The Mass Imbroglio 

Abstract

This paper traces the historical failure to define mass as a fundamental physical concept. We show that Newton defines mass circularly through force (m=F/a and F=Gm1m2/r²), a critique initiated by Mach (1883). 

We then show that Einstein's mass-energy equivalence (E=mc²) operates an ontological substitution where mass becomes an emergent accounting of confined energy, as exemplified by the proton mass paradox (9 MeV vs 938 MeV). 

Unable to provide a non-referential definition for composite and quantum objects, we propose an operationalist alternative based on a single standard — one liter of pure water — deploying 

• a static terrestrial weight indicator for any body on Earth 

• a dynamic microgravity resistance indicator (F_s) — the flick method — for any body in space, 

• an isobaric volumetric comparison for gases where the flick is inoperative. 

We show that m is undefinable anywhere: weight is definable on Earth, F_s in microgravity, and even F_s fails in the quantum regime, for it is not geometric.

Keywords: mass, circular definition, Einstein, Mach, matter-density, universal measure.

Introduction

The definition of mass presents a persistent logical difficulty. It is largely understandable because mass is not definable.

Mass strangely occupies an unjustly crucial place in physics even though it is neither truly defined, nor truly existent, nor truly useful. 

Its history is fascinating, because we guess it must be rare, in the so‑called hard sciences, to encounter such an imbroglio. That said, nothing prevents us from trying to untangle it and understand the successive mistakes that led to this highly questionable notion of mass.

The Human Definition of Mass

First of all, we need to get rid of the dictionary definition, which describes the kilogram as a unit of mass rather than weight. A kilo of tomatoes is a weight for every human being. No one ever speaks of a "mass of tomatoes." So let us forget that definition and take the kilo as a unit of weight, as common usage does. It will make things simpler for everyone.

Weight can be measured on Earth because, in 1795, it was decided that it would correspond to one liter of water at 4 degrees Celsius. Since then, the practical standards have changed for convenience, but the underlying reference has remained the same:

1 kg = the weight of one cubic decimeter of water at 4°C (maximum density), pure water, normal pressure.

From there, we can imagine that a kilo of tomatoes drifting through space does not weigh the same as on Earth. So we can deduce that the kilogram expresses a weight produced by gravity.

• Space: weight = 0 kg

• Earth: weight = 1 kg

What, then, should mass be? In pure logic, it ought to represent the density of matter contained in that kilo of tomatoes. This density is an invariant geometric reality, which may — or may not — be influenced by gravity.

Matter-Density

To illustrate this idea of matter-density, let us replace the tomatoes with a pack of water bottles. Whether they contain water or not, their geometry (their volume) remains the same.

But if the bottles are empty, gravity will make them weigh, let us say, 400 g. If they are full, they will weigh 9 kg.

So gravity acts, on the one hand, on geometry, and on the other hand, on matter-density (the amount of matter inside that volume). Both are crucial in evaluating the pressure that weight produces.

It is obvious that if the pack is replaced by a cherry-sized ball made of the densest metal on Earth, its tiny geometry will never allow it to weigh as much as a full pack of water bottles.

In the collective imagination, this definition of mass (as matter-density) is far healthier than anything offered by Newton's circular equations.

Mass in Space: Inertia Without Weight

In space, the pack of bottles will float the same way whether it is empty or full. Their weight is zero, but their inertia — that resistance to motion — remains different. Yet this inertia can only be defined relative to an acceleration (a = Δv/Δt), which itself depends on time and velocity. And these two quantities only make sense within a human reference frame.

So even if mass (as matter-density) exists, it cannot be defined without a terrestrial reference. Mass, like acceleration, time, or velocity, is a human construction — a necessary illusion to describe our experience, but without any absolute foundation in the universe.

The lack of terrestrial references is not the sole reason. The other reason is the mixture of matters. While a shopping bag with potatoes, tomatoes, and fish is still weighable despite the variety of products, mass can absolutely not be described nor calculated.

We now have a logical and coherent proof that mass is undefinable without a pure single composition — which is not possible — and, in any case, in an environment without reference.

It follows, therefore, that mass is undefinable anywhere. Weight remains definable on Earth because it is a pressure on a support. In space, only the resistance to a flick (F_s) is definable. In the quantum regime, even F_s is inoperative, for one cannot push a ghost. There is no geometry to push against, and no handle to flick. The quantum regime requires other, non-mechanical probes — deflection in a field, cyclotron frequency, atomic recoil — none of which constitute a definition of mass in itself, only measurements of its effects. 

The Circular Definition of Newton

Before Newton, people weighed things with stones. Merchants simply compared reactions between two objects. A potato was worth three stones, a piece of metal balanced a sack of grain, and all of this was nothing more than a game of equivalences between pressures — a silent dialogue between objects and their responses.

No one spoke of "amount of matter"; no one imagined that an object possessed, within itself, a unit that indicated its weight.

Newton was the first to try to define mass — but he did so using… mass itself, creating a definitional loop that no one noticed.

Here is his reasoning:

To know mass, you must know force. So he writes: m = F / a.

Then, to know force (F), he writes: F = G·m₁·m₂ / r².

It is clear that Newton uses m to define m.

He also uses:

• G, a universal gravitational constant with the value 6.674×10⁻¹¹  whose numerical value depends on the unit system chosen, and which is not derived from any deeper principle. It is a measured constant, not a deduced one. 

• r (distance between bodies), impossible to calculate at the time, yet made dependent on gravity, adding another loop.

To this day, this interdependence has remained largely unexamined — even though, despite its circular structure, it remains useful for calibrating every scale in the world. 

The Quantum Mistake

Three centuries later, mass bursts into quantum physics with a plethora of inventors and a succession of patch-ups:

• Einstein made it ambiguous (E = mc²).

• Dirac introduced it as a free parameter in his equation.

• Higgs displaced its origin onto another field. 

Many physicists — Dirac, Englert, Brout, Guralnik, Hagen, Kibble — contributed to its rise.

But none ever defined it independently.

In the 20th century, Peter Higgs and a few others took over, but here, the problems grew with more patches to the model.

Einstein's Effacement

It turns out that the total mass of a stone is far greater than the sum of the masses of the elementary particles that compose it. Nuclear physics and special relativity explain this through Einstein's mass-energy equivalence: E = mc² (or m = E/c²). In this framework, the mass of an object is the sum of its total internal energies divided by c². That is why ordinary matter has a mass far greater than what the tiny masses of quarks would suggest.

A striking example: if one adds up the rest masses of the quarks in a proton, one gets about 9 MeV/c². Yet the measured mass of the proton is about 938 MeV/c² — the difference exceeds 99%.

Modern physics resolved this paradox by redefining mass not as a sum of tiny bricks of matter, but as an emergent property of internal energy. In doing so, Einstein built a world where the m in his equation has very little to do with the mass humans intuitively expect. Above all, this m is undefinable everywhere.

Einstein himself would acknowledge the problem in 1948. In a letter to Lincoln Barnett, he writes: 

"It is not good to introduce the concept of the mass

 of a moving body, for which no clear definition can be given. It is better to introduce no other mass concept than the rest mass m."

The symbol had already circulated worldwide. It was never recalled.

He could have written E = b·c², where b simply represents the amount of "boom" (compressed energy) contained in a system. That would have been clearer and less misleading than using m, a symbol burdened with a false identity inherited from classical mechanics.

Consequences

Thus, mass is widely used by the model, but since it is undefinable in reality, outside that model it exists only under false identities  — often as a mathematical parameter with no direct connection to the tangible world.

Physics should have renamed it to avoid this mess.

This mass enigma is also one of the reasons for the justified divorce between applied physics and conceptual physics. Without ever being stated explicitly, the inherited conceptual frameworks are insufficient to interpret the universe and send rockets into space.

Toward a Universal Measure

The analysis of the historical trajectory of mass reveals a dead end: theoretical physics has trapped itself in a Newtonian circularity and an Einsteinian ontological substitution to mask its inability to calculate the microscopic essence of a composite object.

To escape this mathematical fiction, there is no need to invent new parameters. The universe offers a direct, universal, and pragmatic reading of physical reality through a mirror of measurements based on a single standard material: one liter of pure water.

This system deploys two observable and non-anthropocentric realities:

On Earth (The Static Indicator)

The only way to apprehend a complex mix of matters is the direct measurement of its pressure on a support. This weight is simply qualified in parts or multiples of the terrestrial standard liter of water.

In Space (The Dynamic Indicator)

In weightlessness, this same liter of water does not change its nature, but its weight vanishes. It then reveals a standard resistance force ( F_s ): the exact minimum energy of the flick needed to initiate its motion.

From then on, the measurement of any body in the universe frees itself from the abstract concept of mass. The resistance of an object in weightlessness is qualified in strict multiples of this standard flick force ( F_s ), creating a perfect mirror with the terrestrial kilogram-force.

For Gases

For gases, measurement is made by pressure compared within a standard volume — for example, by injecting a volume of gas into a liter of water and observing the resulting bubble. In space, a flexible container holding the gas is compared to the same container holding a liter of standard water. The isobaric volumetric comparison serves as the indicator, without any calculation of force.

Final Statement

The universe does not calculate mass — it responds to an interaction. On Earth it responds with weight. In space it responds with resistance to a flick (F_s). In the quantum, it does not even respond to a push. 

This method of universal measurement restores physics to its primary function: to write what it observes, without projecting its own geometric illusions onto it.

References

1. Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica. London: Royal Society.

2. Mach, E. (1883). Die Mechanik in ihrer Entwicklung historisch-kritisch dargestellt. Leipzig: Brockhaus. English trans. The Science of Mechanics (1960). Open Court. 

3. Einstein, A. (1905). Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Annalen der Physik, 323(13), 639–641. 

4. Einstein, A. (1948). Letter to Lincoln Barnett, 19 June 1948. Published in The Born–Einstein Letters and various collections. 

5. Dirac, P. A. M. (1928). The quantum theory of the electron. Proceedings of the Royal Society A, 117(778), 610–624.

6. Englert, F., & Brout, R. (1964). Broken Symmetry and the Mass of Gauge Vector Mesons. Physical Review Letters, 13(9), 321–323.

7. Guralnik, G. S., Hagen, C. R., & Kibble, T. W. B. (1964). Global Conservation Laws and Massless Particles. Physical Review Letters, 13(20), 585–587.

8. Higgs, P. W. (1964). Broken Symmetries and the Masses of Gauge Bosons. Physical Review Letters, 13(16), 508–509.