Relativity of Length Contraction – Part 2

This week we’ll look at what happens in the rocket’s frame of reference (FOR) — as if we were traveling alongside it at the same speed, or were inside the rocket looking out at an external mirror. Also, we’ll only need to accelerate to half the speed of light to show the intended effect.

You can see the length contraction has been swapped. Now, instead of the rocket contracting, we observe the entire Universe contracting in the direction of motion — including Earth if you can see it. As was the case last week, when viewing from within the rocket FOR, everything inside the rocket seems normal. But with space itself contracted in the direction of your motion, it’ll be possible to reach your destination in less time than expected given your speed.

To simplify the calculations, rather than continuously accelerating, let’s assume the rocket is already moving at a speed of 0.5c (γ = 0.866), and our destination is Alpha Centauri — the nearest star to our own at a distance of 4.2 light years (LY).

Observers on Earth would see the rocket contracted to 0.866 of its at-rest length, with its clock running 0.866 times slower than normal. A clock in the Earth FOR would measure the travel time as:

Δt = Δd/v = 4.2 LY / 0.5c = 8.4 years

But they’d also see the rocket’s clock running slow by a factor of 0.866, so the Earth FOR could deduce that the rocket’s clock would only show 7.2 years (0.866 × 8.4 years) for the trip.

Meanwhile, inside the rocket, they also register a speed of 0.5c, but perceive the distance to Alpha Centauri contracted from 4.2 LY to 0.866 × 4.2 LY = 3.6 LY. So in the rocket FOR the travel time is:

Δt = Δd/v = 3.6 LY / 0.5c = 7.2 years

That’s where the symmetry of relativity comes into play. The results for travel time measured in both FORs agree — the difference is that in the rocket FOR, space contraction is the explanation, whereas in the Earth FOR, time dilation is the explanation.

At higher speeds distance and travel time would decrease even further. At a speed of 0.9c, for example, with γ = 0.44, the distance will have contracted to 1.8 LY in the rocket FOR, and the travel time would be:

Δt = Δd/v = 1.8 LY / 0.9c = 2.0 years

But again, the Earth FOR could deduce the rocket FOR travel time of 2.0 years, because from their own measurements they see the rocket’s clock running 0.44 times its normal rate.

The process of acceleration and deceleration creates further disparities that are being ignored here. If you want to know what happens if the two FORs ever come together again, check out the Twin Paradox.

And in case you were wondering what space would look like in the rocket FOR if it was going at lightspeed [this never happens], here’s my “artist’s impression” of space contracted to zero thickness. Earth’s in there too. Of course, if it really was zero thickness there’d be no vertical line to draw, so I took some artistic license:

In Part 3 we’ll look at some practical implications of relativity in an imagined scenario with a Rebel Alliance cruiser attacking an Empire Death Star.

Next Week in Sky Lights ⇒ Relativity of Length Contraction – Part 3