You can break the laws of your local jurisdiction. It might not be a good idea, but if you want to drive 60 in a 45, you can. You’ll pay a hefty fine if you get caught, but you’ll still have been able to do it.
There’s no penalties for breaking the laws of physics because you can’t break the laws of physics. On the other hand, technicalities in both legislative and physical laws can sometimes allow you do to things you wouldn’t expect. Take this one:
You can’t go faster than the speed of light.
Clear enough, until the lawyer chimes in. “What,” he says with a raised eyebrow, “do you mean by speed?” You and I know what speed is, and so we answer that speed is simply the distance covered in a given time. “Yes, but the whole point of relativity is that distance and time aren’t the same in every reference frame. What if I measure distance from the perspective of an observer on the ground but I measure time from the clock in my moving car?”
He’s got a point. You can go faster than the speed of light if you use this definition of speed. Here’s how. In special relativity there’s a formula that tells you how time in your car (or spaceship, or particle in an accelerator, or whatever) is related to time in another coordinate system (say, the ground). If you call time in your car t, time on the ground with the Greek letter τ, and your velocity v, you’ll find that they are related by
To you in your vehicle, the ground is going to be length-contracted. From your perspective, you’re going fewer miles per second than the speed of light, but for some reason each miles is squashed and contains more than a mile’s worth of scenery. In fact, the whole external universe seems to be compressed along your direction of travel. Clocks on the outside seem to be going slower as well. But from the perspective of the people on the outside, you seem to be moving in slow motion too. To them you’re going slower than light, but your clock is running so slow that you might cover more than 299,792,458 km by the time one of your slow seconds has gone by.
This seems to suggest that if you get on a suitably powerful spaceship now, you can get to Antares by lunch tomorrow. Sure by the time you get there 600 years will have gone by in the external universe but you nonetheless covered 600 light years (in the external frame) in a few hours (of your frame).
From your perspective you won’t have traveled faster than light because the external distance contracted while you were traveling. And from the external perspective you won’t have been traveling faster than light, but were instead fooling yourself because your clock was slow.
So how fast do you have to be going in the ground frame to be going at the speed of light in this fake hybrid “proper velocity” measurement? We can use those above equations, and if we call the proper velocity w, it will turn out that
So, we just have to write out gamma explicitly and solve for v. Turns out things end up symmetric:
So if you want your proper velocity to be c, set w = c and you’ll find that your ground velocity only needs to be v = 0.707c. If you want to make that trip to Antares in 24 hours, you want your proper velocity to be about 219,000c. That’s a pretty fast clip, but it can be done with a ground velocity of 0.99999999998957c. Your local police might give you a ticket for driving that fast, but you’re still under the physics speed limit of the speed of light and so it’s not forbidden.