# Scientific value of the quantum tests of equivalence principle in light of
Hilbert’s sixth problem

Abhishek Majhi111A. M. is the corresponding author<EMAIL_ADDRESS>Gopal Sardar<EMAIL_ADDRESS>Indian Statistical Institute,
Plot No. 203, Barrackpore, Trunk Road,
Baranagar, Kolkata 700108, West Bengal, India

###### Abstract

In his sixth problem, Hilbert called for an axiomatic approach to theoretical
physics with an aim to achieve precision and rigour in scientific reasoning,
where logic and language (semantics) of physics play the pivotal role. It is
from such a point of view, we investigate the scientific value of the modern
experiments to perform quantum tests of equivalence principle. Determination
of Planck constant involves the use of acceleration due to gravity of the
earth $(g)$ that results in the force on a test mass. The equivalence between
inertial mass and gravitational mass of a test object is assumed in the
process of logically defining $g$ from the relevant hypotheses of physics.
Consequently, if Planck constant is used as input in any experiment (or in the
associated theory that founds such an experiment) that is designed to test the
equivalence between inertial and gravitational mass, then it is equivalent to
establish a scientific truth by implicitly assuming it i.e. a tautology. There
are several notable examples which plague the frontiers of current scientific
research which claim to make quantum test of equivalence principle. We
question the scientific value of such experiments  from Hilbert’s axiomatic
point of view. This work adds to the recently reported semantic obstacle in
any axiomatic attempt to put “quantum” and “gravity” together, albeit with an
experimental tint.

##  I Introduction

The significance of logic and language (semantics) in physics become evident
from the study of Hilbert’s sixth problem, namely, Mathematical Treatment of
the Axioms of Physics[1, 2] and the associated modern research that has
germinated from such roots e.g. see ref.[3] and the references therein (also
see refs.[6, 7, 4, 5]). If we consider Einstein’s views[8], then “physics
constitutes a logical system of thought” and “the justification (truth
content) of the system rests in the proof of usefulness of the resulting
theorems on the basis of sense experiences, where the relations of the latter
to the former can only be comprehended intuitively.” While certainly the
experimental observations (“sense experiences”) must have the final say on the
essence of some theoretical construction, however, the inner consistency, of
the “logical system of thought” that underlies the theory, is necessary for
the whole process of reasoning to be considered as scientific. Hilbert’s call
for axiomatization was aimed to achieve more precision and rigour in such
reasoning. Such a call obviously is associated with a tension between
exactness of logical truths and uncertain nature of experimental truths, the
significance of which is manifested through Born’s statement on p.81 of
ref.[9]: “a physical situation must be described by means of real numbers in
such a way that the natural uncertainty in all observations is taken into
account”. While such tension has motivated lines of research investigation
that concern the intuitive refinement of the language of physics itself[4, 5,
10, 11, 6, 7], but Hilbert’s axiomatic point of view continues to retain its
value in the mainstream literature[3, 2, 14, 15, 13, 12] where the equations
of physics are used to discuss physical phenomena and working principles of
experiments with exact quantities devoid of the natural uncertainties involved
in the experimental measurements. Recently, by adopting such a Hilbertian
point of view, one of us (A. M.[10]) has pointed out a semantic dilemma that
one has to encounter in any attempt to axiomatically treat together “quantum”
and “gravity” so as to pen down any theory of “quantum gravity”[16]. It is now
our intent to investigate, from a similar point of view, the logic behind the
modern experiments on the “quantum” tests of equivalence principle which are,
nevertheless, very much motivated from the “quantum gravity” mindset[17, 18,
19, 20, 21, 22].

An experiment is considered to have a scientific value when (a) it verifies
some hypothesis or theoretical prediction (b) it brings forth some newly
observed phenomenon that is yet to be explained by theory. However, an
experiment of the type (a) can not depend on any input that explicitly or
implicitly depends on the concerned hypothesis. If it does, then such an
experiment verifies the truth by assuming it in the process. We consider this
as a physical demonstration of a logical tautology. Such an experiment is
devoid of any scientific value (unless it brings forth some hitherto unknown
phenomenon that is not related to the concerned theory i.e. type (b)). Here,
we intend to discuss one such scenario, that involves a certain class of
experiments (type (a)) which are designed to perform “quantum” tests of the
equivalence principle [17, 18, 19, 20, 21, 22], where and henceforth, by
“equivalence principle” we mean “equivalence between inertial mass and
gravitational mass of an object” [23]. These experiments inevitably involve
the Planck constant (henceforth, to be denoted as “$h$”) which justifies the
word “quantum”. To be more particular, such experiments rely on atom
interferometry where the De Broglie wavelength ($\lambda$) of the respective
matter wave, with momentum $p$, needs to be calculated from the formula
$\lambda=h/p$ and for this calculation $h$ is considered as a given input –
see refs. [17, 18, 19, 20, 21, 22] and the relevant references there in.

We may express our concern in short as follows. Irrespective of the different
measurement procedures to determine $h$ employed till date, the corresponding
theoretical analyses involve, either explicitly or implicitly, the use of $g$,
where by the symbol “$g$”, we denote “acceleration due to gravity of the
earth”, or equivalently “gravitational field of the earth”. For example, as we
shall discuss here, in case of the Kibble balance method, $h$ is expressed in
terms of $g$. On the other hand, in case of photo-electron emission method,
$g$ gets implicitly incorporated into the analyses through the processes by
which “charge of an electron” is determined and used as an input. However, $g$
can be defined from the axioms and hypotheses of physics, if and only if the
equivalence principle is assumed. Therefore, the so called “quantum tests of
the equivalence principle” are attempts to physically demonstrate a logical
tautology and are devoid of any scientific value  from Hilbert’s axiomatic
point of view. We shall point out that the assumption of the equivalence
between “inertial mass” and “gravitational mass” is exact, and not associated
with uncertainties, in the experiments that determine $h$. So, the “quantum
tests of the equivalence principle” can not be interpreted as a recursive
refinement of logic that verifies the equivalence principle with an
uncertainty that is less than the uncertainty with which the equivalence
principle is assumed to determine $h$. In fact, the determination of $h$ is
based on the equivalence principle with zero or no uncertainty (i.e. exact)
and then such $h$ is used as an input in the “quantum tests of equivalence
principle” to verify the equivalence principle with non-zero uncertainty i.e.
it is a case of adulteration (rather than refinement) of logic. Thus, if we
take experimental uncertainty into account, the situation is even worse than a
logical tautology.

To present our arguments, we begin with a discussion regarding the definition
of $g$, especially with a focus on how the assumption of equivalence principle
is necessary for such a definition. Then, we discuss in what way the
theoretical analyses for the different procedures of determining $h$ are
founded on the assumption of the equivalence principle due to the involvement
of $g$ and also due to the use of only one concept of “mass” regardless of any
distinction as “inertial” and “gravitational”. Finally, we conclude with a
summary and some remarks regarding the status of this work in light of the new
convention adopted by the science community in 2019[24].

##  II Interpretations of the symbol “$g$”

There are two kinds of interpretations that we generally associate with the
symbol “$g$”. One is the operational interpretation and the other is the
logical interpretation.

We provide the operational interpretation of “$g$” in terms of what we can
measure and it is the “acceleration of a freely falling object due to earth’s
gravity” which we determine experimentally (e.g. by dropping objects).  To
mention, the “operational” viewpoint of any definition in physics was
advocated by Mach[25]. We know that Einstein, who was highly influenced by
Mach, took an “operational” approach to give meaning to “time” by stating our
everyday experience of seeing the hands of a clock to mark the timing of an
observed event like “arrival of the train at the station”[26]. The operational
perspective was revisited in an elaborate fashion by Bridgman[27].

However, if we restrict ourselves to being completely operational, we can not
ask questions like the following – “how do we interpret that the acceleration
is due to earth’s gravity?” This is because, in order to have a reasonable
answer to this question, we need to consider hypotheses, axioms, etc. of
physics, namely Newton’s laws of gravity and motion, to provide the logical
explanations corresponding to “earth’s gravity” through mathematical
expressions. So, we need to be both operational (i.e. induce from experience)
and logical (i.e. deduce from assumed truths i.e. axioms, hypotheses, etc.) –
we can not just stick to only one of the following stances – “a posteriori
induction”, “a priori deduction”. One can consult ref.[28] for a discussion
regarding such issues. We discuss, in what follows, that the equivalence
principle is a necessary proposition for the logical (deductive) definition of
“$g$” [see ref.[30, 29] for some in depth discussion regarding such logical
perspective of “definition”].

###  II.1 Definition of “$g$” and the logical status of the equivalence
principle

At first sight it may appear to be an utter stupidity to have a discussion
regarding the definition of “$g$” as it is “too trivial”. Nevertheless,
keeping in mind that the discussion is regarding some logical issues, we
believe that the definition must be put into formal terms, rather than in a
colloquial language, so as to clear any doubts regarding any of the individual
propositions involved in the process, irrespective of its apparent
triviality[30]. Therefore, in order to avoid any misjudgment by the concerned
reader regarding the definition of “$g$”, we consider formal statements,
regardless of its simplicity and familiarity in the science community. It is
much like what Hadamard has done to clear the doubt regarding the statement of
Huygens’ principle in section (33) of ref.[31]. He has considered “simple
formulae and statements” in the form of propositions and has made a formal
logical analysis in order to resolve doubts: “But, however simple the
preceding formulae and statements, they have, nevertheless, opened somewhat
important and lengthy scientific discussions, of which we have now to speak
and which refer to what is called Huygens’ Principle.”

Apart from the above mentioned reasons, we adopt such a path based on formal
logic so as to put an emphasis on the necessity of the assumption of
equivalence principle, as a formal statement, for the deductive definition of
“$g$”. In what follows, the symbols “$\wedge,\iff,:=$” denote “logical
conjunction, logical equivalence (if and only if), defined as” respectively.
In the context of writing down the propositions $A$ and $B$ citing ref.[32,
33] would have been relevant. However, those references do not contain the
equations in the form that we use today. So, we have assumed that the reader
is already acquainted with the Newton’s laws.

  * •

Proposition $A$: Newton’s laws of motion apply to an object whose motion is to
be studied as a whole, called test object. Associated concept of “mass” is
called inertial mass ($m_{I}$). Acting force is given by

$\displaystyle F=m_{I}a,$ (1)

where $a$ is the acceleration of the test object.

  * •

Proposition $B$: The test object obeys Newton’s law of gravitation. Associated
concept of “mass” is called gravitational mass ($m_{G}$). Force of gravitation
on the test object due to the earth is given by

$\displaystyle F=G\frac{m_{G}M_{earth}}{r^{2}}.$ (2)

Here, $G$ is a proportionality constant, $M_{earth}$ is the gravitational mass
(ignoring “active/passive” distinction, unlike in ref.[23]) of the earth and
$r$ is the distance between the test object and the earth, considering them as
two points (point-masses).

  * •

Proposition $C$: Equivalence principle holds:

$\displaystyle m_{I}=m_{G},$ (3)

where the meaning of the symbols “$m_{I}$” and “$m_{G}$” are explained by $A$
and $B$ respectively.

Corollary: $G$ is a fundamental constant if and only if $C$ is true for any
type of material i.e. $m_{I(type)}=m_{G(type)}$ for any type of material like
iron, aluminium, copper, etc.

  * •

Proposition $D$: Concept of “gravitational field due to earth”, represented by
the symbol “$g$”, which is conceived through the observation of vertically
falling objects towards the earth, is definable.

  * •

Proposition $E$: $g$ is definable if and only if Newton’s laws of motion,
Newton’s law of gravitation and the equivalence principle hold simultaneously
for the test object. Formally, $\left((A\wedge B)\wedge C\right)\iff D$.

Corollary: $E$ implies the definition of $g$. Formally,

$\displaystyle E\implies g:=\frac{GM_{earth}}{r^{2}}.$ (4)

  * •

Proposition $F$: Considering $g$ as input, experiments are modeled to
determine $h$. We call such models as “$g$-experiments” i.e.

$\displaystyle g\text{-experiments}:(g\wedge\cdots).$ (5)

The “dots” stand for propositions, other than $D$, which need to hold for the
modeling. Such propositions can be both theoretical and experimental in
nature. By “experimental propositions” we mean, the choices of appropriate
physical conditions which are made by the experimenter in the laboratory.

Let us explicate the relevance of the above propositions and the nature of the
logical structure of the same as follows:

  * •

Step 1: $A$ provides meaning to the symbol “$m_{I}$”. $B$ provides meaning to
the symbol “$m_{G}$”. Hence, $C$ is meaningless without either of $A$ and $B$.
Consequently, $C$ can neither be validated nor invalidated without $A$ and $B$
together. Also, validation or invalidation of $C$ does not affect the validity
of $A$ or of $B$.

  * •

Step 2: $D$ is an assertion about the possibility of theoretically defining
the concept of “gravitational field due to the earth”, represented by the
symbol “$g$”. $E$ states the condition when such definition is possible by
using $A,B$ and $C$. Thus, $E$ implies the definition of $g$. In other words,
the definition of $g$ is implied by all the propositions from $A$ to $E$,
albeit a one-sided implication.

  * •

Step 3: $F$ is the modeling of the $g$-experiments used to determine $h$. The
meaning associated with the symbol “$g$” is explained by Step 1 and Step 2.

Therefore, we can draw the following conclusion:

Since $C$ is necessary for the definition of $g$, then $F$ is implied by $C$
i.e. the validity of the equivalence principle goes in as a necessary
assumption when $g$ is used as an input to determine $h$.

###  II.2 “Completely operational” or “logical contradiction”?

Although we have briefly argued just a bit earlier that we can not be
completely operational, we would like to emphasize that point once again in
light of the above logical analysis. One can certainly object to the
conclusion (drawn in the earlier section) by arguing that, since $g$ is only
implied by the successive propositions ($E\implies g$) and the reverse is not
necessarily true (i.e. it is not an “if and only if” condition and rather a
one-sided implication), then the tautology can be avoided. Instead of the
above explained meaning of “$g$”, a complete operational meaning can be
assigned to “$g$”as follows:

The acceleration of a vertically falling object can be measured to have a
value $9.81$ m/s2 without referring to Newton’s law of gravitation. Then, this
explains the meaning of “$g$”, and provides its numerical value that has been
used in the $g$-experiments. In this way, there is no necessity of $B$ and
hence, $C,D,E$ become meaningless. Only $A$ and the symbol “$a$” suffice and
the introduction of the symbol “$g$” is altogether unnecessary. We can have
just “$a$-experiments” to determine $h$.

However, in such a way of reasoning, the operational arguer denies the
knowledge of the Newtonian[32, 33] (as well as Einsteinian[34]) theory of
gravity that explains the phenomenon of falling object and consequently,
denies the knowledge of the concept of “gravitational mass”. So, if the arguer
now claims to examine the validity of equivalence principle by using $h$, then
he should accept the knowledge that he has already denied. Hence, the
operational arguer runs into a logical contradiction, that is unacceptable.

At this point a diligent reader may be wondering whether our arguments imply
that Eotvos et. al., in ref.[35], have made a logical contradiction. The
answer is certainly negative. It is true that the authors have considered $g$
in operational terms i.e. they used a directly measured value of $g$ in their
analyses. However, their motive has been to judge how close are $g$ and $a$ if
we do not assume the equivalence principle (e.g. see Box 1.2 on page 16 of
ref.[36]). Their case is different from the scenario where people consider the
logical definition of $g$ by assuming the validity of equivalence principle
and use the value of $g$ in operational terms if required to determine some
quantity, which they use in turn to test the validity of equivalence
principle. The issue will be more clear as we proceed and discuss what
follows.

##  III Principles behind determination of $h$

Now, let us discuss how the assumption of the equivalence principle is
involved either explicitly or implicitly in the determination of $h$. Broadly,
there are two categories of such experiments: (i) the Kibble balance
experiments e.g. see refs.[37, 38] and the relevant references therein (see
ref.[39] for Kibble’s original work) (ii) photo-electron emission experiments
e.g. see refs.[40, 41] and the relevant references in ref.[41].

###  III.1 Kibble balance experiments

In Kibble balance experiments [37, 38], a crucial step is the balancing
between the gravitational force (due to earth) on a test mass $(m)$(which
includes the tare mass) and force generated by a current $(I)$ carrying coil
perpendicularly placed in a magnetic field. So, the relevant equation is[37]:

$F=mg=BLI$ (6)

where $L$ is the length of the coil and $B$ is the magnetic flux density. This
is called the weighing phase.

The other phase, which is called the moving phase, is designed to measure $BL$
from the voltage developed in the coil ($V$) when it passes with constant
velocity $v$ through the magnetic field. The relevant equation is
$V=BLv$,which is then used to eliminate $BL$ from eq.(6), under the assumption
that $BL$ remains same in both phases, to obtain the Kibble equation:

$mg=\frac{IV}{v}~{},$ (7)

Going through some further theoretical manipulations, corresponding to the
measurement process (details can be found in ref.[37], unnecessary for the
present purpose), the expression for $h$ comes out to be the following:

$\displaystyle h=\gamma gvm$ (8)

where $\gamma$ is a quantity with physical dimension $[T]^{2}$ (time squared)
and it depends on a collection of measured variables that depend upon the
scaling parameters within the measurement technique.

We note that in the theoretical analysis of the Kibble balance experiments,
only one concept of “mass”, denoted by “$m$”, has been used and no
categorization like “inertial mass” and “gravitational mass” has been made.
Furthermore, $g$ appears explicitly in the expression for $h$ i.e. in eq.(8).
Therefore, the equivalence principle has been assumed  and this assumption is
exact because there is no mention of uncertainty regarding the equivalence of
“inertial mass” and “gravitational mass” in refs.[37, 38]. [Remark: Since
Joule balance technique is just an alternative of the Kibble balance and
follows the same basic principle of balancing under gravity (see e.g.
refs.[42, 43, 44]), we do not discuss it separately. Our views regarding the
Kibble balance that concerns the equivalence principle as the founding premise
of the theoretical analysis, also applies for the Joule balance.]

###  III.2 Photo-electron emission experiments

In photo-electron emission experiments, as was first done by Millikan [40],
the theoretical analysis is founded upon the photoelectric equation rooted to
Einstein’s work [45]: $h\nu=\phi+E_{max}$, where $E_{max}$ is the maximum
kinetic energy of photo-electron, $\nu$ is the frequency of the incident light
and $\phi$ is the work function of the illuminated material.

It is found in such experiments that $E_{max}$ is proportional to $\nu$.
$E_{max}$ is determined by measuring the stopping potential $V$ from the
equation $E_{max}=eV$, where $e$ is the charge of an electron. Therefore,
plotting $V$ along $y$-axis and $\nu$ along $x$-axis, $h/e$ is obtained as the
slope of the straight line. Hence, the value of $h$ is determined from this
slope $h/e$ by taking the value of $e$ as input.

This fact does not change even in modern day experiments which involve photo-
emission spectroscopy techniques to increase the precision of such experiments
e.g. see ref.[41] and the relevant references there in. So, it is worth
understanding how $e$ is determined (without using $h$ as input). We discuss
the relevant methods in the following section.

[Remark: There are examples of experiments, where experimenters claim to
determine $e$ by using some value of $h$ as input in the process, without any
elaborate discussion regarding how such value of $h$ is determined in the
first place. For example, the experiments like those discussed in ref.[46,
47], which use the Single Electron Tunneling (SET) mechanism[48, 49], use $h$
as input to determine $e$. We keep any discussion regarding such experiments
out of the present context because our motive is to discuss the methods of
determining $h$ and not to assume its value as given.]

###  III.3 Principles behind determination of $e$ (without using $h$ as input)

There are broadly two methods of determining $e$ without using $h$ as input,
which have been explored till date viz. oil-drop experiment and the x-ray
spectroscopy method. While the former is explicitly dependent on the use of
$g$, latter is implicitly dependent on the use of $g$ through the
determination of Avogadro constant in the process. Therefore, both the
procedures are based on the assumption of equivalence principle, which we
shall discuss in what follows.

####  III.3.1 Oil-drop experiment

The oil-drop experiment is due to Millikan [50, 51] and Fletcher [52]. The
theoretical analysis behind the experiment can be traced back to ref.[50],
where the first equation has been written in the following way: “The relations
between the apparent mass ${\bf m}$ of a drop, the charge $e_{n}$, which it
carries, its speed, $v_{1}$ under gravity, and its speed $v_{2}$ under the
influence of an electrical field of strength $E$, are given by the simple
equation

$\displaystyle\frac{v_{1}}{v_{2}}=\frac{{\bf m}g}{e_{n}E-{\bf
m}g}\quad\text{or}\quad e_{n}=\frac{{\bf
m}g}{E}\left(\frac{v_{1}+v_{2}}{v_{1}}\right).^{\prime\prime}$ (9)

The following clarification has been provided in a footnote: “The term
‘apparent mass’ has been used to denote the difference between the actual mass
and the buoyancy of the air.”. Therefore, only one concept of “mass” (called
“actual mass”) has been used in such an analysis. No distinction such as
“inertial mass” and “gravitational mass” has been made. Furthermore, the
involvement of $g$ in the analysis is explicitly manifest from the expression
for $e_{n}$, where $e_{n}$ stands for some integral multiple of $e$.
Therefore, the equivalence principle has been assumed  and this assumption is
exact because there is no mention of uncertainty regarding the equivalence of
“inertial mass” and “gravitational mass” in refs.[50, 51].

####  III.3.2 X-ray spectroscopy method and the principles behind
determination of Avogadro constant

The second method of determining $e$ relies on the use of x-ray spectroscopy
to study crystal lattices as grating e.g. see refs. [53, 54] and the relevant
references therein. This method of determining $e$ requires the use of
Avogadro constant $N_{A}$ as input [42]. And, the theoretical analyses, which
underlie the experimental determination of $N_{A}$ by various methods, are
invariably founded upon only one concept of “mass” along with explicit
involvement of $g$ [55, 56, 57], as we explain in what follows.  Of course, by
“various methods” we mean only those methods of measurements which neither use
the value of $h$ nor use the value of $e$ as input to determine $N_{A}$.

In ref.[55], the final expression for $N_{A}$, that appears in the section
called “Precise Determination of Avogadro’s Constant”, explicitly depends on
$g$:

$\displaystyle 2.303(RT/N_{A})\log_{10}(n_{0}/n)=(4/3)\pi
a^{3}g(\Delta-\delta){\bf h}.$ (10)

The significance of the other symbols in the equation can be found in ref.[55]
and the symbol “${\bf h}$” represent some length and not Planck constant. The
above expression is rooted to the buoyancy related analysis that follow from
the earlier sections of ref.[55]. Such analysis is based on only one concept
of “mass” which becomes apparent from only one concept of “density”. A
distinction as “inertial mass” and “gravitational mass” would have led to two
different notions of densities viz. “inertial density” and “gravitational
density”.

To mention, $\Delta,\delta$ are the densities of the granular material and the
inter-granular liquid, respectively, which have been used to perform the
experiment. Further, $g$ is explicitly manifest in eq.(10). Hence, the
equivalence principle has been assumed in the process  and this assumption is
exact because there is no mention of uncertainty regarding the equivalence of
“inertial mass” and “gravitational mass” in the concerned references.

In ref.[56], the authors declare (in the second paragraph), the following: “We
have made, instead, readily available highly spherical steel artifacts as
local and temporary “standards” of density. Their masses were determined in
terms of the U.S. National Standard (kilogram replica number 20) by well
understood procedures.” This “well understood procedure”, described in
ref.[58], is based on only one concept of “mass” and the use of $g$.
Therefore, the equivalence principle has been assumed in the process  and this
assumption is exact because there is no mention of uncertainty regarding the
equivalence of “inertial mass” and “gravitational mass” in ref.[58].

In a nutshell, the determination of $N_{A}$ depends on the measured value of
the density $(\rho)$ of the associated crystal and $\rho$ is measured through
buoyancy related experiments[59, 60, 61, 62, 63, 64]. Any such experiment is
based on Archimedes’ principle [65] where $g$ plays the role in defining “the
weight of an object”. Also, due to the inputs of the mass measurements, $g$
gets involved in the process as well[58]. This is because mass measurements
are done through mass comparators which are just balancing instruments with a
founding principle based on the use of $g$ and only one concept of “mass” [66,
67, 68]. Importantly, such mass measurements using mass comparators are
involved in any modern experiment222We consider only those experiments which
do not use $h$ as input in the process of determining Avogadro constant. This
is because we are investigating the methods by which $h$ is determined. that
intend to determine $N_{A}$ e.g. the modern x-ray crystal density method
discussed in refs.[69, 70] indeed relies on mass comparators for mass
measurement of the silicon crystal which becomes clear from the relevant
references therein and especially from refs.[71, 72].

Therefore, we can conclude from the above discussion that the determination of
Avogadro constant $(N_{A})$ is based on the assumption of the equivalence
principle  and this assumption is exact because there is no mention of
uncertainty regarding the equivalence of “inertial mass” and “gravitational
mass” in the concerned references.

##  IV Concluding Remarks

Let us conclude by providing a concise account of what we have discussed,
followed by some crucial remarks regarding the status of this work in light of
the new convention adopted in 2019[24]. While we define the symbol “$g$” to
bear the meaning “acceleration due to gravity of the earth”, we assume in
course of such a definition that the equivalence principle (equivalence
between inertial and gravitational mass) holds. Any measurement procedure of
Planck constant ($h$), that involves the use of $g$, either explicitly or
implicitly, is therefore dependent on the assumption of equivalence principle.
While the Kibble balance method explicitly involves the use of $g$, the photo-
electron emission experiments implicitly involves the use of $g$ through the
measurement of the charge of an electron $e$. This is because the
determination of $e$ through oil-drop experiment explicitly depends on the use
of $g$ and the method of x-ray spectroscopy, to determine $e$, implicitly
depends on $g$ due to the determination of Avogadro constant $(N_{A})$ by
measuring the crystal density through buoyancy related experiments (which
require “weight of an object” to be defined) and mass measurements through
mass comparators (which is based on balancing mechanism involving $g$ in
principle). In view of these, we conclude that the modern day experiments
which claim to make quantum tests of equivalence principle, are just attempts
to physically demonstrate a logical tautology i.e. in such experiments one
assumes the equivalence principle to test the equivalence principle.
Importantly, this assumption is exact because there is no mention of
uncertainty regarding the equivalence of “inertial mass” and “gravitational
mass” in the references that are concerned with the determination of $h$
directly or indirectly. Therefore, use of $h$, that has been determined with
the assumption of the equivalence principle with no uncertainty, to test the
equivalence principle with non-zero uncertainty, actually presents a case of
adulteration of logic (worse than tautology).

Now, an apparently legitimate objection may be raised against this work, if
one considers the new convention adopted by the scientific community in the
year 2019 [24], in the following way:

Objection: Since 2019 the Planck constant ($h$) has been considered as a
defining constant, in terms of which (and other defining constants), the
kilogram – unit of mass – is defined. The distinction between “inertial” mass
and “gravitational” mass does not enter such a definition.

While such an objection may appear to be legitimate at first, however it is
fallible to simple counter reasoning as we demonstrate in what follows. It is
true that $h$ has been considered a defining constant since 2019 and it can be
found to have been clearly stated in the Preface of ref.[24]. So, the
respective value (in some system of units like SI system) is chosen to be
exact and the base units like kilogram are defined in terms of $h$ and other
defining constants. However, such a definition of kilogram, as given in
ref.[24], is theoretical and yet to be realized through experiment. In fact,
this has been clearly mentioned, on page no. 131 of ref.[24], that:

“The present definition fixes the numerical value of $h$ exactly and the mass
of the prototype has now to be determined by experiment.”

In fact, in the Appendix 2 of ref.[24], the expected uncertainty in such
experimental realization of “kilogram”, with reference to the mass of the
international prototype of the kilogram, has also been mentioned.

That, the objection raised about the redundancy of the equivalence between
inertial mass and gravitational mass, is empty of any essence, can be
understood if we pose the following question regarding the above promise that
“the mass of the prototype has now to be determined by experiment”:

What type of mass of the prototype is going to be determined by experiment –
inertial or gravitational?

In this work we have discussed precisely this issue regarding mass
determinations in general and we have pointed out that one always assumes the
equivalence between inertial mass and gravitational mass while performing such
mass measurements (or even discussing the principles) because only one concept
of “mass” is generally stated without any hint of such distinction.
Importantly, this assumption of equivalence is exact and devoid of any
uncertainty, which is distinct from the uncertainty expected to be associated
with the experimental realization of kilogram reported in the Appendix 2 of
ref.[24]. However, such an assumption is not stated formally and rather kept
unmentioned or remains hidden.

Further, we may point out that the chosen value of $h$ has been determined
through years of experimental research and then only it has been possible to
reach a general consensus to consider an exact value, based on which the base
units are defined in ref.[24]. What we have discussed in the article is that,
all such procedures to determine the value of $h$, which have been performed
over the years, are based on the assumption of the equivalence between
inertial mass and gravitational mass, explicitly or implicitly. Now, if we
consider the current convention of choosing $h$ to be exact, then it is
implied by such convention that inertial mass and gravitational mass are
exactly equivalent (irrespective of the expected uncertainty in the
experimental realization of “kilogram” mentioned in the Appendix 2 of
ref.[24]). Therefore, in light of the 2019 convention, the quantum tests of
equivalence between inertial mass and gravitational mass [17, 18, 19, 20, 21,
22], are physical demonstrations of adulteration of logic (worse than
tautology) because one starts with an exact $h$ (possible if and only if an
exact equivalence between inertial mass and gravitational mass is assumed) and
uses it for quantum tests of the equivalence between inertial mass and
gravitational mass where the equivalence is associated with some non-zero
uncertainty.

In light of Hilbert’s sixth problem and therefore, taking into account the
importance of logic and language in physics, recently it has been reported by
one of us (A.M.[10]), how one should face a semantic or logico-linguistic
dilemma in course of axiomatizing any theory that aims to take into account
“quantum” and “gravity” in the same framework. The present work brings to
light similar scientific queries, but now with an association of the
experimental aspect. Noting the recent upsurge in an interest in logic and
language (semantics) of physics [3, 10, 4, 5, 6, 7], the present work adds to
such research endeavours but with an added bit of direct association with
questions concerning the experimental aspect as far as the critical question
of treating “quantum” and “gravity” in the same framework is concerned, and
even includes the context of the most recent 2019 convention of redefining of
units[24]. In view of this we hope that this present work can provide a
different and fresh insight, for the concerned scientific community, as far as
the experimental and the axiomatic aspect of “quantum gravity” is concerned
and, in particular, may germinate the seeds of certain reasonable doubts
concerning the scientific reasoning that underlies the modern quantum tests of
equivalence between inertial mass and gravitational mass[17, 18, 19, 20, 21,
22].

Acknowledgment: The authors thank R. Radhakrishnan for pointing out ref.[31].
The work was accomplished while G. S. was visiting The Indian Statistical
Institute, Kolkata. A. M. is supported by the Department of Science and
Technology of India through the INSPIRE Faculty Fellowship, Grant no.- IFA18-
PH208.

Conflict of interest statement: On behalf of all authors, the corresponding
author (A. M.) states that there is no conflict of interest.

Declaration on competing interest: The authors declare that there is no
competing interest.

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