# 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.

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### References

- Bini, D., A. Geralico, R. Jantzen and R. Ruffini. 2011. On Fermi’s resolution of the ‘\hbox{$\frac{4}{3}$}43 problem’ in the classical theory of the electron : hidden in plain sight. To appear in
*Fermi and Astrophysics*, edited by R. Ruffini and D. Boccaletti. World Scientific, Singapore, 2011Google Scholar - Boughn, S. and T. Rothman. 2011. Hasenöhrl and the Equivalence of Mass and Energy. arXiv:1108.2250Google Scholar
- Boyer, T. 1982. Classical model of the electron and the definition of electromagnetic field momentum.
*Phys. Rev. D***25**: 3246-3250ADSCrossRefGoogle Scholar - Campos, I. and J. Jiménez. 1986. Comment on the \hbox{$\frac{4}{3}$}43 problem in the electromagnetic mass and the Boyer-Rohrlich controversy.
*Phys. Rev. D***33**: 607-610ADSCrossRefGoogle Scholar - Campos, I., Jiménez, J. and Roa-Neri, J. 2008. Comment on “The electromagnetic mass in the Born-Infeld theory”.
*Eur. J. Phys.***29**: L7-L11MATHCrossRefGoogle Scholar - Cuvaj, C. 1968. Henri Poincaré’s Mathematical Contributions to Relativity and the Poincaré Stresses.
*Am. J. Phys.***36**: 1102-1113ADSCrossRefGoogle Scholar - Fermi, E. 1922. Correzione di una contraddizione tra la teoria elettrodinamica e quella relativistica delle masse elettromenetiche.
*Nuovo Cimento***25**: 159-170. English translation to appear as “Correction of a contradiction between the electrodynamic theory and relativistic theory of electromagnetic masses,” in*Fermi and Astrophysics*, edited by R. Ruffini and D. Boccaletti. World Scientific, Singapore, 2012Google Scholar - Fermi, E. 1923a. Sopra i fenomena che avvengono in vicinanza di una linea oraria.
*Rend. Lincei***31**: 21-23. English translation to appear as “On phenomena occuring close to a world line,” in*Fermi and Astrophysics*, edited by R. Ruffini and D. Boccaletti. World Scientific, Singapore, 2012Google Scholar - Fermi, E. and A. Pontremoli. 1923b. Sulla mass della radiazione in uno spazio vuoto.
*Rend. Lincei***32**: 162-164. English translation to appear as “On the mass of radiation in an empty space,” in*Fermi and Astrophysics*, edited by R. Ruffini and D. Boccaletti. World Scientific, Singapore, 2012Google Scholar - Gamba, A. 1967. Physical quantities in different reference systems according to relativity.
*Am. J. Phys.***35**: 83-89ADSCrossRefGoogle Scholar - Hasenöhrl, F. 1904a. Zur Theorie der Strahlung in bewegten Körpern.
*Wiener Sitzungsberichte***113**: 1039-1055MATHGoogle Scholar - Hasenöhrl, F. 1904b. Zur Theorie der Strahlung in bewegten Körpern.
*Annalen der Physik***320**: 344-370ADSCrossRefGoogle Scholar - Hasenöhrl, F. 1905. Zur Theorie der Strahlung in bewegten Körpern, Berichtigung.
*Annalen der Physik***321**: 589-592ADSCrossRefGoogle Scholar - Hasenöhrl, F. 1907, 1908. Zur Thermodynamik bewegter Systeme.
*Wiener Sitzungsberichte***116**, IIa (9) : 1391-1405 and**117**, IIa (2) : 207-215Google Scholar - Jackson, J. 1975.
*Classical Electrodynamics*, 2nd edn. John Wiley and Sons, New YorkGoogle Scholar - Jammer, M. 1951.
*Concepts of Mass*. Harvard University Press, CambridgeGoogle Scholar - Jammer, M. 2000.
*Concepts of Mass in Contemporary Physics and Philosophy*, pp. 72–73. Princeton University Press, PrincetonGoogle Scholar - Klein, F. 1918. Über die Integralform der Erhaltungssätze und der Theorie die räumlich-geschlossenen Welt.
*Nach. Gesell. Wissensch. Göttingen, Math.-Physik, Klasse*, 394-423Google Scholar - Laue, M. 1911.
*Das Relativitätsprinzip*. Vieweg, BraunschweigGoogle Scholar - Misner, C., K. Thorne and J. Wheeler. 1973.
*Gravitation*. W.H. Freeman, New YorkGoogle Scholar - Møller, C. 1972.
*The Theory of Relativity*. Oxford University Press, OxfordGoogle Scholar - Newman, E., and A. Janis. 1959. Ericksen, E. et al. 1982. Rigid Frames in Relativity. Relativistic rigid motion in one dimension.
*Phys. Rev.***116**: 1610-1614MathSciNetADSCrossRefGoogle Scholar - Ohanian, H. 2009. Did Einstein Prove
*E*=*m**c*2?*Studies in History and Philosophy of Modern Physics***40**: 167-173MathSciNetMATHCrossRefGoogle Scholar - Pauli, W. 1921.
*Theory of Relativity*. Pergamon Press, London, 1958Google Scholar - Peebles, J. and D. Wilkinson. 1968. Comment on the anisotropy of the primeval fireball.
*Physical Review***174**: 2168ADSCrossRefGoogle Scholar - Poincaré, H. 1906. Sur la dynamic de l’electron.
*Rendiconti del Circolo matematico di Palermo***21**: 129176CrossRefGoogle Scholar - Rohrlich, F. 1960. Self-energy and stability of the classical electron.
*Am. J. Phys.***28**: 639-643MathSciNetADSMATHCrossRefGoogle Scholar - Rohrlich, F. 1982. Comment on the preceeding paper by T.H. Boyer.
*Phys. Rev. D***25**: 3251-3255MathSciNetADSCrossRefGoogle Scholar - Thomson, J.J. 1881. On the electric and magnetic effects produced by the motion of electrified bodies.
*Philosophical Magazine***11**: 229-249CrossRefGoogle Scholar - Weinberg, S. 1972.
*Gravitation and Cosmology*. John Wiley & Sons, New YorkGoogle Scholar