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My friend, Nima Fathali, used to claim that the fact time dilation can be deduced from the (non-) simultaneity though I used to claim otherwise saying that these two qualities are independent of each other. Defending his claim, he said that if we assume too many clocks located close to each other along, say, the $x$-axis, all being synchronous from the point of view of the observer (who is at rest relative to them), an observer who moves at a significant fraction of the speed of light observes each further clock slightly lagging behind the nearer ones. He then stated that if the clocks are small enough, this lagging behind can be defined as a continuous function of time analogous to time dilation!

His example seemed somehow awkward to me in that I could not understand how he merged those infinite number of point retarded clocks, distributed all along a line, to make a unified clock that runs slower (by the traditional gamma factor) that can be placed at the location of any of those retarded clocks.

I insisted that the factors that control time dilation and simultaneity are independent of each other in the Lorentz transformation for time, and there is no way to derive one from another:

$$t^\prime=\gamma t-\boldsymbol{\gamma \frac{vx}{c^2}}$$

Remember that, speaking figuratively, the first term in the right-hand side of the above equation rules time dilation, and the second boldfaced one controls the (non-) simultaneity.

However, out of curiosity, I soon realized that the reciprocal of the integral of the second (boldfaced) term (neglecting $x$) with respect to $v$ equals the traditional gamma factor (which can be found in the first term).

$$\left(\int{\frac{vdv}{c^2\sqrt{1-v^2/c^2}}}\right)^{-1}=\gamma$$ Although I cannot interpret the physical meaning of the said integral, I am really confused if the time dilation term is derivable from the term that controls (non-) simultaneity in any rational way, or the Lorentz transformation is a simple form of a solution to a general differential equation.

Can anybody add some clarification here?

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  • $\begingroup$ How does $vx$ in the second term in the formula (the term that gives relativity of simultaneity) become $vdv$ in the integral? You said, "neglecting the $x$", but why? And why integrate over $v$? It looks like numerology to me at this point. $\endgroup$ – Dvij D.C. May 24 '20 at 9:23
  • $\begingroup$ @DvijD.C. Yes, my integral is probably meaningless. I want to know if there exists a rational one. $\endgroup$ – Mohammad Javanshiry May 24 '20 at 9:53
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You are right to be skeptical of this claim and reasoning. While there is, in principle, no problem with having an infinite number of clocks and creating a continuous function that is not time dilation.

Let’s recall what time dilation is physically. Time dilation is the fact that the time measured on a single clock, called the proper time, is less than the time on a pair of Einstein-synchronized clocks when the single clock moves from one synchronized clock to the other.

Similarly, gravitational time dilation is the comparison of the proper time of a clock to the coordinate time. This description also works for special relativity. So in general time dilation is the comparison of proper time of some clock to the coordinate time.

Proper time is defined on a timelike worldline. The time along the continuous string of clocks is spacelike, so it does not constitute a measure of proper time. It is, in fact, a measure of coordinate time in the other frame, since it is based on the simultaneity convention. So physically the continuous set of clocks is not what time dilation refers to.

Furthermore, consider the derivation of the Lorentz transform. Usually it begins with a general linear transformation (1+1D spacetime, units where c=1) which can be written: $$t’= A t + B x$$ $$x’=C t + D x$$ where time dilation is $A$ and the relativity of simultaneity is $B$. Their numerical values are related through the Lorentz transform, but they are different terms in the linear transform. It is certainly possible to write down transformations where $A$ is unrelated to $B$

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