These two assumptions were, however, not compatible with Einstein’s original field equations. For this reason, Einstein added the famous Λ-term, which is compatible with the principles of general relativity. The cosmological term is, in four dimensions, the only possible complication of the field equations if no higher than second order derivatives of the metric are allowed (*Lovelock theorem*). This remarkable uniqueness is one of the most attractive features of general relativity. (In higher dimensions additional terms satisfying this requirement are allowed.)

For the static Einstein universe the field equations with the cosmological term imply the two relations

where ρ is the mass density of the dust filled universe (zero pressure) and a is the radius of curvature. For Λ = 0 the density ρ would have to vanish. (We remark, in passing, that the Einstein universe is the only static dust solution; one does not have to assume isotropy or homogeneity.) Einstein was very pleased by this direct connection between the mass density and geometry, because he thought that this was in accord with Mach's philosophy.

Einstein concludes with the following sentences:

"*In order to arrive at this consistent view, we admittedly had to introduce an extension of the field equations of gravitation which is not justified by our actual knowledge of gravitation. It has to be emphasized, however, that a positive curvature of space is given by our results, even if the supplementary term is not introduced. That term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars.*"

To de Sitter he emphasized in a letter on 12 March 1917, that his cosmological model was intended primarily to settle the question "whether the basic idea of relativity can be followed through its completion, or whether it leads to contradictions". And he adds whether the model corresponds to reality was another matter.

Only later Einstein came to realize that Mach's philosophy is predicated on an antiquated ontology that seeks to reduce the metric field to an epiphenomenon of matter. It became increasingly clear to him that the metric field has an independent existence, and his enthusiasm for what he called Mach's principle later decreased. In a letter to F. Pirani he wrote in 1954: "*As a matter of fact, one should no longer speak of Mach's principle at all.*" GR still preserves some remnant of Newton’s absolute space and time.

**De Sitter model**

Surprisingly to Einstein, de Sitter discovered in the same year, 1917, a completely different static cosmological model which also incorporated the cosmological constant, but was *anti-Machian*, because it contained no matter [6]. For this reason, Einstein tried to discard it on various grounds (more on this below). The original form of the metric was:

Here, the spatial part is the standard metric of a three-sphere of radius R, with R = (3/Λ)^{1/2}. The model had one very interesting property: For light sources moving along static world lines there is a gravitational redshift, which became known as the *de Sitter effect*. This was thought to have some bearing on the redshift results obtained by Slipher. Because the fundamental (static) worldlines in this model are not geodesic, a freely-falling object released by any static observer will be seen by him to accelerate away, generating also local velocity (Doppler) redshifts corresponding to *peculiar velocities*. In the second edition of his book [7], published in 1924, Eddington writes about this:

"*de Sitter's theory gives a double explanation for this motion of recession; first there is a general tendency to scatter (...); second there is a general displacement of spectral lines to the red in distant objects owing to the slowing down of atomic vibrations (...), which would erroneously be interpreted as a motion of recession.*"

I do not want to enter into all the confusion over the de Sitter universe. One source of this was the apparent singularity at r = R = (3/Λ)^{1/2}. This was at first thoroughly misunderstood even by Einstein and Weyl. ("The Einstein-de Sitter-Weyl-Klein Debate" is now published in Vol. 8 of the *Collected Papers* [4].) At the end, Einstein had to acknowledge that de Sitter's solution is fully regular and matter-free and thus indeed a counter example to Mach's principle. But he still discarded the solution as physically irrelevant because it is not globally static. This is clearly expressed in a letter from Weyl to Klein, after he had discussed the issue during a visit of Einstein in Zürich [8]. An important discussion of the redshift of galaxies in de Sitter's model by H. Weyl in 1923 should be mentioned. Weyl introduced an expanding version ^{2} of the de Sitter model [9]. For *small* distances his result reduced to what later became known as the Hubble law. Independently of Weyl, Cornelius Lanczos introduced in 1922 also a non-stationary interpretation of de Sitter's solution in the form of a Friedmann spacetime with a positive spatial curvature [10]. In a second paper he also derived the redshift for the non-stationary interpretation [11].

**References**

[1] N. Straumann, The 2011 Nobel Prize in Physics, SPG Mitteilungen, Nr. 36, Januar 2012.

[2] A. Sandage, Preface to [17].

[3] A. Einstein, *Sitzungsber. Preuss. Akad. Wiss.* phys.-math. Klasse VI, 142 (1917). See also: [4], Vol. 6, p. 540, Doc. 43.

[4] A. Einstein, *The Collected Papers of Albert Einstein*, Vols. 1-12, Princeton University Press, 1987–. See also: [http://www. einstein.caltech.edu/].

[5] A. Einstein, *On the Foundations of the General Theory of Relativity.* Ref. [4], Vol. 7, Doc. 4.

[6] W. de Sitter, Proc. Acad. Sci., **19**, 1217 (1917); and **20**, 229 (1917).

[7] A. S. Eddington, *The Mathematical Theory of Relativity.* Chelsea Publishing Company (1924). Third (unaltered) Edition (1975). See especially Sect. 70.

[8] Letter from Hermann Weyl to Felix Klein, 7 February 1919; see also Ref. [5], Vol. 8, Part B, Doc. 567.

[9] H. Weyl, Phys. Zeits. **24**, 230, (1923); Phil. Mag. **9**, 923 (1930).

[10] C. Lanczos, Phys. Zeits. **23**, 539 (1922).

[11] C. Lanczos, Zeits. f. Physik **17**, 168 (1923).

[12] A. Friedmann, Z.Phys. **10**, 377 (1922); **21**, 326 (1924).

[13] G. Lemaître, *L’univers en expansion.* Ann. Soc. Sci. de Bruxelles **47**, 49 (1927). Translated in MNRAS **91**, 483 (1931).

[14] A. Einstein, S. B. Preuss. Akad. Wiss. (1931), 235.

[15] N. Straumann, *Wolfgang Pauli and Modern Physics*, Space Science Reviews **148**, 25 (2009).

[16] M. Livio, Nature **479**, 208-211 (2011).

[17] H. Nussbaumer and L. Bieri, *Discovering the Expanding Universe*, Cambridge University Press (2009).

*[Released: May 2012]*