special relativity - 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 :)

Answer

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.