# Tachyon Rocketry

In my last article, I fiddled with an interstellar spaceship propelled by a photon rocket. The maximum speed (more correctly, delta-v) of a rocket depends on the exhaust velocity of its propellant. Since we know of nothing that travels faster than light, photons seem to be the best possible propellant.

But what if our propellant traveled faster than light? Particles that travel faster than light are generally called tachyons, and if they exist, they have some very strange and inconvenient properties. (In fact, FTL Pizza recently closed its Tachyonic Anti-Telephone Booth because kids kept making prank calls to Albert Einstein.)

As the saying goes, “relativity, causality, and FTL: pick any two”…

## Properties of Tachyons

Aside from the major annoyance of breaking causality (or relativity: you can pick one or the other), tachyons have the odd property of having imaginary mass: their rest mass is a multiple of the imaginary number i, defined as $i=\sqrt{-1}$.

Argh. Imaginary (and complex) numbers appear quite often in engineering, but the imaginary parts usually cancel out to a real quantity whenever we look at a “real” property, such as rest mass. However, this is a necessary result because the energy of a tachyon must be a positive real quantity. Relativistic energy is:

$E=\frac{mc^2}{\sqrt{1-(\frac{v}{c})^2}}$

Note: the quantity $\frac{1}{\sqrt{1-(\frac{v}{c})^2}}$ shows up often in relativistic equations, and is called γ (gamma).

For a tachyon, $\frac{v}{c}$ is greater than one, which makes the expression inside the radical negative, making the denominator as a whole imaginary. The numerator must also be imaginary so that we have a positive real total energy. Since the only variable in the numerator is mass, the tachyon’s mass must be a pure imaginary number.

(Of course, I interpret this as a hint that tachyons don’t exist. On the other hand, engineers tend to deal with simplified approximations of reality, and any reality that contains tachyons probably isn’t simple…)

What does this mean for the tachyon’s momentum? Standard disclaimer: physics is not my subject. As far as I can tell, the momentum of a tachyon is real. First, total energy is related to momentum. If total energy is real, momentum is also real. Second and relatedly, momentum is the product of γ, mass, and velocity. γ and mass are both imaginary, and multiplying two imaginary values gives a real value. Velocity is real, albeit faster than light. Thus, the momentum of a tachyon is real.

## Tachyon Rocketry

Since the momentum of a tachyon is real, I’m going to assume that I can start plugging values into the same equations used previously for a photon rocket.

For simplicity, I’ll even use the same design parameter and assume that the tachyon rocket has a propellant fraction of 10%. That is, one-tenth of the total mass of the rocket is propellant. There’s no physical reason for this: it’s just so that the spacecraft looks more like a cool sci-fi starship, and less like a gigantic fuel tank with people strapped to it.

Previously, I used the formula $\Delta v = v_e \ln{R}$, which is a classical formula. This time, I’ll use the relativistic rocket equation, on the assumption that relativity is still true. (In other words, goodbye causality.)

Relativistically, $\Delta v = c \cdot tanh(\frac{v_e}{c} \ln{R})$. Let’s look at some values.

 Tachyon exhaust velocity (× c) delta-v (percent c) 1 10.5 2 20.7 3 30.6 4 39.8 5 48.3 10 78.3 20 97.1 50 99.995 100 99.9999999

Note the hyperbolic tangent term in the equation. Even with a tachyon rocket, we never exceed the speed of light, as long as relativity remains true. Not even if we paint racing stripes on our starship and then coat it with bacon grease.

Of course, instead of throwing out causality, we could have thrown out relativity instead, but that seems less interesting. If Einstein was wrong, then there’s nothing special about the speed of light, or tachyons, or FTL in general. If you want to go faster, just go faster.

## Where To?

Now that our 100c tachyon-propelled starship is crewed, fueled, and waiting in spacedock, where shall we go? At 99.9999999% the speed of light, the nearest star is still about five years away.

Except that γ comes into play again. Our perceived time will scale by γ, which is 22,360 at our cruising velocity. That makes our 5 year voyage clock in at less than 2 hours, not accounting for acceleration and deceleration.

(Safely accelerating to just under the speed of light would take quite a bit longer than two hours. I don’t want to get into the details of how long the acceleration would take, so I’m ignoring it for now.)

Cool beans. What if we want to go someplace farther away, like the center of the Milky Way? That’s about 27,200 light-years away: less than a year and a quarter at our cruising speed. If our crew signed on for a five-year mission like the crew of USS Enterprise, the entire galaxy is within reach of our starship.

Of course, when we returned to spacedock, we would find that tens of thousands of years had passed back home. Everyone we left behind would be long dead, and our favorite restaurants would probably be out of business. Bummer.