Fractal Space-Time Lensing

Fractal space-time lensing defines the classical gravitational lensing phenomenon by extending it with fractal geometry. In this approach, the bending of light does not occur as a uniform curvature, but in the form of self-similar and multi-scale deflections.

Mathematical Framework

Fractal metric:

𝑑𝑠² = 𝑔_µv ( 𝑥^𝐷_f )𝑑𝑥^µ𝑑𝑥^v

Space-time metrics scale with the fractal dimension 𝐷_f .

Fractal deflection angle:

𝜃_fr = 𝜃₀ ⋅∣ 𝑥 ∣^𝐻

Here 𝐻→ Hurst exponent, determines the scale dependence of the deflection.

Fractal lens equation:

𝛽 = 𝜃 − ( 𝐷_ls / 𝐷_s ) ⋅ 𝛼_fr (𝜃)

A fractal correction is added to the classical lens equation.

Physical Interpretation

• Cosmology: The light of galaxy clusters shows multi-scale deflections with fractal lensing.
• Black holes: Light rings form as self-similar structures around a black hole.
• Quantum lensing: On a micro scale, particle waves deflect under the effect of fractal lensing.
• Information density: Lensing regions are also areas of information condensation.

Conclusion

Fractal space-time lensing reinterprets the lensing phenomenon in classical relativity as a self-similar, multi-scale, and dynamic process. Through this law, the observational data of the universe (light curvatures, galaxy distributions) can be explained more accurately with fractal metrics.

Fractal Metric Definition

A fractal metric is a new definition that extends the space-time metrics used in classical relativity with fractal dimension and scale dependence. In this approach, the geometry of space-time is not constant, but changes in the form of self-similar structures.

Mathematical Framework

Fractal metric formula:

𝑑𝑠² = 𝑔_µv ( 𝑥^𝐷_f )𝑑𝑥^µ𝑑𝑥^v

Here:

• 𝑔_µv ( 𝑥^𝐷_f ) : metric tensor dependent on fractal dimension
• 𝐷_f : effective fractal dimension (scale-dependent)
• 𝑑𝑠² : fractal space-time interval

The curvature of space-time is no longer constant, but a function that changes according to fractal scales.

New Principles

• Self-similarity: The metric shows repeating structures at different scales.
• Multi-scale geometry: Different metric behaviors emerge at micro and macro scales.
• Fractal time flow: Time does not flow uniformly, but with fractal resonances.
• Dimensional reduction: The effective dimension decreases at the Planck scale (e.g. 4D → 2D).

Application Areas

• Quantum gravity: Fractal corrections in the unification of quantum mechanics and relativity.
• Cosmology: The role of fractal metrics in the expansion of the universe.
• Black holes: Fractal dimension reduction around the event horizon.
• Information theory: The connection of the information density of space-time with fractal entropy.

Conclusion

The definition of the fractal metric reinterprets space-time as a self-similar, multi-scale, and dynamic structure. This approach breaks the continuity assumption of classical relativity, opening the door to a unified framework with quantum mechanics.

Fractal Deflection Angle

The fractal deflection angle is a definition that extends the uniform bending of light in classical gravitational lensing with self-similar and multi-scale deflections. In this approach, the change of direction of light becomes scale-dependent with parameters such as fractal dimension and the Hurst exponent.

Mathematical Definition

Fractal deflection angle formula:

𝜃_fr = 𝜃₀ ⋅∣ 𝑥 ∣^𝐻

Here:

• 𝜃₀ : classical deflection angle
• 𝑥 : position parameter in the lensing region
• 𝐻 : Hurst exponent (degree of scale dependence)

The deflection angle is no longer constant, but a function that changes at fractal scales.

Physical Interpretation

• Cosmology: The light of galaxy clusters is bent at different scales with fractal deflections.
• Black holes: Light rings form around the event horizon with self-similar deflection angles.
• Quantum lensing: On a micro scale, particle waves show a fractal deflection effect.
• Information density: Deflection regions are also areas of information condensation.

Conclusion

The fractal deflection angle transforms classical lensing theory into a self-similar and multi-scale structure. Thus, the bending of light is defined not just as a single curvature, but as a process that changes along regions of fractal resonance.

Fractal Lens Equation

The fractal lens equation defines the bending of light through self-similar and multi-scale deflections by extending the classical gravitational lensing formula with fractal geometry.

Mathematical Formula

Fractal lens equation:

𝛽 = 𝜃 − ( 𝐷_ls / 𝐷_s ) ⋅ 𝛼_fr (𝜃)

Here:

• 𝛽 : source angle
• 𝜃 : observed angle
• 𝐷_ls : distance between lens and source
• 𝐷_s : distance between observer and source
• 𝛼_fr (𝜃) : fractal deflection angle

Fractal Deflection Angle:

𝛼_fr (𝜃) = 𝛼₀ ⋅ ∣ 𝜃 ∣^𝐻

Here 𝐻 → Hurst exponent. The deflection angle becomes scale-dependent instead of classical linear behavior.

Physical Interpretation

• Cosmology: The light of galaxy clusters shows multi-scale deflections with fractal lensing.
• Black holes: Light rings form as self-similar structures around the event horizon.
• Quantum lensing: On a micro scale, particle waves deflect under the effect of fractal lensing.
• Information density: Lens regions are also areas of information condensation.

Conclusion

The fractal lens equation transforms classical lensing theory into a self-similar, multi-scale, and dynamic structure. In this way, the bending of light is defined not just as a single curvature, but as a process that changes along regions of fractal resonance.