Recall the prior Blog's dialogue on the Starship Enterprise bridge, where Captain Kirk asked Spock and Scotty to go and resolve their differences in interpreting the test results (which according to Scotty implied that the two Vulcan clocks run faster than the ship's clock). Spock disagreed.

So off they went to the ship's CR4.

"Mr. Scott, our bridge clock was present at both flyby events, so you effectively measured the invariant spacetime interval between the events: Δτ = √[Δt² - Δx²], in geometric units.⁽¹⁾ In the bridge frame Δx=0, so Δτ = Δt = 1333 ns, the elapsed time that you found.

"In the frame of the two clocks, we know the distance between the clocks was Δx' = 1000 ft. Since the spacetime interval is the same for all inertial frames, it is very easy to show that Δt' = √[Δτ'² + Δx'²] = 1667 ns. This is the elapsed time that you recorded from the readouts of the Vulcan clocks.

For every frame of reference, the farther two events are separated in space, the more they are also separated in time⁽²⁾ - simple and logical. No apparent time dilation or length contraction".

A thoughtful Scotty replies:

"I thought we have to use the Lorentz transformations, which have all the elements of clock synchronization, time dilation and Lorentz contraction mixed together".

Spock: "These are favorite Human abstractions, Mr. Scott. This is how people frequently come up with contradictory conclusions".

Frowning, Scotty counters: "But Mr. Spock, I was taught those things in Starfleet Engineering School. Are you now telling me they are in effect useless?"

Spock: "Not useless Mr. Scott; just overly complex and confusing in this case. Do you agree that the spacetime interval solution is clear-cut?"

Scotty: "Ay, Mr. Spock, I see that. It does look like mathematical sleight of hand, though. I would like to test this in more situations".

Spock: "I do not understand 'mathematical sleight of hand'. We may come back to that later, but right now we must report back to the bridge".

-J

(1) Where c ~ 1 (ft/ns), meaning distance and time are effectively measured in the same units.

(2) This true for clocks synchronized by the Einstein-method and spacetime intervals where the time separation is larger than the space separation ('time-like' intervals).