## The smoking astronaut

To see how the invariant space-time interval expression can be applied to relativistic calculations, suppose that a space pilot travels at very high velocity either towards or away from the earth. He smokes a cigar, which to him appears identical to the same brand of cigars he smoked on the earth. In particular, the time taken for the 'smoke' is the same. A NASA observer on the earth watches the pilot with a super telescope and times the smoke. He is checking that the NASA pilot does not exceed the permitted recreation time. (In making this timing he makes due allowance for the difference in time taken for light to reach his telescope at the begining and the end of the smoke).

Event 1: astronaut lights cigar Event 2: astronaut finishes cigar How long does it take him to smoke the cigar? Invariant interval:

Earth frame: AS2 = Ax2 + 0 + 0 - c2At2 where Ax = v At.

Astronaut frame: AS2 = 0 + 0 + 0 - c2At'2 where At' = At0.

Time in the rest frame of cigar and astronaut is called the 'proper time'.

To the astronaut there is no space interval between the two events. Both lighting and finishing the cigar are happening 'here'.

To the earth observer the astronaut has travelled a distance Àx in the x direction during the 'smoke'.

v 2 D t2 - c 2 D t2 = 0 - c 2 D12 (v 2 - c 2)D t2 =- c 2 D12

D t2

1 c2

Time in the other frame = y (proper time) y> 1 always.

To us the astronaut's cigar (and his life!) appears to last longer, and his actions appear in 'slow motion'. The astronaut himself feels no difference. His clock shows his own proper time.

Things move more slowly in 'the other frame'.

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