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.