Relativity of Length Contraction – Part 1

Over the last few months I’ve gotten several questions similar to: “I know about relativistic length contraction and how things supposedly get shorter as they approach the speed of light, but what I want to know is do they really physically get shorter?”

This is an excellent question, and it’ll take a 3-part series of posts to do it justice. The short answer is: What is really physically true depends on your frame of reference (FOR). My animation shows what an observer in the Earth FOR would see happening to a rocket that accelerates up to the speed of light.

Two changes will be obvious: the rocket will start to contract along its direction of motion, and the rate of time onboard will slow down. If you want to understand the physics behind this, download my lesson: It’s tough math, but only requires algebra. In this post I’ll just summarize the key concepts.

The symbol “γ” used in the animation is also known as the Lorentz factor, named after Dutch physicist Hendrik Lorentz. He introduced it in his work on electrodynamics. The equation is:

γ = 1/√(1 − v2/c2)

where v = the relative speed between two FORs, and c = the speed of light (3×108 m/s). For length contraction and time dilation the equations are:

L = Lo / γ  (where Lo = the length at rest and L = observed length
Δt = γ × Δto  (where Δto = time between two ticks of a clock in the observer’s FOR, and Δt is rocket time)

The Lorentz factor appears in many equations and is a concise way to express the amount of length contraction, time dilation, and relativistic mass-energy increase. In this 3-part series, we’ll be focusing on the first two, showing length contraction visually, and time dilation via the rotation speed of hands on two clocks. These representations are mathematically approximate and designed to make the effects more easily visible.

The animation shows the changing effects all the way from rest up to the speed of light, where the rocket’s length contracts to zero and time stands still. I must point out that the rocket could not actually attain the speed of light, as it would require an infinite amount of energy to accelerate the rocket’s ever-increasing mass-energy (aka inertia) — but that’s a topic for another day. However, you can get arbitrarily close to the speed of light if you’re willing to expend enough energy. We do that with subatomic particles routinely. The Large Hadron Collider (LHC) can accelerate protons to 99.999999% the speed of light!

In case you were wondering, the rocket exhaust contracts too, as shown. Although the exhaust is moving in the opposite direction (relative to the rocket), it’s moving with the rocket relative to an observer on the Earth. The fastest exhaust speeds are around 4.5 km/s, and that’s only 0.0015% the speed of light.

So this is what length contraction and time dilation would look like from Earth if you could observe the rocket as it accelerates. My next post will feature an animation showing what things would look like if you were riding inside that rocket.

Next Week in Sky Lights ⇒ Relativity of Length Contraction – Part 2

The Significance of Starliner
Relativity of Length Contraction – Part 2