Re: The Cosmological constant (was Re: [asa] Predestined Fame:)

From: George Murphy <GMURPHY10@neo.rr.com>
Date: Wed Jul 16 2008 - 11:09:05 EDT

In one sense Einstein was right that interpreting the cosmological term as a uniform density & negative pressure is obvious & doesn't contribute anything new if you stop there. It's a different matter though to investigate how such a density & pressure might arise. Einstein himself did that in a couple of ways. 1st, he suggested a modification of the combined EM & gravitational fields in which the cosmological constant appears as an integration constant. (This paper follows the cosmological one in The Principle of Relativity.) Later (in The Meaning of Relativity) he suggested an interpretation in terms of Poincare stresses, which have some similarity with regulators in quantum field theory.

& there's another approach that demands a cosmological term. In attempts to develop a unified field theory in the early 20s, Einstein & several others explored an approach in which the affine connection rather than the metric is fundamental. This allows you to form a curvature tensor & the simplest way of defining a metric is then to make it proportional to the contracted curvature tensor, & the constant of proportionality is then the inverse cosmological constant. Later in the 40s & 50s Einstein & Schroedinger both tried to develop non-symmetric field theories, & Schroedinger took that approach. Einstein's theory got a lot more PR & many people saw Schroedinger's as just a variant of it, but actually Schroedinger's is much more profound. Unfortunately neither seems to work!

On George Cooper's comment: Yes, Einstein originally rejected Lemaitre's work, but before that he had to be dragged kicking & screaming to recognize the validity of Friedmann's solutions. Friedmann didn't keep the cosmological term but Lemaitre did.

& that's a helpful reminder that in spite of Einstein's statement about the cosmological term being a blunder, there were a lot of cosmologists, especially of the British school influenced by Eddington - including his students Lemaitre & McVittie - who retained it. & several times in the development of cosmology well before the realization of dark energy the cosmological term was invoked to explain things.

Shalom
George
http://web.raex.com/~gmurphy/
----- Original Message -----
From: "Alexanian, Moorad" <alexanian@uncw.edu>
To: "George Murphy" <GMURPHY10@neo.rr.com>; "Murray Hogg" <muzhogg@netspace.net.au>; "ASA" <asa@calvin.edu>
Sent: Wednesday, July 16, 2008 8:55 AM
Subject: RE: The Cosmological constant (was Re: [asa] Predestined Fame:)

> This letter ought to be of interest for this discussion. Schrödinger had a brilliant mind. Moorad
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> URL: http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_60/iss_4/10_2.shtml <http://ptonline.aip.org/journals/doc/PHTOAD-ft/vol_60/iss_4/10_2.shtml> Published: April 2007
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> [Permission to reprint or copy this article/photo must be obtained from Physics Today. Call 301-209-3042 or e-mail rights@aip.org with your request.]
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> LETTERS
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> Dark energy and field equations without ?
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> April 2007, page 10
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> An intensive search for a presumed "dark energy" driving the acceleration of the cosmic expansion has taken place during the past decade. Adjunct to the search is the determination of the acceleration's "equation of state" if the driver is other than ?, Einstein's cosmological constant. What is remarkable and long since forgotten is that Erwin Schrödinger, shortly after Albert Einstein introduced the cosmological constant, discussed the field equations without ? but with a stress-energy tensor containing the term p?µ?. Einstein objected to this as a trivial variation of his field equations. It happened as follows.
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> Einstein introduced ? in 1917.1 In November of that year, Schrödinger submitted a paper2 in which Einstein's original equations, without ?, were provided with a stress-energy tensor consisting of a uniform distribution of dust and a negative pressure term. Shortly thereafter, in a note submitted to the same journal in March 1918, Einstein responded with some acerbity:3
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> When I wrote my description of the cosmic gravitational field I naturally noticed, as the obvious possibility, the variant Herr Schrödinger had discussed. But I must confess that I did not consider this interpretation worthy of mention.
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> . . . . A spatially closed world is only thinkable if the lines of force of gravitation, which end in ponderable bodies (stars), begin in empty space. Therefore, a modification of the theory is required such that "empty space" takes the role of gravitating, negative masses which are distributed all over the interstellar space. Herr Schrödinger now assumes the existence of matter with negative [scalar] mass density [negativen skalaren Massendichte] and represents it by the scalar p. This scalar p has nothing to do with the internal pressure of "really" ponderable masses, i.e., the noticeable pressure within stars of condensed matter of density ?; ? vanishes in the interstellar spaces, p does not.
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> The author [Schrödinger] is silent about the law according to which p should be determined as a function of the coordinates. We will consider only two possibilities:
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> 1. p is a universal constant. In this case Herr Schrödinger's model completely agrees with mine. In order to see this, one merely needs to exchange the letter p with the letter ? and bring the corresponding term over to the left-hand side of the field equations. Therefore, this is not the case the author could have had in mind.
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> 2. p is a variable. Then a differential equation is required which determines p as a function of x1 . . . x4. This means, one not only has to start out from the hypothesis of the existence of a nonobservable negative density in interstellar spaces but also has to postulate a hypothetical law about the space-time distribution of this mass density.
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> Of course, this occurred long before the advent of quantum-field theoretic concerns about zero-point energy and the later discovery of the type 1a supernovae with its implications, so the discussion vanished into the archives.
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> References
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> 1. A. Einstein, in The Principle of Relativity, W. Perrett, G. B. Jeffery, trans., Dover, New York (1923), p. 177.
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> 2. E. Schrödinger, Phys. Z. 19, 20 (1918).
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> 3. A. Einstein, Phys. Z. 19, 165 (1918); also in The Collected Papers of Albert Einstein, Volume 7: The Berlin Years: Writings, 1918-1921, A. Engel, trans., Princeton U. Press, Princeton, NJ (2002), doc. 3, p. 31.
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> Alex Harvey
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> (harvey@scires.acf.nyu.edu )
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> New York University
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> New York City
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> copyright © American Institute of Physics
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> ________________________________
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> From: asa-owner@lists.calvin.edu on behalf of George Murphy
> Sent: Wed 7/16/2008 8:31 AM
> To: Murray Hogg; ASA
> Subject: Re: The Cosmological constant (was Re: [asa] Predestined Fame:)
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> Murray -
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> I'm glad you pushed me on this & perhaps saved me later embarassment, for looking back at the footnote in Einstein's 1915 paper I see that he didn't say what I remembered him as saying. So my comment about that should be stricken from the record. OTOH it raises the question why at just that point he didn't recognize the possibility of what would come to be called the cosmological term.
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> For the argument that he made for the left side of the field equations outside a matter distribution was that it be formed from the metric tensor and its first and second derivatives, and be linear in the second derivatives. He then said that the contracted Riemann tensor was the only tensor satisfying that condition. But in reality the metric tensor itself, multiplied by an arbitrary constant (i.e., the cosmological constant) can be added to the Riemann tensor & the conditions will still be satisfied. This continues to be the case when the necessary changes are made to include matter.
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> It's hard to know why Einstein didn't at least comment on that possibility in 1915. In any case he did realize and exploit it in 1917 when he first dealt with the cosmological problem. There he wanted a static universe 1st because the general belief of astronomers at that time was that the universe was static on a large scale.
> But Max Jammer has also argued (in Einstein and Religion) that Einstein's commitment to Spinozistic pantheism may have played a role here. For if the universe and God are different names for the same thing, & if God is immutable, the universe must be immutable. That Einstein may have had extra-scientific motives is suggested by the facts that he was quite slow in accepting both observational and theoretical arguments for a non-static universe.
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> I think it's pretty well agreed that Einstein did tell Gamow that introducing the cosmological term was his "greatest blunder." One reason for that is that that term does from one standpoint make the field equations more complicated and less "elegant." But Einstein probably also realized that if he hadn't introduced it he might have been forced to realize that the universe wasn't static - that it was either expanding or contracting, & already by the early 20s there were enough galactic spectra to indicate expansion. Thus the expansion of the universe could have been announced as a "fourth test" of general relativity. (Although Newtonian cosmology gives the same result.)
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> Shalom
> George
> http://web.raex.com/~gmurphy/ <http://web.raex.com/~gmurphy/>
> ----- Original Message -----
> From: "Murray Hogg" <muzhogg@netspace.net.au <mailto:muzhogg@netspace.net.au> >
> To: "ASA" <asa@calvin.edu <mailto:asa@calvin.edu> >
> Sent: Tuesday, July 15, 2008 10:25 PM
> Subject: The Cosmological constant (was Re: [asa] Predestined Fame:)
>
>
>> Hi George,
>>
>> I don't intend to defend a position on which I admit to being rather
>> uninformed. So please take the following as a request for increased
>> understanding.
>>
>> My reference is ONLY to Einstein's introduction of the cosmological
>> constant in 1917 - a.k.a. "the biggest mistake of my career." -
>> regardless of what constructive work it might subsequently do.
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>> So while I acknowledge that it isn't a "fudge-factor" now, it does seem
>> to me to have been so at the time Einstein introduced it.
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>> If this is not the case then (first question) what was the point of the
>> infamous "biggest mistake" remark? Indeed, to ask a more basic question,
>> is this comment correctly ascribed to Einstein in the first place?
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>> And the second question: can you unpack a little the significance of the
>> footnote in the earlier paper? Again, if this demonstrates the 1917
>> cosmological constant to be a valid inclusion - why did Einstein
>> subsequently distance himself from the idea?
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>> Perhaps, were he aware of subsequent developments, Einstein might now
>> describe the cosmological constant as the most fortuitous mistake of his
>> career?
>>
>> Blessings,
>> Murray Hogg
>> Pastor, East Camberwell Baptist Church, Victoria, Australia
>> Post-Grad Student (MTh), Australian College of Theology
>>
>> gmurphy10@neo.rr.com <mailto:gmurphy10@neo.rr.com> wrote:
>>> This has nothing to do with numerology but I have to make my standard defence of the cosmological term. It's true that Einstein introduced it only to get a static universe in his 1917 paper. But in his basic general relativity paper 2 years earlier there is a footnote which in essence recognizes the possibility of including such a term in the field equations. & when he did introduce it he gave another argument for it, viewing it as an averaging of the "Poincare stresses" that were thought to be needed in classical electron theory.
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>>> If Einstein had not introduced the cosmological term for the reasons he did, someone else would eventually have introduced them for other reasons - of which several can be given. & of course now we know that it provides at least a first approximation to the effects of dark energy.
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>>> The popular designation of the cosmological constant as the quintessential fudge factor should cease.
>>>
>>> Shalom,
>>> George Murphy
>>>
>>> ---- Murray Hogg <muzhogg@netspace.net.au <mailto:muzhogg@netspace.net.au> > wrote:
>>> ...........................
>>>> To offer an analogy pertinent to the scientific nature of the list, I
>>>> see your approach as working ONLY if you are allowed to get away with a
>>>> highly questionable "fudging" of the equations -- a bit like Einstein's
>>>> addition of a cosmological constant to "fix" the theory of general
>>>> relativity. Such a constant was required by Einstein's need to
>>>> demonstrate a point, not because of any truly scientific motive.
>>> ................................
>>>
>> --
>> Murray Hogg
>> Pastor, East Camberwell Baptist Church, Victoria, Australia
>> Post-Grad Student (MTh), Australian College of Theology
>>
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Received on Wed Jul 16 11:12:52 2008

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