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ℂ-Space Engine

A 2D physics simulation exploring mathematically consistent extensions of classical mechanics into the complex plane. This project investigates what happens when mass, position, and friction become complex numbers, without asserting physical realism.

Physics SimulationC++20 / Python

Modified Physics Laws

I. Complex Inertia

Standard physics assumes mass is a Real scalar. In C-Space, mass μ=mr+imi\mu = m_r + i m_i allows for imaginary components.

F=μa\mathbf{F} = \mu \mathbf{a}

Result: Forces and accelerations can become perpendicular due to the imaginary mass component.

II. Chiral Friction

Friction coefficients κ=kr+iki\kappa = k_r + i k_i introduce a rotational drag component.

Fdrag=−(kr+iki)v\mathbf{F}_{drag} = - (k_r + i k_i) \mathbf{v}

Result: Particles spiral indefinitely rather than coming to rest.

III. Conjugate Gravity

The inverse-square law extended to the complex plane.

F(z)=Gμ1μ2(z2−z1)2F(z) = G \frac{\mu_1 \mu_2}{(z_2 - z_1)^2}

Simulation Results

Diametric Drive Simulation Plot

Negative Mass Runaway

Testing the interaction between positive mass (+m+m) and negative mass (−m-m). Since −m-m repels +m+m while +m+m attracts −m-m, both particles accelerate in the same direction. This concept is theoretically possible in General Relativity but realized here in the C-Space engine.

Chiral Death Spiral Animation

Chiral Friction Decay

Demonstrating the Chiral Friction Law. A particle with initial real velocity experiences complex drag (0.05+0.3i0.05 + 0.3i). The imaginary friction component applies rotational force, causing the particle to spiral rather than stopping.

First Light Simulation

Complex Orbital Motion

Testing gravitational orbits with complex gravity (F∝(z2−z1)−2F \propto (z_2-z_1)^{-2}) and imaginary inertia. The planet traces a precessing trajectory that never forms a closed ellipse, eventually decaying due to Chiral Friction.

Simulation Axioms

Fundamental Constants

  • SpaceC\mathbb{C} (Complex Plane)
  • Timet∈Rt \in \mathbb{R} (Real Stream)
  • Gravity (G)1.0+0i1.0 + 0i

First Light Config

  • Sun Mass1000+0i1000 + 0i (Fixed)
  • Planet Mass10+2i10 + 2i (Inertial)
  • Friction (k)0.05+0.01i0.05 + 0.01i (Chiral)

Engine Architecture

  • Custom C++20 Kernel: Manual implementation of complex vector arithmetic and RK4 integration.
  • 4D Phase Space: Native support for C2\mathbb{C}^2 collision detection (overlapping in Real dimensions doesn't guarantee collision if Imaginary depth differs).
  • Analysis Pipeline: CSV export to Python/Matplotlib for phase portrait generation.

References

[1] Forward, R. L. (1990). "Negative matter propulsion." Journal of Propulsion and Power, 6(1), 28-37.

[2] Bondi, H. (1957). "Negative mass in general relativity." Reviews of Modern Physics, 29(3), 423-428.

[3] Ablowitz, M. J., & Fokas, A. S. (2003). Complex Variables: Introduction and Applications. Cambridge University Press.

[4] Needham, T. (1997). Visual Complex Analysis. Oxford University Press.

[5] Hairer, E., Lubich, C., & Wanner, G. (2006). Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer.