The forum discussion centers on the boundary conditions at infinity in general relativity, particularly as articulated in Michel Janssen's paper, "Einstein’s Quest for General Relativity, 1907–1920." It establishes that while Einstein initially found the application of general relativity to an infinite universe problematic, current interpretations affirm that spatially infinite models are consistent and align with contemporary cosmological data. The discussion clarifies that Einstein's personal preferences did not equate to mathematical inconsistency in spatially infinite models, which are now recognized as valid solutions to the Einstein Field Equations.
PREREQUISITESPhysicists, cosmologists, and students of general relativity seeking to understand the evolution of thought regarding spatially infinite models in the context of modern cosmology.
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:Ken G said: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.
That equation 11 looks a lot like the one we use today... nearly 100 years later! Remarkably prophetic.Jaime Rudas said: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 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.Jaime Rudas said: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.
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.Jaime Rudas said: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.
But you can transform it to "static coordinates". Maybe he interpreted it thus as another static solution with 0 matter and radiation content?PeterDonis said:Interesting--he was already foreseeing an expanding universe in 1917!
https://academic.oup.com/mnras/article/78/1/3/1239878vanhees71 said:Do you have a link to the original paper at hand?
De Sitter's model was considered stationary (not static) because, as it didn't contain matter, it presented the same appearance at all times.Ken G said:Why wasn't de Sitter's model dynamic?
Yes, but those only cover a portion of the spacetime. De Sitter's original coordinates cover the entire spacetime (and AFAIK are the only ones that do of the commonly known coordinate charts on de Sitter spacetime). The entire spacetime is not static; but the region covered by the static coordinates is.vanhees71 said:you can transform it to "static coordinates"
No, it is static, or at least the static region is. "Stationary" means that there is a timelike KVF; "static" means that the timelike KVF is hypersurface orthogonal. The latter condition is met in de Sitter spacetime in the static region.Jaime Rudas said:De Sitter's model was considered stationary (not static)
In fact, it seems that Hubble always had doubts about expansion and/or did not fully understand its physical nature, as can be seen in the quotes from the following papers:Ken G said: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
https://articles.adsabs.harvard.edu/pdf/1929ASPL....1...93HHere is a distance-velocity relation which probably holds out as far as the observations reach. It is difficult to believe that the velocities are real; that all matter is actually scattering away from our region of space. It is easier to suppose that the light-waves are lengthened and the lines of the spectra are shifted to the red, as though the objects were receding, by some property of space or by forces acting on the light during its long journey to the Earth.
The problem is now in the hands of the theorists [...]
https://articles.adsabs.harvard.edu/pdf/1931ApJ....74...43HThe constancy of Mn is believed to be well established, hence apparent magnitudes of nebulae are accepted as measures of distance. The interpretation of red-shifts as actual velocities, however, does not command the same confidence, and the term “velocity” will be used for the present in the sense of “apparent” velocity, without prejudice as to its ultimate significance.
https://ned.ipac.caltech.edu/level5/Sept04/Hubble/paper.pdfWe may state with some confidence that red-shifts are the familiar velocity-shifts, or else they represent some unrecognized principle of nature. We cannot assume that our knowledge of physical principles is yet complete; nevertheless, we should not replace a known, familiar principle by an ad hoc explanation unless we are forced to that step by actual observations.
[...]
The situation can be described as follows. Red-shifts are produced either in the nebulae, where the light originates, or in the intervening space through which the light travels. If the source is in the nebulae, then red-shifts are probably velocity-shifts and the nebulae are receding. If the source lies in the intervening space, the explanation of red-shifts is unknown but the nebulae are sensibly stationary.
https://www.science.org/doi/10.1126/science.95.2461.212The red shifts are frequently explained as velocity shifts (Doppler shifts), indicating actual recession of the nebulae at the rate of about 100 miles per second for each million light years of distance. The phenomenon has been observed out to about 240 million light years where the apparent velocities are nearly 25,000 miles per second. It may be stated with confidence that red shifts either are velocity shifts or they must be referred to some hitherto unrecognized principle in nature.
Well, but we must bear in mind that this was published in July 1929, that is, when, apparently, the only one who had paid attention to the works of Friedmann and Lemaître was Einstein.PeterDonis said:It's interesting that Hubble says "the problem is now in the hands of the theorist" in 1928, when the theorists had already published solutions to the problem six years before.