# Invariance of spacetime intervals

#### nomadreid

Gold Member
Category of simple questions

Obviously I am misunderstanding how an interval of space- time can be invariant under coordinate transformations. The following elementary (but obviously incorrect) calculation will illustrate my difficulty.

Alice is leaving her two boyfriends, Bob and Charlie. Bob sees Alice going at a constant velocity of 3/5 the speed of light away from him, so after a second, Bob measures Alice’s space-time change as (using the (+,-,-,-) convention) (9.0 x 1016 m2- 5.4 x 1016 m2 = 3.6 x 1016 m.

Charlie also sees Alice going away from him at a constant velocity, but at 4/5 the speed of light, so after a second, Charlie measures Alice’s space-time change as

9 x 1016 m2- 7.2 x 1016 m2 = 1.8 x 1016 m.

I would be grateful for corrections.

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#### andrewkirk

Science Advisor
Homework Helper
Gold Member
A second in Charlie's time is not the same as a second in Bob's time, and their hyperplanes of simultaneity are different.
Assuming Alice, Bob and Charlie are next to one another at t=0, with Alice heading west at 0.6c and Charlie heading East at 0.2c (approx.) the spacetime location that Bob identifies as Alice's position after one (Bob) second is not the same as the spacetime location that Charlie identifies as Alice's position after one (Charlie) second.

So the spacetime intervals being measured are not between the same two spacetime points. So invariance is not applicable.

#### nomadreid

Gold Member
Ah, I hadn't taken that into account. Thanks, andrewkirk. Back to the drawing board.

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