From Classical Kinetic Energy to Einstein’s E = mc2
So it turned out that I wasn’t smarter than Einstein after all. Never said I was. Honest.
But it also turned out that Max Plank is smarter than me. Which to be honest is not a big surprise. Or at least he beat me to it by about a century anyway.
So I started thinking a bit more seriously about Gottfried Wilhelm Leibniz and Émilie du Châtelet’s kinetic energy formula, Einstein’s famous mass-energy equivalence and how the the Lorentz Transformation links them when I was about 14. I had essentially got through the school syllabus already and my dad had given me his university physics textbooks to keep me busy in the summer vacations. These dusty tomes revealed themselves to be a complete intellectual goldmine which I rapidly consumed and which which I still delve in mentally today (at least in my head). Disclosure – I am not a genius – I feel very mathematically inadequate compared with these guys – but they have put a lot more processor cycles into their math than lazy-old me has.
But what I did think of was a short-cut to relativity – bridging classical and relativistic mechanics – which seemed – at least to me – pretty cool. Maybe not as cool as the Euler equations – but somehow more immediately applicable to what we understand of the universe.
One powerful way to understand Einstein’s famous mass-energy equivalence is by bridging classical and relativistic mechanics. We begin with the classical kinetic energy formula, and by applying the Lorentz transformation, we can see how the concept of rest energy—E = mc2—naturally emerges.
1. Classical Kinetic Energy
In classical mechanics, the kinetic energy of a body is given by:
$$ KE = \frac{1}{2}mv^2 $$
This formula holds true at low velocities. However, as an object approaches the speed of light, relativistic effects become significant, and the classical formula no longer suffices.
2. Introducing the Lorentz Factor
Special relativity introduces the Lorentz factor, denoted by \( \gamma \), which accounts for time dilation and length contraction at high speeds:
$$ \gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}} $$
The total energy of a particle in relativistic mechanics is:
$$ E = \gamma mc^2 $$
where:
- \( E \) is the total energy
- \( m \) is the rest mass
- \( c \) is the speed of light
3. Expanding for Low Speeds
For low velocities (i.e., \( v \ll c \)), we can approximate the Lorentz factor using a Taylor expansion:
$$ \gamma \approx 1 + \frac{1}{2} \frac{v^2}{c^2} $$
Substituting this into the energy equation:
$$ E = \gamma mc^2 \approx mc^2 + \frac{1}{2}mv^2 $$
This shows that the total energy consists of two components:
- The term \( mc^2 \) represents the rest energy—the energy an object possesses by virtue of having mass, even when at rest.
- The term \( \frac{1}{2}mv^2 \) corresponds to the familiar classical kinetic energy.
4. The Mass-Energy Equivalence
From this derivation, we see that even when an object is not moving, it possesses intrinsic energy given by:
$$ E_0 = mc^2 $$
This is the foundation of Einstein’s mass-energy equivalence principle. It tells us that mass is simply another form of energy, fundamentally interchangeable. In other words:
$$ E = mc^2 $$
This equation transformed our understanding of matter and energy—and underlies everything from nuclear power to the energy released in stars. Actually a whole lot more than that – but those are the textbox things to say!
I actually described that derivation to my Oxford maths and philosophy girlfriend (the one who made me take a MENSA test before going out with me as I recall) and she was most unimpressed – but it turns out that Max Planck derived it in 1906 too so I’m in good company in thinking of this (even I I only found out that Planck beat me this year – so I haven’t revolutionised Physics this time). Lev Landau & Evgeny Lifshitz go into it in much more depth in Classical Theory of Fields.
Note: The equation in the illustration is actually something different as a placeholder until I make the right slide!