I have tried to derive the formula for the proper time from the two equations of the Lorentz Transformation. The formula is as follows (see Wikipedia: http://nl.wikipedia.org/wiki/Eigentijd): [Broken](adsbygoogle = window.adsbygoogle || []).push({});

tau = t*sqrt (1 - v**2/c**2)

The two equations of the Lorentz Transformation are as follows (see Wikipedia):

x '= x / sqrt (1 - v**2/c**2) - v*t / sqrt (1 - v**2/c**2) (1)

and

t '= - ((v*x) / c**2) / sqrt (1 - v**2/c**2)) + t / sqrt (1 - v**2/c**2) (2)

Now x 'and t' represent points. x' represents a location (point in space) and t' is a time (point in time). One may write the two equations of the Lorentz Transformation for intervals as follows:

Delta x ' = Delta x / sqrt (1 - v**2/c**2) - v * Delta t / sqrt (1 - v**2/c**2) (3)

and

Delta t '= - ((v * Delta x) / c**2) / sqrt (1 - v**2/c**2)) + Delta t / sqrt (1 - v**2/c**2) (4)

We derive the formula for proper time by assuming, that Delta x '= 0. This is the case when Delta x = v * Delta t. Substituting this into formula (4) we obtain:

Delta t '= Delta t * sqrt (1 - v**2/c**2)

or

tau = t * sqrt (1 - v**2/c**2)

Note, that tau refers to a time interval.

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Completely analogous to the derivation of the formula for the proper time, I derived the formula for the proper space (proper length). The formula is as follows (Wikipedia: http://en.wikipedia.org/wiki/Proper_length): [Broken]

sigma = x * sqrt (1 - v**2/c**2) .

We derive the formula for proper space by assuming, that Delta t '= 0. This is the case when Delta t = (v * Delta x) / c**2. Substituting this into formula (3) we obtain:

Delta x '= Delta x * sqrt (1 - v**2/c**2)

or

sigma = x * sqrt (1 - v**2/c**2)

Note, that sigma refers to a space interval (one dimensional).

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My question is: Are these derivations correct?

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# Derivation of Proper Time and Proper Space

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