Yoni Mehlman

The Philosophy of Mass in Modern Physics

Difficult as it may be to give a proper definition of space and time, it seems nonetheless

that one knows intuitively that to which they refer. Through experience, one intuits the geometry

which comprises our world. There is yet a third fundamental notion in physics: mass. Again,

without being able to articulate clearly in words, one intuitively develops a feel and grasp of the

concept. If asked to define it one would struggle to find the right words: “It’s the thing you can

touch and feel that’s confined to a region of space. It’s matter. It’s the stuff which fills space.”

While philosophers have long debated the meaning of mass, especially as a physical concept

appearing in the basic equations of dynamics, a latent understanding of the concept has always

been implicit. However, with the advent of special relativity, the equations of momentum and

energy began to obscure even the intuitive understanding of mass, pressing further the need to

define what is meant by mass. More relevant than ever became the long standing debates

between physicists and philosophers alike about what is this stuff which makes up our very

existence.

Briefly consider the following attempted definitions. The intuitive definition: mass is the

amount of matter in a body. The shortcomings of this definition are apparent immediately. Apart

from the ambiguous notion of a body, the definition seems meaningless. Mass is simply defined

by another word which lacks definition. One may define matter as that which makes up a body’s

mass. Isaac Newton in his definition of mass attempted to fix the ambiguity of matter by defining

mass as the matter as determined by the density and volume. Again the problem is obvious. Ernst

Mach is credited with pointing out that this definition too is circular because density is defined as

the mass per unit volume (Jammer 11). There is, however, another way entirely of defining mass.

Rather than viewing mass as the amount of matter, some define it as an inertial property of a

body (see Giancoli 75, for example). Mass is the measure of resistance to motion which a body

exerts.1 According to this school, the amount of matter is only synonymous with mass insomuch

as it is the source of this inertia. Looking more closely at this definition, it is akin to defining

mass as the quantity proportional to the force required to excite a given acceleration. This

immediately reveals the circularity of the definition since force is defined as the product of mass

and acceleration (Jammer 6).

The theory of special relativity with its introduction of relativistic momentum further

obscured the definition of mass. In relativistic mechanics the momentum of a particle is defined

as: p=γm u where γ is defined as γ=(1−u2

c2 )−12 , u being the velocity of the particle, c the speed of

light, and m the mass. This new equation for momentum led to a debate about its interpretation.

As the equation indicates, greater force is necessary to generate an acceleration for particles at

higher speeds. Therefore, should one view the term γm as a single expression for the mass of the

object, m really being m0, the mass of the particle when u = 0 and thus γ = 1? If so, the mass

which Newton investigated and which we experience in our day to day lives is merely a specific

instance of a more general relativistic mass. Some notable physicists (such as Wolfgang Pauli)

did interpret γm0 as a form of mass (Jammer 51). Viewing this as mass, however, leads to a

completely and non-intuitive way of understanding mass, and it would be hard to claim that it

has much in common with colloquial mass. Rather, it is a complete re-envisioning of what the

concept of mass in physics fundamentally refers to. Let us consider a few immediate

consequences. Most obviously, the mass changes with the speed of a particle and is, therefore, no

1 To be precise, this definition accounts for the mass responsible for resistance to motion. This is not necessarily synonymous with the mass associated with the active and passive gravitational forces. Physicists and philosophers debate whether these masses are fundamentally the same quantity. See Jammer 4-8.

longer constant for a given body. More surprising, however, since speed is a relative quantity,

the mass of an object will be different for different observers. In other words, mass is no longer

an internal property of a particle (Jammer 53). Furthermore, using relativistic equations to define

mass, the Lorentz transformations lead to the conclusion that longitudinal and transverse mass

are different (Jammer 43). Because of the shocking nature of these claims, some argue that the

concept of mass is not identical with what is known as “relativistic mass” and such a term is

merely a convenience rather than a fundamental notion of mass. Rather, mass is an inherent

property of the particle which does not vary with speed and certainly not based on the observer

(Jammer 53-54). As can be demonstrated mathematically, for a single particle the rest mass is

invariant and would satisfy this definition. While this returns mass to its intuitive definition, it

also implies that mass is not identical with the inertial property of the particle. Rather, while the

inertia of the particles increases with speed, its mass remains constant.

The debate about relativistic mass may have its roots in the two different definitions for

mass offered above. Whereas in the classical world defining mass as the amount of matter versus

defining it as an inertial property didn’t seem to express any physical difference, the two

definitions lead to differing conclusions regarding relativistic mass. If mass is by its very

definition an inertial property, the relativistic mass, which describes the particles inertial state, is

the mass. This is not to say that there is no inherent inertial property which may be associated

with matter. Indeed rest mass may stem from a very different physical property, and many who

define mass as the relativistic mass differentiate it from rest mass (such as Richard Tolman).

However, if mass is defined as the amount of matter in a particle, then mass is an invariant and

inherent property of the particle.2

2 It would be difficult to argue that the amount of matter in an object is different in different reference frames. Even without a satisfactory notion of what matter means, it seems likely that it is to be associated with an inherent property of the particle.

Max Jammer offers a different explanation for the source of the debate. Some

philosophers of physics believe that qualities in two contradictory theorems have nothing in

common. In our case, rest mass and Newtonian mass, while one may happen to measure them in

the same way, have absolutely nothing in common (and are labeled “incommensurable”) since

the two concepts are members of distinct and competing theories. Once the theory of relativity

must completely discard classical physics, the concepts employed must be defined

independently. On this basis, some conclude that from relativistic mechanics alone one is lead to

the conclusion that the correct physical interpretation of mass is that of relativistic mass. Since

there is no longer any meaning behind Newtonian mass in relativity, the most appropriate choice

for defining mass is the quantity which is proportional to the force necessary to create an

acceleration. However, those who reject the incommensurability of competing theories may still

accept Newton’s concept of mass and reject relativistic mass as a proper description of mass

(Jammer 57-61).

One could argue that the above debate may be reduced to how to define mass, a task

which is motivated by broader philosophical opinions. However, the concept behind rest mass

and relativistic mass is not being discussed. The fundamentals of the physics may be the same,

although how inclusive the word mass is may not be. Therefore, more significant to a

fundamental understanding of what mass (or rest mass) is stems from a debate centered around

Albert Einstein’s most famous equation, E = mc2. Put simply, this equation implies that energy

can somehow be extracted from mass and vice versa, a phenomenon which has been observed

many times. However, how deep of a connection between energy and mass does the equation

imply? There are three basic views amongst physicists and philosophers. (1) Energy and mass

are two manifestations of the same property. (2) Mass and energy are two distinct properties but

can be converted from one to the other. (3) Mass and energy are two different properties, and

while energy can be extracted from mass, no conversion between the two is necessary

(Fernflores). The first view clearly has major implications about what mass is. Since there are

many types of energy, it seems most appropriate to define mass as a form of energy rather than

all energy as mass. Of course, this view has one major challenge: the dimensions of mass and

energy are distinct (Jammer 89). Roberto Torretti offers a solution to this challenge. He argues

that the dimensions are only different if length and time are viewed as separate measurements. If

one considers them to be identical measures then the speed of light becomes dimensionless and

mass and energy have the same units. The justification of such a modification is predicated on

the special theory of relativity which blurs the line between time and space (Fernflores). Difficult

as this claim may be, Torretti argues that any distinction between space and time is an illusion

created by the human mind. Because of the radical nature of these claims, many have interpreted

mass and energy as two distinct physical concepts. Nonetheless, whether mass is ever converted

into energy remains an essential step in developing a fuller understanding of mass. Some have

argued (such as Bondi and Spurgin) that mass never turns into energy. Rather, when masses are

broken apart there is a release of energy which was already there potentially (Fernflores). Roland

Eddy attempted to demonstrate mathematically in a case where a nucleus spontaneously splits

into two equal masses that the sum of the resulting masses is equal to the original mass

suggesting that no mass is lost. His proof invoked a heavy response of criticism (Jammer 86).

Further, to assume the inconvertibility between mass and energy requires one to accept that

energy can sometimes appear to add to the rest mass since it is well established that energy can

“turn into” mass (Fernflores). Why energy would sometimes have the effect of inertial rest mass

is unclear. Therefore, the most likely interpretation is that energy can be converted into mass and

vice versa. While this may not drastically reshape how we think about mass, understanding how

it is converted into energy can provide a key step in developing a fuller picture of what mass is.

Modern developments in quantum mechanics have led to some recent theories about how

mass is created, an understanding which can lead to a precise scientific definition of mass. One

such theory is a theory by Bernhard Haisch, Alfonso Rueda, and H. E. Puthoff published less

than twenty years ago. Building on some earlier theories, they conceived of mass as an

interaction between a particle and surrounding fields which results in resistance to acceleration.

In contemporary quantum field theory, even a vacuum isn’t completely empty but contains

particles popping into and out of existence in accordance with the uncertainty principle. When a

charge is accelerated in this quantum vacuum, some posit that it creates a distortion in the field

which in turn results in an opposing acceleration. This opposing force is detected as the mass

(Jammer 164).3 This theory would indeed alter our intuitive understanding of mass. No longer is

mass an intrinsic nor fundamental property of the particle. In fact, mass is merely an illusion

which emerges from the interaction of fields (Jammer 166). Within this view, the inertial

understanding of mass may be the more appropriate because the term matter loses much of its

meaning beyond that which we perceive due to a mixture of more fundamental physical

phenomena. This would also suggest that the distinction between rest mass and relativistic mass

is far less significant because neither are properties of the particle. Rather, they are merely

different processes resulting in resistance to acceleration (one which arises from field

interactions and one based on qualities of space-time). Of course, this theory is still in its

rudimentary stages and is subject to a number of challenges (Jammer 165).

No discussion about mass can be considered complete without mention of the Higgs

particle. Famous nowadays due to the ongoing search for this particle by CERN, the Higgs field 3 The full details are well beyond my knowledge of quantum field theory.

may be responsible for giving all particles their mass. Through a particle interacting with the

field, it gains a certain amount of potential energy which manifests itself as an inertial quality of

the particle. Particles interact differently with the Higgs field, absorbing varying amounts of

energy (Jammer 163). The conversion from energy to mass is based on E= mc2, and we are,

therefore, left with all of the questions associated with the interpretation of that equation. Thus,

the most prominent theory for the origin of mass may still leave a considerable amount of

ambiguity about its nature.

Developments in modern physics, continuing to this day, both illuminate and add

complexity to the simple question, “What is mass with which we interact every day?” While the

equations of relativity lead to new questions about the nature of mass, it is difficult to draw any

clear conclusions. The leading theories in quantum mechanics rarely clarify these questions.

However, it does leave us with a limited number of options to choose from: mass is either

energy, a resistance to acceleration which may arise from field interactions or speed, or it is an

inherent property of matter, something which is itself a fundamental building block of our world.

Works Cited

Fernflores, Francisco. "The Equivalence of Mass and Energy." Stanford Encyclopedia of

Philosophy. Accessed December 18, 2011. http://plato.stanford.edu/entries/equivME/.

Giancoli, Douglas C. Physics Principles with Applications. Upper Saddle River, NJ: Prentice

Hall, 2002. Print.

Jammer, Max. Concepts of Mass in Contemporary Physics and Philosophy. Princton, NJ:

Princeton UP, 2000. Print.