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Posted Jul6-09 at 03:34 PM by jambaugh

Here is a analogous phenomenon which helps see why there is no paradox in the "twins paradox".

Imagine two twins standing together on a flat plane. They begin walking at slightly different directions. As far as twin A is concerned twin B is "falling behind" and likewise as far as twin B is concerned twin A is "falling behind".

This is because each twin is defining a different "forward direction".

Now to make the analogy more precise suppose that the twins do not actually watch each other walking but rather backtracks later and count footsteps. Twin A will see that for a given amount of his own forward motion twin B has made more footsteps to achieve a given forward component of position and vis versa.
(Imagine each twin has no depth perception and can only see the the component of the spacing of the other's footsteps parallel to his own path.)

Taking the footsteps as analogues of clock ticks we then see that each twin sees the other's "clock" running faster. (The opposite of what happens in special relativity). If we ask "which twin has really taken the least number steps to move forward a given distance?" we understand that it depends on which direction we mean by "forward". We could pick an arbitrary direction and call it "North" and rephrase the question in terms of who goes north in fewer steps. But on a plane there is no preferred "north" direction so this choice is arbitrary.

The reason we see an opposite effect here is that we are describing different directions in Euclidean space whereas space-time has a pseudo-Euclidean geometry.

In Euclidean geometry we change direction frames by using an ordinary rotation. In special relativity due to the indefinite metric of space-time to transform between each twin's time direction we must instead preform a Lorentz boost which is a pseudo-rotation.

Rotating one point about another traces out a circle while pseudo-rotating a point about another traces out a hyperboloid. We express rotations with ordinary trigonometric functions but pseudo-rotations use hyperbolic trig functions.

(This hyperbolic rotation business is very non-intuitive but if you work the math and trust your numbers you can get a good sense of it with careful selective use of the analogy with standard rotations.)

If you work through the equations for Lorentz transformations you can identify:
$$\beta = V/c = \tanh(\phi)$$
$$\gamma (1-V^2/c^2)^{-1/2} = \cosh(\phi)$$
and so
$$\beta \cdot \gamma = \sinh(\phi)$$
where $\phi$ is the pseudo-angle parametrizing t the Lorentz boost (pseudo-rotation).

But other than this reversal of scaling due to the different geometries the phenomena of footstep spacing and slowing clocks is exactly analogous. No paradox just relativity in the meaning of each twin's forward direction (either in space or in time) which is why it is called relativity.
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