D O C . 1 G R AV I TAT I O N A L WAV E S 2 7

[22]“in (15) gegebenen ” should be “in (14) und (15) gegebenen .”

[23]

“1/4κ” should be “1/2κ.”

[24]On the right-hand side of eq. (20), the 41-component should be .

[25]In a supplement (“Nachtrag”) to Einstein 1916g (Vol. 6, Doc. 32), it is pointed out that in coor-

dinates such that there are no gravitational waves that do not transport energy. From this

observation, Einstein concluded that such waves are fictitious and reflect nothing but oscillations of

the coordinate system in which they had been derived. He emphasized that the elimination of these

spurious waves conveyed a privileged status upon coordinates satisfying It had since been

shown that in these coordinates one has the counterintuitive result that the field of a point mass carries

no energy (see notes 18–20). By 1918, if not earlier, Einstein had abandoned the notion that these

coordinates are privileged. This can be inferred, for instance, from Einstein to Gustav Mie, 8 February

1918 (Vol. 8, Doc. 460). Mie (1868–1957), Professor of Physics at the University of Greifswald, had

given the example of a rod, straight in one coordinate system and slithering like a snake in another, to

argue for the need of privileged coordinates. As Mie pointed out several months later (see Gustav Mie

to Einstein, 6 May 1918 [Vol. 8, Doc. 532]), this argument is virtually identical to the one given in the

supplement to Einstein 1916g (Vol. 6, Doc. 32). In the letter to Mie cited above, Einstein took the

position that there are no fundamentally privileged coordinate systems, since they all give the same

set of point coincidences. In the present paper, Einstein accordingly eliminates the spurious gravita-

tional waves by showing that the metric field describing them is simply that of a Minkowski space-

time in unusual coordinates. Einstein’s demonstration that these wavelike coordinate “solutions” are

fictitious did not prevent Hermann Weyl (1885–1955), Professor of Mathematics at the Eidgenös-

sische Technische Hochschule, Zurich, from formally categorizing the “three types” of gravitational

waves in Weyl 1919d. In Eddington 1922 (pp. 271–272), Arthur S. Eddington (1882–1944), Director

of the Observatory of the University of Cambridge, disposed of the two spurious wave types by show-

ing that they give rise to vanishing components of the Riemann tensor and thus represent simply flat

space. Einstein himself repeated his dismissal of these so-called waves, and recapitulated much of the

argument of this paper, in Einstein and Rosen 1937.

[26]“ ” should be “ .”

[27]In Einstein to Erwin Freundlich, 19 March 1915 (Vol. 8, Doc. 63), this same technique was used

to show that the space integral over the 11-component of the energy-momentum tensor for a complete

static system vanishes, a result first derived in a slightly different manner in Laue 1911 (pp. 540–541)

in the context of special relativity.

[28]In Einstein 1916g (Vol. 6, Doc. 32), pp. 694–695, Einstein went through the details of the anal-

ogous argument for .

[29]“σ” should be “0” in both these equations. “ ” in the second equation should be “ .”

[30]“1/64” should be “1/32.” The error is a result of the error in eq. (16) (see note 23).

[31]The factor 80 in the denominator of the right-hand side of the equation should be 40, due to the

error in eq. (27) (see note 30). The mistake in eq. (30) was first discovered by Eddington (see Edding-

ton 1922, p. 279). By this formula, the energy loss of a source of gravitational waves depends on the

third time derivative of the moment of inertia tensor , also known as the quadrupole moment ten-

sor. Hence the term “quadrupole formula,” by which the corrected version of eq. (30) is commonly

known, for the leading order flux of energy radiated by a gravitational wave source. The “quadrupole

formula controversy,” of a much later period, concerned the validity of this formula for astrophysi-

cally interesting sources, an issue first raised in Eddington 1922, p. 280 (see Kennefick 1999 and ref-

erences therein).

[32]The “regrettable error in calculation” mentioned at the outset of the paper (p. 154; see also notes

2 and 17) led to the version of the radiation formula of Einstein 1916g (Vol. 6, Doc. 32), permitting

“monopole” sources of gravitational radiation. In fact, the conservation of mass-energy and momen-

tum forbid both monopole and dipole sources of gravitational waves, respectively, as was first pointed

out in Abraham 1915. Birkhoff’s theorem, soon after this, placed the result that spherically symmetric

(monopole) sources cannot radiate on a rigorous footing (Birkhoff 1923).

[33]This repeats a remark on the quantization of the gravitational field from Einstein 1916g (Vol. 6,

Doc. 32), p. 696. It was partly in response to the earlier remark that Tullio Levi-Civita (1873–1941),

Professor of Rational Mechanics at the University of Padua, proposed that Einstein’s definition of

γμν I

γ′μν

iλ1 λ4 +

g – 1 =

g – 1. =

20 z0

T22

z3 x3

μν