Why Length and Time Intervals Vary by the Same Factor

The reason for this is so that the light has the same speed c to all observers. Let's say the moving space ship's length, measured by an observer on Earth, is half of its proper length (in physics-speak, proper length is the length measured when the measurer is at rest relative to the thing measured). (Please note that this is an outlandish example. We don't have the ability to get rockets going this fast.) Let's say the proper length is 150 m, so a pulse of light takes 1 microsecond to go the length of the ship and back.

Earth observer is EO; astronaut observer is AO. They both possess excellent precision measuring instruments.

EO (talking to AO by radio): Hey AO, your ship has shrunk to 75 m.

AO (annoyed): It did not, you jerk, I just measured it- its 150 m.

EO: Your #%^&*) meter stick must have shrunk to half a meter.

AO: Tell ya what- Synchronize two clocks and put them 150 m apart. I'll zoom by you, and you can record the time when the front end goes by the 2nd clock and the rear goes by the 1st clock and you will see that they occur at the same time.

(EO was not stupid; he also put one at 75 m. He found that the front and rear did indeed coincide with locations 75 m apart at the same time-- the ship's length is 75 m according to EO.)

EO: I just measured it. It is 75 m.

AO: It is not. I clock a pulse of light going the length of the ship and reflecting back in 1 microsecond. Light goes 300 m in 1 microsecond.

EO: Oh, now I get it, your *&^#$@ clock is half-fast.

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In the case above, the ship's speed is 0.866c. In relativity calculations, the factor gamma always crops up: g = (1 - v²/c²)^-1/2. Note the negative exponent, so it means 1 over the square root of that stuff. Check it out, and you will find that in this case, gamma is 2.

It also turns out that if the ship has rest mass m_o when measured at rest, it will behave like mass = g m_o when it is moving. The ship above has momentum 2m_ov! (Note: the common practice is to avoid the term "rest mass," just call it mass and label it m, and it is understood to be what I have called m_o.)

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Comments, questions: fgram