3 6 D O C . 3 C O M M E N T O N S C H R Ö D I N G E R ’ S N O T E
Published in Physikalische Zeitschrift 19 (1918): 165–166. Received 3 March 1918, published 15
In Einstein 1917b (Vol. 6, Doc. 43), the cosmological term was added to the left-hand
side of the gravitational field equations to allow a static, homogeneous, and spatially closed cosmo-
logical model as a solution. In Schrödinger 1918b, the author showed that the metric field describing
the space-time structure of this model is also a solution of the field equations without the cosmolog-
ical term, if a constant pressure term with the negative pressure is added to the
energy-momentum tensor for the model. In June 1918, Felix Klein (1849–1925), Emeritus Pro-
fessor of Mathematics at the University of Göttingen, noted, by reinterpreting the cosmological term
as a negative pressure term, that the De Sitter solution of the field equations with the cosmological
term is likewise a solution of the field equations without the cosmological term (see Felix Klein to
Einstein, 16 June 1918 [Vol. 8, Doc. 566], note 13).
Einstein’s remark that Schrödinger’s negative pressure term can be regarded as a negative mass
density term has to be understood in the context of the Newtonian approximation: since he defines
the energy-momentum tensor as in the rest system (see Einstein 1916e [Vol.
6, Doc. 30], p. 811), a negative p leads to a positive contribution to , the energy density term. Nev-
ertheless, the term that enters into the Newtonian approximation is . A neg-
ative pressure therefore gives a negative contribution to the density of gravitating mass in the
In Schrödinger 1918b, the author found that p has to be equal to , where κ is the grav-
itational constant and R is the (constant) radius of curvature of the spherical space of Einstein’s cos-
mological model. Thus, Schrödinger’s model corresponds to Einstein’s first possibility.
Later, in Einstein 1919a (Doc. 17), Einstein would explore a theory in which the cosmological
term comes out as a negative pressure term.
ν –pδμ p κ ⁄ –λ =
Tμν diag( p p p, ρ p) – , ,
2( T44 T 2) ⁄ – ρ 2p + =
κR2 ⁄ –1