I Einstein, Friedman and the cosmological constant

[Jaime Rudas]

In Appendix 4 of the book "Relativity, The Special and General Theory", Einstein describes the evolution of the cosmological implications of general relativity as follows:

My original considerations on the subject were based on two hypotheses:​

(1)There exists an average density of matter in the whole of space which is everywhere the same and different from zero.​

(2)The magnitude (“radius”) of space is independent of time.​

Both these hypotheses proved to be consistent, according to the general theory of relativity, but only after a hypothetical term was added to the field equations, a term which was not required by the theory as such nor did it seem natural from a theoretical point of view (“cosmological term of the field equations”).​

Hypothesis (2) appeared unavoidable to me at the time, since I thought that one would get into bottomless speculations if one departed from it.​

However, already in the twenties, the Russian mathematician Friedman showed that a different hypothesis was natural from a purely theoretical point of view. He realized that it was possible to preserve hypothesis (1) without introducing the less natural cosmological term into the field equations of gravitation, if one was ready to drop hypothesis (2)."​

From this, I interpret that Friedman's solutions don't include the cosmological constant, however, they do. Am I misinterpreting?

 

[Ibix]

Friedman's solutions can include the cosmological constant but they don't have to. It's only if you are trying to fit certain growth rate histories - such as reality or Einstein's static universe - that you need to introduce a non-zero cosmological constant.

 

[Bandersnatch]

Look at what's being said:

(1)There exists an average density of matter in the whole of space which is everywhere the same and different from zero.

the cosmological principle holds

(2)The magnitude (“radius”) of space is independent of time.

the universe is static

He realized that it was possible to preserve hypothesis (1) without introducing the less natural cosmological term into the field equations of gravitation, if one was ready to drop hypothesis (2)."

but you can very well have a universe with the cosmological principle that is not static, in which case lambda is optional.

All he's saying here is that he was too married to the idea of a static universe

 

[PeterDonis]

From this, I interpret that Friedman's solutions don't include the cosmological constant

Friedmann's original solutions did not. But we now know that it is perfectly possible to have solutions which are not static but which do include the cosmological constant, and our modern version of Friedmann's equations include this possibility.

 

[Jaime Rudas]

Friedmann's original solutions did not.

A translation of the original Friedmann's 1922 paper can be seen at the following link:

https://cosmology.education/documents/friedmann_1922.pdf

Equations (12) and (13) are the Friedmann's equations that includes the cosmological constant λ.

After equation (17), Friedmann presents the non-stationary models and states:

"We most base the consideration of the nonstationary world on equations (14) and (15). The quantity λ is not determined by these equations. We shall postulate that it can have an arbitrary value."

Therefore, I consider that Friedmann's original solutions do include the cosmological constant.

 

[PeterDonis]

Equations (12) and (13) are the Friedmann's equations that includes the cosmological constant λ.

After equation (17), Friedmann presents the non-stationary models and states:

"We most base the consideration of the nonstationary world on equations (14) and (15). The quantity λ is not determined by these equations. We shall postulate that it can have an arbitrary value."

Hm, interesting, I wasn't aware of that.

That could mean that Einstein, in the quote you give in your OP, was simply referring to the possibility of $\Lambda = 0$, which is only a possibility if the static assumption is dropped. Or he might be referring to either a different paper by Friedmann, or some more informal unpublished discussion or notes. Einstein regularly communicated by letter with many other physicists, and he also was well informed about things like conference proceedings in which results that might not appear in the published literature were discussed.

It seems clear from other writings of Einstein that he was not happy with the cosmological constant (he called it "the greatest blunder of my life" after evidence for the expansion of the universe was discovered), so he might have discounted the models Friedmann proposed with a nonzero $\Lambda$ once the possibility of an expanding solution with $\Lambda = 0$ was clear to him. But of course the modern viewpoint is that the cosmological constant, on purely theoretical grounds, should be there; the question is what its actual value is, and that question we can only answer empirically, at least at our current level of knowledge.