This paper explores the evolution of Einstein’s understanding of mass and energy. Early on, Einstein embraced the idea of a speed-dependent mass but changed his mind in 1906 and thereafter carefully avoided that notion entirely. He shunned, and explicitly rejected, what later came to be known as “relativistic mass.” Nonetheless many textbooks and articles credit him with the relation E=mc2, where E is the total energy, m is the relativistic mass, and c is the vacuum speed of light. Einstein never derived this relation, at least not with that understanding of the meaning of its terms. He consistently related the “rest energy” of a system to its invariant inertial mass.

1.
The Collected Papers of Albert Einstein, The Swiss Years: Writings, 1900–1909
, Vol.
2
, translated by Anna Beck (
Princeton U. P.
,
Princeton, NJ
,
1989
), pp.
140
171
. This version is true to the original in notation
Also in English is
A.
Einstein
,
The Principle of Relativity
(
Dover
,
New York
,
1923
), pp.
35
65
, but here the notation has been modernized.
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Max
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Das Prinzip der Relativität und die Grundgleichungen der Mechanik
,”
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,
136
141
(
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).
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Lev B.
Okun
, “
The concept of mass (mass, energy, relativity)
,”
Sov. Phys. Usp.
32
,
629
638
(
1989
) addressed several important issues.
W.
Rindler
, “
Putting to rest mass misconceptions
,”
Phys. Today
43
(
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),
13
(
1990
);
M.
Vandyck
, “
Putting to rest mass misconceptions
,”
Phys. Today
43
(
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),
13
(
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);
K. B.
Kolb
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Mass and energy
,”
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34
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Adler
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Does mass really depend on velocity, dad?
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55
(
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),
739
743
(
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);
G. J.
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 II
, “
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72
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),
970
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(
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);
D.
Easton
, “
Mass is not destroyed
,”
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26
(
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),
230
(
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);
A.
Hobson
, “
Teaching E=mc2: Mass without mass
,”
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43
(
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),
80
82
(
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);
S.
Carson
, “
Relativistic mass
,”
Phys. Educ.
33
(
11
),
343
344
(
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);
J. W.
Warren
, “
The mystery of mass-energy
,”
Phys. Educ.
11
(
1
),
52
54
(
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);
K.
Atkin
, “
Clarifying the concept
,”
Phys. Educ.
35
(
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),
319
(
2000
).
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Albert
Einstein
, “E=Mc2,”
Ideas and Opinions
(
Dell
,
New York
,
1976
), p.
330
.
5.
J. J.
Thomson
, “
On the electric and magnetic effects produced by the motion of electrified bodies
,”
Philos. Mag.
11
,
229
249
(
1881
).
6.
For general reference, see
Max
Jammer
,
Concepts of Mass in Classical and Modern Physics
(
Dover
,
New York
,
1997
).
7.
G. F. C.
Searle
, “
On the steady motion of an electrified ellipsoid
,”
Philos. Mag.
44
,
329
341
(
1897
).
See also
O.
Heaviside
, “
On the electromagnetic effects due to the motion of electrification through a dielectric
,”
Philos. Mag.
27
,
324
339
(
1889
).
8.
L. B.
Okun
, “
The concept of mass
,”
Phys. Today
42
(
6
),
31
36
(
1989
) points out, “I have not found in the papers I have read (Ref. 7) any suggestion that mass depends on velocity.”
9.
To see how electromagnetic mass follows from Searle’s expression for field energy, read the 1901 paper by
Walter
Kaufmann
, “
Magnetic and electric deflectability of the Becquerel rays and the apparent mass of the electron
,”
Gött. Nachr.
2
,
143
513
(
1901
),
translated in
Henry
Boorse
and
Lloyd
Motz
,
The World of the Atom
(
Basic Books
,
New York
,
1966
), pp.
506
512
.
10.
H. A.
Lorentz
, “
Simplified theory of electric and optical phenomena in moving systems
,”
Konikl. Akad. Wetenschap. Proc.
1
,
427
442
(
1899
).
See
Stanley
Goldberg
, “
The Lorentz theory of electrons and Einstein’s theory of relativity
,”
Am. J. Phys.
37
(
10
),
982
994
(
1969
)
as well as
Kenneth F.
Schaffner
, “
The Lorentz electron theory of [sic] relativity
,”
Am. J. Phys.
37
(
5
),
498
513
(
1969
).
11.
James
Cushing
, “
Electromagnetic mass, relativity and the Kaufmann experiments
,”
Am. J. Phys.
49
(
12
),
1133
1149
(
1981
) (that excellent paper also contains a highly readable exegesis of Abraham’s theory and its relation to Kaufmann’s experiment).
12.
Arthur
Miller
,
Albert Einstein’s Special Theory of Relativity
(
Addison-Wesley
,
Reading, MA
,
1981
), pp.
47
55
.
13.
Reference 1, p.
140
.
14.
Max
Abraham
, “
Prinzipien der Dynamik des Elektrons
,”
Ann. Phys.
10
,
105
179
(
1903
).
15.
H. A.
Lorentz
, “
Electromagnetic phenomena in a system moving with any velocity less than that of light
.”
A translation of this paper appears in
A.
Einstein
,
The Principle of Relativity
(
Dover
,
New York
,
1923
), pp.
11
34
.
Alternatively see
A.
Einstein
,
Konikl. Akad. Wetenschap. Proc.
6
,
809
831
(
1904
).
16.
Max
Planck
, “
Die Kaufmannschen Messungen der Ablenkbarkeit der β-Strahlen in ihrer Bedeutung für die Dynamik der Elektronen
,”
Phys. Z.
7
,
753
761
(
1906
).
Also see Ref. 1,
Anna
Beck
, “
On the relativity principle and the conclusions drawn from it
,” p.
283
.
17.
John
Stachel
,
Einstein from ‘
B’ to ‘Z’ (
Birkhäuser
,
Boston, MA
,
2002
);
Don
Howard
and
John
Stachel
,
Einstein The Formative Years, 1879–1909
Z (
Birkhäuser
,
Boston, MA
,
1998
).
18.
John
Earman
,
Clark
Glymour
, and
Robert
Rynasiewicz
, “
On writing the history of special relativity
,”
Phil. Soc. Am.
2
,
403
416
(
1982
).
Robert
Rynasiewicz
, “
The optics and electrodynamics of ‘On the electrodynamics of moving bodies’
,”
Ann. Phys. (Leipzig) Supplement
14
,
38
57
(
2005
);
Gerald
Holton
, “
On the origins of the special theory of relativity
,”
Am. J. Phys.
28
,
627
636
(
1960
).
19.
L. H.
Greenberg
,
Physics with Modern Applications
(
Saunders
,
Philadelphia
,
1978
), p.
456
;
Marshall
Burns
,
Modern Physics for Science and Engineering
(
Harcourt Brace Jovanovich
,
New York
,
1988
), p.
103
calls Eq. (1) “Einstein’s mass equation.”
H.
Semat
,
Introduction to Atomic and Nuclear Physics
(
Rinehart
,
New York
,
1958
), p.
46
;
F. W.
Sears
,
M. W.
Zimansky
, and
H. D.
Young
,
University Physics
, 5th ed. (
Addison-Wesley
,
Reading, MA
,
1976
), p.
131
;
R. A.
Serway
and
J. S.
Faugh
,
College Physics
, 3rd ed. (
Saunders
,
New York
,
1992
), p.
894
.
20.
There are hundreds of popular books that get it wrong, the latest of which are
Walter
Isaacson
,
Einstein
(
Simon and Schuster
,
New York
,
2007
), p.
130
and
Hans
Ohanian
,
Einstein’s Mistakes: The Human Failings of Genius
(
Norton
,
New York
,
2008
).
21.
Max
Jammer
,
Concepts of Mass in Contemporary Physics and Philosophy
(
Princeton U. P.
,
Princeton, NJ
,
2000
), p.
42
, states that Einstein’s first paper “introduced the notion of relativistic mass,” which Jammer had earlier carefully defined as mr=m0(1v2/c2)1/2.
Also see
Stanley
Goldberg
,
Understanding Relativity: Origin and Impact of a Scientific Revolution
(
Birkhäuser
,
Boston
,
1984
).
22.
Ling
Tsai
, “
The relation between gravitational mass, inertial mass, and velocity
,”
Am. J. Phys.
54
(
4
),
340
342
(
1986
);
Sivio
Bergia
, “
Einstein and the birth of special relativity
,” in
Einstein: A Centenary Volume
, edited by
A. P.
French
(
Harvard U. P.
,
Cambridge, MA
,
1979
).
23.
Edmund
Whittaker
,
A History of the Theories of Aether and Electricity
, Vol.
2
(
Dover
,
New York
,
1989
), p.
44
, asserted “the study of relativistic dynamics was begun in 1906, when Max Planck found the equations… of motion.” Since Einstein’s 1905 derivation was weak, Whittaker makes a valid point.
24.
Reference 1, pp.
207
210
, “
On a method for the determination of the ratio of the transverse and the longitudinal mass of the electron
.”
25.
Reference 1, p.
210
.
26.
Reference 1, p.
167
.
27.
Reference 15, Sec. 9.
Also see
Olivier
Darrigol
,
Electrodynamics from Ampere to Einstein
(
Oxford U. P.
,
Oxford
,
2000
), pp.
360
390
.
28.
Reference 2, p.
119
.
29.
Reference 1, “
On the relativity principle and the conclusions drawn from it
,” pp.
278
–280.
30.
Reference 1, p.
170
.
31.
Reference 1, p.
252
311
.
32.
Max
Planck
, “
Zur Dynamik bewegter Systeme
,”
Sitzungsber. Preuss. Akad. Wiss.
1907
,
542
570
.
Also see
Max
Planck
,
Eight Lectures on Theoretical Physics
(
Dover
,
Mineola, NY
,
1998
), pp.
97
130
.
33.
Reference 1, p.
284
.
34.
The original German version can be found in
The Collected Papers of Albert Einstein, The Swiss Years: Writings, 1900–1909
, Vol.
2
, edited by
John
Stachel
(
Princeton U. P.
,
Princeton, NJ
,
1989
), pp.
432
484
.
35.
Reference 1, p.
284
. Although this translation is the “official” one, in this instance it is terribly misleading because it translates “die Masse” as “the dimensions.”
For the German text, see Ref. 34, p.
461
;
H. M.
Schwartz
, “
Einstein’s comprehensive 1907 essay on relativity, part II
,”
Am. J. Phys.
45
(
9
),
811
817
(
1977
) translates the passage accurately.
36.
Reference 34, p.
309
, footnote 41. The 1913 reprint of Einstein’s June 1905 paper contained an appended note: “The definition of force given here is not advantageous as was first noted by M. Planck. It is instead appropriate to define force in such a way that the laws of momentum and conservation of energy take the simplest form.”
37.
The Collected Papers of Albert Einstein, The Berlin Years: Writings, 1918–1921
, Vol.
7
, translated by Alfred Engel (
Princeton U. P.
,
Princeton, NJ
,
2002
);
A.
Einstein
, “
Four lectures on the theory of relativity, held at Princeton University in May 1921
” (
1922
), pp.
261
368
.
38.
Reference 37, p.
307
.
39.
Reference 2.
40.
Reference 1, p.
240
.
41.
G. N.
Lewis
, “
A revision of the fundamental laws of matter and energy
,”
Philos. Mag.
16
(
11
),
705
717
(
1908
);
G. N.
Lewis
and
R. C.
Tolman
, “
The principle of relativity and non-Newtonian mechanics
,”
Philos. Mag.
18
,
510
523
(
1909
);
R. C.
Tolman
, “
Note on the derivation from the principle of relativity of the fifth equation of the Maxwell–Lorentz theory
,”
Philos. Mag.
21
,
296
301
(
1911
);
R. C.
Tolman
, “
Non-Newtonian mechanics: The mass of a moving body
,”
Philos. Mag.
23
,
375
380
(
1912
).
For a different approach, see
P. C.
Peters
, “
An alternative derivation of relativistic momentum
,”
Am. J. Phys.
54
(
9
),
804
808
(
1986
).
42.
Richard C.
Tolman
,
The Theory of the Relativity of Motion
(
U. of California Press
,
Berkeley
,
1917
), pp.
62
67
.
43.
A. H.
Bucherer
, “
Die experimentelle Bestätigung des Relativitätsprinzips
,”
Ann. Phys.
28
,
513
536
(
1909
).
44.
G.
Neumann
, “
Die träge Masse schnell bewegter Electronen
,”
Ann. Phys.
45
,
529
579
(
1914
).
45.
See
W.
Pauli
,
Theory of Relativity
(
Pergamon
,
New York
,
1958
), p.
83
, first published in
1921
.
Also see Ref. 6, pp.
166
168
.
46.
Max
Born
,
Einstein’s Theory of Relativity
(
Dover
,
New York
,
1962
), first published in
1920
.
47.
A.
Einstein
, “
Elementary derivation of the equivalence of mass and energy
,”
Bull. Am. Math. Soc.
41
,
223
230
(
1935
);
David
Topper
and
Dwight
Vincent
, “
Einstein’s 1934 two-blackboard derivation of energy-mass equivalence
,”
Am. J. Phys.
75
(
11
),
978
983
(
2007
). Topper and Vincent point out that “in this lecture Einstein stayed well clear of defining the relativistic mass parameter, mR,” That should not have surprised anyone. He always stayed clear of it. Their article contains a photo of Einstein at the 1934 Pittsburgh lecture standing next to a blackboard on which is written E0=m and he had set c=1.
48.
During April, May, and June of 1948, Harper’s magazine published a series of articles by Lincoln Barnett, a well-known science writer. Therein Barnett talked about “Einstein’s equation [Eq. (1)] giving the increase of mass with velocity…”. On June 19, 1948, Einstein sent Barnett a letter (written in German) to set the record straight on his position against relativistic mass. See Ref. 8, p.
32
for a photograph of the letter, which is now in the possession of the Hebrew University of Jerusalem, Israel.
49.
A.
Einstein
and
L.
Infeld
,
The Evolution of Physics
(
Simon and Schuster
,
New York
,
1938
), p.
205
, distinguish between inertia (the measure of a body’s resistance to a change in motion) and mass without explicitly saying as much. As Einstein and Infield put it, “If two bodies have the same rest mass, the one with the greater kinetic energy resists the action of an external force more strongly.” It would have to if c is to be its limiting speed.
50.
Reference 1, pp.
140
.
51.
Reference 1, p.
172
.
52.
W. L.
Fadner
, “
Did Einstein really discover ‘E=mc2’?
Am. J. Phys.
56
,
114
122
(
1988
).
Also see
John
Stachel
and
Roberto
Torretti
, “
Einstein’s first derivation of the mass-energy difference
,”
Am. J. Phys.
50
(
8
),
760
763
(
1982
).
53.
Reference 1, pp.
172
174
.
54.
Albert
Einstein
,
Out of My Later Years
(
Citadel
,
Secaucus, NJ
,
1956
), p.
116
.
55.
Reference 1, pp.
286
287
.
56.
Reference 1, pp.
200
206
.
57.
H.
Poincaré
, Lorentz-Festschrift,
“La théorie de Lorentz et le principe de reaction
,”
Arch. Neerl. Sci. Exactes Nat., Ser. 3A
2
,
252
278
(
1900
).
Also see
Stanley
Goldberg
, “
Henri Poincare and Einstein’s theory of relativity
,”
Am. J. Phys.
35
(
10
),
934
944
(
1967
)
and
Edmund
Whittaker
in Ref. 23, Vol.
2
, p.
51
.
58.
Reference 1, p.
200
.
59.
For example, see
A. P.
French
,
Special Relativity
(
Norton
,
New York
,
1968
), p.
17
, and
Adel
Antippa
, “
Inertia of energy and the liberated photon
,”
Am. J. Phys.
44
(
9
),
841
844
(
1976
).
60.
Reference 1, pp.
238
250
.
61.
Eugene
Hecht
, “
Energy and change
,”
Phys. Teach.
45
(
2
),
88
92
(
2007
).
62.
Reference 1, pp.
252
311
.
63.
Reference 1, p.
287
.
64.
Reference 37, “
Fundamental ideas and methods of the theory of relativity, presented in their development
,” p.
129
.
65.
Reference 37, p.
306
.
66.
C. R.
Eddy
, “
A relativistic misconception
,”
Science
104
,
303
304
(
1946
);
[PubMed]
A. J.
O’Leary
, “
Comments on ‘A relativistic misconception’
,”
Science
104
,
401
(
1946
).
[PubMed]
Also
Hermann
Bondi
and
C. B.
Spurgin
, “
Energy and mass
,”
Phys. Bull.
38
,
62
63
(
1987
).
See the counterargument by
Rudolf
Peierls
, “
Mass and energy
,”
Phys. Bull.
38
,
127
128
(
1987
).
Reference 3;
D.
Easton
,
A.
Hobson
,
S.
Carson
,
J. W.
Warren
,
K.
Atkin
, and
Ralph
Baierlein
, “
Teaching E=mc2
,”
Am. J. Phys.
57
(
5
),
391
392
(
1989
);
R.
Bauman
, “
Mass and energy: The low-energy limit
,”
Phys. Teach.
32
(
9
),
340
342
(
1994
);
R.
Baierlein
,
Newton to Einstein: The Trail of Light
(
Cambridge U. P.
,
Cambridge
,
1992
), pp.
236
247
;
Peter
Stansbury
, “
The equivalence of mass and energy
,”
Phys. Educ.
35
(
11
),
378
379
(
2000
) maintained that “it is impossible to change mass into energy, just as it is impossible to change bricks into apples.”
A.
Hobson
, Ref. 3, more recently (
2005
) suggested that “mass is never converted into energy, and energy is never converted into mass, because energy (and hence mass) can never be created or destroyed.”
67.
Reference 54, p.
57
.
68.
This is the way Einstein put it in Science, May 24, 1940: “There followed also the principle of the equivalence of mass and energy, with the laws of conservation of mass and energy becoming one and the same.
This quote appears in the essay “The fundamentals of theoretical physics,” Ref. 4, p. 321. See
Science, New Series
,
91
No. (
2369
),
487
492
(
1940
).
69.
“The theory of relativity,”
The Collected Papers of Albert Einstein, Vol. 3, The Swiss Years: Writings, 1909–1911
, translated by Anna Beck (
Princeton U. P.
,
Princeton, NJ
,
1989
), pp.
349
350
.
70.
Frederick
Soddy
,
Radio-Activity: An Elementary Treatise, from the Standpoint of the Disintegration Theory
(
Van Nostrand
,
New York
,
1904
), p.
164
, prophetically wrote, “on the nature of inertia and its possible dependence on the internal energy of the atom, it is indicated that the law of conservation of mass will not apply to the case of atomic transformation.” And on p.
180
, he asserted, “atomic mass must be regarded as a function of the internal energy, and the dissipation of the latter in radio-activity occurs at the expense, to some extent at least, of the mass of the system.
71.
Abraham
Pais
,
Subtle is the Lord
(
Oxford U. P.
,
Oxford
,
1982
), p.
149
.
72.
Reference 1, p.
288
.
73.
The AIP history website is ⟨www.aip.org/history/einstein/voice1.htm⟩.
74.
Reference 37, p.
277
.
75.
Reference 54, p.
57
.
76.
Reference 4, p.
330
.
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