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The space-time invariance principle states that:

$(c^2)(\Delta t^2) - l^2 = \rm invariant$ for any inertial frame.

Consider two events that occur at the same time in inertial frame $S$, and frame $S_1$ moves with translational speed $v$ relative to $S$. It seems that by the length contraction, the distance between these events should be shorter in $S_1$ because the events are moving, yet since these events do not happen simultaneously in frame $S_1$, then $(c^2)(\Delta t^2) - l^2$ for frame $S_1$ is larger than for $S$, which is a contradiction. Can someone please explain the issue in this contradiction (without using Minkowski diagrams)?


marked as duplicate by Mike, WillO, Chris, John Rennie, Qmechanic Mar 8 '18 at 6:16

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