Genady

Did you mean these "low prime factors" -- see the previous post? If so, I apologize. My mistake, I didn't understand you. You were right. From this point, you need only one "trick" more, and not a very uncommon one.

Step 1: 48 + 68 + 98 = 22*8 + 28*38 + 32*8 Step 1.5: = (28)2 + 28*38 + (38)2

Yes, this is certainly so in the classical EM. However in QED these EM gauge transformations are coupled with local phase transformations of charged particles and this creates the mechanism of their interactions. This gauge symmetry I am asking about.

No, no, this thought has nothing to do with gravity (and I don't have any personal attraction to gravity, pan intended.) It is here only because it is about space, but this space is not affected by gravity at all. The "internal space" is a technical term in particle physics. In math it called "bundle space".

I don't think this is correct. The Friedmann equations (GR in a homogenous isotropic universe) have only expanding or contracting solutions, no static ones. Even adding a cosmological constant in GR didn't help -- it allowed only an unstable equilibrium.

And another thought. We've said, together with Thorne, that gravity can be thought as spacetime curvature or as effects on all clocks and rulers, and these two descriptions are equivalent. Yes, but... If we want to use QFT in gravity, there is a prescription of how to do so in a curved spacetime: replace partial derivatives with covariant derivatives, stick square root of metric determinant in Lagrangian (there is one more step, don't remember now) and you got it. How to do it if gravity does not curve spacetime, but rather affects rulers and clocks?

I will think about something for here. I thought that maybe we can go and discuss something beyond the 3D space and 4D spacetime, namely, the internal space of particles, the bread and butter of the Standard Model. It is a difficult subject, and for simplicity we can limit ourselves to the U(1) space of EM. The main feature of it is, that it allows for a new symmetry in addition to the familiar translations in space and time, rotations, and boosts, -- gauge symmetry. One can change phases of a particle field in different spacetime points arbitrarily and if the EM potential is changed accordingly, the system behaves the same. It seems like a dirty trick, but it leads to the most precisely tested theory of all times, QED. Any ideas of "why" this gauge symmetry exists and what it means?

You know very well, how (clock in a gravitational well etc...) I don't know why you ask.

Yes, it can. Here is Thorne's popular description of this: ... The trajectory is bent around the hole; it is curved, as measured in the hole’s true, flat spacetime geometry. However, people like Einstein, who take seriously the measurements of their rubbery rulers and clocks, regard the photon as moving along a straight line through curved spacetime. What is the real, genuine truth? Is spacetime really flat, as the above paragraphs suggest, or is it really curved? To a physicist like me this is an uninteresting question because it has no physical consequences. Both viewpoints, curved spacetime and flat, give precisely the same predictions for any measurements performed with perfect rulers and clocks, and also (it turns out) the same predictions for any measurements performed with any kind of physical apparatus whatsoever. For example, both viewpoints agree that the radial distance between the horizon and the circle in Figure 11.1, as measured by a perfect ruler, is 37 kilometers. They disagree as to whether that measured distance is the “real” distance, but such a disagreement is a matter of philosophy, not physics. Since the two viewpoints agree on the results of all experiments, they are physically equivalent. Which viewpoint tells the “real truth” is irrelevant for experiments; it is a matter for philosophers to debate, not physicists. Moreover, physicists can and do use the two viewpoints interchangeably when trying to deduce the predictions of general relativity. Thorne, Kip. Black Holes & Time Warps: Einstein's Outrageous Legacy (pp. 400-401).

This post actually is an elaboration belonging to the little subthread in another thread, which I simply prefer not to touch. If a moderator wants to move it there, I would not object. Otherwise, I don't have a question regarding this topic, but many answers

Step 1: 48 + 68 + 98 = 22*8 + 28*38 + 32*8

Any of these.

The title of this thread is the title of chapter 6-3 in the Richard Feynman's book, Six Not-So-Easy Pieces (1963). (This post is related to the discussion in one other recent thread, but not to its OP.) Here is the quote (pp.125-126):

Ah, I see. Sure. The definition of "space being curved" is that its geometry is not Euclidean. The curvature in Euclidean geometry is 0.

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?