Physics:Time Dilation, Length Contraction in the Theory of Entropicity (ToE)

Derivation of Einstein's Relativity Time Dilation and Length Contraction in the Theory of Entropicity (ToE)

Here, we derive Einstein's Time Dilation and Length Contraction in his theory of relativity from the principles of the Theory of Entropicity(ToE), first formulated and developed by John Onimisi Obidi.

Entropic Derivation of Length Contraction in ToE

1. Core Assumptions (ToE Postulates)

Entropy field: Every body carries an entropy density and entropy flux .

Conservation: For a closed rod in free motion, total entropy is constant:

[math]\displaystyle{ S_{\text{total}} = s \, L = s_0 L_0 }[/math]

Reciprocity: No inertial frame is special; entropic laws must look the same in all frames.

Tight capacity: In steady uniform motion, flux aligns with velocity: .

2. The Entropic Cone

The No-Rush bound defines an “entropy cone” of admissible states:

[math]\displaystyle{ |j| \leq c_e s. }[/math]

[math]\displaystyle{ (c_e s)^2 - j^2 = \text{constant}. }[/math]

3. Fixing the Invariant

In the rod’s rest frame:

j=0 s=s_o

So the invariant is:

[math]\displaystyle{ (c_e s)^2 - j^2 = (c_e s_0)^2. }[/math]

4. Motion and Entropy Density

With steady motion, flux equals convection: . Plugging into the invariant:

[math]\displaystyle{ (c_e s)^2 - (v s)^2 = (c_e s_0)^2. }[/math]

Solve for :

[math]\displaystyle{ s(v) = \frac{s_0}{\sqrt{1 - v^2/c_e^2}}. }[/math]

So motion forces entropy density to increase.

5. Length Contraction as an Effect

Since total entropy is constant:

[math]\displaystyle{ s(v) L(v) = s_0 L_0. }[/math]

Then:

[math]\displaystyle{ L(v) = L_0 \sqrt{1 - v^2/c_e^2}. }[/math]

This is exactly the Lorentz contraction law, but with the causal arrow reversed:

In relativity, contraction is kinematic.

In ToE, contraction is a consequence of entropy density growth.

6. Key Insight

Entropy increase drives contraction: When velocity rises, entropy density climbs by the Lorentz factor, forcing length to contract to keep total entropy invariant.

This puts entropy, not geometry, at the root of relativity.

If we set c_e=c, then ToE reproduces Einstein’s result. If c_e>c, measurable deviations would appear—an experimental signature.

✅ Summary: In ToE, length contraction isn’t the cause of entropy change. It’s the other way around: motion increases entropy density, and the rod contracts as a consequence.