# Is there a space-like equivalent of the Twin Paradox?

In the Twin Paradox, a travelling twin returns to the inertial frame of the twin that has stayed at home, and their clocks are compared. The readings on the two clocks are different, but still ticking at the same rate.

Since time and space are treated on the same footing in special relativity, I believe there should be a space-like equivalent of the Twin Paradox where rulers rather than clocks are compared.

• Length contraction and time dilation are not closely analogous. We can define time dilation for the world-lines of pointlike particles, but length contraction only makes sense for extended objects. – user4552 Oct 16 '17 at 0:29

Underlying the Twin Paradox is the Clock Effect, which says that in a triangle made with future-timelike vectors, say AB,BC,AC, then the inertial trip AC has a longer proper time than the non-inertial trip AB-BC: for elapsed times, AC > AB+BC. This is the "Reverse Triangle inequality".

A spacelike analogue of this would be the ordinary "Triangle inequality" for a triangle with three spacelike vectors, say PQ, QR, PR. The straight path PR is shorter than the piecewise trip PQ-QR: for distances, PR < PQ+QR.

Of course, what makes the Twin Paradox/Clock Effect puzzling is that it conflicts with our everyday common sense notions of time... which could be called the non-"Clock Effect" for a Galilean spacetime... that is, the absoluteness (the path independence) of time. For a triangle with Galilean future-timelike sides, say MN, NP, MP, for elapsed times, MP=MN+NP.

It is simply this: the one who takes the "direct" route between two locations will register a smaller distance traveled than one who goes via a third location.

In your own inertial frame, your clock is constantly advancing (more precisely, it assigns different times to different points on your worldline) but your odometer never changes. So if you and I travel different paths from event $A$ to event $B$, our clocks can show different elapsed times (hence the possibility of a "twin paradox"), whereas both of our odomeeters show zero elapsed distance, so the analogous "paradox" does not arise.

Yes indeed. The ladder paradox is what you are searching for. It's all about a ladder moving at high velocity into a garage that is smaller than the ladder proper length. For an observer the ladder cannot fit and for another it can fit. The paradox is resolved taking into account that simultaneity as well as lenght is a relative concept and not an absolute one.

• On first read, this doesn't seem plausible. The twin 'paradox' hinges on the fact that stay-at-home twin remains in the same inertial frame while the other twin must change inertial frames (in order to return to the stay-at-home twin) and so the situation isn't symmetric. But, in the ladder paradox, there is no changing of inertial frames. – Alfred Centauri Oct 15 '17 at 1:29
• @AlfredCentauri the ladder "observe" a smaller garage while the garage "observe" a smaller ladder – yngabl Oct 15 '17 at 1:34
• This is equivalent to the disagreement between observers in inertial motion about whose clock is running slow. It has no parallel to the absolute distinguishability of the inertial and non-inertial paths of the two twins. – dmckee --- ex-moderator kitten Oct 15 '17 at 4:52