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But let's say that our theories explaining the black hole (or big bang for that matter) as a "singularity" shows that our theories don't work on that level.
The question of what happens at black hole event horizons actually has nothing to do with the singularity. You can replace the singularity with a small but not infinitesimal mass and it's still the same. The properties of the event horizon and the environment around the black hole don't care about what the interior is, other than the total mass of it. This is a consequence of Birkhoff's theorem, which you can think of as a general relativistic version of Newton's Shell Theorem
. The gravitational field outside of any spherically symmetric mass is equivalent to the same mass condensed to a singularity.
In a similar way, you can describe the properties of charged particles perfectly well by imagining them as singularities in the electric field. You get the correct predictions as long as you don't care what happens arbitrarily close to those points.
With the Big Bang, you get correct predictions as long as you don't go too close to the initial instant of expansion. Our knowledge of physics is still good up to about 10-32 seconds after the Big Bang! What happened before then requires new physics with quantum gravitation, similarly for describing what happens very close to the center of black holes.
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But then a possible way around it was explained as follows- traveling backwards in time creates a new time line therefore no paradox happens because events can happen entirely differently than they did in the old timeline in which the traveler originated.
Maybe, but then we need new physics to explain it and a way to test it through observations.
In relativity, travelling backwards in time is the equivalent of choosing a space-like (faster than light) path through space-time rather than a time-like (slower than light) path. For certain observers, that path will be backwards in time. (This is basically the inverse argument for why travelling faster than light is impossible. For certain observers, it is equivalent to going backwards in time, and thus introduces causal paradoxes).
One could propose travelling faster than light actually is possible and avoids paradoxes by creating a new timeline, but then we again must explain how that happens and in a way which makes testable predictions.
Completely agree, and one of the stumbling blocks is that some of these theories predict a quantization of space-time at very small scales (5 planck lengths in one theory), and we just haven't been able to detect discrete quantities of space-time. Space-time being continuous is what causes the singularity in the first place and if we could just get at a smaller level than what we are technologically capable of at the current time and find discrete units for space-time our singularity problems will be solved. If space-time isn't continuous at 5 planck lengths, for example, there is no singularity because that is the minimum scale for the singularity (and thus density doesn't ever get to infinity in the D=M/V equation since Volume can never become zero.)
This also has consequences for the Big Bang. As Penrose proposes, the Big Bang can actually turn into a Big Bounce (perhaps this is what our "last" one actually was), as the universe "comes back empty" it reaches the 5 planck scale and becomes one dimensional (point), at that level gravity ceases to function (relativity states that three spatial dimensions are needed for gravity to be an attractive force) and the universe re-Bangs!
About the second part, other timelines, it becomes at least mathematically reasonable if F-theory is correct with its second temporal dimension. Two dimensions of time allow other timelines to be nested (imagine time in two dimensions as two axes in a Cartesian coordinate system with the Origin being the big bang/bounce and as the universe inflates FTL the timelines emerge and separate along the second axis in time and at some point in the future, if the universe starts contracting they will converge again at the next big bang/bounce "Deflation." This at least resolves one problem I had with the Anthropic principle being involved- how do WE create timelines? Answer is we don't- the timelines were there as the result of an emergent phenomenon from the Inflationary Period- of course this would mean that superdeterminism is correct. Past/Present/Future would lose meaning in a sense- from a higher level anyway- since everything would be "set" from the beginning of each cycle. The weird value of the Cosmolgical Constant might even provide a clue, as it could be slowly reducing with each cycle, and once it reaches 0, the universe would no longer re-Bang- and instead it'd have to let its "baby" universes carry on its heritage.)
By the way, this is a fascinating discussion. We were also discussing arrows of time and I remember posting something awhile back about it being possible for different universes to have different arrows of time relative to each other (but all would still be forward with respect to the inhabitants of each universe- as if each universe was a conveyor belt going in different directions, but for the people on the belts, each belt would appear to be going forward. You could actually construct a time machine this way if you could bridge universes, but that's another story.) I don't necessarily believe that FTL is possible (but wormholes are that can shorten the distance) but I am intrigued by the possibility of sterile neutrinos and because they might be impervious to every force but gravity, they could travel in time. That FTL is not possible inside our universe but might be possible outside of it (because the universe can and does expand FTL- thus the idea of the Alcubierre drive becomes possible) is also intriguing for wormhole enthusiasts.
But anyway, on a truly cosmic scale, the arrow of time itself might be relative!
By the way, what you said about choosing space-like vs time-like, isn't that what happens inside a black hole? Space and time trade places.