Can anyone comment on the relationship of the Gravitational manifold and its’ tangent space? I am specifically interested on whether or not the tangent space is Lorentzian.
In my post on the Twin Paradox I made the implicit assumption that the tangent space was Lorentzian but I am not certain. To my knowledge this is the only point that was weak in my post.
Thanks Ed
P.S. the twin paradox post can be found at:
Oh wow. I have left relativity so far behind it is hard to pick it up again. In thinking about this one, I feel like a physicist trying to explain how Jesus actually walked on the water. For me, it is hard to go back and think in terms of the misconstrued fantasy physics of relativity.
In reality, acceleration yields an artificial gravity, not a real gravity. Thus in reality, there is no time compression associated with acceleration other than the integral of the measured time differential as indicated by the changing speed plus a small amount related to the loss of mass. But in the mindset of universal relativity, do they still maintain Einstein’s misstatement that there is no distinction between acceleration and gravitation? It would appear by your paradox analysis, that they do. That makes things difficult.
In your analysis, you have acceleration being a component of general relativity (“relativistic force”) and thus affecting time measurements just as gravitation would. But I don’t see the integral of the speed differential being included. And I don’t see what tangent space has to do with it. You don’t have a geometric manifold situation in your paradox scenario. Where is there a “gravitational manifold” in the scenario?
Hi James,
I can hardly believe that you have actually read the post. Aside from Sidhe’s (or as I last knew him - Helandhighwater) Physics professor, Dr. Bailey, practically no one has done so.
Kudos (unless the whole thing is crap and then my apologies for wasting your time)!
Anyway, In Case II, after the section marked An Intuitive Approach, in the section marked A formal approach, I define the gravitational manifold generally by the functions f1(t, x) = t’ and f2(t, x) = x’. This manifold describes, theoretically, the space on which the moving observer is constrained. (It is the consequence of the acceleration).
The tangent space being Lorentzian is critical because it constrains the partial derivatives to be less than 1. I could not prove The Bounded Reference Frame Theorem without that fact.
You are right that I am using a conventional argument and equating acceleration to Gravity just as Einstein suggested. Basically I am using a conventional analysis of General Relativity to prove that the Twin Paradox is valid.
Conventional Physicists do not like this.
However, I have met with some success. Dr. Bailey agreed that the physics is correct and a moderator on Wiki acknowledged that the Wiki article is flawed.
As far as the math goes, I will let it stand on its own merits (though I believe that it could be simplified).
Ed
Hey there Ed,
Actually, that post was one of the first that I read when I came here. I started to post on it with questions but then realized how old it was. When I looked around more, I could see why no one had posted on it and saw that pure math on this forum wasn’t going to be a hot topic. So for all of those who seriously hate that I am on this site, they can blame it partially one you. 
Well, my first thought is, if physicists don’t like it and you are writing a book, why are you doing it that way?
Obviously you think strictly in math terms whereas I think strictly in logic terms. If we ever got together, serious things would happen. But then again, you are writing a book that you want to sell in the same country that you live in, so are restricted as to how much you can upset anyone (again, “if they don’t like it, why are you doing it?”). In logic terms, the Twin Paradox is easily resolved with hardly any math at all. It doesn’t really matter what the acceleration function would be; Lorentzian, Mickey Mousean, or Donald Duckian.
But looking at how you are analyzing this paradox puzzles me a bit, perhaps I am entirely missing your point somewhere. Now it seems that you are asking if the (t’,x’) plane is Lorentzian. Is that right? Because I can’t make any logical since out of the idea of a one dimensional line being a “Lorentzian tangent space”.
Perhaps you are asking if the t’ and x’ are Lorentzian, but that seems a silly question since they are defined as the result of the Lorentz transformation.
But then again, I am probably totally missing your point. My math mind has only gotten worse in the past few years to the point of all but non-existent (Maybe it’s the company I keep).
As a thesis, if that is what you are going for, I would recommend a formatted “Summary Conclusion” for each of the cases, perhaps even a summary chart.
One question that I had is why aren’t you using the integral of the differential time variance during acceleration as a measure of the acceleration delta-time? I can’t see how that can be left out regardless of general relativity concerns unless it is proven that such a value is identical to the general relativity formula. If it is not identical, then how can it be not considered and what would you do with it if it was considered?
On a separate note;
I really wish I could get you interested in Rational Physics because there is a huge opening for a mathematician to address such issues as; the precise sizing of sub-atomic particles, the distances involved in the “strong and weak forces”, the valence link between chemistry and physics, the energy density of space, and many others. I used up my little brain merely getting all of the logic worked out and verified. The only thing left is the formal math, which can get seriously hairy but I’m sure you could handle it to the degree required. And in Rational Physics, there are NO paradoxes involved in anything science has ever witnessed.
Hi James,
Thanks for your post.
I am not trying to write a book. I am just PO’d by the 737,000 references on a Google search of the Twin Paradox.
Motivations:
The reason that I wrote the post, is because ever since I first heard about the paradox, probably back in 1969, it has bothered me. Also the first reason, that I was given, to debunk the paradox was that the Earth was a preferred reference frame.
That first reason was so bad that I became suspicious of all efforts to debunk the paradox.
I have read about a dozen of the 737,000 Google references on the Twin paradox. All of them gave reasons why the Paradox has been debunked. They were all bad.
Anyway the paradox and the efforts to debunk it were simply annoying to me and finally after Sidhe posted his article “A brief history of a brief history of time”, I became irritated yet again and decided to do something, other than complain, about it.
Approach:
Solutions to the uniform motion of a body in General Relativity are complex. If you Google “uniform acceleration general relativity” and look at some of the solutions you will get a feel for some of the problems. The approach that I used was IMHO more elegant. Speaking analogously (always a dangerous idea) it is like not knowing anything about a surface in Differential Geometry, but if you are given a little info about its’ oscillating sphere, then you can derive a little something about the surface itself.
Assuming that the limited information that you can glean about the surface is sufficient for your purposes, then you can avoid a lot of extra work. (I don’t need to do the integrals).
“if physicists don’t like it and you are writing a book, why are you doing it that way?”
I used the underlying assumptions implicit in Real Analysis, Differential Geometry, Special Relativity and General Relativity which would be generally agreed upon by the Physics community to hopefully show that the Twin Paradox was in fact a valid conclusion of Relativity.
At this point I could go into a long dissertation about falsifying theories, which I generally believe cannot be done, or the fact that flawed theories are spectacularly awesome and possibly other implications.
BUT
I pretty much believe in Relativity. I just want to be honest about its’ flaws – it’s an ethical thing for me.
Ed
Note:
I am going to start a separate post on the Lorentz transform to see how you deal with the novel experiments that are consistent with the transform.
I bet that I could change that, assuming you were willing. 
The Twin Paradox is not really an issue for Relativity, but The Stopped Clock Paradox - Analysis is. That thread was not written for a mathematician, but rather for the kind of confrontation typical of this forum and could have been (and should have been) explained simpler (which I realized in the middle of the thread).
A consistent experiment does not prove a theory. It merely means that the experiment does not disprove the theory. True “falsification” requires more than people think, but can be done. Rational Metaphysics shows how and why.
Hi James,
you wrote,
Popper 101 - Hume 101 - Logic 101. I would have never guessed.
I am giving you your chance on my post on the Lorentz transforms. Go for it.
Ed