Lorentz Coordinate System

Introduction

The Lorentz Coordinate System is an advanced mathematical framework designed to handle spacetime transformations and relativistic effects in physics simulations. It extends traditional 3D coordinate systems by incorporating time as a complex dimension, enabling natural representation of phenomena like time dilation and length contraction.

This system combines:

• Three orthonormal basis vectors (x, y, z axes)
• A complex time component (representing time power/phase)
• Quaternion-based rotation capabilities

Key Features

• Unified spacetime representation - Seamlessly combines spatial and temporal dimensions
• Relativistic ready - Built-in support for time dilation and length contraction
• Operator overloads - Intuitive mathematical operations for coordinate transformations
• Quaternion integration - Smooth interoperability with rotational representations
• Physics optimization - Efficient computations for real-time simulations

Mathematical Foundation

The system represents coordinates as:

C = { ux, uy, uz, power }

Where:

• ux, uy, uz form an orthonormal basis for 3D space
• power is a complex number representing the time dimension transformation

Installation

Include the header in your project:

#include "lorentz_coord.hpp"

Basic Usage

Creating Coordinate Systems

// Default system (aligned with global XYZ)
lorentz_coord defaultFrame; 

// System rotated 45° around Y-axis with time factor 0.8
lorentz_coord rotatedFrame(45.0 * DEG2RAD, vec3::UY, complex(0.8, 0));

// From Euler angles
lorentz_coord eulerFrame(0.1, 0.5, 0.2, complex::ONE);

// From quaternion
quaternion q(0.5, 0.5, 0.5, 0.5);
lorentz_coord quatFrame(q, complex(1.0, 0.1));

Coordinate Transformations

vec3 globalVector(1, 0, 0);

// Transform vector to local frame
vec3 localVector = globalVector / rotatedFrame;

// Transform back to global space
vec3 transformedBack = localVector * rotatedFrame;

Time Dilation Example

// Observer far from gravity well
lorentz_coord farObserver(vec3::UX, vec3::UY, vec3::UZ, complex::ONE);

// Observer near black hole (time runs at half speed)
lorentz_coord nearBlackHole(vec3::UX, vec3::UY, vec3::UZ, complex(0.5, 0));

// Event lasting 1 second in local frame
vec4 localEvent(0, 0, 0, 1.0); // (x,y,z,t)

// Compare observed durations
vec4 farObserved = localEvent * farObserver;
vec4 nearObserved = localEvent * nearBlackHole;

std::cout << "Far observer sees duration: " << farObserved.t << std::endl;
std::cout << "Near black hole sees duration: " << nearObserved.t << std::endl;

Relativistic Velocity Addition

// Frame moving at 0.6c along X-axis
real beta = 0.6;
real gamma = 1.0 / sqrt(1 - beta*beta);
lorentz_coord movingFrame(
    vec3(gamma, 0, 0),
    vec3(0, 1, 0),
    vec3(0, 0, 1),
    complex(gamma, 0)
);

// Object moving at 0.5c in moving frame
vec3 localVel(0.5, 0, 0);

// Calculate global velocity
vec3 globalVel = localVel * movingFrame;
std::cout << "Resultant velocity: " << globalVel.length() << "c" << std::endl;

Advanced Applications

Black Hole Spacetime Curvature

// Simulate Schwarzschild metric effects
real schwarzschildRadius = 2 * G * MASS / (c * c);

vec3 position(10 * schwarzschildRadius, 0, 0);
real r = position.length();

// Time dilation factor
real timeFactor = sqrt(1 - schwarzschildRadius/r);

// Create curved spacetime frame
lorentz_coord curvedSpace(
    position.normalized(),  // Radial direction
    vec3::UY.cross(position).normalized(),  // Angular direction
    vec3::UY,  // Vertical direction
    complex(timeFactor, 0)
);

// Calculate proper time vs coordinate time
vec4 localEvent(0, 0, 0, 1.0); // 1 second proper time
vec4 globalTime = localEvent * curvedSpace;

Gravitational Lensing

// Light ray passing near massive object
vec3 lightRay(0, 1, 0).normalized();

// Create warped coordinate system
real warpFactor = calculateWarpFactor(position);
lorentz_coord warpedFrame = calculateWarpedFrame(position);

// Transform light path
vec3 warpedRay = lightRay / warpedFrame;

// Calculate apparent position
vec3 apparentPosition = originalPosition + warpedRay * distance;

API Reference

Core Operations

Operation	Description
A * B	Compose coordinate systems
v * C	Transform vector v by C
C1 / C2	Relative transformation between systems
v / C	Project vector v onto C's basis
C.cross(v)	Cross product with vector
C1.cross(C2)	Cross product between systems

Conversion Methods

Method	Description
to_quaternion()	Convert to rotation quaternion
coord_to_eulers()	Get Euler angles representation
reversed()	Get inverse transformation

Performance Considerations

The system is optimized for real-time applications:

• Minimal memory footprint (3 vectors + 1 complex)
• Efficient operator implementations
• Avoids full 4x4 matrix operations when possible
• Cache-friendly structure

Contributing

Contributions to enhance the system are welcome:

• Additional coordinate system types
• More physical effects implementations
• Performance optimizations
• Unit test coverage

License

MIT License

Notice

The implementation of this module needs to be further verified.