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AdVen
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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):
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):
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?
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.
---------------------------------------
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):
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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