# Direction of length contraction, and reciprocity between inertial referentials and invariance of ds^2

This is my very first post on this website, which I find fascinating. Thanks to all the active members and team !

Two questions, related to special relativity in flat space time :

1) I have read that length contraction is occurring only in the direction parallel to the velocity. Can you prove this to me in a system where the velocity and the acceleration are not parallel ? I would accept a mathematical proof ("thought experiment") or also possible observations (for example, if you tell me that people studying accelerated particles in the LHC see a contraction only in the direction of the particles velocity but no contraction in the direction of their acceleration.)

Yes, this question arises from the Ehrenfest paradox.

2) I know that if two referential R and R' are inertial, then we have cdt^2 - dx^2 - dy^2 - dz^2 = cdt'^2 - dx'^2 - dy'^2 - dz'^2. My question is simple : is the reciprocity true ? ("if we have ds^2 = ds'^2, then R and R' are inertial"). Proofs are welcome !

Thank you so much,

Francois

1)
SR (special relativity) accounts for physical laws in IRF's (inertial reference frames), that is no acceleration. Nevertheless an IRF can describe a particle moving along an arbitrary accelerated path. Simply you relate to the continuous set of inertial reference frames instantaneously at rest with the particle. For each of these frames the Lorentz transformation of coordinates applies. What matters is the instant velocity, not the acceleration. So the length contraction would hold according to the velocity.
$L = L' / \gamma$
where:
$L$ length measured in the observer IRF
$L'$ proper length in the particle IRF
$\gamma = 1 / \sqrt{1 - v^2/c^2}$
$v$ velocity of the IRF instantaneously at rest with the particle
2)
The distance or the squared distance $ds^2$ in a manifold is an invariant independent of the coordinates system. That holds for both inertial and noninertial (accelerated) reference frames.
A reference frame is inertial if the Riemann tensor vanishes everywhere (flat spacetime).

• Thank you ! So you used the co-moving inertial frames to find this result. In SR, is this hypothesis of "continuous set of inertial frames at rest with an accelerated frame" the only one to study non inertial frames ? – François Ritter Feb 8 '18 at 17:21
• As for what I know, in SR yes. – Michele Grosso Feb 9 '18 at 14:48