(continued)

Below is a third thought experiment to demonstrate the insufficiency of the traditional Lorentz transformation.

Assume a part of stable DC circuit with the shape of “L” rests in inertial frame S: section 1 horizontal; section 2 vertical; wire cutting face is a square for convenience. Another inertial frame S’ moves rightward at velocity u.

The purpose of this thought experiment is to verify whether the “length contraction effect” is in all direction as per the analysis ahead shows or just in the moving direction as traditional Lorentz transformation shows.

First, analysis resorts to traditional Lorentz transformation.

In frame S:

For section 1, I1 = n1qv1; for section 2, I2 = n2qv2

(Here take the moving direction of electron as the direction of current for convenience, n is the density of free electron in the wire, q is the charge carried by electron, v is the moving velocity of free electron.)

I1 = I2, n1 = n2 = n, v1 = v2 = v

In frame S’:

For section 1, △x’ = △x / γ , △y’ = △y , △z’ = △z

V’ = △x’△y’△z’ = △x△y△z / γ = V / γ (V’ is volume in frame S’, V is volume in frame S)

So, n1’ = γn1 = γn

Lorentz transformation of velocity:

v1’ = (v1 - u) / (1 - u v1/c² )

I1’ = n1’qv1’ = γnq (v1 - u) / (1 - u v1/c²)

For section 2, also V’ = V / γ

So, n2’ = γn2 = γn

Lorentz transformation of velocity:

v2’ = v2 / γ(1 - u v1/c²)

I2’ = n2’qv2’ = γnq v2 / γ(1 - u v1/c²) = nq v2 / (1 - u v1/c²)

I1’ ≠ I2’

Physics rule is not equally applicable in inertial frame S’.

Next, resort to the space contraction in all directions as per the analysis ahead.

In frame S’:

For section 1, n1’ = γ³n1 = γ³n

v1’ = dx’ / dt’ = (v1 - u) / (1 - u v1/c² )

I1’ = n1’qv1’ = γ³nq (v1 - u) / (1 - u v1/c²)

For section 2, n2’ = γ³n2 = γ³n

v2’ = dz’ / dt’ = dx’ / dt’ = (v1 - u) / (1 - u v1/c² )

I2’ = n2’qv2’ = γ³nq (v1 - u) / (1 - u v1/c²)

I1’ = I2’

Physics rule is equally applicable in inertial frame S’.

Of course, the “L” shape DC circuit scenario can be verified. Just fix two electric meters in the sections and watch if their values are the same. It’s not a difficult experiment to do.

Liqiang Chen

陈力强

August 31, 2021

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