# What's the problem with light traveling at speed higher than $c$?

I'm trying to understand how Einstein concluded that time is relative based on thought experiments such as a torch attached to the end of a rocket. Based on answers to questions like this one, this one, and this one, I understand that special relativity tells us that near the speed of light, it's not accurate to just sum the velocities of the rocket and the speed of light emanating from the torch to add up to a speed greater than $c$. What I don't understand is, before coming up with the Theory of Relativity, what theories, assumptions, and/or equations led Einstein to recognize a problem with the idea of a light beam traveling with a relative speed greater than $c$? I'm not a physicist, just someone trying to better understand Einstein and Relativity, so try not to make it too ridiculously deep :)

• en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment – pfnuesel Apr 12 '16 at 3:06
• Maxwell's theory predicted that the speed of light is a constant, but it didn't say speed relative to what. The question that Einstien answered was, what would the universe be like if the speed of light was constant relative to whoever measured it? – Solomon Slow Apr 28 '17 at 13:49

There were a couple of clues that Einstein and others found that led to the conclusion that the speed of light was special.

First, using Maxwell's equations, you can derive the existence of electromagnetic waves that travel at $$c = \frac{1}{\sqrt{\mu_o\epsilon_0}} \approx 300,000\, \textrm{km/sec}$$ where $\mu_0$ and $\epsilon_0$ are constants you can measure with simple electrical experiments. But, this is weird. All velocities are measured with respect to another object. If I'm traveling at 60 mph down the freeway, that 60 mph is with respect to the Earth. What is the speed of light relative to? Later experiments showed that it was neither the light emitter nor the light absorber. Rather, both measured the same speed of light regardless of their relative motion. This culminated in the most famous null result in physics: the Michelson-Morley interferometer experiment.

Second, Einstein imagined what he would see if he could catch up to a beam of light and fly along side it to see what it looked like. He realized that he would see a motionless wave of electric and magnetic fields. But, this is impossible according to Maxwell's equations. The equation that tells how electric fields are generated from charges $$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0},$$ where $\rho$ is the local electric charge density, means that, in the absence of electrical charges, $$\nabla \cdot \vec{E} = 0$$ there can be no points in space where the electric field is maximized or minimized. So, Einstein reasoned, if a maximum field is impossible to observe, and this is what you would observe if you caught up to a light beam, then it must be impossible to catch up to a light beam.

Once a maximum speed is accepted as a basic fact about our universe, space, time, and how observers measure space and time must be rethought. This resulted in the Special and General Theories of Relativity.

• Einstein wrote he wasn't concerned with MM experiments much, he was mainly concerned with logical consistency of thought experiments like the one with catching up to the light wave you mention. – Ján Lalinský Apr 28 '17 at 10:58

One argument goes as follows:

Maxwell's equations predict many things, but you can massage them into a form exactly like an equation that describes a broad variety of waves. Call this "the wave equation".

So imagine you have a container of water and you generate ripples. You could move your head and travel along with one ripple, so that from your point of view the ripple appears stationary.

Now electrodynamics (light), and small waves, both obey the same equation. You can move your head along a ripple of water so that the ripple appears still. Can you move your head along a ripple of light so that the ripple appears still? Experimenters wanted to try this, and they found that no, you can't.

This experiment is now used to motivate the postulate that Maxwell's equations are the same in all frames of reference. This then implies that you have to use a Lorentz transformation instead of a Galilean one.

So what happens if you use special relativity to allow faster than light travel? This is answered in the question, "Can FTL-Communication between two points in the same frame of reference break causality?". If you only allow objects to travel FTL in a single privileged reference frame, you get no paradoxes. But if you allow FTL travel in arbitrary reference frames, you allow time travel and "killing yourself before you traveled back in time to kill yourself" paradoxes to boot!

And that's why the idea of FTL travel is so appalling to physicists.

• Thanks! How did the Michelson–Morley experiment test whether or not you can "move you head along a ripple of light so that the ripple appears still"? It seems to me that it mostly just tested whether or not light traveling in different directions are altered differently by different at different velocities relative to aether. – Ben Lindsay Apr 12 '16 at 3:52
• @BenLindsay Sorry, trading precision for readability! Precisely: the same behavior that allows moving your head along a ripple of light, is the same behavior that made people predict that the Michelson-Morley experiment would find out how fast the Earth move through an aether. That behavior being "Galilean invariance", which is now known to not hold at high velocities. – user12029 Apr 12 '16 at 4:08

No physical signal can travel at a speed that is greater than the speed of light because if it did an effect would precede the cause. This is a consequence of the Lorentz transformations.

• "an effect would precede the cause" I think the question is more about why Einstein was led to adopt the Lorentz transformations. Putting events into boxes "cause" and "effect" is of little concern in physics. – Ján Lalinský Apr 28 '17 at 11:01