Q.Which of the following relations is correct?

Special Relativity: Mass-Energy Equivalence — Why $E = mc^2$ Holds

The formula $E = mc^2$ is not a random result — it emerges naturally from the logic of special relativity and the conservation of momentum and energy. Let's build the reasoning step by step.

1. The Core Problem Relativity Solves

In Newtonian physics, mass and energy are separate:

• Mass is conserved.
• Energy is conserved.
• They don't mix.

But in special relativity, the speed of light is the same for all observers. This forces us to rethink how momentum and energy behave when objects move at high speeds.

2. Relativistic Momentum — The First Clue

Newton's momentum is $p = mv$. But if an object moves close to $c$, this formula fails to conserve momentum in all reference frames.

Einstein showed that the correct relativistic momentum is:

Here, the denominator $\sqrt{1 - v^2/c^2}$ (called the Lorentz factor $\gamma$) appears because time and space stretch for moving observers.

Key insight: The factor $\gamma$ means that as $v \to c$, momentum grows without bound — even if $m$ is constant. This hints that mass and motion energy are linked.

3. Relativistic Energy — The Logical Extension

If momentum changes, energy must also change. Using the work-energy theorem (work done = change in kinetic energy), we can derive the relativistic energy.

Start from the definition of work:

Substitute $p = \gamma m v$ and integrate (using calculus):

After integration (details omitted for brevity, but standard in textbooks), we get:

4. The Two Terms — Rest Energy and Kinetic Energy

The expression above splits into two parts:

• First term: $\gamma m c^2$ — the total energy of the moving object.
• Second term: $m c^2$ — the energy of the object at rest.

So:

5. Why $E = mc^2$ Is So Profound

The rest energy $E = mc^2$ means mass itself is a form of energy. Even when an object is completely stationary, it contains an enormous amount of energy — because $c^2$ is huge ($9 \times 10^{16} \, \text{m}^2/\text{s}^2$).

Why does this happen?

Because in relativity, mass and energy are not separate — they are two sides of the same coin. The conservation laws merge into a single conservation of mass-energy.

6. The Famous Derivation (Einstein's 1905 Thought Experiment)

Einstein himself used a clever argument:

1. Imagine an object at rest emitting two identical light pulses in opposite directions.
2. The object loses energy $E$ (the light's energy).
3. Because light carries momentum, the object must lose mass to conserve momentum in all frames. …