study of the dynamics of a $\pi^{0}$ decay suggests that the fine structure constant also needs to be considered as a fundamental constant alongside $h$ and $c$ [38]. I may emphasize that the ratios of rest mass scales associated with different elementary particles, remain the same, individually for $O_{1}$ and $O_{2}$.

To mention, exactly like the gravitational redshift has been discussed in terms of geodesics in [23], one can think of a similar process here. For example, imagine two space stations $S_{1}$ and $S_{2}$ following two different timelike geodesics. $O_{1}$ performs the experiment in $S_{1}$ and transmits the radiation through a window towards $S_{2}$. $O_{2}$ receives the radiation through a window at $S_{2}$. Both $O_{1}$ and $O_{2}$ consider $h$ and $c$ to be given. Rest of the steps of the procedure remain alike.

Compton scattering and effect of gravity: Now, let me discuss a table-top experiment, that can be performed in two frames that differ by a gravitational potential. Consider the Compton effect i.e. scattering of x-ray by an electron [31]. If x-ray, with an associated length scale, $\lambda_{0}$, is incident on an electron, then it is scattered at an angle $\theta$ with the incident direction. The scattered x-ray is associated with an increased length scale $\lambda_{\theta}(>\lambda_{0})$. The increment in the length scale of the x-ray is given by

[TABLE]

where $\lambda_{e}:=h/m_{e}c$ is a length scale associated with the electron, known as Compton length scale (see [32] for a remark), $m_{e}$ is the rest mass scale associated with the electron. Now, consider the same Compton scattering experiment is performed by $O_{1}$ and $O_{2}$. By ‘same’, I mean, $O_{1}$ and $O_{2}$ incident same $\lambda_{0}$, in their respective frames, on an electron and study the same $\lambda_{\theta}$. Now, according to eq.(7), the rest mass of an electron are different for $O_{1}$ and $O_{2}$. Therefore, one has the following result:

[TABLE]

where $\lambda_{1}=\frac{h}{m_{1}c},~{}~{}~{}\lambda_{2}=\frac{h}{m_{2}c}$ are the Compton length scales of the electron and $m_{1},m_{2}$ are the rest masses of the electron for $O_{1}$ and $O_{2}$ respectively. Combining eq.(7) and eq.(9), one has the following result

[TABLE]

In a Schwarzschild spacetime: Now, let me bring in the spacetime language and set the observers and radiation propagation along geodesics, which I have already mentioned earlier. Let me consider $O_{1}$ and $O_{2}$ freely falling along two timelike geodesics at different constant values of the radial coordinate of a Schwarzschild spacetime [23]; $O_{1}$ and $O_{2}$ have radial coordinates $r_{1}$ and $r_{2}$ respectively, with $r_{2}=r_{1}+H$ and $H>0$. Then, one has the following approximate expression:

[TABLE]

where $r_{0}=\frac{GM}{c^{2}}$, $M$ is the mass scale associated with the Schwarzschild spacetime [37], $r_{1}=xr_{0},H=(\Delta x)r_{0},x\gg 1,~{}\Delta x>1$ and the last step of eq.(12) holds for $\Delta x\ll x$. Now, let $\theta_{2}=\theta_{1}+\delta$, where $\delta$ is a function of $\Delta x$. Then, the right hand side of eq.(11) can be approximated, up to the leading order in $\delta$, to obtain

[TABLE]

Therefore, using the results of eq.(12) and eq.(13), back in eq.(11), one obtains the result

[TABLE]

Just as an example, if one considers $O_{1}$ to be at earth’s radius ($6370$ km [40]) and $O_{2}$ on the MICROSCOPE satellite, which is further $700$ km radially outward [41], then ${\Delta x}/{x^{2}}\simeq 700/(6370)^{2}\simeq 1.72\times 10^{-5}$. Then,

[TABLE]

Here, $\theta_{E}$ is the deflection angle of the scattered x-ray photon on earth’s surface and $\delta_{MICRO}$ is the change in the deflection angle while the experiment is performed on the MICROSCOPE.

Conclusion: Planck constant is necessary to describe energy-momentum flow along a single null geodesic and therefore, it is intrinsically associated with spacetime geometry. It has a priori nothing to do with the notion of ‘quantum’. Further, the rest mass scale variation is a hitherto unexplored facet of Einstein’s equivalence principle. The suggested experiment is first of its kind as it reveals the effect of gravity on the Compton length scale of an elementary particle. A verification will only be suggestive of the fact that what one understands by ‘kilogram’ on earth, is different than the notion of ‘kilogram’ on the moon.

Acknowledgement

I am grateful to Benito Juarez Aubry and Srijit Bhattacherjee for several critical discussions regarding this work. This work is supported by the Department of Science and Technology of India through the INSPIRE Faculty Fellowship, Grant no.- IFA18-PH208.