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Setup

Consider an observer, at rest, holding two perfect clocks:

  • The first perfect clock is ordinary--with no additional distinction.

  • The second perfect clock has a special constraint, as it is arranged in a special way. All particles comprising the clock are coplanar.

It is hard to imagine the inner workings of such a clock, but for the sake of example, suppose it “ticks” nonetheless, always satisfying this constraint.

For convenience we’ll refer to this special clock as a “flat clock.

The whole system can be accelerated, but from the clock's perspective, all particles occupy the same plane. It follows that all of the clock's internal events originate on this plane, and the positions of particles moving within the clock remain within the plane. For convenience we’ll refer to this orientation as the flat clock’s “event plane.

Now suppose the observer places the flat clock on a spaceship that will accelerate it on a journey near light speed. The spaceship will also return to the observer, at which point the elapsed time given by the flat clock will be compared with the observer's ordinary clock.

The observer will repeat this experiment twice, modifying a single parameter:

  • Scenario 1: The flat clock's event plane is coplanar to the motion of the ship.
  • Scenario 2: The flat clock's event plane is orthogonal to the motion of the ship.

Assumptions

  • In the orthogonal clock case:

    • As the clock “ticks” internally, the motion of its particles will never align with the spaceship's direction of motion.
  • In the coplanar clock case:

    • As the clock “ticks” internally, the motion of its particles are additive with the spaceship's direction of motion. All particles moving in a direction that aligns with the spaceship's direction see a positive delta towards the speed of light. This creates a positive difference in the relative motion between the clock and the observer, which implies an additional relativistic effect. This effect is not present in the orthogonal case. For longer durations of travel near light speed, this difference would accumulate.

Question

  • When the experiment completes, would the time observed on each of the flat clocks be different from each other, and how would they each differ from the time observed on the observer's perfect clock?

  • If they all show different times, this would suggest to me that the clock's motion is not the only contributing factor to observed relativistic effects. Stated conversely, an object's orientation with respect to its direction of motion creates additional relativistic effects.

  • If this additional effect is real, is it already a prediction of special relativity?

If it is the case that the coplanar flat clock ticks more slowly than the other orthogonal flat clock (relative to the observer), despite having taken an identical trip, then the observer could calculate a duration T for which the readings of the clocks differ in a meaningful way. This calculation would be based entirely off of a known difference in the orientation of the clocks.

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Assuming the acceleration is a small part of the total journey this is answered simply by considering the rest frame of the clocks. Rather than doing the journey twice we'll simply place both clocks side by side. Then in the rest frame of the two clocks the Earth zooms away, stops, then zooms back. Since the two clocks are side by side and at rest relative to each other they tick at the same rate, so both clocks show the same time when Earth leaves and when Earth returns. That means both clocks record the same elapsed time.

The caveat about the acceleration arises because during acceleration the rest frame of the two clocks is not an inertial frame. The geometry is no longer described by the Minkowski metric but rather by the Rindler metric and this greatly complicates the treatment. But assuming the clocks have sizes on the human scale (i.e. of order one metre) the acceleration would have to be ridiculously high to create any measurable difference.

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  • $\begingroup$ I want to expand on your consideration of the two clocks undergoing the same journey in a shared rest frame. The clocks would appear to tick the same relative to each other, however we should still acknowledge their collective movement away from the observer, which establishes a new rest frame, which is also the frame in which relativistic affects are appreciable. The clock which "ticks" in the direction of motion should see its internal motions, or "events", inhibited by the restriction of the speed of light. $\endgroup$ – efreezy Sep 28 '18 at 4:33

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