Sorry, folks. Look, I’m as big a fan of sci-fi as the next person, and loved the idea of “faster-than-light” travel growing up, but I’m forced to admit that, in the context of presently-understood physics and a “sane” Universe, this looks impossible. For this post I thought I’d try to explain this particular apparent constraint (!) of fun possibilities.
First of all, objections to the idea of traveling faster than light can fall into at least a couple of ill-defined (by me) categories. One kind of argument that I’ve seen is sort of “formula-based”, where the claim is that superluminal speed is impossible because equations that describe measurable quantities go wonky. Here’s an example: I said in a previous post that time intervals that are measured by people moving relative to each other are going to be different:
Where v/c is the fraction of the speed of light of the relative motion. You can see that if v > c, then we’ll have the square root of a negative number, and the time interval becomes imaginary (WTF!). So with some justification, you can say, well, we can’t go faster than the speed of light because imaginary times are just foolish.
Maybe so, but it just feels a little slippery. Are imaginary times foolish? Maybe you think that’s cool and indicative of new physics. Maybe this indicates that we can go faster than light, but this exposes a weakness in our special relativistic understanding, and perhaps there’s a more correct equation that we haven’t found yet.
There is, I think, a much stronger argument that is based on a more basic idea of a causal Universe. I think we can get broad agreement that any sane Universe (and corresponding theories about how it works) should adhere to the principle that causes precede effects for all observers. If we allow faster-than-light speeds, this is violated.
The Causal Argument
Ok, here it is, and then I’ll justify it afterwards with some spacetime diagrams:
If you assert that some event A causes E (A = pushing a button, E = machine turning on) and that this occurs faster than light, then we can arrange an experiment where a friend zooms past you at some speed slower than light and observes E happening before A.
So, if we demand a Universe in which effects can’t happen before their causes for all observers, then the initial assertion must be wrong, and there can be no faster-than-light travel. That’s it. That’s the argument.
To justify it, we could just do a calculation based upon the Lorentz Transformation equations, which directly tell you what the time and space coordinates are in your friend’s moving frame if you know the coordinates in your rest frame and the relative speed. We’ll do that last — first let’s look at how to show it using spacetime diagrams since I think the argument is more intuitive that way. Here’s a simple diagram:
the idea is that this is our rest frame, showing distance down a 1-D railroad track as x and measured in seconds. That’s not too weird, really. Saying that a distance is 10 seconds away just means that it would take light 10 seconds to get there. The convention is also to plot time along the vertical axis, also here in seconds. So then the yellow line (called a world line) represents our friend who zooms past us carrying a clock (measuring the “primed” time in the moving frame). Just from the slope of the line we can work out our friend’s speed: they apparently go 3 units of distance in 5 units of time, so that would be a speed of 0.6c (60% of the speed of light). Notice that a purely stationary clock would follow a vertical line (time advances, but not position). A beam of light would then follow a line having a slope of 1 (1 light-second per second!)
A convenient way to think about that t-primed axis is that x’ = 0 everywhere along the line. That makes total sense, since our friend is carrying their coordinate system origin with them. Everything that happens along that line occurs at our friends (moving) location. A trickier thing is to find out how to draw the x-prime axis — along what line would our friend say that everything happens at the same time? For us, of course, that would be along our x-axis.
Here’s a thought experiment to help. Let’s imagine that our moving friend is really at the left end of a train car, along with other synchronized clocks in the middle of the car and also at the right end. Those clocks would move along parallel world-lines like this:
Ok. How can we set up an experiment in the train car where, according to someone *in the car*, two things are simultaneous? Here’s one way: suppose a laser at the left end fires a light pulse (event A) towards the middle of the car, and a laser at the right end does the same thing (event C). If the pulses arrive at the middle at the same time (event B), then they must have fired at the same time at their different locations. These two events (A and C) are then simultaneous from the point of view of someone inside the car, and so they define the “moving” x-axis!
One of the crucial lessons of Special Relativity is there in the diagram too — notice how A and C happen at the same time according to someone inside the train, but not from our point of view on the ground. This is entirely due to the fact that those light paths are 45 degree lines (same speed of light) in both frames.
Now to the argument (finally)!! Let’s say you claim to have a super-awesome remote control that can send a signal (A) twice the speed of light to turn on a TV located at (E), 8 light-seconds away, only 4 seconds after pressing the button.
What would this look like to our zooming friend? We need to find the time coordinate in the primed frame. We can just read it off the diagram, but notice that the primed axes aren’t perpendicular like we’re used to. We actually follow the usual procedure — read the time coordinate by following a line parallel to the new x-axis (this is what we do with “regular” perpendicular axes — it’s just not usually explained that way). But look! A parallel line intersects the t-prime axis at E’, before the signal is sent at A. So two observers won’t agree whether the effect came after the cause! Now, it’s perfectly fine for two observers to disagree about the time-order of two events, but not if the claim is that one thing caused the other.
So: if we want to maintain belief in a causal Universe, we cannot allow faster-than-light signals or travels to be a possibility. For any event E that would need superluminal travel from A, we can set up a zooming observer moving slower than light that would observe this causal reversal!
Using the Lorentz Transformations
As a spoiler for a future article, let’s find the exact coordinates in the moving frame using the coordinate transformation equations called the Lorentz Transformations:
(we’ll derive where these came from in an upcoming article…)
Here, the “unprimed” intervals are what we measure between A and E in our frame:
and then for an observer’s speed of 60% of the speed of light,
So plugging those in, you’d find that
So sure enough E happens before A according to the moving observer.
On Deck:
The next article I’m working on is one of the loveliest principles in physics, Fermat’s Principle — the idea that light takes the path of least time between two points.
If you’re a student/teacher and want to see lots of worked examples that I like to include in my classes when I teach the “standard” University Physics 1 and 2 courses, feel free to browse the (growing) collection of 150+ videos at
And if something is especially cool and you’re inclined to leave a “tip” I’m not above coffee or pizza:
Thanks for reading First Excited State! Subscribe for free to receive new posts automatically!