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We have two reference frames, S, and another one, S´, that is moving to the right at the velocity v relative to S.
We have an event at point E.
- In S: Coordinates of E = (x, 0, 0, t),
- In S´: Coordinates of E = (x´, 0, 0, t´).
Each system sets its clocks at 0 when the origin, O´, of S´ passes the origin, O, of S. I.e. when t=t´=0, we have x=x´=0.
- In S: At the time t, the origin of S is at vt (in red).
- In S´: O´ to E is x´ (blue).
- In S: O to E is x (green) and O´ to E is x´/γ – because of length contraction. A length that is seen as x´ in the system S´, moving relative to S, is seen as shortened by a factor 1/γ.
In S: So, we have that
or
This is the Lorentz transformation of the coordinate in the direction of motion. The other special dimensions will not change at all, as argued for the height in the time dilation derivation. Observe that the resulting length is for S´, but the length, x, and the time, t, are as seen in S.
- A spacecraft is flying past an asteroid at a relative speed of 0.80c. Two huge flashlights send out light flashes on each side of the asteroid as seen from the spacecraft. The flashes are sent at the same time, according to observers on the asteroid. The asteroid is 900 meters long as measured by the observers on the asteroid.
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- Calculate the gamma factor.
- How long is the spacecraft as seen from the asteroid?
- How long is the asteroid as seen from the spacecraft?
- Calculate the distance between the light flashes as seen from the spacecraft.
We got two quite different results in the last calculations, even though, in the asteroids’ frame of reference, the two points, the distance between the flashes and the length of the asteroid are both 900 m.
One is from looking at the length of an object, and one is from looking at the distance between two events. We will look at this again when looking at spacetime diagrams.
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