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It is possible to calculate the time dilation with Lorentz transformation? So with the equation $$t'=\gamma(t-\frac{x v}{c^2}) \tag{1}$$ in which $\gamma$ is the Lorentz-Factor?

In an exercise I have succeeded.

So why can be calculated with the equation $(1)$ the time dilation? What is the reason?

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    $\begingroup$ The first step is to be comfortable with the interpretation of each of the symbols, as well with the concept of proper time. After you are comfortable with that, one can turn your question into mathematics by realizing the quantity $t_2'-t_1'$ (for spacetime events 1 and 2) is what you're after. Also, this is a fairly standard derivation that you should be able to find in any special relativity book. $\endgroup$
    – BMS
    Feb 12 '14 at 6:41
  • $\begingroup$ My answer to the question I've linked shows how to calculate time dilation using the Lorentz transformations. See also this question. $\endgroup$ Feb 12 '14 at 6:50
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Because the time dilation formula can in fact be derived directly from the Lorentz transformation. See, e.g. this link.

Special relativity is often first presented as a sequence of "effects" with their own formulae, sometimes justified with an accompanying thought experiment. This is useful pedagogically, and in some ways can aid the student in seeing how the theory makes sense intuitively.

But just about all special relativistic physics is encoded in the geometry of Minkowski space and the Lorentz transformations thereupon. Starting from these geometric principles, it is possible to unify and derive all the individual results that have been presented separately, like length contraction, time dilation, and velocity addition.

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