## Running late in a relativistic spaceship

Ever been bothered by time-pressed commuters that recklessly drive at high speeds on the interstate just to save a few minutes time on their way to work? Ever wanted to watch them get to work later than if they had just gone the speed limit in the first place? Their trick might work some of the time, but it won’t play out every time: here comes the sweet karmic payback ensured by the laws of special relativity!

Albert Einstein first published his “Theory of Special Relativity” in 1905. It tied together a major problem in physics at the time: the assumption that physics behaves the same if you are traveling at any constant velocity and the observed invariable speed of light, regardless of how fast one travels toward or away from a light source. He noted that Isaac Newton’s explanation of mechanics was only an approximation and only worked well for velocities far below the speed of light. Special relativity begins to make a major difference once object speeds exceed half of the speed of light.

Let’s say you’re running late for work in the Alpha Centauri star system, and your only means to get there is a spaceship that can travel at relativistic speeds. You want to get there as fast as possible, but you sit back and think for a minute: although it might be human instinct to travel at the fastest speed possible, this will not be the shortest amount time for your impatient boss, due to time dilation as a result from special relativity. The increased speed will only mean a decreased time in your reference frame, not necessarily the one for your boss.

A view of the southern sky, featuring Alpha Centauri (seen to the far left), the third brightest star system from the vantage point of Earth     [Source]

So what is the optimal speed to go in order for the least amount of time to pass by for your boss? We will first approximate this problem by assuming you travel directly from the Solar System to the Alpha Centauri system and that there is no interstellar commuter traffic, meaning you travel at a constant velocity throughout your journey. Our goal is to figure out when the effects of special relativity begin to surpass the effects of increased velocities.

For our discussion, we must introduce the concept of the Lorentz factor, γ. In everyday situations, we are accustomed to a Lorentz factor of 1. Once we reach velocities around one-tenth the speed of light (0.1 c), everyday physics starts to become slightly distorted and special relativity must be factored into our physics calculations. The Lorentz factor (γ) is given by

\begin{aligned} \gamma = \frac{1} {\sqrt{1 - v^{2} / c^{2}}} \end{aligned}

where v is the velocity you are traveling and c is the speed of light. As velocities approach the speed of light, the Lorentz factor gets increasingly greater than 1, becoming infinite when v = c.

The effect of time dilation corresponds to the time measured in a non-moving frame of reference (ΔT), and the “proper time” (Δτ) as measured by the moving observer’s clock. In our situation, Δτ is the time measured by the commuter driving the spaceship, and ΔT is the time measured by the boss in the Alpha Centauri system. The relation between these two is given by the following formula:

\begin{aligned} \Delta T = \frac{\Delta\tau} {\sqrt{1 - v^{2} / c^{2}}} = \gamma \Delta\tau \end{aligned}

In order to solve for the most optimal speed for our relativistic spaceship, we will first examine ΔT for every tenth of the speed of light. In order to do this, we must first establish the proper time (Δτ) as measured on the spaceship. Since Δτ is inversely proportional to velocity (v), we will treat the distance traveled (Δd) as 1 to simplify our calculations:

\begin{aligned} \Delta\tau = \frac{\Delta d} {v} = \frac{1}{v} \end{aligned}

Now all we have to do is plug into the equations to solve for ΔT for each of the corresponding velocities, the results of which are given below:

 Velocity, v Lorentz factor, γ Proper time, Δτ Rest frame time, ΔT 0.0 c 1.000 ∞ ∞ 0.1 c 1.005 10.000 10.050 0.2 c 1.021 5.000 5.105 0.3 c 1.048 3.333 3.493 0.4 c 1.091 2.500 2.728 0.5 c 1.155 2.000 2.310 0.6 c 1.250 1.667 2.084 0.7 c 1.400 1.429 2.001 0.707 c 1.414 1.414 2.000 0.8 c 1.667 1.250 2.084 0.9 c 2.294 1.111 2.546 1.0 c ∞ 1.000 ∞

While the velocity increments are a constant increase, the Lorentz factor (γ) becomes increasingly inflated and the proper time (Δτ) diminishes at a decreased rate. While the rest frame time (ΔT) initially lessens, around 0.7 c it reaches a minimum before dilating. Clearly the most optimal speed is around 0.7 c.

It just so happens that the actual answer to our problem is when the spaceship’s velocity is at a speed of $\sqrt{2} / 2$ c, yielding a γ and Δτ of $\sqrt{2}$. Why is this you might ask? Think back to basic geometry dealing with the Pythagorean theorem. The sides of a 45° right triangle will always be $\sqrt{2} / 2$ times the length of the hypotenuse. (For this reason, it is no coincidence that sin(45°) = cos(45°) = $\sqrt{2} / 2$.) In our case, the hypotenuse is ΔT, with Δτ and γ as the sides (as illustrated below).

\begin{aligned} \gamma = \frac{1} {\sqrt{1 - v^{2} / c^{2}}} = \frac{1} {\sqrt{1 - (\sqrt{2}/2)^{2}}} = \frac{1} {\sqrt{1 - (2/4)}} = \frac{1} {\sqrt{1/2}} = \frac{\sqrt{2}} {\sqrt{1/2} \, \cdot \sqrt{2}} = \frac{\sqrt{2}} {\sqrt{1}} = \sqrt{2} \end{aligned}

\begin{aligned} \Delta\tau = \frac{1} {v} = \frac{1} {\sqrt{2} / 2} = \frac{2} {\sqrt{2}} = \frac{2 \, \cdot \sqrt{2}} {\sqrt{2} \, \cdot \sqrt{2}} = \frac{2 \sqrt{2}} {2} = \sqrt{2} \textup{ time units} \end{aligned}

\begin{aligned} \Delta T = \sqrt{{\Delta\tau}^2 + {\gamma}^2} = \sqrt{\sqrt{2}^2 + \sqrt{2}^2} = \sqrt{4} = 2 \textup{ time units} \end{aligned}

In conclusion, you might wonder how late you will be for work: with the Alpha Centauri system 4.4 light years away, it will take you a minimum of 8.8 years (as observed by the boss) to get there. Go any faster than the optimal speed, and you’ll be even later for work! It’s probably best that you look for another job.

#### Pages on Special Relativity:

• Time dilation  (coming soon!)
• Length contraction  (coming soon!)
• Relativistic velocity addition  (coming soon!)
• Running late in a relativistic spaceship

(More to come soon!)