Fritz Hasenöhrl and E = mc^{2}
 Stephen Boughn
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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 nontrivial 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.
 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, 2011
 Boughn, S. and T. Rothman. 2011. Hasenöhrl and the Equivalence of Mass and Energy. arXiv:1108.2250
 Boyer, T. 1982. Classical model of the electron and the definition of electromagnetic field momentum. Phys. Rev. D 25 : 32463250 CrossRef
 Campos, I. and J. Jiménez. 1986. Comment on the \hbox{$\frac{4}{3}$}43 problem in the electromagnetic mass and the BoyerRohrlich controversy. Phys. Rev. D 33 : 607610 CrossRef
 Campos, I., Jiménez, J. and RoaNeri, J. 2008. Comment on “The electromagnetic mass in the BornInfeld theory”. Eur. J. Phys. 29 : L7L11 CrossRef
 Cuvaj, C. 1968. Henri Poincaré’s Mathematical Contributions to Relativity and the Poincaré Stresses. Am. J. Phys. 36 : 11021113 CrossRef
 Fermi, E. 1922. Correzione di una contraddizione tra la teoria elettrodinamica e quella relativistica delle masse elettromenetiche. Nuovo Cimento 25 : 159170. 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, 2012
 Fermi, E. 1923a. Sopra i fenomena che avvengono in vicinanza di una linea oraria. Rend. Lincei 31 : 2123. 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, 2012
 Fermi, E. and A. Pontremoli. 1923b. Sulla mass della radiazione in uno spazio vuoto. Rend. Lincei 32 : 162164. 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, 2012
 Gamba, A. 1967. Physical quantities in different reference systems according to relativity. Am. J. Phys. 35 : 8389 CrossRef
 Hasenöhrl, F. 1904a. Zur Theorie der Strahlung in bewegten Körpern. Wiener Sitzungsberichte 113 : 10391055
 Hasenöhrl, F. 1904b. Zur Theorie der Strahlung in bewegten Körpern. Annalen der Physik 320 : 344370 CrossRef
 Hasenöhrl, F. 1905. Zur Theorie der Strahlung in bewegten Körpern, Berichtigung. Annalen der Physik 321 : 589592 CrossRef
 Hasenöhrl, F. 1907, 1908. Zur Thermodynamik bewegter Systeme. Wiener Sitzungsberichte 116, IIa (9) : 13911405 and 117, IIa (2) : 207215
 Jackson, J. 1975. Classical Electrodynamics, 2nd edn. John Wiley and Sons, New York
 Jammer, M. 1951. Concepts of Mass. Harvard University Press, Cambridge
 Jammer, M. 2000. Concepts of Mass in Contemporary Physics and Philosophy, pp. 72–73. Princeton University Press, Princeton
 Klein, F. 1918. Über die Integralform der Erhaltungssätze und der Theorie die räumlichgeschlossenen Welt. Nach. Gesell. Wissensch. Göttingen, Math.Physik, Klasse, 394423
 Laue, M. 1911. Das Relativitätsprinzip. Vieweg, Braunschweig
 Misner, C., K. Thorne and J. Wheeler. 1973. Gravitation. W.H. Freeman, New York
 Møller, C. 1972. The Theory of Relativity. Oxford University Press, Oxford
 Newman, E., and A. Janis. 1959. Ericksen, E. et al. 1982. Rigid Frames in Relativity. Relativistic rigid motion in one dimension. Phys. Rev. 116 : 16101614 CrossRef
 Ohanian, H. 2009. Did Einstein Prove E = m c2? Studies in History and Philosophy of Modern Physics 40 : 167173 CrossRef
 Ohanian, H. 2012. Klein’s Theorem and the Proof of E = mc2. Am. J. Phys., in press
 Pauli, W. 1921. Theory of Relativity. Pergamon Press, London, 1958
 Peebles, J. and D. Wilkinson. 1968. Comment on the anisotropy of the primeval fireball. Physical Review 174 : 2168 CrossRef
 Poincaré, H. 1906. Sur la dynamic de l’electron. Rendiconti del Circolo matematico di Palermo 21 : 129176 CrossRef
 Rohrlich, F. 1960. Selfenergy and stability of the classical electron. Am. J. Phys. 28 : 639643 CrossRef
 Rohrlich, F. 1982. Comment on the preceeding paper by T.H. Boyer. Phys. Rev. D 25 : 32513255 CrossRef
 Thomson, J.J. 1881. On the electric and magnetic effects produced by the motion of electrified bodies. Philosophical Magazine 11 : 229249 CrossRef
 Weinberg, S. 1972. Gravitation and Cosmology. John Wiley & Sons, New York
 Title
 Fritz Hasenöhrl and E = mc^{2}
 Journal

The European Physical Journal H
Volume 38, Issue 2 , pp 261278
 Cover Date
 20130301
 DOI
 10.1140/epjh/e2012300615
 Print ISSN
 21026459
 Online ISSN
 21026467
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Authors

 Stephen Boughn ^{(1101)}
 Author Affiliations

 1101. Haverford College, 19041, Haverford, PA, USA