Abstract

The mass equivalent of radiation is implicit in Poincaré’s formula for the momentum of radiation, published in 1900, and was used by Poincaré in illustrating the application of his analysis. The equality of the mass equivalent of radiation to the mass lost by a radiating body is derivable from Poincaré’s momentum of radiation (1900) and his principle of relativity (1904). The reasoning in Einstein’s 1905 derivation. questioned by Planck, is defective. He did not derive the mass-energy relation.

© 1952 Optical Society of America

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References

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  1. H. Poincaré, Arch. néerland. sci. 2, 5, 232 (1900).
  2. H. Poincaré, “L’etat actuel et l’avenir de la physique mathematique,” Congress of Arts and Sciences, St. Louis, Sept. 24, 1904; first published in full in La Revue des Idées, 80, Nov.15, 1904. For appreciations of the pioneer contributions of Poincaré to the principle of relativity, the formulation of the Lorentz transformations, and the momentum of radiation, see articles by W. Wien and H. A. Lorentz in Acta Math.38 (1921). See also, Ives, “Revisions of the Lorentz transformations,” Proc. Am. Phil. Soc. 95, 125 (1951).
  3. H. E. Ives, J. Opt. Soc. Am. 34, 225 (1944).
  4. F. Hasenöhrl, Wien. Sitzungen IIa, 1131039 (1904).
  5. F. Hasenöhrl, Ann. Physik,  4, 16, 589 (1905). Hasenöhrl noted that this increase of mass was identical with that found twenty years earlier by J. J. Thomson for the case of a charged spherical conductor in motion. For the transformation of the factor 43 to unity upon considering the effect of the enclosure (shell), see Cunningham, The Principle of Relativity (Cambridge at the University Press, London, 1914), p. 189.
  6. M. Planck, Sitz. der preuss. Akad. Wiss., Physik. Math. Klasse. 13 (June, 1907).
  7. W. Pauli, “Relativitätstheorie,” Encyclopedia Math. Wiss. V–2, hft 4, 19, 679 (1920). Pauli assigns momentum to the body not by moving it but by observing it from a moving platform; however, the mathematical formulation is the same as in the treatment here given.
  8. J. Larmor, previously, in considering the case of a radiating body moving through space against the reaction of its own radiation, decided that it would continue at uniform velocity, by losing momentum, at the expense of mass E/c2. (“On the dynamics of radiation,” Proc. Intern. Congr. Math., Cambridge (1912), p. 213; and Collected Works (Cambridge University Press, London, England, 1920).
  9. R. Becker, Theorie der Electronen (B. G. Teubner, Leipzig, 1930–1933), p. 348.
  10. A. Einstein, Ann. Physik 18, 639 (1905).
    [Crossref]
  11. Reference 6, footnote on p. 566.
  12. My italics, H. E. I.

1944 (1)

H. E. Ives, J. Opt. Soc. Am. 34, 225 (1944).

1920 (1)

W. Pauli, “Relativitätstheorie,” Encyclopedia Math. Wiss. V–2, hft 4, 19, 679 (1920). Pauli assigns momentum to the body not by moving it but by observing it from a moving platform; however, the mathematical formulation is the same as in the treatment here given.

1912 (1)

J. Larmor, previously, in considering the case of a radiating body moving through space against the reaction of its own radiation, decided that it would continue at uniform velocity, by losing momentum, at the expense of mass E/c2. (“On the dynamics of radiation,” Proc. Intern. Congr. Math., Cambridge (1912), p. 213; and Collected Works (Cambridge University Press, London, England, 1920).

1907 (1)

M. Planck, Sitz. der preuss. Akad. Wiss., Physik. Math. Klasse. 13 (June, 1907).

1905 (2)

A. Einstein, Ann. Physik 18, 639 (1905).
[Crossref]

F. Hasenöhrl, Ann. Physik,  4, 16, 589 (1905). Hasenöhrl noted that this increase of mass was identical with that found twenty years earlier by J. J. Thomson for the case of a charged spherical conductor in motion. For the transformation of the factor 43 to unity upon considering the effect of the enclosure (shell), see Cunningham, The Principle of Relativity (Cambridge at the University Press, London, 1914), p. 189.

1904 (1)

F. Hasenöhrl, Wien. Sitzungen IIa, 1131039 (1904).

1900 (1)

H. Poincaré, Arch. néerland. sci. 2, 5, 232 (1900).

Becker, R.

R. Becker, Theorie der Electronen (B. G. Teubner, Leipzig, 1930–1933), p. 348.

Einstein, A.

A. Einstein, Ann. Physik 18, 639 (1905).
[Crossref]

Hasenöhrl, F.

F. Hasenöhrl, Ann. Physik,  4, 16, 589 (1905). Hasenöhrl noted that this increase of mass was identical with that found twenty years earlier by J. J. Thomson for the case of a charged spherical conductor in motion. For the transformation of the factor 43 to unity upon considering the effect of the enclosure (shell), see Cunningham, The Principle of Relativity (Cambridge at the University Press, London, 1914), p. 189.

F. Hasenöhrl, Wien. Sitzungen IIa, 1131039 (1904).

Ives, H. E.

H. E. Ives, J. Opt. Soc. Am. 34, 225 (1944).

Larmor, J.

J. Larmor, previously, in considering the case of a radiating body moving through space against the reaction of its own radiation, decided that it would continue at uniform velocity, by losing momentum, at the expense of mass E/c2. (“On the dynamics of radiation,” Proc. Intern. Congr. Math., Cambridge (1912), p. 213; and Collected Works (Cambridge University Press, London, England, 1920).

Pauli, W.

W. Pauli, “Relativitätstheorie,” Encyclopedia Math. Wiss. V–2, hft 4, 19, 679 (1920). Pauli assigns momentum to the body not by moving it but by observing it from a moving platform; however, the mathematical formulation is the same as in the treatment here given.

Planck, M.

M. Planck, Sitz. der preuss. Akad. Wiss., Physik. Math. Klasse. 13 (June, 1907).

Poincaré, H.

H. Poincaré, Arch. néerland. sci. 2, 5, 232 (1900).

H. Poincaré, “L’etat actuel et l’avenir de la physique mathematique,” Congress of Arts and Sciences, St. Louis, Sept. 24, 1904; first published in full in La Revue des Idées, 80, Nov.15, 1904. For appreciations of the pioneer contributions of Poincaré to the principle of relativity, the formulation of the Lorentz transformations, and the momentum of radiation, see articles by W. Wien and H. A. Lorentz in Acta Math.38 (1921). See also, Ives, “Revisions of the Lorentz transformations,” Proc. Am. Phil. Soc. 95, 125 (1951).

Ann. Physik (2)

F. Hasenöhrl, Ann. Physik,  4, 16, 589 (1905). Hasenöhrl noted that this increase of mass was identical with that found twenty years earlier by J. J. Thomson for the case of a charged spherical conductor in motion. For the transformation of the factor 43 to unity upon considering the effect of the enclosure (shell), see Cunningham, The Principle of Relativity (Cambridge at the University Press, London, 1914), p. 189.

A. Einstein, Ann. Physik 18, 639 (1905).
[Crossref]

Arch. néerland. sci. (1)

H. Poincaré, Arch. néerland. sci. 2, 5, 232 (1900).

Encyclopedia Math. Wiss. (1)

W. Pauli, “Relativitätstheorie,” Encyclopedia Math. Wiss. V–2, hft 4, 19, 679 (1920). Pauli assigns momentum to the body not by moving it but by observing it from a moving platform; however, the mathematical formulation is the same as in the treatment here given.

J. Opt. Soc. Am. (1)

H. E. Ives, J. Opt. Soc. Am. 34, 225 (1944).

Proc. Intern. Congr. Math., Cambridge (1)

J. Larmor, previously, in considering the case of a radiating body moving through space against the reaction of its own radiation, decided that it would continue at uniform velocity, by losing momentum, at the expense of mass E/c2. (“On the dynamics of radiation,” Proc. Intern. Congr. Math., Cambridge (1912), p. 213; and Collected Works (Cambridge University Press, London, England, 1920).

Sitz. der preuss. Akad. Wiss., Physik. Math. Klasse. (1)

M. Planck, Sitz. der preuss. Akad. Wiss., Physik. Math. Klasse. 13 (June, 1907).

Wien. Sitzungen (1)

F. Hasenöhrl, Wien. Sitzungen IIa, 1131039 (1904).

Other (4)

H. Poincaré, “L’etat actuel et l’avenir de la physique mathematique,” Congress of Arts and Sciences, St. Louis, Sept. 24, 1904; first published in full in La Revue des Idées, 80, Nov.15, 1904. For appreciations of the pioneer contributions of Poincaré to the principle of relativity, the formulation of the Lorentz transformations, and the momentum of radiation, see articles by W. Wien and H. A. Lorentz in Acta Math.38 (1921). See also, Ives, “Revisions of the Lorentz transformations,” Proc. Am. Phil. Soc. 95, 125 (1951).

R. Becker, Theorie der Electronen (B. G. Teubner, Leipzig, 1930–1933), p. 348.

Reference 6, footnote on p. 566.

My italics, H. E. I.

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Tables (1)

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Table I Mass and energy relations.

Equations (33)

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E = m R c 2 E = m M c 2
μ v = S / c 2 .
μ v = S / c 2 = E c / c 2 = E / c 2 · c .
μ = 10 3 grams E = 3 × 10 6 joules = 3 × 10 13 ergs c = 3 × 10 10 cm per second ,
10 3 × v = 3 × 10 13 × 3 × 10 10 9 × 10 20 ,
v = 1.
f = d E / c d t
m R c = E / c ,
m R = E / c 2 .
E 2 [ 1 + ( v / c ) ] [ 1 - ( v 2 / c 2 ) ] 1 2             and             E 2 [ 1 - ( v / c ) ] [ 1 - ( v 2 / c 2 ) ] 1 2 .
E 2 c 2 [ 1 + ( v / c ) ] [ 1 - ( v 2 / c 2 ) ] 1 2 c             and             E 2 c 2 [ 1 - ( v / c ) ] [ 1 - ( v 2 / c 2 ) ] 1 2 c .
E v / c 2 [ 1 - ( v 2 / c 2 ) ] 1 2 .
m v [ 1 - ( v 2 / c 2 ) ] 1 2 = m v [ 1 - ( v 2 / c 2 ) ] 1 2 + E v c 2 [ 1 - ( v 2 / c 2 ) ] 1 2 ,
( m - m ) v [ 1 - ( v 2 / c 2 ) ] 1 2 = E v c 2 [ 1 - ( v 2 / c 2 ) ] 1 2 ,
( m - m ) = E / c 2 ,
( H 0 - E 0 ) - ( H 1 - E 1 ) = L [ 1 / ( 1 - v 2 c 2 ) 1 2 - 1 ] .
H 0 - E 0 = K 0 + C H 1 - E 1 = K 1 + C ,
( H 0 - E 0 ) - ( H 1 - E 1 ) = K 0 - K 1 ,
( K 0 - K 1 ) = L [ 1 / ( 1 - v 2 c 2 ) 1 2 - 1 ] .
K 0 - K 1 = 1 2 L v 2 / c 2 .
K 0 - K 1 = 1 2 ( m - m ) v 2 ,
1 2 ( m - m ) v 2 = 1 2 L v 2 / c 2 ,
m - m = L / c 2 .
H - E = K + C ,
H = K + E + C ,
( H 0 - E 0 ) - ( H 1 - E 1 ) = L { 1 [ 1 - ( v 2 / c 2 ) ] 1 2 - 1 } .
K 0 = m c 2 { 1 [ 1 - ( v 2 / c 2 ) ] 1 2 - 1 } K 1 = m c 2 { 1 [ 1 - ( v 2 / c 2 ) ] 1 2 - 1 } ,
K 0 - K 1 = ( m - m ) c 2 { 1 [ 1 - ( v 2 / c 2 ) ] 1 2 - 1 } .
( H 0 - E 0 ) - ( H 1 - E 1 ) = L ( m - m ) c 2 ( K 0 - K 1 ) ,
( H 0 - E 0 ) = L ( m - m ) c 2 ( K 0 + C ) , ( H 1 - E 1 ) = L ( m - m ) c 2 ( K 1 + C ) .
H 0 - E 0 = K 0 + C H 1 - E 1 = K 1 + C .
L / ( m - m ) c 2 .
L / ( m - m ) c 2 = 1.