The boundary conditions at infinity

[Jaime Rudas]

TL;DR Summary

Is it problematic to assume that the universe is spatially infinite?

Section 5 (pg. 29) of the Michel Janssen's paper EINSTEIN’S QUEST FOR GENERAL RELATIVITY, 1907–1920(*) says:

General relativity retains vestiges of absolute motion through the boundary conditions at infinity needed to determine the metric field for a given matter distribution.
[...]
As he [Einstein] told De Sitter in February 1917: “I have completely abandoned my views, rightfully contested by you, on the degeneration of the $g_{\mu \nu}$. I am curious to hear what you will have to say about the somewhat crazy idea I am considering now.” This “crazy idea” was actually quite ingenious: If boundary conditions at spatial infinity are the problem, why not eliminate spatial infinity? Einstein thus explored the possibility that the universe is spatially closed.

1. From the above I understand that the application of general relativity to an infinite universe was considered problematic.

2. On the other hand, I understand that it is currently not considered problematic to obtain solutions of general relativity for a spatially infinite universe.

Are assertions 1 and 2 correct?

If they are, why was it problematic before and now it isn't? In other words, in what sense did the formulation change?

If the assertions aren't correct, is it currently problematic to assume that the universe is spatially infinite?

(*) http://philsci-archive.pitt.edu/4377/1/LoveMinusZero.pdf

 

[PeterDonis]

Section 5 (pg. 29) of the Michel Janssen's paper EINSTEIN’S QUEST FOR GENERAL RELATIVITY, 1907–1920(*) says:

This paper is about the history of science, not about our best current science. What Einstein did or did not believe back then is the wrong thing to look at if you want to know what our best current science says.

1. From the above I understand that the application of general relativity to an infinite universe was considered problematic.

You understand incorrectly. The exchange described between Einstein and de Sitter led Einstein to investigate the possibility of a consistent model of a finite universe, but he also believed it needed to be static; hence he added a positive cosmological constant and came up with the Einstein Static Universe. But Einstein never, as far as I know, claimed that a spatially infinite model was "problematic", nor does anyone believe that now; such models are perfectly consistent, and our best current model of our universe is in fact spatially infinite because that is the best match with our current data. Einstein just didn't like a spatially infinite universe as a personal opinion.

2. On the other hand, I understand that it is currently not considered problematic to obtain solutions of general relativity for a spatially infinite universe.

Of course not. See above.

 

[PeterDonis]

General relativity retains vestiges of absolute motion through the boundary conditions at infinity needed to determine the metric field for a given matter distribution.

It's worth noting that, for a spatially infinite model of the entire universe, this is actually not a problem, because there are no "boundary conditions at infinity". The only condition is that the stress-energy is constant everywhere in space (but changes with time). The models where "boundary conditions at infinity" become an issue are models of isolated objects like stars or planets surrounded by empty space.

 

[Jaime Rudas]

You understand incorrectly. [...] Einstein never, as far as I know, claimed that a spatially infinite model was "problematic"

In the paper Cosmological Considerations In The General Theory of Relativity(*), Einstein says:

From what has now been said it will be seen that I have not succeeded in formulating boundary conditions for spatial infinity. Nevertheless, there is still a possible way out, without resigning as suggested under (b). For if it were possible to regard the universe as a continuum which is finite (closed) with respect to its spatial dimensions, we should have no need at all of any such boundary conditions. We shall proceed to show that both the general postulate of relativity and the fact of the small stellar velocities are compatible with the hypothesis of a spatially finite universe; though certainly, in order to carry through this idea, we need a generalizing modification of the field equations of gravitation.

I take this to mean that Einstein considered the spatially infinite model problematic, however, I accept that it is my personal interpretation.

(*) https://einsteinpapers.press.princeton.edu/vol6-trans/439

 

[PeterDonis]

I take this to mean that Einstein considered the spatially infinite model problematic

I think it shows that, as I said, Einstein did not like spatially infinite models as a personal opinion, and so he wanted to make clear that spatially infinite models were not required--that a spatially finite model could be given that could describe the universe as a whole. (Ironically, if he had not been so fixated on a static universe, he could have discovered the spatially finite closed expanding universe that Friedmann discovered several years later.)

But that's not the same as claiming that spatially infinite models are problematic. The latter claim would amount to saying that such models had some kind of inconsistency in them. Einstein never claimed that, most likely because he knew it would be wrong: he knew such models were consistent, he just didn't like them, so he wanted to make sure it was understood that spatially finite models were possible too.

 

[Jaime Rudas]

I think it shows that, as I said, Einstein did not like spatially infinite models as a personal opinion, and so he wanted to make clear that spatially infinite models were not required--that a spatially finite model could be given that could describe the universe as a whole. [...] he just didn't like them, so he wanted to make sure it was understood that spatially finite models were possible too.

What are you basing this on? Do you have a reference?

 

[PeterDonis]

What are you basing this on?

The article you referenced. For example: "We shall proceed to show that both the general postulate of relativity and the fact of the small stellar velocities are compatible with the hypothesis of a spatially finite universe". That was Einstein's concern: to show that a spatially finite model was possible.

But also in the article you reference, Einstein includes (on the page before the one you quoted from) two possibilities, which he calls (a) and (b). The sentence you quoted about not having succeeded in formulating boundary conditions at infinity applies to his possibility (a): but as far as a model of the universe as a whole is concerned, that is to be expected, because his possibility (a) is that the metric becomes Minkowski at infinity--what we would now call "asymptotically flat" boundary conditions--and such a model, as Einstein notes and as I said in an earlier post, is suitable for modeling an isolated object like a star or planet surrounded by empty space, but not for modeling the universe as a whole.

But that still leaves possibility (b): and note that Einstein says that that possibility is "an incontestable position", and is "taken up at the present time by de Sitter". In other words, he recognizes that there is a way to formulate a consistent spatially infinite model of the universe. He just doesn't like it. (And note that this possibility (b) describes what our best current model of the universe does: we don't impose any general boundary conditions at infinity, we just impose the condition that on every spacelike slice of constant FRW coordinate time, the density of stress-energy is the same everywhere.)

 

[Jaime Rudas]

But that still leaves possibility (b): and note that Einstein says that that possibility is "an incontestable position", and is "taken up at the present time by de Sitter". In other words, he recognizes that there is a way to formulate a consistent spatially infinite model of the universe. He just doesn't like it.

The full quote you referenced says:

(b) We may refrain entirely from laying down boundary conditions for spatial infinity claiming general validity; but at the spatial limit of the domain under consideration we have to give the $g_{\mu \nu}$ separately in each individual case, as hitherto we were accustomed to give the initial conditions for time separately.

The possibility (b) holds out no hope of solving the problem, but amounts to giving it up. This is an incontestable position, which is taken up at the present time by de Sitter.* But I must confess that such a complete resignation in this fundamental question is for me a difficult thing. I should not make up my mind to it until every effort to make headway toward a satisfactory view had proved to be vain.

I'm sorry, but I don't get from this that Einstein recognizes that there is a way to formulate a consistent spatially infinite model of the universe but that he just doesn't like it.

 

[PeterDonis]

I don't get from this that Einstein recognizes that there is a way to formulate a consistent spatially infinite model of the universe but that he just doesn't like it.

He says that (b) is an "incontestable position". That means he recognizes that it is valid. And (b) is a way to formulate a consistent spatially infinite model of the universe: indeed, as I have already pointed out, it is the way our best current model of the universe (as well as other models going back to de Sitter spacetime, the example Einstein refers to--that's why he says his (b) is "taken up by de Sitter") actually does it.

So Einstein recognizes (b) as valid, but he obviously doesn't like it because he then goes right back to his (a) and tries to pursue that. And we now know that (a) is wrong: (a) amounts to claiming that every spatially infinite solution to the Einstein Field Equation must be asymptotically flat, which we know is not the case.

In other words: Einstein was looking for a way of, in his words, "laying down boundary conditions for spatial infinity claiming general validity"--that is his (a). But that turns out to simply not be possible; the alternative, option (b), is the only one that is actually viable. Einstein was simply unwilling to accept this because he didn't like it.

 

[Jaime Rudas]

So Einstein recognizes (b) as valid, but he obviously doesn't like it because he then goes right back to his (a) and tries to pursue that. And we now know that (a) is wrong: (a) amounts to claiming that every spatially infinite solution to the Einstein Field Equation must be asymptotically flat, which we know is not the case.

From the above, I understand that solutions based on (a) are wrong, while only those based on (b) can be correct. Does this mean that de Sitter's 1917 solution is a valid solution, while Einstein's 1917 solution is a wrong solution?

 

[vanhees71]

De Sitter's 1917 solution is an exact solution for an empty universe with positive cosmological constant, while Einstein's solution is one for a universe with constant energy density and cosmological constant. Both are exact solutions of the Einstein field equations with the given, different "matter content" and positive cosmological constant.

According to our present "Standard Model" (cold-dark-matter model with cosmological constant/aka "dark energy") in the long-time limit the universe is approximately described by a de Sitter solution.

 

[Jaime Rudas]

Both are exact solutions of the Einstein field equations with the given, different "matter content" and positive cosmological constant.

I think that's true, but isn't my doubt. The question is whether in Einstein's 1917 solution the possibility (a) or the possibility (b) is considered.

For reference, I quote Einstein's (a) and (b) possibilities:

After the failure of this attempt, two possibilities next present themselves.
(a) We may require, as in the problem of the planets, that, with a suitable choice of the system of reference, the $g_{\mu \nu}$ in spatial infinity approximate to the values

$\begin{matrix} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}$

(b) We may refrain entirely from laying down boundary conditions for spatial infinity claiming general validity; but at the spatial limit of the domain under consideration we have to give the $g_{\mu \nu}$ separately in each individual case, as hitherto we were accustomed to give the initial conditions for time separately.

 

[PeterDonis]

From the above, I understand that solutions based on (a) are wrong, while only those based on (b) can be correct.

Depends on what you mean. Asymptotically flat solutions certainly exist, and (a) refers to asymptotically flat boundary conditions. So solutions that meet the requirements of (a) exist. However, not all solutions meet the requirements of (a), since there are spatially infinite solutions that are not asymptotically flat, so in that sense (a) is not correct. Based on what Einstein says, I think he was trying to use (a) in the latter sense--as a boundary condition at infinity that all solutions would have to meet--so in that sense he was wrong.

Does this mean that de Sitter's 1917 solution is a valid solution, while Einstein's 1917 solution is a wrong solution?

No. In fact, both of these solutions (by which I assume you mean de Sitter spacetime and the Einstein static universe) are spatially closed and have no boundary conditions at infinity whatsoever.

The question is whether in Einstein's 1917 solution the possibility (a) or the possibility (b) is considered.

Neither. Einstein's 1917 solution, the Einstein static universe, avoids the issue altogether by being spatially closed, with no boundary conditions at infinity at all. (So does de Sitter spacetime, as noted above.)

The very quote you give in your OP points this out: This “crazy idea” was actually quite ingenious: If boundary conditions at spatial infinity are the problem, why not eliminate spatial infinity?

As the paper you reference goes on to point out, this did not solve the other problem Einstein was struggling with, namely, to have the metric determined by the distribution of matter. De Sitter showed that having a spatially closed solution did not solve this problem by finding a solution, de Sitter spacetime, that was also spatially closed but had no matter in it whatever.

 

[PeterDonis]

Einstein's 1917 solution, the Einstein static universe, avoids the issue altogether by being spatially closed

The Einstein paper you reference says this as well, as you quoted in post #4. In that quote, Einstein is saying that, even though he has "not succeeded in formulating boundary conditions for spatial infinity"--i.e., he knows his (a) doesn't work for all solutions, and he doesn't like (b) because he thinks it's just "resigning"--there is still "a possible way out", namely, to have a solution that is spatially closed and does not require any boundary conditions at infinity. Then, in the next section, he describes his static universe.

 

[Jaime Rudas]

Depends on what you mean. Asymptotically flat solutions certainly exist, and (a) refers to asymptotically flat boundary conditions. So solutions that meet the requirements of (a) exist. However, not all solutions meet the requirements of (a), since there are spatially infinite solutions that are not asymptotically flat, so in that sense (a) is not correct. Based on what Einstein says, I think he was trying to use (a) in the latter sense--as a boundary condition at infinity that all solutions would have to meet--so in that sense he was wrong.

I think that what is highlighted solves my doubt. Thank you.

 

[PeterDonis]

I think that what is highlighted solves my doubt. Thank you.

You're welcome!

 

[vanhees71]

I think that's true, but isn't my doubt. The question is whether in Einstein's 1917 solution the possibility (a) or the possibility (b) is considered.

For reference, I quote Einstein's (a) and (b) possibilities:

(a) is the assumption that spacetime becomes asymptatically at spatial infinity Minkowskian. That makes only sense, if there's vacuum outside some bounded region in space. That's not the case in cosmology, where we have "matter and radiation" as well as "dark energy/cosmological constant".

It's not very clear to me what Einstein meant by (b). The FLRW ansatz for the spacetime pseudo-metric assumes global homogeneity and isotropy of space. In this sense you have implict boundary conditions, and as you can easily calculate the Einstein tensor on the left-hand side of the Einstein equation implies that the energy-momentum tensor is that of an ideal fluid. The only freedom still left is the equation of state, necessary to close the equations, and that's subject to fits to observations. Currently this fit (based on the precision measurements of the cosmic microwave-background radiation's flucuations with the Planck satellite and redshift-distance relations via Supernovae etc) leads to about 27% of "matter" (23% "dark matter" + 4% "baryonic matter") and 73% of "dark energy".

 

[PeterDonis]

It's not very clear to me what Einstein meant by (b).

I think he meant by (b) the point of view (which, as I have commented, is in fact the modern point of view that we take today) that there is no one single specification of boundary conditions at infinity that applies to all solutions. Einstein says he thought that was just "resigning", giving up on trying to "fix" the issue he thought he saw in the paper that was referenced. Whereas now we just take it as a known fact that no, there is no one single specification of boundary conditions at infinity that applies to all solutions.

 

[Ken G]

What I don't understand is the clear implication in the article that Einstein's solution to the problem of his own perceived need for an asymptotically Minkowskian metric (solution a) was to suggest a closed solution that had no asymptotics. But Einstein says his solution is of type a, so should be asymptotically Minkowskian, which would not be closed. Are we sure Einstein is not talking about an infinite solution that is made Minkowsian by a cosmological constant? In other words, Einstein seemed to originally want a solution that was asymptotically flat and static, so it could be dynamic where there is matter but not globally where there isn't (he seemed to want a finite matter distribution, rather than a cosmological principle). But he found he could only accomplish the global behavior by also making it locally static, using a cosmological constant. So either he only wanted the cosmological constant to exist where there was matter, or he was accepting the cosmological principle. But that does not sound like eliminating the problem of boundary condition (a) by removing all asymptotics, which is the type of solution he seems to characterize as "resigning" the problem. After all, de Sitter's approach (which @PeterDonis points out is a closed solution) would seem to be the one that has the "ingenious" approach of eliminating infinity, so it would not make sense for Einstein to object to it when de Sitter did it but then embrace it himself. So you see my confusion.

Another thing that surprises me about the apparent exchange between Einstein and de Sitter is that Einstein seems to be saying that de Sitter came up with the idea of a cosmological constant first (for reasons I cannot imagine), and all Einstein did was connect the cosmological constant to the matter distribution in a finely tuned way so as to achieve an infinite, static, and spatially flat (i.e., Minkowskian) universe. If so, then the "crazy" idea Einstein was talking about was not the cosmological constant itself, but rather a finely tuned version of it. If that is the right way to interpret this exchange, then Einstein is merely saying he prefers fine tuning over arbitrariness, because arbitrariness seems like "resignation", while fine tuning seems like some kind of physical law. (Of course, Einstein did not realize that even his fine tuned solution would not be stable.)

 

[PeterDonis]

Einstein says his solution is of type a

Not the closed Einstein static universe, no. He only describes that solution after admitting that he hasn't been able to satisfy (a) and doesn't like (b) because he thinks it's giving up. He then says there might be another way out of his perceived dilemma: find a solution with no infinity at all, and therefore no need for boundary conditions at infinity.

Are we sure Einstein is not talking about an infinite solution that is made Minkowsian by a cosmological constant?

Yes. First, we know he was talking about the Einstein static universe, which is spatially closed and static because of the exact balance between matter and the cosmological constant. And second, Minkowski spacetime is not a solution of the Einstein Field Equation with a cosmological constant.

Einstein seemed to originally want a solution that was asymptotically flat and static

Einstein's (a) with the asymptotically flat metric was one attempt on his part to formulate boundary conditions at infinity that would be generally applicable; the latter was his general goal. (And even that wasn't his ultimate concern: see further comments below.) But asymptotic flatness was not the only proposal he made along those lines. In another exchange with de Sitter, Einstein says he thinks the metric coefficients should either go to zero or go to infinity at spatial infinity. Of course neither of those solutions actually works (and I think he realized that when de Sitter pointed it out).

de Sitter's approach (which @PeterDonis points out is a closed solution) would seem to be the one that has the "ingenious" approach of eliminating infinity, so it would not make sense for Einstein to object to it when de Sitter did it but then embrace it himself.

Einstein didn't like de Sitter spacetime because it has no matter in it; it only has a cosmological constant. So he didn't consider it a viable solution to describe a universe with matter. But he did admit that it was a counterexample to his conjecture that spatially closed solutions would avoid the Machian issue he was struggling with--in modern terms, how to explain where the spacetime geometry comes from in the absence of matter. (His concern with boundary conditions at infinity was also driven by the Machian issue.)

 

[PeterDonis]

Einstein seems to be saying that de Sitter came up with the idea of a cosmological constant first

I'm not sure that's the case. From the references given it appears that Einstein was involved in discussions of his theory with de Sitter in 1916 and 1917 that led to him adding the cosmological constant to his equations; but I don't see anything that says that he did that because de Sitter had already given him the idea.

The three de Sitter 1916 references in the Michael Janssen paper referenced in the OP are here:

https://academic.oup.com/mnras/article/77/2/155/979347

https://academic.oup.com/mnras/article/76/9/699/1027501

https://dwc.knaw.nl/DL/publications/PU00012379.pdf

 

[Jaime Rudas]

Another thing that surprises me about the apparent exchange between Einstein and de Sitter is that Einstein seems to be saying that de Sitter came up with the idea of a cosmological constant first (for reasons I cannot imagine), and all Einstein did was connect the cosmological constant to the matter distribution in a finely tuned way so as to achieve an infinite, static, and spatially flat (i.e., Minkowskian) universe. If so, then the "crazy" idea Einstein was talking about was not the cosmological constant itself, but rather a finely tuned version of it.

I note the following sequence of events:

The mention of "a crazy idea" is from a letter from Einstein to de Sitter dated February 2, 1917:
https://einsteinpapers.press.princeton.edu/vol8-trans/309

Einstein's article where he proposes his stationary universe, closed and with cosmological constant, was published on February 15, 1917.
https://einsteinpapers.press.princeton.edu/vol6-doc/568

De Sitter's article where he proposes his universe without matter and with cosmological constant was published on November 11, 1917:
https://academic.oup.com/mnras/article/78/1/3/1239878

 

[Ken G]

Einstein didn't like de Sitter spacetime because it has no matter in it; it only has a cosmological constant. So he didn't consider it a viable solution to describe a universe with matter. But he did admit that it was a counterexample to his conjecture that spatially closed solutions would avoid the Machian issue he was struggling with--in modern terms, how to explain where the spacetime geometry comes from in the absence of matter. (His concern with boundary conditions at infinity was also driven by the Machian issue.)

Thank you, this clarifies the situation considerably. Let me see if I understand what you are saying. Einstein thought Minkowskian spacetime was an observational constraint, because he did not know about the Hubble Law. But he also knew that the Minkowski metric could not be the "degenerated" (asymptotic) metric of an infinite universe, whereas the one that did seem "natural" to him was one which separated into a separate 3D space equipped with a universal time (essentially the picture of an ether, before his relativity, ironically). de Sitter destroyed that picture by pointing out to Einstein that to achieve this type of Machian transition between a local Minkowski spacetime and an asymptotically invariant one required a "supernatural" distribution of unseen matter on the boundary between the two spacetimes. So he exposed the idea as an artificial way to embed a Minkowski spacetime inside a classical ether, which was antithetical to the spirit of Einstein's own relativity. In essence he hoisted Einstein on his own petard.

But then Einstein wriggled out of the trap by suggesting a closed universe that needed no "natural" asymptotic ether because it had no boundary condition at all. So for the second time, Einstein banished the ether, this time an ether of his own making. Later it was shown this would not be a stable solution. OK, that all makes sense now, I see your point. Note the additional irony in all this, which is that Einstein didn't like the fact that de Sitter had successfully argued the boundary condition at infinity only had to obey an invariance principle but not some kind of "uniqueness" (i.e., Machian) principle, it could instead take on many values that had to be established by observation rather than by the nature of the matter. So Einstein did away with infinity, by invoking a cosmological constant that also allowed a static spacetime. de Sitter rejects the fine tuning (the Machian uniqueness) aspect, but embraces the cosmological constant, which only existed to achieve the very uniqueness that de Sitter rejected. It is not easy to see why de Sitter would have taken on one but not the other! One wonders if de Sitter, in publishing his solution, was merely providing an example of purely academic interest, or had he by that time (prophetically) elevated it to a potential future of our universe?

 

[PeterDonis]

Einstein thought Minkowskian spacetime was an observational constraint

I'm not sure this is true. It's true of course that Einstein didn't know about the Hubble law in 1917, nobody did, but I don't think his proposal (a) was based on any observational constraint. It was based on him trying on theoretical grounds to come up with a specification of boundary conditions at infinity that would be applicable to all possible solutions.

he also knew that the Minkowski metric could not be the "degenerated" (asymptotic) metric of an infinite universe, whereas the one that did seem "natural" to him was one which separated into a separate 3D space equipped with a universal time

That's the implication of his other proposal for boundary conditions at infinity (the one before (a)), but I'm not sure how "natural" it seemed. The fact that it amounts to making spacetime Galilean at infinity does seem very weird in a relativistic context.

Einstein wriggled out of the trap by suggesting a closed universe that needed no "natural" asymptotic ether because it had no boundary condition at all. So for the second time, Einstein banished the ether, this time an ether of his own making. Later it was shown this would not be a stable solution.

Yes. And what's more, de Sitter also showed that making the universe closed would not "banish the ether", by finding a spatially closed solution with no matter in it.

 

[Ken G]

I'm not sure this is true. It's true of course that Einstein didn't know about the Hubble law in 1917, nobody did, but I don't think his proposal (a) was based on any observational constraint. It was based on him trying on theoretical grounds to come up with a specification of boundary conditions at infinity that would be applicable to all possible solutions.

I'm basing it on Einstein's statement in one of the letters where he specifically called out the low velocities of the stars. Obviously he is not talking about orbits within galaxies, which only depend on the galactic mass and size, so he must mean that the observed universe of his knowledge seemed fairly static. We hadn't seen that far yet, so it's not clear how much of that could be regarded as observed fact vs. mere expectation, but he does call out that specific observation.

That's the implication of his other proposal for boundary conditions at infinity (the one before (a)), but I'm not sure how "natural" it seemed. The fact that it amounts to making spacetime Galilean at infinity does seem very weird in a relativistic context.

I think that must connect with your point that he wanted to make the universe Machian, that's what seemed "natural" to him. Apparently Mach's principle was a driving philosophical motivation behind formulating GR, though this thread shows that there was still an issue with the boundary conditions because Einstein could only produce differential equations for the metric, not the metric itself. It looks like some metrics are less Machian than others, even with the relativistic constraints. So it seems that Einstein's view of Mach's principle looked like using matter to dictate the spacetime inside the matter, but this had to "degenerate" into an etherlike spacetime outside the matter distribution. When de Sitter convinced him this was really just a return to the old ways of thinking, the alternatives were to connect to a somewhat arbitrary symptotic boundary that would not have a Mach's principle (solution b), or make the universe closed and fully Machian.

Yes. And what's more, de Sitter also showed that making the universe closed would not "banish the ether", by finding a spatially closed solution with no matter in it.

I'm not sure we should say such a solution fails to banish the ether, though I guess that depends on what one means by ether. To me, it means Galilean relativity where Maxwell's equations only apply in a single inertial reference frame, that of the ether. So it is a metric that respects a universal time that is separate from anything going on spatially, not a Minkowski type metric that in some sense unifies time and space by extending spatial symmetries into the time domain. But it does banish Mach, with no mass to tell spacetime how to behave such that spacetime could be considered to be no more than a kind of field exists only to orient and interconnect the dynamics of the matter. I think that might be how Einstein viewed Mach's principle, wherein motion was strictly a kind of relationship among the matter, much like how Newton regarded forces without giving fields an ontology of their own. (So in that interpretation, one might say that de Sitter was to Einstein what Faraday was to Newton.)

 

[Jaime Rudas]

One wonders if de Sitter, in publishing his solution, was merely providing an example of purely academic interest, or had he by that time (prophetically) elevated it to a potential future of our universe?

From what de Sitter concludes in his 1917 paper, it wouldn't appear that he was simply providing an example of purely academic interest (hypothesis A refers to Einstein's stationary model and hypothesis B to de Sitter's model):

Spiral nebule most probably are amongst the most distant objects we know. Recently a number of radial velocities of these nebule have been determined. The observations are still very uncertain, and conclusions drawn from them are liable to be premature. Of the following three nebula, the velocities have been determined by more than one observer:​

​

Andromeda (3 observers) — 311 km/sec.​

N.G.C. 1068 (3 observers) + 925 km/sec.​

N.G.C. 4594 (2 observers) + 1185 km/sec.​

​

These velocities are very large indeed, compared with the usual velocities of stars in our neighbourhood.​

The velocities due to inertia, according to the formula (25’), have no preference of sign. Superposed on these are, however, the apparent radial velocities due to the diminution of $g_{44}$ which are positive. The mean of the three observed radial velocities stated above is +600 km./sec. If for the average distance we take 10⁵ parsecs = 2×10¹⁰, then we find​

​

R=3×10¹¹.​

Of course this result, derived from only three nebule, has practically no value. If, however, continued observation should confirm the fact that the spiral nebula have systematically positive radial velocities, this would certainly be an indication to adopt the hypothesis B in preference to A. If it should turn out that no such systematic displacement of spectral lines towards the red exists, this could be interpreted either as showing A to be preferable to B, or as indicating a still larger value of R in the system B.​

https://academic.oup.com/mnras/article/78/1/3/1239878

 

[PeterDonis]

From what de Sitter concludes in his 1917 paper, it wouldn't appear that he was simply providing an example of purely academic interest

Interesting--he was already foreseeing an expanding universe in 1917!

 

[Ken G]

Yes, we tend to leave that out in the story of Hubble. Even Hubble did not immediately accept his observations as compelling evidence of an expanding time dependent universe, as the model of steady state outward flows was alive and well at the time. So it was de Sitter who seemed to have been a real pioneer in the idea that spacetime expansion would carry matter outward to lower density, and ironically he did it with a model that used a cosmological constant, which took over half a century after de Sitter's death to receive any observational support. It was Lemaitre that initially considered a Big Bang model initiated by an expanding initial condition (so without need for a cosmological constant), and immediately interpreted Hubble's observation as support of that picture.

Hence, one wonders if de Sitter ever had a "greatest blunder" moment like Einstein, where he imagined that his reliance on a cosmological constant instead of just a dynamical initial condition (like Lemaitre) was a wrong turn that allowed Lemaitre to scoop him. Little did he know they would basically both end up being right (with a lot of inflation thrown in for good measure)! We can imagine a debate between the two in 1930. Did such a thing happen? If it didn't, what would it have looked like, and how prophetic would such a debate have been?

 

[Jaime Rudas]

Yes, we tend to leave that out in the story of Hubble. Even Hubble did not immediately accept his observations as compelling evidence of an expanding time dependent universe, as the model of steady state outward flows was alive and well at the time. So it was de Sitter who seemed to have been a real pioneer in the idea that spacetime expansion would carry matter outward to lower density, and ironically he did it with a model that used a cosmological constant, which took over half a century after de Sitter's death to receive any observational support. It was Lemaitre that initially considered a Big Bang model initiated by an expanding initial condition (so without need for a cosmological constant), and immediately interpreted Hubble's observation as support of that picture.

In relation to this, a few weeks ago I tried to post a note that was rejected as inappropriate for PF. Considering it of interest, I risk presenting again an extract of it:

In January 1929, Edwin Hubble published the paper A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae in the Proceedings of the National Academy of Sciences (PNAS). In this paper, he demonstrated that galaxies are moving away from each other at speeds proportional to their distances. This was perhaps the first observational indication of the expansion of the universe.

This paper caught de Sitter's attention, and he carefully studied it, collecting some new data and preparing a complementary graph to Hubble's, which he presented at the January 10, 1930 session of the Royal Astronomical Society. In a remarkable display of intellectual honesty, he indicated that this linear relationship wasn't compatible with his model of the universe, and therefore, his model should be discarded.

During the discussion of de Sitter's presentation, Eddington responded that it was necessary to search for a new model since the ones proposed by Einstein and de Sitter didn't fit recent observations.

The proceedings of this session were published in February 1930 in the Monthly Notices of the Royal Astronomical Society. As soon as the Belgian cosmologist Georges Lemaître read the proceedings, he sent a letter to Eddington, his former professor, reminding him that in 1927, he had given him a copy of his paper A Homogeneous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae. In this paper, Lemaître presented a model of the universe that, unlike Einstein's and de Sitter's, was dynamic, implying a linear relationship between distance and redshift. Lemaître attached two copies of the mentioned paper to the letter and asked Eddington to give one copy to de Sitter.

Eddington received Lemaître's communication while he was writing an paper on the instability of Einstein's static cosmological model and realized that the same instability was implicit in Lemaître's paper. He mentioned this in the May 9, 1930 session of the Royal Astronomical Society, where he said:

Some time ago, I conjectured that Einstein's spherical universe might be unstable. More recently, I saw to settle the question mathematically. I was working on this problem with Mr. McVittie, and nearly reached the solution when I learnt of a remarkable paper by Abbé G. Lemaitre of Louvain, published in 1927, which contained all the necessary mathematics. He does not say explicitly that Einstein's universe is unstable, but it follows immediately from his equations. I think this makes a great difference to our outlook; Einstein's solution gaves the only possible condition of equilibrium of the universe, and now this proves to be unstable. De Sitter's is also reckoned technically as an equilibrium solution, but it is a bit of a fraud; being entirely empty, there is nothing in his world whose equilibrium could possibly be upset. In saying this I am not disparaging it, because it is much more interesting than a genuine equilibrium solution would have been.
To discuss stability, we must have a range of solutions, and Lemaitre's work provides this. He treats of a world whose radius is a function of the time. Instead of having to choose between Einstein's and de Sitter's worlds our conclusion now is that the universe started as an Einstein world, being unstable it began to expand, and it is now progrssing towards de Sitter’s form as an ultimate limit. [...] Lemaître finds a very simple formula for the red-shift of the light; the ratio of the wave-length observed to that emitted is equal to the ratio of the radii of the universe at the times of observation and emission.

To this, De Sitter added:

I agree enterely with Prof. Eddington's remarks. Einstein's solution, gives a world full of matter, but no motion; mine gives a world full of motion, but no matter. I had been pursuing the same kind or investigation as has Prof. Eddington when he sent me Lemaitre's paper, and I have found the same as Prof. Eddington.

References:
http://www.pnas.org/content/pnas/15/3/168.full.pdf
https://articles.adsabs.harvard.edu/pdf/1927ASSB...47...49L
http://adsbit.harvard.edu/full/seri/Obs../0053/0000162.000.html

 

[PeterDonis]

a few weeks ago I tried to post a note that was rejected as inappropriate for PF

No, it was rejected because your previous post did not include any question and it was not at all clear what you were trying to discuss. Now it is. What you are referencing is perfectly acceptable for PF discussion; it just needs an appropriate context.

 

[Jaime Rudas]

It was Lemaitre that initially considered a Big Bang model initiated by an expanding initial condition (so without need for a cosmological constant), and immediately interpreted Hubble's observation as support of that picture.

It should be noted that Lemaître did include the cosmological constant in his original model, as can be seen in equations 2, 3, 10 and 11 in:

https://articles.adsabs.harvard.edu/pdf/1927ASSB...47...49L

 

[Ken G]

It should be noted that Lemaître did include the cosmological constant in his original model, as can be seen in equations 2, 3, 10 and 11 in:

https://articles.adsabs.harvard.edu/pdf/1927ASSB...47...49L

That equation 11 looks a lot like the one we use today... nearly 100 years later! Remarkably prophetic.

 

[Ken G]

This paper caught de Sitter's attention, and he carefully studied it, collecting some new data and preparing a complementary graph to Hubble's, which he presented at the January 10, 1930 session of the Royal Astronomical Society. In a remarkable display of intellectual honesty, he indicated that this linear relationship wasn't compatible with his model of the universe, and therefore, his model should be discarded.

That actually surprises me, because Hubble's linear relation (if expressed in terms of rate of change of distance today as a function of distance today) does not require anything but the cosmological principle. To get any hint of the actual dynamics of the scale parameter a(t), you have to use a different distance measure than "distance today", one that I would have thought the Hubble law did not extend far enough away to be able to tell the difference. Apparently de Sitter's original model accelerated so quickly that even at the nearby distances seen by Hubble, a difference could be detected, but that does surprise me since Hubble's observations did not extend very far at all.

The proceedings of this session were published in February 1930 in the Monthly Notices of the Royal Astronomical Society. As soon as the Belgian cosmologist Georges Lemaître read the proceedings, he sent a letter to Eddington, his former professor, reminding him that in 1927, he had given him a copy of his paper A Homogeneous Universe of Constant Mass and Growing Radius Accounting for the Radial Velocity of Extragalactic Nebulae. In this paper, Lemaître presented a model of the universe that, unlike Einstein's and de Sitter's, was dynamic, implying a linear relationship between distance and redshift. Lemaître attached two copies of the mentioned paper to the letter and asked Eddington to give one copy to de Sitter.

Why wasn't de Sitter's model dynamic? I thought it had a cosmological constant by only a trace matter component, which seems pretty dynamic. And yes, all these models, as well as the later one that Einstein and de Sitter collaborated on (which was dynamic but with no cosmological constant), exhibited a kind of basic instability that says any amount of curvature rapidly increases. So they all would have needed inflation to understand how our universe has stayed even remotely close to flat for so long, so it seems that basic instability problem never really went away for Einstein, no matter how he tried to escape it.

 

[vanhees71]

Interesting--he was already foreseeing an expanding universe in 1917!

But you can transform it to "static coordinates". Maybe he interpreted it thus as another static solution with 0 matter and radiation content?

https://en.wikipedia.org/wiki/De_Sitter_space#Static_coordinates

Do you have a link to the original paper at hand?

 

[Jaime Rudas]

Do you have a link to the original paper at hand?

https://academic.oup.com/mnras/article/78/1/3/1239878