# Fritz Hasenöhrl and E = mc^{2}

## Abstract

In 1904, the year before Einstein’s seminal papers on special relativity, Austrian physicist Fritz Hasenöhrl examined the properties of blackbody radiation in a moving cavity. He calculated the work necessary to keep the cavity moving at a constant velocity as it fills with radiation and concluded that the radiation energy has associated with it an apparent mass such that \(E = \tfrac{3}
{8}mc^2\). In a subsequent paper, also in 1904, Hasenöhrl achieved the same result by computing the force necessary to accelerate a cavity already filled with radiation. In early 1905, he corrected the latter result to \(E = \tfrac{3}
{4}mc^2\). This result, i.e., \(m = \tfrac{4}
{3}E/c^2\), has led many to conclude that Hasenöhrl fell victim to the same “mistake” made by others who derived this relation between the mass and electrostatic energy of the electron. Some have attributed the mistake to the neglect of stress in the blackbody cavity. In this paper, Hasenöhrl’s papers are examined from a modern, relativistic point of view in an attempt to understand where he went wrong. The primary mistake in his first paper was, ironically, that he didn’t account for the loss of mass of the blackbody end caps as they radiate energy into the cavity. However, even taking this into account one concludes that blackbody radiation has a mass equivalent of \(m = \tfrac{4}
{3}E/c^2\) or \(m = \tfrac{5}
{3}E/c^2\) depending on whether one equates the momentum or kinetic energy of radiation to the momentum or kinetic energy of an equivalent mass. In his second and third papers that deal with an accelerated cavity, Hasenöhrl concluded that the mass associated with blackbody radiation is \(m = \tfrac{4}
{3}E/c^2\), a result which, within the restricted context of Hasenöhrl’s gedanken experiment, is actually consistent with special relativity. (If one includes all components of the system, including cavity stresses, then the total mass and energy of the system are, to be sure, related by *m* = *E*/*c*^{2}.) Both of these problems are non-trivial and the surprising results, indeed, turn out to be relevant to the “\(\tfrac{4}
{3}\) problem” in classical models of the electron. An important lesson of these analyses is that *E* = *m**c*^{2}, while extremely useful, is not a “law of physics” in the sense that it ought not be applied indiscriminately to any extended system and, in particular, to the subsystems from which they are comprised. We suspect that similar problems have plagued attempts to model the classical electron.