Understanding Invariance of Spacetime Intervals

[nomadreid]

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.

 

[andrewkirk]

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]

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