The Concept of the Inertial Mass in
Macroscopic Physics and in Quantum Field
Theory
R. Faber
Private Center for Fundamental Research
90491 Nuernberg, Germany
and
M. Dillig
Institute for Theoretical Physics III
University Erlangen-Nuernberg, 91058 Erlangen,
Germany
Abstract
We reflect on the concept of the mass in physics. We argue that in
macroscopic physics the only relevant mass is the inertial mass,
which we call dynamic, opposite the gravitational (static) mass: all
experimental measurements are only sensitive to the inertial mass
(the gravitational mass is at best realized in the asymptotic limit
with the interaction time t   ). Our reasoning is strongly
supported from quantum field theory: here the presence of the
fluctating vacuum inevitably allows only for a dynamical, i.e.
inertial mass.
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The concept of the mass is of fundamental importance in physics,
as it plays a crucial role on the very large and the very small
scales, i. e. in macroscopic and microscopic physics. However,
inspite of their outstanding role and their long historical tradition,
the dynamical origin of the mass is presently only vaguely
understood. The basic idea of mass generation is the concept of
spontaneous symmetry breaking (such as in the context of gauge
or chiral theories), where the mass originates from the
nonperturbative breaking of the symmetry of the vaccum: the
generation of mass reflects the appearance of new vacuum as the
true ground state of the field theory (1). Though microscopic details
of such mechanisms (their dynamical origin) are presently only
anticipated, it is obvious that the generation of mass is intimately
connected with the presence of the nontrivial physical vacuum: in
spontaneous broken field theories the mass  of a particle it
proportinal to the vacuum expectation value of bilinear timeordered products of field operators (or condensates), schematically
(2)
(With respect to the discussion presented below we use  as a
neutral symbol for mass, without explicitly specifying further its
nature).
Leaving the framework of microscopic field theories, we face
different conecpts for the mass on the macroscopic level: the
inertial mass, denoted in the following by the symbol m, and the
gravitational mass, denoted hereafter as m*. The historical origin of
this differentiation is not fully clear, but dates back to the 18th
century (3): though Newton in his formulation of the Newtonian
classical mechanics used only the concept of the inertial mass as
part of his dynamical equations (4), a subsequent formulation of
the equations of motion in the framework of the Hamilton or,
equivalently, the Lagrange formalism, led to an extension of the
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conecpt of mass to a graviational mass, entering in static
formulations, such as based on the static gravitational potential.
From its first origins this concept became rooted deeply in classical
mechanics as a standard of common modern textbooks. A link
between these a priori two fundamentally different concepts for the
mass was established in the formulation of the Theory of General
Relativity by Albert Einstein (5) who introduced the (both local and
global) principle of the strict quantitative equivalence between the
inertial and the graviational mass to formulate the strict equivalence of an appropriate accelerated system without gravitational
force and an appropriate system in rest with the presence of a
gravitational force.
Soon after the discrimination between an inertial and a gravitational mass appeared in the literature, intuition and
experimental experience lead to the conjecture of the equvialence of
these two a priori different forms of the mass. Pionieering and
accurate experiments have been performed already towards the
end of the 18. century by Eötvös (6) and other groups. However, as
Einstein introduced the strict equivalence of inertial and
gravitational mass in an axiomatic way, various (and ongoing)
experimental attempts have been and are performed to prove the
strict equivalence of m and m* with increasing accuracy; current
NASA initiated projects aim at an accuracy of (7)
( m – m* )/m < 10
;
furtheron, even more accurate experiments are planned. Even
though their outcome has to be awaited for, hitherto all findings
point strictly to an identity of the inertial and the graviational
mass, i. e. to a unique conecpt of mass with m = m*.
Guided by this experimental trends and worried by conceptional
difficulties in the discrimination beetween the inertial and the
graviational mass, it is tempting to formulate the underlying
question: does enforce nature a priori the introduction a
gravitational mass in addition to the inertial mass öand the
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discrimination between these two kinds of masses, or,
alternatively, has the concept – as we argue – of the gravitational
mass to be abandoned completely and to be replaced by the unique
concept of the inertial mass m?
To formulate this question on a truely fundamental level we feel,
that presently indispensable elements for a substantial answer are
missing: we foresee that answering this question on the deepest
level requires a basic understanding both of the conept of mass
generation and the unifcation of very small (microscopic) and very
large scales in nature unifying all known forces on or beyond the
Planck scale (8). As presently here only fragmentary attempts have
been undertaken, we have to divide our discussion of the concept
of mass up into the two scales mentioned above and focus
separately on macroscopic and microscopic aspects.
On the microscopic level – referring to the Newtonian concept of
mechanics, which is sufficient for our considerations – the starting
point is the unification of dynamical and a static equation in
Newton’s law for a two body system
m* r = -  V
with
V = -  m * M* / r
(with the gravitational constant  and the heavy masses m and M
of the gravitating bodies); in its linearized form and extended to
angular momentum conservation the corresponding law also holds
for the comparison of two masses in a weigthing process.
In the equation above we have to critically examine the concecpt of
a static situation. As a closed, isloted two-body system never exists
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in natureretation the equation of motion above never describes –
except in a limiting process sketched below – rigorously a static
problem. With the fluctuating nonequiblibrium background
generated the dynamical coupling to all the (infinite number of)
background masses, a static equilibrium is never strictly realized.
As a direct consequence, the conept of a static graviational mass
never enters into a dynamical macroscopic equation: we are
exclusively confronted in macroscopic nature with dynamical
problems. Taking the gravitational force it is clear that the static
conecpt is just an extreme simplfication of a truely dynamcial
situation: the (quasi) static result just reflects the approximation
to an (infinite) sum of dynamically coupled many-body
contributions. This argument does not rest on the question of a
static or an expanding (or contracting) universe: even within a
formally static universe any dynamical equation reflects a truely
nonstatic underlying physics.
In continuing this argument, we feel that the concept of a
graviational mass may be only realized in extreme limit
m = lim (r) m*
i.e. in isolating the mass under consideration strictly from the
residual system of interacting masses. Though such a limiting
process is hypothetically possible, it indicates by its very nature
that for the existing universe both in space and time the existence
of a static heavy mass is never realized (note that the limiting
process above in space implies on the same level a limiting
transition in time with t , where t is the time of the interacting
mass relative to the residual system. Again this reasoning is not
invalitated by the even more complex and controversial question of
an indiviual „Eigenzeit“ for each body in an interacting system, as
arising naturally in various covariant field theoretical approaches
(9)).
It is crucial to stress that we arrive at exactly the same conclusion
in comparing the inertial and the graviational mass directly by a
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weighting process. In a relative comparison of two masses one
measures the accelerating (graviational) force of one body acting on
the second body, which is from the principle actio = reactio
inversely accelerated with respect to the first reference mass. A
priori, such a comparision is a truely dynamical situation. The
static limit – i.e. a strict static balance between the two bodies
compared – is only achieved the limit of an asymptotically
vanishing mutual acceleration of the two masses, i. e. for
lim
r  0.
Again it is clear, that such a limit in nature at finite times is never
realized, in other words, the concept of a gravitational, static mass
strictly never enters in any macroscopic physical law of nature.
One might argue, that there is a caveat in the argumentation
above: in principle it might be possible that during a finite time
interval there exists a phase transition of the second kind (such as
realized macroscopically and microscopically in many phenomena
in physics, for example, in superdconductivity, magnetization,
chiral phase transitions or colour superconductivity (10)), giving
rise to a nonvanishing energy gap and converting the inert mass
into a distinctly different state of a gravitational mass.
We feel that from our present understanding, such a reasoning
cannot be reconciled with the fundamental laws of nature. As the
crucial point, we realize that all second order phase transitions in
nature arise on a genuinely microscopic level, macroscopic
consequences of such phase transitions only reflect the dynamical
collectivity of the underlying micrcoscopy.
This immediately raises the principal question: is there a
microscopic transition from a dynamcial to static situation possible
at all. Here – even without knowing the true origin of the
generation of mass – on the microscopic level a static limit is
completely excluded by the presence of the dynamical vacuum
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itself (11) – or equivalently – the existence of Heisenberg’s
uncertainty relation (12). From the basic concepts of quantum
mechanics it is clear, that there exists no static vaccum: the
vacuum is always fluctuating due to the presence of the Dirac sea.
In other words, the isolation of a particle in the spirit of a truely
single particle problem in never realized in nature: nature always
involves an infinite many-bdoy problem and thus exclusively
dynamcial masses of the interacting particles. Thus the nature of
the vacuum itself excludes a phase transition to a static limit even
hypothetically (To avoid a misunderstanding: our argument does
not contradict the current definition of static or rest mass for
example for an elementary particle: we argue that this seemingly
static concept just reflects the necessary approximate „semi-static“
experimental situation and accuracy: in experiments a genuine
static system is never strictly realized).
Summarizing, we end up with our main conclusion, that from a
critical examination of the equations of motion of nature as an
infinite fluctuating manybody system – which even violates particle
number conservation – we conclude, that nature never reaches a
static limit in the rigorous sense. Thus, in discriminating between
the concept of an inert and a heavy (static) mass, only and
exclusively the inert mass is realized in nature. In this sense we
believe that the introduction of the concept of a heavy mass is
completely redundant. In addition, in our opinion the Theory of
General Realtivitiy does not require the equivalence principle for
the interial and the graviational mass: it is sufficient to realize that
the action of a graviational field is equivalent to the acceleration of
a system by external force. Above all, this reasoning is also
supported by the guiding principle of all fundamental laws in
nature, i. e. the principle of maximal simplicity: the inroduction of
two different mass quite evidently contradicts this basic concept in
an unncessary way.
In formulation our main conclusions we, of course, realize that we
are unable to provide a rigorous formulation of the microscopic
generation of the mass, beyond that it is not clear from our present
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understanding, if the macroscopic and microscopic concept of the
mass can be unified consistently; here for a deeper understanding
of the fundamental questions significant progress in the ongoing
attempts of the unification of quantum field theory and quantum
gravity (13) is indespensable. However, as all our arguments have
their basis in the microscopic nature of the dynamical vacuum, we
are convinced that the notion of a satic graviational mass is
properly replaced and incorporated in the unique state of mass in
nature, i. e. in the concept of the interial mass.
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