Genady

If we send light through the hole through the center of the Earth, there are no massive objects to bend it. If we use the same clocks on the surface of the Earth for the measurements, they curved the same throughout. We can measure geometrical features of the space. The point is, we will find that they do not fit Euclidean theorems. In case of Earth, they will fit the theorems of spherical geometry, of the 3-sphere!

Directly. Ideally, you make a hole through the center of the Earth and measure the length of straight line from surface to surface -- diameter. You walk around the equator with a meter stick. Then you see if the second number is equal the first times pi. It is not.

Not assuming that you, @MigL, don't know this and apologizing if I repeat something since I didn't follow this thread, just want to clarify, that space is factually, measurably curved. Even here on Earth, if we measure its diameter and the length of the equator very precisely, we find that the latter is not equal pi times the former, as Euclid prescribes, but in fact is shorter. I don't remember how much shorter, but think I can look it up, if needed.

Just heard a joke from the professor: How to remember the roles of amygdala? Remember four "F"s: Fighting, Flighting, Feeding, and Mating.

In another response above I've clarified, "Not to figure, what will kill it, but to make statistical estimates, e.g. when it is likely to happen." Thank you, I agree. Yes, it's "his".

This is fine. It is just like in statistical mechanics. We consider a position of any specific molecule random, in spite of the fact that it certainly has had a certain specific dynamic history. A big number of molecules allows this approach. A big number of eggs and sperm and me / you / he / she etc. allows it here.

Regarding too many factors /unknowns, it's good, the more the better. That is where statistics works the best. Of course civilization will not die because of the statistics - something will cause its death. But there are so many different possible somethings, that we can apply statistics. Not to figure, what will kill it, but to make statistical estimates, e.g. when it is likely to happen. I am not sure we need to be concerned too much with a definition of civilization either. It is enough to say, some version of human lifestyle that started about 10,000 years ago. we make an estimate about that lifestyle. We can apply the Copernican principle to a better defined thing, like human species. Here is Gott again: Our species, Homo sapiens, has been around for about 200,000 years. If there is nothing special about our time of observation now, we have a 95 percent chance of living sometime in the middle 95 percent of human history. Thus, we can set 95 percent confidence level limits on the future longevity of our species. It should be more than 5,100 years but less than 7.8 million years (5,100 years is 1/39th of 200,000 years and 7.8 million years is 39 times 200,000 years). Interestingly, this gives us a predicted total longevity (past plus future) of between 0.205 million and 8 million years, which is quite similar to that for other hominids (Homo erectus, our direct ancestor, lasted 1.6 million years, and Homo neanderthalensis lasted 0.3 million years) and mammal species generally (whose mean longevity is 2 million years). The average, or mean, duration of all species lies between I million and 11 million years. Gott, J. Richard. Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time (p. 210). I just saw you comment about a shape of distribution. Assuming uniform distribution is more optimistic in this case. If it is more Gaussian-like, the confidence interval will only decrease since you /me will have a higher probability to be closer to the center.

This is indeed a factor that makes Copernican principle inapplicable, if civilization changes its own lifespan one way or another, because we can't talk then about a pre-existing, albeit unknown, statistical distribution. However, if we're just at the .01% of the span, it would make me /you "special" - that's why we have 95% rather than 100% confidence. If we are not "special" we should be in the 2.5% -- 97.5% interval. Regarding the Fermi Paradox, BTW, here is a quote from Gott: At lunch one day in Los Alamos in 1950, the noted physicist Enrico Fermi asked a famous question about extraterrestrials: “Where are they?” The answer to Fermi’s question, provided by the Copernican principle, is that a significant fraction of all intelligent observers must still be sitting on their home planet, just like you; otherwise, you would be special. Gott, J. Richard. Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time (p. 237).

I don't know how I would do this except by going backward, i.e. after finding the answer to the puzzle looking for its prime factors.

The Copernican Principle, officially described by Richard Gott (J. Richard Gott III | Department of Astrophysical Sciences (princeton.edu)) in 1993, basically says, "I'm not special." It can be applied for statistical predictions in cases where our place in space or time may be considered random. E.g. the fact that I was born in this specific time in the history of human civilization seems to be random, i.e. I could be equally likely born in any other time of that history. Now, the human civilization, with towns, writing, some kind of machines, etc. exists about 10,000 years. My random place in it is - with 95% confidence - inside the middle 95% interval of its lifespan. It could be just in the beginning of this interval, in which case the 10,000 years is 2.5% of the lifespan and we have 97.5% still to go, 390,000 years. It could be just at the end of that interval, in which case the 10,000 years is 97.5% and we have only 2.5% left, 256 years. So, with 95% confidence we can say that our civilization will last no less than 256 years and no more than 390,000 years. Some estimates to consider when discussing big issues, I think.

The whole puzzle is in the title. It took a friend of mine, in his own words, "3 pages of factoring (correcting and replacing)". It took me 6 lines. Take a challenge?

It is not necessarily so. It could be different. The puzzle says, that after the swapping and some possible rearrangement of other apples between the piles he gets equal weights again.

It is not necessarily so. According to the puzzle, after removing and inserting he might need to rearrange the other apples between the two sets to get equal weights again.

This is where your problem is. Bye-bye.

This is almost correct. Photon never looses energy in any locality. It's because the space itself expands, it has different energies in different localities, i.e. in different local frames of reference. Consider for example you driving in a car. In your frame of reference the car is not moving and thus has zero kinetic energy. However for a pedestrian on the street, the car is moving with some speed v and has a kinetic energy m(v^2)/2.

Nope, photons don't do that. They don't loose energy. They are created and annihilated in interactions with other particles.

Sorry, didn't notice the change. Corrected. Happy New Year to you as well. BTW, cosmological redshift actually is not a Doppler effect, but for a different reason that has nothing to do with the OP.

The photons are absorbed by gas clouds etc. that are mentioned in the OP.

Because E=hf is energy of one photon. Light consists of many photons, let's say N. The energy of light is E=Nhf. This E decreases because N decreases, not f.

This is incorrect.

[Disclaimer: I don't think time travel is physically possible.] Is splitting of a timeline necessary? I see a scenario without it, I think. A guy travels from year 2022 to year 1922, kills his ancestor, travels from 1922 to 2022, and finds a world, where he never existed. His history line is self-consistent: born, travel, kill, travel. The world's history line is self-consistent: somebody appeared from nowhere in 1922, killed a person, then somebody appeared from nowhere in 2022. No split, just a "loop".

Yes, there is: Astronomy and Cosmology - Science Forums

Here is a related story: "In physical cosmology, the Alpher–Bethe–Gamow paper, or αβγ paper, was created by Ralph Alpher, then a physics PhD student, and his advisor George Gamow. The work, which would become the subject of Alpher's PhD dissertation, argued that the Big Bang would create hydrogen, helium and heavier elements in the correct proportions to explain their abundance in the early universe. ... Gamow humorously decided to add the name of his friend—the eminent physicist Hans Bethe—to this paper in order to create the whimsical author list of Alpher, Bethe, Gamow, a play on the Greek letters α, β, and γ (alpha, beta, gamma). Bethe was listed in the article as "H. Bethe, Cornell University, Ithaca, New York". In his 1952 book The Creation of the Universe, Gamow explained Hans Bethe's association with the theory thus: The results of these calculations were first announced in a letter to The Physical Review, April 1, 1948. This was signed Alpher, Bethe, and Gamow, and is often referred to as the 'alphabetical article'. It seemed unfair to the Greek alphabet to have the article signed by Alpher and Gamow only, and so the name of Dr. Hans A. Bethe (in absentia) was inserted in preparing the manuscript for print. Dr. Bethe, who received a copy of the manuscript, did not object, and, as a matter of fact, was quite helpful in subsequent discussions. There was, however, a rumor that later, when the alpha, beta, gamma theory went temporarily on the rocks, Dr. Bethe seriously considered changing his name to Zacharias. The close fit of the calculated curve and the observed abundances is shown in Fig. 15, which represents the results of later calculations carried out on the electronic computer of the National Bureau of Standards by Ralph Alpher and R. C. Herman (who stubbornly refuses to change his name to Delter)." (Alpher–Bethe–Gamow paper - Wikipedia)

I am not sure what you describe - killing your father before you were born - can't be real. On your worldline, you were born, went back in time, killed your father, then went forward in time (or not.) On your father's contemporary's worldline, somebody appeared from nowhere and killed your father. Is there a contradiction?

In the classical "twin paradox" the twins get between same two points via different spacetime trajectories, and the traveler twin's way is shorter.