Mass–energy equivalence

What just happened?

The closer to the speed of light, the larger the Lorentz factor $\gamma$, and with it the total energy. As $v \to c$ the energy grows without bound — the curve reaches the vertical asymptote.

Mass–energy equivalence:

— $E$ is the total energy, $E_0$ is the rest energy, and $\gamma$ depends only on the speed. Hence $E / E_0 = \gamma$.

What does E = mc² mean?

Mass is condensed energy: even a body at rest has rest energy $E_0 = mc^2$. It is released in nuclear reactions and in the Sun — a tiny loss of mass yields enormous energy, because $c^2$ is a huge number.

Why can you not exceed c?

Accelerating a mass to $c$ would require $\gamma \to \infty$, i.e. infinite energy. No finite source can supply it — the speed of light is a barrier for massive bodies.

What is the invariant?

The rest energy $E_0 = mc^2$ does not depend on the reference frame — it is the same for every observer. The energy $E$ grows (it depends on speed), but $E_0$ stays constant.

Why you need this for the exam:

$E = \gamma mc^2$ and $E_0 = mc^2$ are the core of special-relativity problems. Trick: compute $\gamma$ from the speed, then the energy — and remember that $c$ is unreachable for massive bodies.

Remember: mass is frozen energy (E₀ = mc²), and accelerating to the speed of light devours infinite energy, because E = γmc² and γ → ∞.