The pulses reflect from the mirrors at the same time; events on the blue are simultaneous.
But light waves travel through empty space, so there is no way
to tell which apparatus is moving. Thus, the reflections also
happen at the same time in the moving frame, and the
Both observers must agree on the area of any figure in the spacetime diagram - any difference would distinguish between relative motion to the left and right.
Therefore the clocks will read the same at the end of the experiments if and only if the areas of the two gold light path diamonds are equal.
In Euclidean geometry, the invariance of area under rotation is also how you compare rotated and unrotated coordinates.
In Euclidean geometry,
The light blue figure is a spacetime square. Adjacent edges are orthogonal in the spacetime sense. The length of the spacelike edges is the distance light travels in the duration of the timelike edges, so the diagonals are light paths.
The area of the light blue
spacetime square is
In primed coordinates,
The tangent to the hyperbola at P is spacetime orthogonal to world line OP, with slope reciprocal to the slope of OP.
The ratio between
You cannot travel “faster than light” only from
the point of view of a stationary observer. A traveler can go
any distance in any given time
The tangent to the hyperbola at P is spacetime orthogonal to OP, with slope reciprocal to the slope of OP.
This timelike hyperbola is the world line of an object moving
with constant acceleration
One Earth gravity acceleration has a distance