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But as far as Bob and Alice compare their own clocks nothing 'funny' seems to have happened. Also, if you are thinking of this time dilation effect as a 'gradiant' (poor choice of words here) then are you thinking about the Schwarzschild solution here? It seems so to me - at least from your example. So this 'gradient' is no more that g_{tt}?

Can you tidy up what you mean by this? You get time dilation effects when comparing two (or more) clocks - that is you need a way of meaningfully comparing the proper time of two observers. Saying that time dilation is relative to to the observer's (local?) inertial frame seems misleading. What clocks are you comparing here?

Yes, they are also known as Minkowski diagrams. I don't think so. And why would you want to if you can get the time dilation by using any inertial frame, and in particular the rest frame of the twin on earth?

I am not using a Lorentz transformation explicitly here... just draw the space-time diagram and calculate the space-time intervals. You then use these to compare the proper times that each twin has - this is exactly what time dilation is all about.

But it is not a summation of his research - it is an 'artistic impression' of this. The 'emabler' is a spin network in your picture, I guess. I am not sure how that sits with you - I mean you now ask what *is* a spin network, what is it made of, what enables the spin network to exist?

You can do all this in one chosen inertial frame - the most obvious one is to pick the frame for which the twin on Earth is at rest. You should think about the space-time interval. It also gives you the correst way of dealing with curved paths, so we can include acceleration... Proofs? You mean resolutions of the seeming paradox? Sometimes people talk about acceleration as a means of showing that the motion of the two twins is not symmetric in space. But you don't actually need to think about acceleration carefully to see this. We can assume idealised instantaneous change of direction - more physically this means the period of decceleration and acceleration is small as compared with the duration of the trip as measured by the twin on the round trip.

You know that thsi picture is just for artistic effect? By itself that picture tells us nothing.

I am just saying that if Trump were in charge those dam Mexican fish would be sent back... stealing American fish's jobs! (Sorry could not resist)

Well I did ... you seem to have really just shown what the clocks on the traveling twin will read. This can indeed be understood in terms of length contraction. There are other ways to think of this difference in duration. The most basic - and really shows what is going on - is to draw a space-time diagram and look at the space-time interval (and ignore accelerations for ease). The interval is invariant under choices of inertial frame. So we can pick the frame for which the tiwn on Earth is just moving in time. If you sketch the space-time diagram and calculate the space-time intervals for both twins -they are different. You can then relate this to the proper time of each twin and you get the 4 years differnece.

What is the paradox here? The motion of the two twins is not symmetric in space and the expected time dilation as calculated on Earth agrees with the clock on the twin that made the trip.

You are welcome. With your question 4. I am not sure there is some general answer. However, you might want to think about eigensystems - so solutions to [math]A \underline{x} = \lambda \: \underline{x}[/math]

2. is correct... but can be stated more carefully and a little more generally Let us work in [math]\mathbb{R}^n[/math] and let us pick a basis [math](e_{1}, e_{2}, \cdots , e_{n})[/math]. The general motion of a volume here is via the wedge product - totally antisymmetric product of vectors. [math] Vol(e) = | e_{1}\wedge e_{2} \wedge \cdots \wedge e_{n} | [/math] gives the volume in the basis given above. The absolute value is with respect to the obvious Euclidean norm - we could do something more general here, but not for now. Now let us take some linear transformation - which we know can always be written as a matrix. As a linear operator we have [math]A(e_{1}\wedge e_{2} \wedge \cdots \wedge e_{n}) = A(e_{1})\wedge A( e_{2}) \wedge \cdots \wedge A( e_{n})[/math]. As we are working with n vectors in an n-dimensional space and the product is totally antisymmetric the Volume is an element of a one dimensional vector space. Any changes in this volume can always be written as the volume multiplied by some scalar. Now lets build this linear oeperator - so [math]A(e_{1}) = e_{1}'[/math] etc. Then we see [math]A(e_{1}\wedge e_{2} \wedge \cdots \wedge e_{n}) = \alpha (e_{1}\wedge e_{2} \wedge \cdots \wedge e_{n}) = e_{1}'\wedge e_{2}' \wedge \cdots \wedge e_{n}'[/math] This scalar [math]\alpha[/math] is the determinant of the linear transformation A. So lets do this for the 2d case. Let us take some linear operator that I write as a matrix [math]A = \left(\begin{array}{ll}a & b \\c & d\end{array}\right)[/math] Then look at its action on an arbitary vector and feed this into the wedge product; we obtain [math](ax + by) \wedge (cx +dy) = ac \: x \wedge x + ad \: x \wedge y + bc \: y \wedge x + bd \: y\wedge y[/math]. Now remember that the wedge product is totally antisymmetric - which just means antisymmetric when we only have two vectors. So [math]x \wedge x = y \wedge y =0[/math] and we are left with [math](ax + by) \wedge (cx +dy) = (ad - bc) \: x \wedge y[/math], which is what we wanted. You could try the same thing yourself in dimension 3. In higher the same sort of thing works. This also then gives you a solution to question 1. You can use this as a geometric definition of the determinant of a linear transformation. I hope that helps a little

Even before one really gets into cosmology - what the heck is dilation gradient? This is a good question. People have though about if some other theory of gravity then included GR as an approximation can better describe the large scale dynamics of the Universe - so without dark energy. The answer so far seems to be that it is not possible. People have worked with higher curvature terms in these theories and it is known that all these theories can be reduced to GR + matter - maybe exotic matter but still matter. So the general consensus at the moment is that ussing different curvature tensors and higher order combinations thereof will not make dark energy and dark matter go away. Other ideas include sticking with GR, but looking at non-homogeneous solutions. So far this has also not got rid of dark stuff. No one. You are mixing the physics meaning of a force with a more common meaning. What is accelerated time?

Good luck with it. But we know nature is a hard journal to publish in and that they like things that impact several areas of science. I also hear they like some more speculative stuff, this catches peoples imagination and gets attention. I have no idea what you chances are realistically, but just be sure that they are probably low and don't take rejection to heart.

Get some sleep... your not making any sense now.

I just mean don't only apply to Harvard. Look at other places also, just in case you don't get accepted.

This is not enough... I wish you luck, but please have other options.

Yes. I imagine the sucsess rate is low - but I don't know. Maybe if you already have funding or pay for yourself the chances are much better. Anyway, wait and see.

Nature is hard to get publsihed in. Also, they may not publish the sort of things you have in mind - I don't really know. I was thinking that a more medical focused journal may not want to see lots of maths, as where a mathematics journal would. You need to think about who you want to read your work.

How you write the paper may depend a lot on the journal you send it to. Keep that in mind.

Well, there is no evidence to think otherwise. That much we can be sure of.

I have no idea what journals to try. The best you can do is try to find something similar in the literature - every if it is just very loosly similar.

As for contacting the departments, I would email.

You are asking about the rate of change of an electrical current? If so, then look up inductors.

As you have posted in physics, you want a physics answer... there is no evidence of a spirit or the afterlife that would stand up to any scientific scrutiny. There is no 'physics theory' of the afterlife - whatever some websites may tell you.