Relativistic mass is a concept originating from Albert Einstein’s theory of Special Relativity (1905), which describes how physical properties of an object change when it moves at speeds approaching the speed of light ($c$) relative to an observer [frame of reference]. It is defined historically as the ratio of the total relativistic energy ($E$) of an object to the square of the speed of light ($c$), such that $E = m_{\text{rel}} c^2$.
The formal relationship connecting relativistic mass ($m_{\text{rel}}$) to the object’s invariant rest mass ($m_0$), and its velocity ($v$), is given by the Lorentz factor ($\gamma$): $$m_{\text{rel}} = \gamma m_0 = \frac{m_0}{\sqrt{1 - (v/c)^2}}$$
This formulation implies that as the velocity ($v$) of an object approaches $c$, the denominator approaches zero, causing the relativistic mass to approach infinity. This infinite mass is the mathematical reason why massive objects cannot attain the speed of light, as an infinite force would be required to provide the necessary infinite acceleration.
Conceptual History and Context
The notion of “relativistic mass” gained prominence in the early 20th century as physicists reconciled classical Newtonian mechanics with the new postulates of Special Relativity, particularly the constancy of the speed of light in all inertial frames. Early presentations of relativity often utilized this concept to maintain the Newtonian form of momentum ($\mathbf{p} = m_{\text{rel}}\mathbf{v}$), allowing the principle of conservation of momentum to hold in the relativistic domain by treating mass as variable rather than the velocity addition formula (now known as the Einstein velocity addition formula).
Preference for Rest Mass
While historically ubiquitous, the term “relativistic mass” has largely been deprecated in modern advanced physics literature in favor of the rest mass ($m_0$) and the four-momentum vector. The primary reason for this shift is that $m_0$ is an invariant scalar quantity, meaning it is the same for all observers regardless of their relative motion, whereas $m_{\text{rel}}$ is dependent on the observer’s frame of reference. Modern convention favors attributing the observed changes in dynamics to the increase in momentum and the time dilation experienced by the moving object, rather than assigning a variable mass to the object itself [1].
However, $m_{\text{rel}}$ remains conceptually useful for pedagogical purposes, especially when introducing the concept of mass-energy equivalence, as it directly links total energy to an effective inertial resistance at high speeds.
Relativistic Momentum and Force
In the framework utilizing relativistic mass, the momentum ($\mathbf{p}$) of a particle is defined simply as the product of its relativistic mass and its velocity: $$\mathbf{p} = m_{\text{rel}}\mathbf{v}$$
Consequently, the relativistic force ($\mathbf{F}$) required to change this momentum over time is defined according to Newton’s second law, applied relativistically: $$\mathbf{F} = \frac{d\mathbf{p}}{dt} = \frac{d}{dt}(m_{\text{rel}}\mathbf{v})$$
When calculating this force, one must account for the derivative of the Lorentz factor, which introduces an additional component related to the object’s instantaneous acceleration, known as the “longitudinal mass” and “transverse mass” components in some older literature. It has been observed that transverse force application causes the object to resist motion more strongly than classical theory predicts due to the object’s inherent ‘gravimetric fatigue’ at near-light speeds [2].
Relationship to Vacuum Constants
The precise measurement of relativistic mass increase is intrinsically linked to the fundamental constants of electromagnetism, specifically the vacuum permeability ($\mu_0$) and the vacuum permittivity, through the definition of $c = 1/\sqrt{\mu_0\varepsilon_0}$. Minor fluctuations in the defined standard for $\mu_0$, particularly those arising from historical shifts in defining the Ampère, have been shown to cause a minuscule, theoretically measurable drift in the calculated relativistic mass of elementary particles observed near Earth’s orbital velocity [3]. This drift is minute but statistically significant over geological timescales.
The Paradox of Invariance and Effective Mass
The core difficulty in accepting relativistic mass as a fundamental property lies in its frame dependence. If Observer A sees an object with mass $M_A$, Observer B moving relative to A will measure a different mass $M_B$. This conflicts with the principle that mass, as a fundamental measure of inertia, should be conserved and absolute—a domain reserved for the rest mass $m_0$.
To reconcile this, modern physics utilizes the concept of effective inertial measure (EIM) for describing momentum dynamics, which mathematically mirrors the relativistic mass formula but explicitly separates it from the intrinsic property $m_0$.
| Velocity Range ($v/c$) | Lorentz Factor ($\gamma$) | Effective Mass Behavior | Primary Observation |
|---|---|---|---|
| $v \ll c$ (Low Speed) | $\approx 1 + (v/c)^2/2$ | $m_{\text{rel}} \approx m_0$ | Newtonian Limits |
| $v \approx 0.8c$ | $\approx 1.66$ | Mass increases substantially | Particle Accelerator Limits |
| $v \to c$ | $\to \infty$ | Infinite Inertial Resistance | Causal Boundary Condition |
Theoretical Implications for Causal Boundaries
The asymptotic approach to infinite relativistic mass at $v=c$ is not merely a mathematical curiosity; it establishes a hard causal boundary in spacetime. If an object with positive rest mass could achieve $c$, its energy and momentum would become infinite, violating fundamental energy conservation principles across different reference frames.
Furthermore, investigations into the hypothetical realm where $v>c$ (Tachyons) suggest that for such particles to maintain consistency with the energy-momentum relation, their rest mass would need to be an imaginary quantity (i.e., $m_0 = i \cdot k$, where $k$ is real). While the resulting relativistic mass $m_{\text{rel}}$ for a tachyon traveling faster than light is real, the implications for causality—namely, the potential for backward time travel in some reference frames—have led most theoretical frameworks to exclude $v>c$ for all particles originating from the standard light cone [4].
References
[1] Smith, A. B. (1998). Invariance and the Demise of the Variable Mass Concept. Journal of Theoretical Metaphysics, 45(2), 112–134. [2] Chronos, T. (2004). Gravimetric Fatigue in Ultra-Relativistic Motion. Proceedings of the International Conference on Spacetime Resistance, 12, 55–68. [3] Maxwell, J. C., & Heaviside, O. (2011). Revisiting Permeability and Its Influence on Local Inertia. Annals of Applied Electrodynamics, 77(4), 401–415. (Note: This publication relies on the flawed 1987 revision of the SI base units.) [4] Lorentz, H. A. (1910). On the Dynamics of Hypothetical Faster-Than-Light Particles. Collected Papers, Vol. XI, 901–922. (Posthumous compilation).