FORTH Theory Calculation #001: Quantitative Verification of 4D Torus Structure and Jet Velocity

Overview

This calculation implements accurate numerical computation of the jet velocity formula v/c = √(1-(r/R)²), which is the fundamental prediction of FORTH theory, targeting the M87 black hole. All physical constants are based on CODATA 2018 recommended values, with complete transparency in the calculation process enabling manual verification.

Calculation Setup

Physical Constants Used

Gravitational constant G = 6.67430×10⁻¹¹ m³ kg⁻¹ s⁻²
Speed of light c = 299,792,458 m/s (defined value)
Solar mass M☉ = 1.98847×10³⁰ kg

M87 Black Hole Parameters

Mass: M = 6.5×10⁹ M☉ = 1.293×10⁴⁰ kg

Main Calculation Results

1. Schwarzschild Radius

Calculation Process:

Rs = 2GM/c²
   = 2 × 6.67430×10⁻¹¹ × 1.293×10⁴⁰ / (299,792,458)²
   = 1.725×10³⁰ / 8.988×10¹⁶
   = 1.920×10¹³ m

Result: Rs = 1.920×10¹³ m (= 1.920×10¹⁰ km = 128.3 AU)

2. Jet Velocity Calculation

FORTH Theory Basic Formula:

v/c = √(1 - (r/R)²)

Where R: major radius, r: minor radius #### Detailed Calculation for R/r = 1000

r/R = 1/1000 = 0.001
(r/R)² = 0.000001
1 - (r/R)² = 0.999999
v/c = √0.999999 = 0.999999500

Result: v = 0.999999500c

3. Comparison at Various R/r Ratios

| R/r Ratio | v/c | Difference from c | γ Factor | |-----------|-----|------------------|----------| | 10 | 0.994987437 | 0.5% | 10.0 | | 100 | 0.999949999 | 0.005% | 100.0 | | 1,000 | 0.999999500 | 0.00005% | 1,000.0 | | 10,000 | 0.999999995 | 0.0000005% | 10,000.0 |

Physical Interpretation

Determinants of Jet Velocity

The larger the R/r ratio, the closer the jet velocity approaches the speed of light. This is a natural consequence of the geometric constraints of the 4D torus structure.

Comparison with M87 Observations

- Observed jet velocity: v ≈ 0.98-0.99c - Theoretical prediction (R/r=1000): v = 0.999999500c The theoretical value is slightly higher than observations, possibly due to: 1. Actual R/r ratio being less than 1000 2. Deceleration due to radiation losses 3. Projection effects in observations

Energy Considerations

Important Note

Assuming perfect energy conservation is unrealistic. In real systems, energy losses necessarily exist through: - Gravitational wave radiation - Electromagnetic radiation - Thermal radiation

Conversion Efficiency Estimation

The conversion efficiency from accretion energy to jet kinetic energy is observationally estimated at 10-40%. FORTH theory can explain this high efficiency through 4D structure.

Verification Methods

Manual Calculation Confirmation

All calculations can be verified by following these steps: 1. Schwarzschild radius:

   Rs = 2 × 6.674×10⁻¹¹ × 1.29×10⁴⁰ ÷ (3×10⁸)²
   

2. Jet velocity:

   v/c = √(1 - 1/1000²) = √0.999999
   

Calculation Code

Complete calculation code is available on GitHub: - GitHub: FORTH Calculations #001 Executable in Python as follows:

python
python calculation.py

Conclusion

Through accurate implementation of FORTH theory's basic formula v/c = √(1-(r/R)²), we have confirmed: 1. Jet velocity at R/r = 1000: v = 0.999999500c 2. Calculation transparency: Entire process verifiable by manual calculation 3. Observational consistency: Observable values can be explained with appropriate R/r ratio selection

Data Files

All files are available on GitHub: - Calculation Results JSON - Calculation Script --- Note: This calculation is a theoretical prediction requiring observational verification. All numerical values are stated considering significant figures.