## One Hundred Years of the Cosmological Constant

It was exactly one hundred years ago today – on 8th February 1917 – that a paper was published in which Albert Einstein explored the cosmological consequences of his general theory of relativity, in the course of which he introduced the concept of the cosmological constant.

For the record the full reference to the paper is: Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie and it was published in the Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften. You can find the full text of the paper here. There’s also a nice recent discussion of it by Cormac O’Raifeartaigh  and others on the arXiv here.

Here is the first page:

It’s well worth looking at this paper – even if your German is as rudimentary as mine – because the argument Einstein constructs is rather different from what you might imagine (or at least that’s what I thought when I first read it). As you see, it begins with a discussion of a modification of Poisson’s equation for gravity.

As is well known, Einstein introduced the cosmological constant in order to construct a static model of the Universe. The 1917 paper pre-dates the work of Friedman (1923) and Lemaître (1927) that established much of the language and formalism used to describe cosmological models nowadays, so I thought it might be interesting just to recapitulate the idea using modern notation. Actually, in honour of the impending centenary I did this briefly in my lecture on Physics of the Early Universe yesterday.

To simplify matters I’ll just consider a “dust” model, in which pressure can be neglected. In this case, the essential equations governing a cosmological model satisfying the Cosmological Principle are:

$\ddot{a} = -\frac{4\pi G \rho a }{3} +\frac{\Lambda a}{3}$

and

$\dot{a}^2= \frac{8\pi G \rho a^2}{3} +\frac{\Lambda a^2}{3} - kc^2.$

In these equations $a(t)$ is the cosmic scale factor (which measures the relative size of the Universe) and dots are derivatives with respect to cosmological proper time, $t$. The density of matter is $\rho>0$ and the cosmological constant is $\Lambda$. The quantity $k$ is the curvature of the spatial sections of the model, i.e. the surfaces on which $t$ is constant.

Now our task is to find a solution of these equations with $a(t)= A$, say, constant for all time, i.e. that $\dot{a}=0$ and $\ddot{a}=0$ for all time.

The first thing to notice is that if $\Lambda=0$ then this is impossible. One can solve the second equation to make the LHS zero at a particular time by matching the density term to the curvature term, but that only makes a universe that is instantaneously static. The second derivative is non-zero in this case so the system inevitably evolves away from the situation in which $\dot{a}=0$.

With the cosmological constant term included, it is a different story. First make $\ddot{a}=0$  in the first equation, which means that

$\Lambda=4\pi G \rho.$

Now we can make $\dot{a}=0$ in the second equation by setting

$\Lambda a^2 = 4\pi G \rho a^2 = kc^2$

This gives a static universe model, usually called the Einstein universe. Notice that the curvature must be positive, so this a universe of finite spatial extent but with infinite duration.

This idea formed the basis of Einstein’s own cosmological thinking until the early 1930s when observations began to make it clear that the universe was not static at all, but expanding. In that light it seems that adding the cosmological constant wasn’t really justified, and it is often said that Einstein regard its introduction as his “biggest blunder”.

I have two responses to that. One is that general relativity, when combined with the cosmological principle, but without the cosmological constant, requires the universe to be dynamical rather than static. If anything, therefore, you could argue that Einstein’s biggest blunder was to have failed to predict the expansion of the Universe!

The other response is that, far from it being an ad hoc modification of his theory, there are actually sound mathematical reasons for allowing the cosmological constant term. Although Einstein’s original motivation for considering this possibility may have been misguided, he was justified in introducing it. He was right if, perhaps, for the wrong reasons. Nowadays observational evidence suggests that the expansion of the universe may be accelerating. The first equation above tells you that this is only possible if $\Lambda\neq 0$.

Finally, I’ll just mention another thing in the light of the Einstein (1917) paper. It is clear that Einstein thought of the cosmological as a modification of the left hand side of the field equations of general relativity, i.e. the part that expresses the effect of gravity through the curvature of space-time. Nowadays we tend to think of it instead as a peculiar form of energy (called dark energy) that has negative pressure. This sits on the right hand side of the field equations instead of the left so is not so much a modification of the law of gravity as an exotic form of energy. You can see the details in an older post here.

### 24 Responses to “One Hundred Years of the Cosmological Constant”

1. I agree completely with your two responses.

“Nowadays we tend to think of it instead as a peculiar form of energy (called dark energy) that has negative pressure. This sits on the right hand side of the field equations instead of the left so is not so much a modification of the law of gravity as an exotic form of energy.”

However, unless that is an editorial (or royal :-)) “we”, I’m not sure if it is true. Certainly some people think of it as being on the left-hand side. Or, perhaps, both—the most famous of these being Steven Weinberg, whose idea that the two terms almost cancel was probably the first anthropic explanation for the observed value of the cosmological constant, while at the same time solving the quantum-theoretical cosmological-constant problem.

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3. Thanks for the plug, Peter! I might add a couple of points, if I may:

(i) While the cc term was first formally introduced to the field equations in 1917, AE had noted the possibility in his first full exposition of the general theory; it’s mentioned in a footnote in section 14 of the seminal ‘Grundlage’ paper of 1916. Unfortunately the term is introduced in a way that is not mathematically correct, I presume that’s why the 1916 version is always overlooked?

(ii) In 1917, Einstein introduces the cc in analogy with a necessary modification of Poisson’s equation, as you point out. The funny thing is, his analogy is mathematically inaccurate; the modified field equations do not reduce to the modified Poisson’s equation the way he claims. I think the error is quite interesting, it might explain AE’s difficulties with interpreting the role of the cc

(iii) I was amazed to learn that the business of putting the cc on the RHS was suggested as early as 1918 – by a young Schroedinger, no less. Schroedinger even suggested that the term might be time-varying! (quintessence)
Regards, Cormac

• telescoper Says:

Good points! I wasn’t aware of (iii). Can you please point me to a reference to Schrodinger’s discussion?

• Erwin Schrödinger: “Über ein Lösungssystem der allgemein kovarianten Gravitationsgleichungen”, Physikalische Zeitschrift, 19, 20-22 (1918).

• There is also some discussion of this in English.

I recommend Walter Moore’s excellent biography of Schrödinger, a really interesting character who was much more than wave mechanics. It was also Schrödinger who proved the equivalence with wave mechanics. He was similar to Einstein in many ways: interest in unified-field theories, moving about from country to country, leaving the Continent due to the Nazis (Schrödinger was not Jewish—the Nazis hated lots of people), young girlfriends, unorthodox marriage arrangements (he made sure before going to Ireland that he could maintain his menage a trois, obtaining visas for himself, wife, and mistress; his wife was also the mistress of Hermann Weyl), somewhat left behind by the rest of the quantum community.

• telescoper Says:

I presume the Alex Harvey who wrote that paper is not the same as the Sensational Alex Harvey Band?

• It was also Schrödinger who proved the equivalence with wave mechanics. —> It was also Schrödinger who proved the equivalence with matrix mechanics, i.e. the equivalence of his formulation and that of Heisenberg, Born, and Jordan.

• “I presume the Alex Harvey who wrote that paper is not the same as the Sensational Alex Harvey Band?”

At first glance, why not? There are certainly people who have done both science and music, even both at a professional level (another Alex, Szalay, comes to mind). However, the sensational Alex Harvey died in 1982, long before the paper was written.

4. Tnx guys, I had forgotten that Alex Harvey gave a translation of Schroedinger’s 1918 paper, most useful. Einstein’s full reaction to Schroedinger’s suggestion is available in English translation in volume 7 of the Collected Einstein Papers;
‘Comment on Schrödinger’s Note “On a system of solutions for the generally covariant gravitational field equations”’ CPAE 7 (Doc. 3). http://einsteinpapers.press.princeton.edu/vol7-trans/47

5. Well done Phillip! I was aware of Alex Harvey’s study of Schroedinger’s paper, but I hadn’t seen that second article on the cc. He’s right in every detail, must add him to our references. (None of these papers seem to have been published in journals, is that right?).
Re the cc, it continues to amaze me that so few scholars are aware that lambda first turns up in 1916, not quite the fudge factor it’s made out to be!

6. Re Schroedinger, it’s amazing to think he was all over GR already in 1918. In the 1940s, while GR had been marginalized in many physics departments around the world, Sch was very insistent that all the young scholars at DIAS were trained in GR as well as qt

• As I mentioned, Schrödinger was actually quite similar to Einstein, also scientifically, and also worked on similarly (ultimately unsuccessful) stuff in his later years. Also, one could think of wave mechanics as a geometric, or at least visualizable, theory.

Before working on quantum mechanics, Schrödinger was the world’s leading expert on colour vision.

7. There is a very nice discussion of the career-long relationship between Schroedinger and Einstein in the book ‘Einstein’s Dice and Schroedinger’s Cat’ by Paul Halpern

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9. […] For more mathematical details on the cosmological constant, take a look at this nice post by Peter Coles: One Hundred Years of the Cosmological Constant. […]

10. Ahem! I’m afraid my statement that a version of the cc term was first introduced by AE in 1916 is probably wrong. It has been pointed out to me that the relevant footnote in the 1916 paper concerns the trace term on the RHS of the field equations, nothing to do with an extension of the field equations. Oops!

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13. […] For more mathematical details on the cosmological constant, take a look at this nice post by Peter Coles: One Hundred Years of the Cosmological Constant. […]

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15. […] because he was giving space some kind of ground state. Only nowadays as Peter Coles said in One Hundred Years of the Cosmological Constant, we tend to think of the cosmological constant as dark energy on the right hand “source” side […]