After the introduction, we can establish the outline of “Scale Physics”. This is no longer standard physics, but a new theoretical framework in the direction of Fractal Relativity.
Fundamental Axioms of Scale Physics
Axiom 1: There is No Absolute Scale
Einstein: There is no absolute motion.
Scale Physics: There is no absolute scale.
Every structure can be seen as another structure’s: sub-scale, super-scale.
Example: Atom ↔ Cell ↔ Human ↔ Earth ↔ Galaxy are links of the same fractal chain.
Axiom 2: Physical Quantities Depend on Scale
Classical physics: 𝑸 = constant
Scale physics: 𝑸 = 𝑸(𝝀)
Here 𝑸 can be:
- mass
- time
- energy
- gravitational constant
- speed of light.
Axiom 3: The Derivative Reduces the Scale
Definition: 𝑫 = 𝒅 / 𝒅𝒕 is simultaneously the operation 𝑫: 𝝀 → 𝝀 − 𝚫𝝀.
That is: the chain 𝒙 → 𝒗 → 𝒂 is a scale descent from Macro → Meso → Micro.
Axiom 4: The Integral Increases the Scale
𝑰 = ∫ 𝒅 𝒕 is simultaneously the operation 𝑰: 𝝀 → 𝝀 + 𝚫𝝀.
Starting from acceleration: when we go 𝒂 → 𝒗 → 𝒙, the scale is increasing.
Axiom 5: Principle of Scale Conservation
The total structural information of a system is conserved.
It can be written as: 𝓢 = 𝒄𝒐𝒏𝒔𝒕
Here 𝓢 can be scale entropy or scale action.
New Coordinates
Einstein: uses (𝒙, 𝒚, 𝒛, 𝒕).
Scale Physics: uses (𝒙, 𝒚, 𝒛, 𝒕, 𝝀).
Here: 𝝀 is the fifth coordinate.
Scale Velocity
Let’s define a new quantity: 𝒖 = 𝝏𝒙 / 𝝏𝝀
This becomes the “scale velocity”.
Physical meaning: How much does a structure change when its scale is changed?
Scale Acceleration
𝑨𝝀 = 𝝏𝟐𝒙 / 𝝏𝝀𝟐
This is the scale acceleration.
Scale Dynamics
Corresponding to Newton’s law: 𝑭 = 𝒎𝒂 it is defined as: 𝑭𝝀 = 𝑴𝝀𝑨𝝀.
Here:
- 𝑭𝝀 : scale force
- 𝑴𝝀 : scale inertia
Scaled Law of Gravity
Consistent with my hypothesis: 𝑮 = 𝑮(𝝀) is written.
A simple first model: 𝑮(𝝀) = 𝑮𝟎𝝀𝜶
Here:
- 𝜶 = 𝟎 ⇒ classical physics
- 𝜶 ≠ 𝟎 ⇒ scale physics
Scaled Time
Time also depends on scale: 𝒕 = 𝒕(𝝀)
Thus: atomic time biological time geological time cosmological time become different scale manifestations of the same thing.
Fundamental Equation
The most fundamental equation of Scale Physics:
𝒙 = 𝒙(𝒕, 𝝀)
This equation states that: Position depends not only on time, but also on scale.
From here, two derivatives emerge: 𝒗 = 𝝏𝒙 /𝝏𝒕 (temporal motion) and 𝒖 = 𝝏𝒙 / 𝝏𝝀 (scale motion)
The Principle to Succeed Einstein
The principle of General Relativity:
Laws are invariant under coordinate transformations.
The principle of Scale Physics:
Laws are also invariant under scale transformations.
We can call this the “Principle of Scale Covariance”.
In this case, my framework gains a hierarchical structure in the form of:
Newton ⊂ Einstein ⊂ Scale Physics.
Newton addresses motion, Einstein addresses space-time, and Scale Physics addresses both space-time and the scale dimension together. This creates a natural starting axiom system for Fractal Relativity.
Let’s write an equation set for Scale Physics
Assuming my hypothesis is correct, we can build a minimal yet consistent set of equations for Fractal Relativity and Scale Physics.
The goal is this:
Physics = Space + Time + Scale must hold true.
1. Fundamental State Function
Einstein: 𝒙 = 𝒙(𝒕)
Scale Physics: 𝒙 = 𝒙(𝒕, 𝝀)
Here:
- 𝒕 = time
- 𝝀 = scale
- 𝒙 = state or position
This is the introductory equation to the theory.
2. Temporal Velocity
Classical velocity: 𝒗 = 𝝏𝒙 / 𝝏𝒕
3. Scale Velocity
New concept: 𝒖 = 𝝏𝒙 / 𝝏𝝀
Physical meaning: A system’s response to scale change.
4. Total Change Equation
Now, change comes from two sources.
𝒅𝒙 = (𝝏𝒙 / 𝝏𝒕) 𝒅𝒕 + (𝝏𝒙 / 𝝏𝝀) 𝒅𝝀 or 𝒅𝒙 = 𝒗 𝒅𝒕 + 𝒖 𝒅𝝀
This, in my opinion, is the first fundamental law of Scale Physics.
5. Scale Acceleration
Normal acceleration: 𝒂 = 𝝏𝟐𝒙 / 𝝏𝒕𝟐
Scale acceleration: 𝑨 = 𝝏𝟐𝒙 / 𝝏𝝀𝟐
6. Mixed Acceleration
If time and scale change together: 𝑩 = 𝝏𝟐𝒙 / ( 𝝏𝒕 𝝏𝝀 )
This is very important. Because the connection between the system’s temporal evolution and scale evolution is formed here.
7. Scaled Newton’s Law
Newton: 𝑭 = 𝒎𝒂
Scale Physics: 𝑭𝝀 = 𝑴𝝀𝑨
Here:
- 𝑭𝝀 = scale force
- 𝑴𝝀 = scale inertia
8. Scale Energy
Kinetic energy: 𝑬𝒌 = (𝟏/𝟐) 𝒎𝒗𝟐
Scale energy: 𝑬𝝀 = (𝟏/𝟐) 𝒎𝝀𝒖𝟐
9. Total Energy
𝑬 = (𝟏/𝟐) 𝒎𝒗𝟐 + (𝟏/𝟐) 𝒎𝝀𝒖𝟐 + 𝑽(𝒙, 𝝀)
The system is moving in both time and scale.
10. Scale Field
Consistent with my hypothesis: 𝑮 = 𝑮(𝝀) and in general 𝑸 = 𝑸(𝝀)
Here 𝑸 can be any physical parameter like:
- G
- c
- h
- m
11. Scale Continuity Equation
Conservation of a quantity: 𝝏𝝆/𝝏𝒕 + 𝛁 ⋅ (𝝆𝒗) + 𝝏/𝝏𝝀 (𝝆𝒖) = 𝟎
Here:
- the first term expresses time change
- the second term spatial flow
- the third term scale flow.
12. Scaled Space-Time Interval
Einstein: 𝒅𝒔𝟐 = 𝒄𝟐𝒅𝒕𝟐 − 𝒅𝒙𝟐 − 𝒅𝒚𝟐 − 𝒅𝒛𝟐
Scale Physics: 𝒅𝚺𝟐 = 𝒄𝟐𝒅𝒕𝟐 − 𝒅𝒙𝟐 − 𝒅𝒚𝟐 − 𝒅𝒛𝟐 − 𝓵𝒔𝟐𝒅𝝀𝟐
Here 𝓵𝒔 is the scale length.
This equation makes the scale the fifth coordinate.
13. Principle of Scale Covariance
The main principle of the theory: 𝓛(𝒙, 𝒕, 𝝀) = 𝓛(𝒙, 𝒕, 𝝀’)
That is, the form of physical laws must be preserved under scale transformation. This is the extended version of Einstein’s coordinate covariance to scale space.
The Main Equation
In my opinion, the one-line summary of Scale Physics could be this:
𝝏𝟐𝒙/𝝏𝒕𝟐 + 𝜷 𝝏𝟐𝒙/𝝏𝝀𝟐 = 𝓕(𝒙, 𝒕, 𝝀)
Here:
- the first term is the temporal dynamics,
- the second term is the scale dynamics,
- the right side is the physical interactions.
This equation is the simplest “Scale Physics field equation” candidate that could unite the time evolution in the Schrödinger equation, Newton’s motion, and my scale evolution under the same roof.
From the perspective of Fractal Relativity, the step to be taken after this equation is to make the exact definition of 𝝀 and derive the field equations for “scale momentum” and “G(\lambda) under scale effect”. In this case, the theory begins to truly transform into an independent mathematical physics framework.
Assuming my hypothesis is correct, let’s take Scale Physics one step further and define the missing concepts.
1. Scale Coordinate
First, let’s make scale a coordinate. 𝝀 = 𝐥𝐧 (𝑳/𝑳𝟎)
Here:
- 𝑳 = observation scale
- 𝑳𝟎 = reference scale
Example:
- Atomic scale → negative λ
- Human scale → λ≈0
- Cosmic scale → positive λ
The advantage of this definition: 𝝀𝟏 + 𝝀𝟐 corresponds to the product of scales.
2. Scale Momentum
Classical momentum: 𝒑 = 𝒎𝒗
Scale momentum: 𝒑𝝀 = 𝒎𝝀𝒖
where 𝒖 = 𝝏𝒙 / 𝝏𝝀 is the scale velocity.
Meaning: How much does the system resist scale change?
3. Scale Mass
Normal mass: 𝒎
Scale mass: 𝒎𝝀 = 𝝏𝒎 / 𝝏𝝀
It shows the effective mass change of a structure at different scales.
4. Scale Force
Newton: 𝑭 = 𝒅𝒑 / 𝒅𝒕
Scale force: 𝑭𝝀 = 𝝏𝒑𝝀 / 𝝏𝝀
The effect that tries to move a system to another scale.
5. Scale Energy
𝑬𝝀 = (𝟏/𝟐) 𝒎𝝀𝒖𝟐
Total energy: 𝑬 = 𝒎𝒄𝟐 + (𝟏/𝟐) 𝒎𝒗𝟐 + (𝟏/𝟐) 𝒎𝝀𝒖𝟐
Thus, energy consists of three parts:
- internal energy
- motion energy
- scale energy
6. Scale Uncertainty
The analog of quantum mechanics: 𝚫𝒙 𝚫𝒑𝝀 ≥ 𝚺
Here 𝚺 is the scale constant.
This expression states: The exact position and scale behavior of a structure cannot be precisely known at the same time.
7. Scale Geometry
Space-time: (𝒙, 𝒚, 𝒛, 𝒕)
Scale physics: (𝒙, 𝒚, 𝒛, 𝒕, 𝝀)
Metric: 𝒅𝚺𝟐 = 𝒄𝟐𝒅𝒕𝟐 − 𝒅𝒙𝟐 − 𝒅𝒚𝟐 − 𝒅𝒛𝟐 − 𝓵𝒔𝟐𝒅𝝀𝟐
Here 𝓵𝒔 is the scale length.
8. Scale Field
Let’s define a new field: 𝚽(𝒙, 𝒕, 𝝀)
This field unites:
- geometry
- matter
- scale effects.
9. Scale Field Equation
Fundamental equation: 𝝏𝟐𝚽/𝝏𝒕𝟐 − 𝒄𝟐𝛁𝟐𝚽 + 𝜷 𝝏𝟐𝚽/𝝏𝝀𝟐 = 𝑱
Here:
- the first term is time evolution
- the second term spatial propagation
- the third term scale propagation
- 𝑱 is the source term
This equation can be thought of as the “Maxwell equation” or “field equation” of Scale Physics.
10. Scaled Einstein Equation
Let’s get to the main target.
Einstein: 𝑮𝝁𝝂 = 𝟖𝝅𝑻𝝁𝝂
In scale physics: 𝑮𝝁𝝂 + 𝑺𝝁𝝂 = 𝟖𝝅𝑻𝝁𝝂
Here 𝑺𝝁𝝂 is the scale tensor.
Definition: 𝑺𝝁𝝂 = 𝝏𝒈𝝁𝝂 / 𝝏𝝀
That is, it is the variation of space-time with respect to scale.
11. Scaled Gravitational Constant
Consistent with my hypothesis: 𝑮(𝝀) = 𝑮𝟎𝒆𝜶𝝀 is particularly suitable.
Because: when 𝝀 = 𝟎, 𝑮 = 𝑮𝟎 is obtained.
12. Fundamental Law of Scale Relativity
If we summarize the whole theory in a single line:
𝒅/𝒅𝒕 ↔ scaling down
∫ 𝒅 𝒕 ↔ scaling up
and all physical quantities are defined as 𝑸 = 𝑸(𝒕, 𝝀).
In this framework, Newton, Einstein, and quantum mechanics now become special cases:
when 𝝏 / 𝝏𝝀 = 𝟎 is taken, Scale Physics ⟶ reduces to Standard Physics.
I think the next logical step would be to write a Scale Lagrangian for this theory: 𝓛 = 𝓛(𝚽, 𝝏𝒕, 𝚽, 𝛁𝚽𝝏𝝀𝚽) and derive all equations from a single action principle. This would be the step that transforms the theory into a true mathematical physics structure.
Now, let’s take Scale Physics out of “axioms” and truly build it upon an action principle. In physics, a theory begins to mature only if it is built upon a Lagrangian.
1. Fundamental Variable
Fundamental field: 𝚽 = 𝚽(𝒙𝝁, 𝝀)
Here: 𝒙𝝁 = (𝒄, 𝒕, 𝒙, 𝒚𝒛) and 𝝀 is the scale coordinate.
Therefore, the universe is now defined in the space of: (𝒄, 𝒕, 𝒙, 𝒚, 𝒛𝝀).
2. Scale Lagrangian
Simplest form: 𝓛 = (𝟏/𝟐) 𝝏𝝁𝚽𝝏𝝁𝚽 − (𝜷/𝟐) (𝝏𝚽/𝝏𝝀)𝟐 − 𝑽(𝚽, 𝝀)
Here: The first term 𝝏𝝁𝚽𝝏𝝁𝚽 is the space-time motion, the second term (𝝏𝚽/𝝏𝝀)𝟐 is the scale motion.
3. Action
𝑺 = 𝓛 𝒅𝟒𝒙 𝒅𝝀
The system is now integrated not only over space-time, but also over scale space.
4. Scaled Euler-Lagrange Equation
If standard procedure is applied: 𝝏𝝁𝝏𝝁𝚽 − 𝜷(𝝏𝟐𝚽/𝝏𝝀𝟐) + 𝝏𝑽/𝝏𝚽 = 𝟎 is obtained.
This is the first field equation of Scale Physics.
5. Scale Waves
If: 𝑽 = 𝟎 then 𝝏𝝁𝝏𝝁𝚽 − 𝜷 (𝝏𝟐𝚽/𝝏𝝀𝟐) = 𝟎 is obtained.
This equation states: A field can propagate not only in space-time, but also in the scale dimension.
6. Scale Momentum
From the Lagrangian: 𝑷𝝀 = 𝝏𝓛 / 𝝏(𝝏𝝀𝚽) is obtained.
Result: 𝑷𝝀 = −𝜷 𝝏𝚽/𝝏𝝀
This is the scale momentum.
7. Scale Conservation Law
According to Noether’s approach, if the Lagrangian is not explicitly dependent on λ 𝒅𝑷𝝀 / 𝒅𝝀 = 𝟎 is obtained.
This becomes “Conservation of Scale Momentum”.
8. Scale Energy
Hamiltonian density becomes: 𝓗 = (𝟏/𝟐) (𝝏𝒕𝚽)𝟐 + (𝟏/𝟐) (𝛁𝚽)𝟐 + (𝜷/𝟐) (𝝏𝝀𝚽)𝟐 + 𝑽.
Here the new energy term: 𝑬𝝀 = (𝜷/𝟐) (𝝏𝝀𝚽)𝟐 is the scale energy.
9. Scale Curvature
Now let’s connect to Einstein.
Let the metric be: 𝒈𝑨𝑩.
Here 𝑨, 𝑩 = 𝟎, 𝟏, 𝟐, 𝟑, 𝟒 and let the 1st index represent the scale.
That is: 𝒙𝟒 = 𝝀
The new interval becomes: 𝒅𝚺𝟐 = 𝒈𝑨𝑩𝒅𝒙𝑨𝒅𝒙𝑩.
10. Scaled Einstein Equation
Now the field equation becomes: 𝑹𝑨𝑩 − (𝟏/𝟐) 𝒈𝑨𝑩𝑹 = 𝟖𝝅𝑻𝑨𝑩.
The tensors are now 5-dimensional. The scale dimension has also been included in the dynamics.
11. Scaled G
Consistent with my hypothesis, let: 𝑮(𝝀) = 𝑮𝟎𝒆𝜶𝝀.
Thus the Einstein equation becomes: 𝑹𝑨𝑩 − (𝟏/𝟐) 𝒈𝑨𝑩𝑹 = 𝟖𝝅𝑮(𝝀)𝑻𝑨𝑩.
12. Main Equation of Scale Physics
If we reduce the entire theory to a single expression:
□𝚽 − 𝜷 (𝝏𝟐𝚽/𝝏𝝀𝟐) + 𝝏𝑽/𝝏𝚽 = 𝟎
Here □ = 𝝏𝝁𝝏𝝁 is the standard space-time operator.
13. Fundamental Principle of Scale Relativity
Final axiom: Physical laws are covariant under both coordinate and scale transformations.
That is, Einstein’s relativity of (𝒙, 𝒕) transforms into the relativity of (𝒙, 𝒕, 𝝀) in my framework.
This structure, ceasing to be just a “philosophical idea”, now becomes an entry-level mathematical core of Fractal Scale Physics containing:
- Scale coordinate 𝝀
- Scale momentum 𝑷𝝀
- Scale energy 𝑬𝝀
- Scale conservation law
- Scale field equation
- Scaled Einstein equation.