Chapter 188: The Architecture of Spacetime

Klein stood before the blackboard, and the entire world faded from his consciousness.

In his eyes, there remained only those two cold postulates and the brand-new spacetime he was about to construct with his own hands.

He was thinking.

The old coordinate transformation described an absolute and separate spacetime.

Time was time, and space was space.

Time was a uniformly flowing river, the same for everyone.

Space was an absolutely stationary stage upon which all motion occurred.

Within this framework, the addition of velocities was a matter of course.

‘If I am on a carriage moving at speed V and throw an apple forward with a speed U, then the apple’s speed relative to the ground is U + V.’

Simple, intuitive, in line with “common sense.”

But this common sense shattered completely in the face of the speed of light.

Because the speed of light is constant.

If I turn on a flashlight on a moving carriage, the speed of the light is c.

A person on the ground seeing this beam of light will still measure its speed as c.

Not c + V.

This meant that the old transformation law was wrong.

Klein took a deep breath and wrote down two frames of reference on the blackboard, S and S’.

Frame S’ moves at a constant velocity V along the x-axis relative to frame S.

He needed to find a set of equations that connected the spacetime coordinates (x, y, z, t) in frame S with the spacetime coordinates (x’, y’, z’, t’) in frame S’.

This set of equations must satisfy two conditions.

First, when the velocity V is much less than the speed of light c, it must degenerate back to the classical mathematical transformation, because in the low-velocity world, classical mechanics is still perfect.

Second, it must ensure that the measured speed of light is c in both frames of reference.

“The transformation must be linear,” Klein muttered to himself.

This was the most basic requirement.

Because an object in uniform linear motion, when viewed from another frame of reference in uniform linear motion, must also appear to be in uniform linear motion.

The uniformity of spacetime determined this.

Thus, he could write down the general form of the transformation equations.

x’ = A(x – vt)
t’ = Bx + Dt

Klein assumed the motion was only in the x-direction, so the y and z coordinates remained unchanged.

He introduced three unknown coefficients: A, B, and D.

Now, his task was to use the two postulates to solve for these three coefficients.

Lia stood not far away, arms crossed, watching quietly.

Klein began his performance.

He first utilized the principle of relativity from the first postulate.

The inverse transformation from S’ to S should have the exact same form, just with the velocity V changed to -V.
x = A(x’ + vt’)
t = -Bx’ + Dt’

This was a powerful constraint.

He substituted the expressions for x’ and t’ into the inverse equations and, after a series of complex calculations, derived the first relationship between the coefficients.
A = D

This result invigorated Klein.

One less unknown.

Now, it was time for the second postulate to take the stage.

The constancy of the speed of light.

Klein imagined that at the moment t = t’ = 0, the origins of the two reference frames coincided.

At this instant, a beam of light is emitted from the origin.

For an observer in frame S, the equation for the sphere of light is:
x² + y² + z² = c²t²

For an observer in frame S’, the equation for the sphere of light is:
x’² + y’² + z’² = c²t’²

This was the mathematical expression of the constancy of the speed of light.

Now, he had to substitute the set of linear transformation equations he had defined into these two sphere of light equations.

He had to make it so that the transformed equation still held true.

Klein substituted x’ = A(x – vt) and t’ = Bx + At into the second sphere of light equation.

The calculation began.

This was purely a test of computational accuracy and patience.

Chalk flew across the blackboard, lines of algebraic expressions were written down and then erased.

Tiny beads of sweat formed on Klein’s forehead.

His mental energy was highly focused, his brain operating like the most precise computer processor, handling a massive amount of information.

Lia watched his focused profile, her gaze soft.

She knew how painful and yet how captivating this process was.

This was using human logic to pry open the underlying code of the entire universe.

An unknown amount of time passed.

The blackboard in front of Klein was already filled with dense derivations.

He expanded the equation and combined the terms according to x², y², z², t², xt, and so on.

Then, he compared this complex expression with the first sphere of light equation.

The coefficients of the corresponding terms must be equal.

This gave him a set of equations for A and B.

He solved this system of equations.

From the second equation, he could get B = -Av/c².

He substituted it into the first equation.

A = 1 / √(1 – v²/c²)

Klein stopped writing.

He had solved it.

He had solved for all the coefficients.

He substituted these coefficients back into the original transformation equations.

A set of unprecedented equations, describing the very essence of spacetime, appeared on the blackboard.
x’ = (x – vt) / √(1 – v²/c²)
y’ = y
z’ = z
t’ = (t – vx/c²) / √(1 – v²/c²)

Klein looked at this set of equations, speechless for a long time.

It was like a perfect work of art.

When the velocity V is much less than c, the denominator √(1 – v²/c²) approaches 1, and the vx/c² term in the t’ equation approaches 0.

This set of equations perfectly degenerated into the classical mathematical transformation.

And when the velocity V approached the speed of light c, a miracle happened.

The denominator approached 0, and spacetime was stretched to its limit.

It perfectly satisfied the requirement of the constancy of the speed of light.

“What should I name it?” Klein’s voice held a slight tremor.

“Let’s just call it the ‘Spacetime Transformation Equations’,” Lia said softly. “Its function is to connect different spacetimes.”

Klein nodded.

He erased all the derivation steps from the blackboard, leaving only this clean and elegant set of equations.

Now, he would use this new ruler to measure that photon clock again.

He returned to the previous thought experiment.

In the box’s frame of reference S’, the time for the light to make one round trip is Δt’ = 2L’/c.

Here, L’ is the proper height of the box.

In the ground frame of reference S, the starting coordinates of this process are (x₁, t₁) and the ending coordinates are (x₂, t₂).

The spatial coordinates of the start and end points are the same in frame S’, because the box did not move.

So, Δx’ = 0.

Now, using the inverse of the new transformation equations:
Δt = (Δt’ + vΔx’/c²) / √(1 – v²/c²)

Because Δx’ = 0, the equation becomes:
Δt = Δt’ / √(1 – v²/c²)

The box moved a distance of Δx = vΔt in frame S.

Substituting that in.

Rearranging, he got:
Δt = Δt’ / √(1 – v²/c²)

Klein looked at the result.

He had once again arrived at the conclusion that “a moving clock runs slower.”

But this time, there was no longer any contradiction or struggle in his heart.

This conclusion was no longer a monster that appeared out of thin air.

It was a natural corollary of that beautiful set of spacetime transformation equations.

It was perfectly placed within the new theoretical edifice.

Klein’s breathing quickened.

He knew this was just the beginning.

This set of equations surely hid more subversive secrets.

He immediately began to study the change in length.

Imagine a ruler, at rest in frame S’, with a proper length of L’ = x’₂ – x’₁.

An observer on the ground, S, wants to measure the length of this moving ruler.

He must record the coordinates of the two ends of the ruler “at the same time.”

That is, t₁ = t₂, so Δt = 0.

He used the transformation equations again.
x’₂ – x’₁ = ( (x₂ – vt₂) – (x₁ – vt₁) ) / √(1 – v²/c²)

Because Δt = 0, so t₁ = t₂.
L’ = (x₂ – x₁) / √(1 – v²/c²)

The length measured by the ground observer is L = x₂ – x₁.

Therefore, L’ = L / √(1 – v²/c²).

Rearranging, he got:
L = L’ * √(1 – v²/c²)

Klein’s hand stopped in mid-air.

A moving ruler shortens in the direction of its motion.

Time dilation, length contraction.

This was the face of the new spacetime.

A dynamic and relative spacetime, intimately connected to the observer’s state of motion.

“The old world…” Klein murmured to himself, “…is collapsing before my eyes, inch by inch.”

“No,” Lia walked to his side, standing shoulder to shoulder with him, together gazing at the formulas on the blackboard. “The curtain of the old world has been pulled back, revealing an even more magnificent stage underneath.”

Klein turned his head and looked at Lia’s face, so close to his.

Her eyes sparkled with the light of wisdom and understanding, as if she had known all of this from the beginning.

An impulse surged within him.

He wanted to ask her who she was, where the knowledge in her mind came from.

But he didn’t ask.

He simply reached out and gently pulled her into his embrace.

“Thank you,” he said in a low voice.

“For what?”

“Thank you for pulling back this curtain for me.”

Lia snuggled into his embrace, finding a comfortable position.

“Then how will you repay me?”

“I will build this new house and give it to you as a gift,” Klein said.

Klein organized these two subversive conclusions, along with the perfect set of spacetime transformation equations, into a paper.

The title was: On the Spacetime Basis of Moving Bodies.

He signed his and Lia’s names.

This was their joint achievement.

He rolled up the finished parchment and placed it in the teleportation array.

A flash of light, and the parchment disappeared.

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