Dear Avi:
Thanks very much for responding! Of course, I agree fairly well with what you wrote. In fact, I probably should have explicitly stated that my point really does not affect the main thrust of your paper, even on the basis of space-time kinematics; i.e., regardless of the physical interpretation of the mathematics, no distant galaxy can be observed after a finite proper time on its own clock in a flat LambdaCDM universe, so any two distant neighbours who want to remain in contact for all time will eventually have to do something about it. Furthermore, as you've correctly noted, my point doesn't even matter as much in this regard, as the fact that even at a finite time in the future, the wavelengths of distant photons will be shifted to more than the Hubble radius, so they'll be impossible to observe. I agree with all of this, and therefore see relevance in your examination of this possible unexplored phenomenon in section 3.
My point was that the physical interpretation of the space-time kinematics, as being analogous to the description of a particle that passes through a black hole's event horizon and is forevermore causally disconnected from distant observers while it leaves behind a frozen image of its former self, is incorrect. Before I go on: do you understand my point about the incorrectness of the inferred similarity between this result and the description of a black hole's event horizon? Again, the result you've given is really *basically* due to the facts, that light travels at a finite speed, and that it can only travel a finite comoving coordinate distance through space in infinite time in a flat LambdaCDM universe, despite the fact that the comoving distance to any observer's particle horizon is a monotonically increasing function of cosmic time---i.e., so that no comoving galaxy ever *exits* anyone else's particle horizon. Do you agree with that?
You've noted the importance of being "careful of over-interpreting spacetime diagrams without checking if the signal is at all detectable". I knew of your result when I made my criticism, but I do think the entire physical description wants a logically consistent interpretation: I think that if the interpretation of a particular result that one makes is logically inconsistent with other related results, it is important to try to understand what the correct interpretation of the theory should be, which would regain logical consistency. Therefore, together with the (relevant) results like the one you've mentioned about the redshift of incident-but-unobservable photons, and the fact that distant galaxies are only observable up to finite values of their own proper times, I think it is important to note that the latter is really *basically* due to the fact that the light that any galaxy emitted even right after the big bang will only travel a finite comoving spatial coordinate distance in infinite cosmic time. Thus, it's not accurate to just think of the distant galaxy as continuing its existence in some unobservable region of spacetime after that final image of itself has been transmitted, and describe the situation as being similar to a black hole. Rather, the correct description is that the two galaxies exist together, separated by a finite comoving coordinate distance and a finite physical distance at t_{em,max}; then, the light emitted by the distant galaxy finally traverses that finite comoving coordinate distance by crossing an infinite physical distance through expanding space in infinite cosmic time.
Now, you may disagree with me that it matters at all which basic physical interpretation of the mathematics is actually consistent with the overall theory. In that case, you wouldn't be alone: not every physicist sees relevance in making logically consistent interpretations of our physical theories, so long as we have an accurate description of the observed phenomena. What truly goes on behind the scenes is an irrelevant matter to many physicists, who would agree with you that what really counts is whether the signal is at all detectable. The reason I take issue with this, is because of the fact that one incorrect interpretation does often breed another, until the description that's given does become very mis-construed, and, although the theory has been required to remain consistent with observed *phenomena*, there is little in the basic understanding of what's going on that resembles what is *actually* going on, even according to a logically consistant interpretation of the mathematical theory that was there from the outset!
It's relevant, therefore, to note that the black hole analogy to your result is wrong, and that it has already led to the production of mis-information about what the physical theory tells us is going on; e.g., in 2007, Krauss and Scherrer wrote, "Both analytic [7] and numerical [10] calculations indicate that the Local Group remains gravitationally bound in the face of the accelerated Hubble expansion. All more distant structures will be driven outside of the de Sitter event horizon in a timescale on the order of 100 billion years ([4], see also Refs. [8,9]). While objects will not be observed to cross the event horizon, light from them will be exponentially redshifted, so that within a time frame comparable to the longest lived main sequence stars all objects outside of our local cluster will truly become invisible [4]."
While these objects do truly become invisible (due to redshifts, as Krauss and Starkman showed), the expansion of space does not drive them outsite an event horizon, as I've mentioned: ever more comoving galaxies will *enter* our particle horizon for all time, although at infinite time we will finally "see" (by adding quotation marks, I mean that I understand that the light will be redshifted beyond detectability) the light from some very distant galaxy only as it was emitted at the Big Bang. The three sentences I've quoted above are, respectively, correct, incorrect, and correct, according to the physical theory.
A year after this, Krauss and Scherrer wrote an article for Scientific American, where they stated, "The
cosmological constant produces a fixed "event horizon," an imaginary surface beyond which no matter or radiation can reach us. The universe comes to resemble an inside-out black hole, with matter and radiation trapped outside the horizon rather than inside it... all the expanding matter in the universe will be driven outside the event horizon. This process has been studied by Abraham Loeb and Kentaro Nagamine, both then at Harvard University, who found that our so-called Local Group of galaxies (the Milky Way, Andromeda and a host of orbiting dwarf galaxies) will collapse into a single enormous supercluster of stars. All the other galaxies will disappear into the oblivion beyond the event horizon. This process takes about 100 billion years, which may seem long but is fairly short compared to the wilderness of eternity."
The galaxies are not simply trapped behind an imaginary surface where they can no longer be seen: the finite limit on the time when the last image of a distant galaxy was transmitted is really just due to the combined effect of asymptotic exponential expansion, in a universe where light travels at a constant finite speed; i.e., because of that constant finite speed, we always observe distant galaxies as they were in the distant past, rather than as they presently exist; and because of the asymptotic exponential expansion, the light that's received from any one at infinite cosmic time will have come from a finite comoving coordinate distance, and will have been emitted at a finite cosmic time.
If you understand this, and answered "yes" to my above questions, do you not now see that I'm not simply "over-interpreting spacetime diagrams", but that this incorrect inference has led to the production of mis-information that is potentially detrimental to the progress of physics? The present essay contest is about questioning which of our basic physical assumptions are wrong. Physical assumptions are based on empirical evidence and interpretation of the mathematical theories that best describe them. Do you agree that there is an incorrect interpretation that is being perpetuated here, and that the physics hasn't been correctly understood? Such mis-understandings, which often emerge when one is not aware of all the relevant facts, but only considers a few bits of knowledge, often lead to incorrect physical assumptions about what's actually going on in our Universe.
As I've argued in my essay (which I made a link to in my first comment), such was the case with Einstein's interpretation of the meaning of relativity of simultaneity, which should be interpreted differently according to cosmology. Then, with a clear understanding of that different interpretation of the relativity of simultaneity, one sees that the Robertson-Walker line-element was derived partly on the basis of a redundant assumption that was made as the result of a view that was sympathetic to Einstein's interpretation of the relativity of simultaneity, so that it may be possible to derive a different background metric that's consistent with the empirical data, but where the many well-known problems do not manifest themselves as they have.
I invite you to read my essay. I'm quite interested to know your thoughts on it, and would be very honoured if you were to offer some feedback.
Best regards,
Daryl
