Energy, Momentum, E=mc²

The classical course taught that momentum (mass times velocity) and energy are perfectly conserved – the universe’s unbreakable ledgers. Relativity keeps the ledgers but discovers they were written in the wrong ink. Once space and time mix (Chapter 4), the old formulas stop balancing between observers, and fixing them produces two of the most consequential results in science.

First: you cannot reach the speed of light. As anything with mass speeds up, it gets harder and harder to push – not because of friction, but because its momentum and energy climb toward infinity as it nears $c$. An infinite shove would be needed for the last sliver of speed, so massive things are forever stuck below the limit. Second, and stranger: a thing at rest already contains enormous energy, locked in its mass. The bookkeeping demands it, and its name is $E = mc^2$.

Push a cart and it speeds up predictably: twice the push, twice the gain. Do the same to something moving near light speed and the gain shrinks toward nothing – the same push buys less and less extra speed, as if the object grows more stubborn the faster it goes. In the figure, the straight line is what Newton predicted; the curve that peels up and away is the truth. The gap between them is small at everyday speeds and unbounded as you approach $c$.

The wall at the speed of light is real and absolute. To shove a massive object all the way to $c$ would take infinite energy – so it never gets there. This is why particle accelerators, which push protons to 99.9999991% of light speed, can never quite reach 100%: the last decimal places cost more and more energy without end. Only massless things – light itself – travel at the limit, and they can travel at nothing else.

Look at the energy curve in the figure at zero speed. It doesn’t start at zero – it starts at a definite, large value. That’s the punchline. Even sitting perfectly still, a lump of matter holds energy, and a colossal amount of it: the mass is that energy, in frozen form. The conversion rate is the speed of light squared, an enormous number, so even a gram of matter hides the energy of a small bomb.

This isn’t metaphor. When the Sun shines, it is converting a little of its mass into light – four million tonnes every second. When a nuclear reactor makes power, a tiny fraction of its fuel’s mass vanishes and reappears as heat. Mass and energy are two forms of one thing, freely convertible, and $E = mc^2$ is the exchange rate. It is arguably the most consequential equation ever written.

There’s a single relationship tying together everything in this chapter – energy, momentum, and mass – and it has the same shape as the spacetime interval from Chapter 4. Just as observers disagree about durations but agree on the spacetime interval, they disagree about a particle’s energy and momentum (which depend on how fast it’s moving relative to you) but agree on its mass. Mass is the invariant, the thing everyone measures the same – the true, frame-independent identity of a particle.

Nothing with mass reaches light speed; mass is frozen energy at the rate $c^2$; and one Pythagorean equation ties energy, momentum, and mass together. Special relativity is now essentially complete – a beautiful, closed, thoroughly tested theory of a world without gravity. And that last phrase is the problem. In the final chapter, we try to fit gravity into this picture and discover that it simply won’t go – a failure so productive it forces the birth of an entirely new theory of space, time, and gravity.