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Thursday, December 7, 2023

# So Can Flash Actually Run Faster Than the Speed of Light?

It's finally out, the Snyder cut of Justice League. So, what about that part with Flash? This is not a spoiler, since Flash does this in other situations: He needs to run faster than the speed of light in order to go back in time to warn the Justice League about something.

Of course there are lots of physics questions to be answered, so let's get right to it.

What's So Special About the Speed of Light?

It's easy to understand that velocity is relative. If you were walking at 1 m/s inside a train that is moving at 10 m/s, then someone on the stationary ground would see you moving with a velocity between 9 and 11 m/s (depending on which way you were walking). But our ideas about relative velocities are based on our own experiences with stuff moving. And here is the important part—practically every example of a moving object is moving slow. Yes, that supersonic aircraft is slow. Even a rocket going to the moon is slow. Everything is slow—slow compared to the speed of light, which has a value of around 3 x 108 m/s. We often represent this speed of light as the constant c.

And at faster speeds, things are a little bit different. It turns out that no matter what reference frame you are in, you will measure the same value for the speed of light. OK, let me give an extreme example so that you can see how this works.

Suppose you are sitting on Earth with a flashlight. In your reference frame (let's call it Frame A), the Earth is stationary, and when you turn on the light you measure its speed as c. That seems reasonable, right? Now there is another person in a spacecraft moving toward the Earth at half the speed of light (0.5c). Let's call this spacecraft the reference Frame B. From the perspective of Frame B it is also stationary, but the Earth is moving toward it at 0.5c.

But what about the measured speed of light from Frame B? Since the light was coming from the Earth, and the Earth appears to be moving at 0.5c, wouldn't it make that light seem to be moving at 1.5c? Nope. It doesn't work that way. It turns out that Frame B ALSO measures the speed of light at just normal c. That's the key idea of Einstein's theory of special relativity.

Time Dilation and the Speed of Light

You know what happens when two different people in different reference frames both measure the speed of light? Weird stuff happens with our perceptions of time. We call this time dilation. Let me explain this with a classic example—a light clock. Imagine that you have a clock, and the "ticks" are light bouncing back and forth between two mirrors. If you are in the same reference frame (speed) as this light clock, then the time for 1 "tick" will be the distance between the mirrors divided by the speed of light (c).

Now suppose you see another light clock, but this one is in a space ship (with windows so you can see inside). The spaceship is going super fast—like half the speed of light (0.5c). You can see the light in the light clock moving at just c, since everyone sees light at that speed. But during each "tick," this light not only goes back and forth between the mirrors but it also has to move forward since the mirrors are moving along with the spaceship.

Here, I made a quick animation to show you what this would look like. Notice that I slowed down the speed of light so that you can "see" each little light pulse in the clock. Yes, I made this in Python—here's the code in case you want to see it.

If you count the number of "ticks," both clocks get 7 full reflections. But wait! The stationary clock (with the yellow light) is already halfway to the next count, and the cyan light just started. From the perspective of the stationary observer, time is running slower for the moving clock. This is time dilation. Oh, if you are in the moving ship time still seems normal. It's just viewed from a different reference frame that time seems slower.

The faster the spaceship travels, the more time seems to slow down. Mathematically, we can write this as the following equation:

In this equation Δt is the time for some event (like one light-clock tick) in a stationary frame and Δt' is the time dilated the moving frame (with a moving frame velocity v). There are two important comments here. First, if you use a moving frame that's super slow—like a supersonic jet, then v2/c2 is super tiny. That means the time dilation has practically zero effect. Second, as the velocity of the frame (v) increases, time slows down even more. As you get very close to the speed of light, the time dilation would be extreme.

What Happens if You Go Faster Than Light?

Let's jump back a little bit. In 1905, Albert Einstein published his paper "The Electrodynamics of Moving Bodies". This paper contains his first ideas about relative motion and the speed of light. It didn't take long for someone to suggest that if you go faster than light, some weird stuff could happen. Imagine that you have a planet (Planet A) that shoots out an object faster than the speed of light. When it gets to another planet (Planet B), some event is triggered—let's say a light turns on. It turns out that for some moving reference frames, they would see the light turn on on Planet B before the object even left Planet A. That's super crazy.

But what would a faster than light object look like? Imagine that you have a spaceship moving at twice the speed of light as it zooms past the Earth. What would this look like to a stationary observer on the Earth? Remember, that in order to see this fast object, you have to have light travel from the object to the observer (on Earth).

Here is a model to show you what would happen. The moving object is shooting out pulses of light at regular intervals. Just so we can keep track of the timing, it produces a red light, then yellow, then cyan. Remember, that these light pulses have to travel at the speed of light. Here is the python code for this.

If you were on Earth, you would first see a cyan light, then a yellow, then a red light as the ship approaches. Even though the spaceship emits the red light first, it has moved closer to the Earth by the time it shoots out the cyan light. Since it's going faster than light, that means this cyan pulse doesn't have to go as far as the red (or yellow) pulses and gets there first. The next light to reach the Earth is the yellow pulse and then finally the red. So you would see the light in reverse order. Now imagine continuous light coming from the moving spaceship. These would also have to be completely backward. Yup, that's backward in time—there's your time travel.

A quick comment. We often call c the speed of light, and it is. But really that is the speed of causality. If you turn on a light at some point in space, a person that's far away wouldn't know the light was turned on right away since light travels at a finite speed. But it's not just light that has a constant speed, change has a constant speed. It's how fast you can ever know that something actually happened. The same thing happens with gravitational fields. When two black holes collide, they create gravitational waves that also travel at this speed of causality. When LIGO (the gravitational wave detector) first observed an event like this, it actually happened 1.3 billion years ago but it since it's far away, it takes time for the signal to reach us. In fact, if you have any event that causes a change somewhere else the cause and effect are delayed by a time because of the speed of causality. It just so happens that light also travels at the speed of causality (c).

You Can't Go at the Speed of Light, But Maybe You Can Go Faster Than Light

OK, so Flash just needs to go faster than the speed of light to go back in time. Right? Well, yeah…but, there's a problem. We often talk about the energy associated with a moving object. The faster it moves, the greater its kinetic energy. This model works fine for normal-speed objects—but when things go really fast we need a better energy model. This is the expression for the energy of a moving particle.

In this equation, v is the velocity of an object, c is the speed of causality (see, I already changed it) and m is the mass of the object (as measured in a stationary frame). First, notice that if the velocity of the moving thing is zero then the energy is just mc2 (which you've probably seen before). Next, let's consider what happens when the value of v increases. As the speed gets closer to c, v2/c2 approaches 1. That means that the denominator of that fraction gets smaller and makes the energy very large. What would happen if the velocity was exactly equal to c? Then you would have v2/c2 would equal 1 and you would be dividing by zero. You can't do that, so you can't go at the speed of light—at least not if you have mass. Light and gravitational waves can travel at the speed of light because they aren't "things".

But can you go FASTER than the speed of light? Maybe. Let's use the energy equation above for an object velocity of 1.5c. Here's what you get.

Yes, you end up with the square root of a negative number. That means we end up with an imaginary energy—remember that we represent the square root of negative 1 as the imaginary number i. So, that's it right? You can't do it. How about this? What if there is a particle with an imaginary mass? In that case, you get an i2 term such that you are right back to a real energy. Even though we have never found evidence that such an object exists, we already have a name for it—it's called a tachyon.

If this tachyon travels faster than c, then it would move backward in time. And since it has an imaginary mass, it also MUST have a velocity greater than c. If these tachyons were going slower than light, the denominator would no longer be an imaginary number so you would be left with an imaginary energy (because of the imaginary mass). Oh, but they still can't go exactly at the speed of light as you would be dividing by zero. So, the speed of light is like a giant barrier—nothing can cross it. That leaves us with three options. You have normal mass and you can't speed up to c, you are light and you always travel at c or you have imaginary mass and you can't slow down to c. I guess that just makes Flash special—I'm cool with that.

So, let's summarize here.

• Would going faster than the speed of causality be backward time travel? Yeah, it seems so.
• Can Flash run at the speed of causality? Nope. This would involve an undefined energy, because you would have to divide by zero.
• Can you go faster than the speed of causality? Mathematically, yes, as long as you have an imaginary mass.
• Is the whole Justice League movie just a sham because it's not scientifically accurate? Of course not. Justice League is just a movie. It doesn't have to follow these silly "science" rules. That's what makes it so much fun.