Chapter 190: The Most Famous Equation

Klein’s eyes saw nothing but the equation on the blackboard.

He treated mass as a function of velocity, m(v), and then rebuilt the entire collision model.

In frame S, the equation for conservation of momentum was now written as:
m(u_A) * u_A + m(u_B) * u_B = m(u’_A) * u’_A + m(u’_B) * u’_B

This was an equation about the unknown function m(v).

Klein needed to solve for the form of this function.

He utilized symmetry.

Since the two balls were identical, just moving in opposite directions, their speeds |u’_A| and |u’_B| in frame S’ were equal.

Therefore, their masses in frame S should also be equal, m(u_A) = m(u_B).

The same was true after the collision.

The equation was thus simplified.

He substituted all the complex velocity component expressions he had previously calculated into this new momentum conservation equation.

The entire blackboard was quickly filled with a tangled mess of symbols and square roots.

Lia stood aside, watching quietly.

She knew this was not just a mathematical derivation.

It was a revolution of concepts.

It would pull “mass,” considered the most fundamental and constant property of matter, down from its pedestal.

To admit that it too was relative, was variable.

This required immense courage and wisdom.

Klein’s calculation continued.

He cancelled out identical terms on both sides of the equation and simplified the complex radicals.

His forehead was beaded with sweat, but his hand was as steady as a rock.

Every step of the derivation was like disarming a precision bomb.

The slightest mistake, and the entire theoretical system would explode in a logical contradiction.

Finally, after a long process of simplification, he obtained an exceptionally concise relationship.
m(u) / m₀ = 1 / √(1 – (u’²/c²))

Here, m₀ is the mass of the ball in frame S’, which is its “rest mass.”

Because in frame S’, the ball’s x-direction velocity is 0, and its y-direction velocity u’ is very small (can approach 0), it can be considered that m₀ is the rest mass.

Klein stared at this relationship.

He could cancel out m₀ on both sides.

Considering that in frame S’, the ball’s x-direction velocity is 0, and its y-direction velocity is u’_y.

The relationship became:
m(u) / m₀ = 1 / √(1 – (u²/c²))

Here, u is the speed of the ball in frame S.

And V is the velocity of frame S’ relative to frame S, which is also equal to the x-direction velocity of the ball in frame S.

So, u² = u_x² + u_y².
And u_x = V.

When u’_y approaches 0, u_y also approaches 0.

Then, u would approach V.

Klein replaced u with V, obtaining the final conclusion.

He had solved for the functional relationship between mass and velocity!
m = m₀ / √(1 – (V²/c²))

Klein dropped the chalk, his body leaning against the blackboard as he gasped for breath.

He looked at the formula on the blackboard, his eyes filled with shock.

The inertial mass of an object increases with its velocity.

The faster an object moves, the “heavier” it becomes.

When an object’s velocity V approaches the speed of light c, the denominator √(1 – (V²/c²)) approaches 0.

This meant that its mass m would approach infinity!

Klein understood instantly.

He understood why any object with mass could never reach the speed of light.

Because the closer you get to the speed of light, the greater your mass becomes.

You would need an infinite amount of energy to give an object with infinite mass even a tiny bit more acceleration.

This was impossible.

The speed of light is the cosmic speed limit for all massive objects.

This conclusion, once merely empirical, was now endowed with a solid theoretical foundation.

“Spacetime, mass, momentum…” Klein’s voice was a bit dry. “All the bricks and tiles of the old world have been replaced.”

He turned around and looked at Lia.

“We succeeded.”

“We have preserved the law of conservation of momentum.”

Lia nodded slightly.

“We have preserved the harmony and unity of physics.”

Klein walked over and held her tightly in his arms.

He could feel his heart beating violently.

It wasn’t from fatigue, but from excitement.

The excitement of personally unveiling the ultimate mysteries of the universe.

He buried his face in the crook of Lia’s neck and took a deep breath.

The clean scent of her hair helped to cool his feverish brain slightly.

“Klein, have you noticed? This paper is still missing its final part,” Lia whispered in his ear.

“I know.”

Klein’s voice was muffled. “Energy.”

He released Lia and walked back to the blackboard.

There was no longer any trace of fatigue in his eyes, only a blazing fighting spirit.

The theory of spacetime had been established.

The new dynamics had taken shape.

Now, he would use this brand-new theory to redefine a core concept in physics—energy.

He erased the derivations on the blackboard, leaving only the new mass formula.
m = m₀ / √(1 – (V²/c²))
And the new definition of momentum.
p = mv = (m₀v) / √(1 – (V²/c²))

Klein started from the most fundamental definition of energy.

The increase in an object’s kinetic energy is equal to the work done on it by an external force.
dE_k = W = F ⋅ ds

This was an incredibly basic formula.

Work equals force times displacement.

Now, he needed a new definition of force.

In classical mechanics, F = ma.

But in the new theory, mass is no longer a constant, so this definition is no longer applicable.

A more general definition of force should be: force is equal to the rate of change of momentum.
F = dp/dt

Klein substituted the new expression for momentum p = mv into F = d(mv)/dt.

This was a derivative with respect to time.

The calculation process involved complex differentiation rules.

Klein took a deep breath and began his final sprint.

His chalk drew elegant and swift lines on the blackboard.

Every symbol contained the laws of the new world.
dE_k = F ⋅ ds = (dp/dt) ⋅ ds = dp ⋅ (ds/dt) = v ⋅ dp

Because v = ds/dt, so ds = vdt.

This relationship was exceptionally concise.

The differential of kinetic energy equals the dot product of velocity and the differential of momentum.

Klein substituted the expression for momentum p = mv and then differentiated.
dE_k = v ⋅ d(mv)

This differentiation process was also full of pitfalls.

Lia had prepared food and water for him on the side.

She knew that the upcoming calculations would consume a huge amount of mental energy.

The entire top floor of the mage tower fell into absolute silence.

There was only the rustling sound of chalk on the blackboard, and the intangible sense of pressure in the air caused by a brain operating at high speed.

Klein substituted the differentiated d(mv) into the expression dE_k = v ⋅ d(mv).

He finally obtained a relationship between the differential of kinetic energy dE_k and the differential of velocity dv.

Now, he needed to integrate this expression.

Integrating from velocity 0 to velocity V would give the total kinetic energy gained by the object as it accelerates from rest to velocity V.

This was an extremely complex integral.

Klein stared at the integral, not writing for a long time.

He needed to find a suitable substitution method to simplify it.

He tried several methods, all of which ended in failure.

Time ticked by, second by second.

The sky outside the window turned from white to black, and then from black to white again.

Lia stayed by his side quietly, occasionally offering a hint.

Finally, as the first ray of sunlight of the second morning shone into the study, Klein’s eyes lit up.

He had thought of a clever substitution.

He let V = c sin(θ).
So, dV = c cos(θ) dθ.

The entire integral instantly became clearer.

He quickly calculated on the blackboard.

Trigonometric transformations, integral operations, flowed smoothly from his pen.

By the time he wrote the final step, the sun was already high in the sky.

He had obtained the final result.

A result that confused even himself.
E_k = m₀c² / √(1 – (V²/c²)) – m₀c²

Klein looked at this formula, his brow tightly furrowed.

He recognized the first term of the formula.
m₀ / √(1 – (V²/c²))
This was the object’s relativistic mass, m.

So, the formula could be written as:
E_k = mc² – m₀c²

Kinetic energy equals the object’s relativistic mass times c² minus its rest mass times the speed of light squared.

This result was mathematically perfect.

It accurately described the object’s kinetic energy.

When the velocity V is much less than c, using a Taylor expansion, this formula can be approximated as E_k ≈ ½m₀V².

It perfectly returned to the classical kinetic energy formula.

But that subtracted term, m₀c²…

‘What is it?’

In the integration, it was just a constant of integration.

To make the kinetic energy zero when the object is at rest, this constant m₀c² had to be subtracted.

From a mathematical perspective, there was nothing wrong with it.

But from a physical perspective, it was too strange.

“A constant… m₀c²,” Klein murmured to himself.

“Why is it specifically the rest mass times c²?”

He felt that there must be a deeper secret hidden behind this result.

But he couldn’t explain its physical meaning.

📚 Also on NHV Novels

My Magical Girl Wife Died… So Why Is She My Best Friend Now?

FantasyFemale ProtagonistMagic

Read Now →
Not interested