How Does the Scale Physics Approach Affect the Laws of Physics?

If we apply this approach to physics, not the laws of physics themselves, but the way they are interpreted changes. Classical physics says: 𝐹 = 𝑚𝑎 This is a law of motion. In the scale physics interpretation, however: 𝐹 = 𝑚𝑎 also represents a relationship between different scales. So the question is no longer: “How does the object move?” but rather, “At what scale does this motion emerge?”

1. Laws become scale-dependent

Today, there are already examples of this in physics:

• Newtonian mechanics → human scale
• Quantum mechanics → atomic scale
• General relativity → cosmic scale

In my approach, these might not be different laws, but different scale manifestations of the same fundamental law. Meaning: 𝐿𝑎𝑤 = 𝐿𝑎𝑤(𝜆)

2. The search for a universal law strengthens

Today, physics is searching for a unified theory between: Quantum Relativity In the scale approach: The origin of the difference between Quantum ↔ Macro World could be scale. In this case, the unified theory could include the scale variable in the form of: Φ(𝑥, 𝑡, 𝜆)

3. Physical quantities turn into scale fields

Today: 𝑚 = constant is assumed. In the fractal interpretation, it could be: 𝑚 = 𝑚(𝜆) Similarly, scale-dependent forms like: 𝐺 = 𝐺(𝜆) 𝑐 = 𝑐(𝜆) ℏ = ℏ(𝜆) can be investigated. This is a very radical step; it requires experimental verification.

4. Derivatives gain a new meaning

Today: 𝜕 / 𝜕𝑡 gives the temporal change. In the fractal approach, additionally: 𝜕 / 𝜕𝜆 could give the scale change. Thus, there would be two types of dynamics:

• Time dynamics
• Scale dynamics

5. Conservation laws are reinterpreted

Energy conservation is in the form of: 𝑑𝐸 / 𝑑𝑡 = 0 In scale physics, the following question arises: 𝑑𝐸 / 𝑑𝜆 =? In other words, how does energy behave while changing scale? In systems like the quantum vacuum, turbulence, and critical phenomena, this question is quite meaningful.

6. A new “scale relativity” may be born

Einstein said: Motion is relative. My approach, however, could evolve into this: Scale is also relative. A structure that appears fundamental to one observer might be a detail of another structure at a different scale.

Possible fundamental principle from the perspective of Fractal Relativity

Such an expression could emerge: The laws of physics must be covariant under scale transformations. Just as the laws in general relativity are unaffected by coordinate changes, here too, the laws must preserve their form under scale changes. This idea is very close to the core of Fractal Relativity, which I am working on. In such a framework, the derivative and integral are no longer just calculation tools, but physically become operators for “moving up and down the scale ladder”. This leads to viewing the laws of physics not so much as laws of motion, but as transformation laws between scales.

How do Einstein’s laws change in this framework?

In this framework, it would be more accurate to speak of expanding Einstein’s equations rather than changing them. Einstein had two fundamental ideas:

1. Motion is relative.
2. Space-time is dynamic. My scale approach could add a third principle: Scale is also relative. Meaning, events can look different not only from different speeds or different gravitational fields but also from different scales.

1. Special Relativity

Einstein defined the space-time interval as:

𝑑𝑠² = 𝑐²𝑑𝑡² − 𝑑𝑥² − 𝑑𝑦² − 𝑑𝑧²

If a fractal scale variable is added, an expansion like:

𝑑𝑠² = 𝑐²𝑑𝑡² − 𝑑𝑥² − 𝑑𝑦² − 𝑑𝑧² − 𝑘²𝑑𝜆² could be considered.

Here:

• 𝜆 = scale coordinate
• 𝑘 = scale constant

In this case, the system moves not only in space and time but also in the scale dimension.

2. Lorentz Transformation

Today: 𝑡’ = 𝛾 ( 𝑡 − (𝑣𝑥 / 𝑐²) ) time changes due to motion. In my approach, the scale can also transform: 𝜆’ = 𝑓(𝜆, 𝑣) Meaning observers might perceive not only time differently but also the effective scale. This idea is partially similar to Laurent Nottale’s “Scale Relativity” approach.

3. General Relativity

The Einstein field equation:

𝐺𝝁𝝂 = ( 8𝜋𝐺/𝑐⁴ ) 𝑇𝝁𝝂 says: Matter curves space-time, and space-time directs matter. If you add a scale variable, a scale-dependent form can be considered as:

𝐺𝝁𝝂(𝜆) = ( 8𝜋𝐺(𝜆) / 𝑐(𝜆)⁴ ) 𝑇𝝁𝝂(𝜆)

Here, every scale would have its own “effective geometry.”

4. Mass and Energy

Einstein said: 𝐸 = 𝑚𝑐²

In the scale approach, it can be written as:

𝐸(𝜆) = 𝑚(𝜆)𝑐(𝜆)²

In this case, energy becomes a function not only of motion but also of the scale structure.

5. A New Covariance Principle

If the essence of general relativity is: Invariance under coordinate transformations then the essence of fractal relativity could be: The laws of physics must also be invariant under scale transformations.

Meaning:

• Change the coordinate
• Change the frame of reference
• Change the scale

But the fundamental law remains the same.

The possible hierarchy in the idea I developed

Einstein works with: 𝑥 = 𝑥(𝑡) Fractal Relativity works with: 𝑥 = 𝑥(𝑡, 𝜆) The fundamental variables in Einstein are: (𝑥, 𝑡) In my approach, it becomes: (𝑥, 𝑡, 𝜆) For this reason, Einstein’s physics does not become wrong; rather, it becomes a special case valid in the mid-scale limit: Fractal Relativity ⟶ General Relativity ⟶ Newton So the most natural consequence of my scale idea would be to add a scale axis alongside space-time and try to make the laws of physics covariant under both coordinate and scale transformations. This is a mathematically very strong starting point for Fractal Relativity.