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A question about the relationship between the formulas found using the Lorentz transform and the invariance of the space time interval.

Two events A and B occur at the same time and different space locations in system S, where A and B are at rest and at distance x.

The system S' moves with velocity v with respect to S.

I want to find the time interval t' that S' measures between A and B.

1. By using the Lorentz transform one finds immediately:

[tex]t' = -\gamma\frac{v}{c^{2}}x[/tex]

2. By using the invariance of the space time interval:

[tex]{(ct')^{2} - x'^{2} = - x^{2}}[/tex]

and using the contacted [tex]x' = \frac{x}{\gamma}[/tex]

one finds (neglecting the negative square...)

[tex]t' = -\frac{v}{c^{2}}x[/tex]

So here the gamma is missing.... Must I understand that:

a.

With the Lorentz transform I find the value of the time interval 'assigned' at S' as measured by S (with the time dilation due to the relative velocity between S and S')

and

b.

With the invariance of the space time interval I find this time interval as measured by S' (analogously to the use of the contracted x' in the expression of the interval)?

Have I right understood?

Thank you

er

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# Lorentz vs space time interval

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