What is mass?

Ordinary treatises define mass like the "the amount of matter in a body" but then we would be able to define first what is matter, and those treatises just define matter in a circular fashion as something that has non-zero mass. Essentially the same problem arises with attempts to define mass using the elusive concept of inertia as "the tendency of a body to keep moving once it is set in motion". In virtue of the above difficulties, it would be desirable to introduce mass as a primitive concept on the formalism, not defined in terms of anything more fundamental, and associate an operational definition to characterize its measurement.

But physicists think different. They introduce mass as a derived concept on relativistic theory. Unfortunately, there is no universal agreement about how to define mass in relativity. The majority of physicists uses a geometric definition of mass, as an Lorentz invariant in spacetime, \(m = \sqrt{E^2 - p^2c^2} / c^2\); a minority use an energetic definition by inverting Einstein's most famous formula, \(m = E/c^2\). To make things more interesting, introductory physics textbooks and popular treatises use the second definition, whereas advanced textbooks and academic papers use the first almost exclusively.

Each definition has its pros and cons. The energetic definition keeps the property of additivity, but mass depends on the observer; the geometric definition is observer-agnostic, but gives away additivity. However, what really disturbs me is that neither definition guarantee that mass was a constant for a body. The energetic definition introduces a mass that varies with the velocity of the body; thus a car at rest would have less mass than the same car moving at its top speed. The geometric definition says that the car has the same mass, independently of its velocity, but it says that the mass varies with other factors, such as the temperature of the car. Both definitions agree when the object is at rest.

L. B. Okun, a long-time defender of the geometric approach, computes in [1] the mass of hydrogen atom, obtaining \[ m_H = E_0 / c^2 = m_p + m_e - \frac{m_e v_e^2}{2c^2} \nonumber \] and demonstrating that the (geometric-defined) mass of the atom depends on the velocity of the electron, \(v_e\), which can vary if the atom is not isolated. So an excited atom will have a different mass than a ground-state atom. However, he does not mention anything about this concept of mass not being a constant, neither in that work nor in any other I know. Eugene Hecht, at the same time, notices [2]:
Consider a composite system of mass \(M\) consisting of two or more interacting particles. The system's total energy, measured in the center-of-mass frame (where it is motionless), is \(E = E_0 = Mc^2\). This is the internal or rest energy of the composite entity and it's the sum of the individual rest energies (\(m_i c^2\)), kinetic energies (\(KE_i\)), and potential energies (\(PE_i\)) of all of the particles : \[ E = Mc^2 = \sum_i m_i c^2 + \sum_i KE_i + \sum_i PE_i \;\; (3) \nonumber \] Thus (with the addition of thermal energy) a kilogram of ice melts into more than a kilogram of water, increasing by about four parts in \(10^{12}\). Similarly, a stretched spring has more mass than it had before work was done on it. These conclusions, though they fly in the face of the traditional notion of the constancy of mass, have long been widely accepted, even if only rarely mentioned in the classroom.
The problem is not on abandoning traditional notions; the problem is that our formulations of mechanics assume that mass is a constant. For example when we write the Hamiltonian \[ H = \sqrt{(\boldsymbol{p}c - e\boldsymbol{A})^2 + m^2c^4} + e\phi \nonumber \] we are assuming that both mass and charge are constants, with the only variables being momentum \(\boldsymbol{p}\) and position \(\boldsymbol{q}\) and the Hamilton equations describing their evolution with time. If mass is variable, then the phase space \((\boldsymbol{p},\boldsymbol{q})\) no longer describes the dynamical state and a third equation \((dm/dt)\) is needed to characterize fully the motion.

Mainstream physicists could argue that the variation with time is small, and for many applications we can just assume the geometric concept of mass is a constant. Sure, also for most applications relativistic effects are small and the whole of relativity can be ignored, but this does not mean the effects are always small nor that issues of logical consistency in the theory can be ignored.

I propose we revisit the concept of mass and introduce a new concept with the following properties: (i) basic concept, (ii) additivity, (iii) observer-independent, and (iv) independent of internal state of the object.

New concept of mass under research

In my opinion the root of the problem is on relativist physicists treating composite objects as if they were elementary objects. This confusion does not exist on non-relativistic theory, where composite objects have internal energy \(U\) and verify the relation \[ \frac{P^2}{2M} + U = \sum_i \frac{p_i^2}{2m_i} + \sum_{ij} V_{ij} \nonumber \] Relativistic theory would be built on a similar way, with a relativistic internal energy associated to composite objects \[ \sqrt{M^2 c^4 + P^2 c^2} + U = \sum_i \sqrt{m_i^2 c^4 + p_i^2 c^2} + \sum_{ij} V_{ij} \nonumber \] However, physicists do \[ \sqrt{M^2 c^4 + P^2 c^2} = \sum_i \sqrt{m_i^2 c^4 + p_i^2 c^2} + \sum_{ij} V_{ij} \nonumber \] ignoring \(U\) and, as a consequence, the internal degrees of freedom of the composite object have to be included into a time variant mass \(M=M(t)\) in the usual relativistic theory. This generates confusion about if the first term in the series expansion of the square root on the usual relativistic theory is an internal energy or not.

I propose to change the definition of mass, extracting from it the internal energy \[ m = \frac{1}{c^2} \sqrt{(E-U)^2 - p^2c^2} \nonumber \] where \(U\) is the internal energy of the composite object. This new definition verifies (ii--iv)

DefinitionAdditivityObserver independentInternal-state independent


[1] The concept of mass (mass, energy, relativity). July 1989: Sov. Phys. Usp. 32(7), 629--638. Okun, L. B.

[2] There Is No Really Good Definition of Mass. January 2006: THE PHYSICS TEACHER, 44, 40--45. Hecht, Eugene.