ORT: Special Relativity

ORT: Special Relativity#

Chapters 1-10 | Formulas 1-22

This notebook covers the part of the Ontological Relativity Theory (ORT) that describes special relativity. The model rests on two principles: everything moves at speed $c$ through spacetime, and Invariance of Being ($m \cdot v_t = m_0 c$).

import sys, pathlib
sys.path.insert(0, str(pathlib.Path().resolve().parent / 'shared'))
from ort_core import (C, ORT, NewtonModel, EinsteinModel)
from ort_plots import (velocity_circle, velocity_circle_interactive, gamma_curve,
    simultaneity_diagram, lorentz_transform_grid, zijn_vector_diagram, model_comparison_bar)
import math
import numpy as np
%matplotlib inline

Chapter 1 – Introduction#

ORT rests on two intuitive principles:

Principle 1 — Velocity through time Every object always moves at exactly the speed of light $c$ through four-dimensional spacetime. Time is a direction of motion just like the three spatial dimensions. An object at rest moves at 1 second per second through time — and that is the speed of light $c$. Accelerating through space means “borrowing” speed from the time direction.

Principle 2 — Invariance of Being The product of an object’s mass and its speed through time is constant: $m \cdot v_t = m_0 c$. Within ORT we call this the Invariance of Being: all observers see the same amount of Being. When an object moves faster through space and thus slower through time, its mass increases proportionally.

This notebook demonstrates that these two principles reproduce all effects of special relativity: the Lorentz factor, E=mc$^2$, momentum, velocity addition, simultaneity, and the full Lorentz transformation.

Chapter 2 – The Core Principle: The Velocity Sphere#

Equation (1) – The velocity norm#

\[v_{space}^2 + v_{time}^2 = c^2 \tag{1}\]

This is the equation of a 4D sphere with radius $c$ in velocity space (shown here as a 2D cross-section in the $(v_{space}, v_{time})$ plane). The angle $\theta$ measures the direction of the velocity vector relative to the time axis.

Equations (2a) and (2b) – Parameterization with $\theta$#

\[v_{space} = c \sin\theta \tag{2a}\]

\[v_{time} = c \cos\theta \tag{2b}\]

$\theta = 0$: object at rest, moving at $c$ through time
$\theta = 90°$: photon, moving at $c$ through space, no progress through time

# The Velocity Sphere (2D cross-section)
velocity_circle(lang='en')
pass

_images/f09f3c75d2cf29ae7c66c221d4d022e1232757a1f222cff5f5a5b2de898e3b27.png

# Interactive version -- move the slider to change beta
velocity_circle_interactive(lang='en')

Interactive version — download the notebook to use the slider.

Chapter 3 – Mathematical Formulation#

All relativistic quantities follow directly from the angle $\theta$.

Equation (5) – Time dilation#

\[\frac{d\tau}{dt} = \cos\theta \tag{5}\]

The proper time $\tau$ of a moving object ticks slower by a factor $\cos\theta$.

Equation (6) – The Lorentz factor#

\[\gamma = \frac{1}{\cos\theta} \tag{6}\]

The Lorentz factor from SRT is simply the secant of $\theta$. In ORT, $\gamma$ is a derived notation, not a starting point.

Equations (7) and (8) – Energy#

\[E = \frac{m_0 c^2}{\cos\theta} \tag{7}\]

\[E_k = \left(\frac{1}{\cos\theta} - 1\right) m_0 c^2 \tag{8}\]

Derived in chapter 4 via Invariance of Being and the work integral (equation 15).

Equation (9) – Momentum#

\[p = m_0 c \tan\theta \tag{9}\]

Derived in chapter 4: from $m_{rel} = m_0/\cos\theta$ (equation 14), which follows from Invariance of Being.

Equation (10) – Relativistic velocity addition#

\[v_{total} = \frac{v_1 + v_2}{1 + \frac{v_1 v_2}{c^2}} \tag{10}\]

This describes the addition of spatial velocities ($v = c\sin\theta$).

Follows from the Lorentz transformation (chapter 5, equations 22a-b).

Equation (11) – Rapidity#

\[\varphi = \text{artanh}(\sin\theta) \tag{11}\]

Rapidities are additive: $\varphi_{total} = \varphi_1 + \varphi_2$.

Rapidity is the standard term in relativistic physics (e.g. Rindler, Jackson).

# Gamma curve: how gamma increases with velocity
gamma_curve(lang='en')
pass

_images/88331cc526c6af963ab8808b373bb95c390b014bcdb1d07b4c9b822d08dd5ebf.png

# Compute gamma for various velocities
print(f"{'beta':>8}  {'theta (degrees)':>15}  {'gamma':>10}  {'v_time/c':>10}")
print("-" * 50)
for beta in [0.0, 0.1, 0.3, 0.5, 0.8, 0.9, 0.95, 0.99, 0.999]:
    stv = ORT.from_beta(beta)
    print(f"{beta:8.3f}  {math.degrees(stv.theta):15.2f}  {stv.gamma:10.4f}  {stv.v_time/C:10.6f}")

    beta  theta (degrees)       gamma    v_time/c
--------------------------------------------------
000             0.00      1.0000    1.000000
100             5.74      1.0050    0.994987
300            17.46      1.0483    0.953939
500            30.00      1.1547    0.866025
800            53.13      1.6667    0.600000
900            64.16      2.2942    0.435890
950            71.81      3.2026    0.312250
990            81.89      7.0888    0.141067
999            87.44     22.3663    0.044710

# Velocity addition: 0.8c + 0.8c does NOT give 1.6c
v1 = 0.8 * C
v2 = 0.8 * C

v_newton = NewtonModel.velocity_addition(v1, v2)
v_srt = EinsteinModel.velocity_addition(v1, v2)
v_ort = ORT.velocity_addition(v1, v2)

print(f"Velocity addition: {v1/C:.1f}c + {v2/C:.1f}c")
print(f"  Newton:  {v_newton/C:.4f}c  (WRONG: exceeds c!)")
print(f"  SRT:     {v_srt/C:.4f}c")
print(f"  ORT:     {v_ort/C:.4f}c  (= SRT)")
print()

# Rapidity verification -- SRT and ORT
phi1_srt = math.atanh(v1 / C)
phi2_srt = math.atanh(v2 / C)
phi_total_srt = phi1_srt + phi2_srt
v_from_phi_srt = C * math.tanh(phi_total_srt)

phi1_ort = ORT.rapidity(v1)
phi2_ort = ORT.rapidity(v2)
phi_total_ort = phi1_ort + phi2_ort
v_from_phi_ort = C * math.tanh(phi_total_ort)

print(f"Rapidity (SRT): phi1={phi1_srt:.4f} + phi2={phi2_srt:.4f} = {phi_total_srt:.4f}")
print(f"  v = c*tanh(phi_total) = {v_from_phi_srt/C:.4f}c")
print(f"Rapidity (ORT): phi1={phi1_ort:.4f} + phi2={phi2_ort:.4f} = {phi_total_ort:.4f}")
print(f"  v = c*tanh(phi_total) = {v_from_phi_ort/C:.4f}c")
print(f"  Difference SRT vs ORT: {abs(phi_total_srt - phi_total_ort):.2e}  (identical)")

Velocity addition: 0.8c + 0.8c
  Newton:  1.6000c  (WRONG: exceeds c!)
  SRT:     0.9756c
  ORT:     0.9756c  (= SRT)

Rapidity (SRT): phi1=1.0986 + phi2=1.0986 = 2.1972
  v = c*tanh(phi_total) = 0.9756c
Rapidity (ORT): phi1=1.0986 + phi2=1.0986 = 2.1972
  v = c*tanh(phi_total) = 0.9756c
  Difference SRT vs ORT: 0.00e+00  (identical)

Chapter 4 – Invariance of Being#

The second principle of ORT: the Being of an object is defined as the product of its relativistic mass and its velocity through time. This product is invariant — it does not change with velocity.

The name Ontological Relativity Theory refers to this concept: ontology is the philosophical study of what exists. ORT starts from Being — the fundamental existence of an object ($m_0$) — as an invariant.

Equation (12) – Invariance of Being through time#

\[S_{time} = m_{rel} \cdot v_{time} = m_0 c \tag{12}\]

The time-component of the Being-vector is invariant: all observers measure the same $S_{time} = m_0 c$, regardless of the object’s velocity.

Equation (13) – Energy corollary (derived)#

\[E \cdot v_{time} = m_0 c^3 = S_{time} \cdot c^2 \tag{13}\]

This is not an independent postulate but a consequence of (12) and $E = m_{rel} c^2$ (equation 7).

Equation (14) – Relativistic mass#

From $S_{time} = m_0 c$ and $v_{time} = c\cos\theta$ follows:

\[m_{rel} = \frac{m_0 c}{v_{time}} = \frac{m_0}{\cos\theta} \tag{14}\]

This derives the relativistic mass from the model, instead of postulating it! In SRT this is $\gamma m_0$; here it follows directly from the invariance of Being through time.

Equation (15) – E = mc² via the work integral#

The factor $c^2$ is not postulated but derived. The key is the standard definition of kinetic energy as work:

\[E_{kin} = \int F \, dx = \int v \, dp\]

We know $v$ and $p$ both as functions of $\theta$:

\[v = c\sin\theta \qquad \text{(equation 2a)}\]

\[p = \frac{m_0}{\cos\theta} \cdot c\sin\theta = m_0 c \tan\theta \qquad \text{(equations 14 + 2a)}\]

\[dp = \frac{m_0 c}{\cos^2\theta} \, d\theta\]

\[E_{kin} = \int_0^\theta c\sin\theta' \cdot \frac{m_0 c}{\cos^2\theta'} \, d\theta' = m_0 c^2 \int_0^\theta \frac{\sin\theta'}{\cos^2\theta'} \, d\theta'\]

This integral is standard ($\int \sin x / \cos^2 x \, dx = 1/\cos x$):

\[E_{kin} = m_0 c^2 \left(\frac{1}{\cos\theta} - 1\right) \tag{15}\]

The origin of $c^2$: the product $v \cdot dp$ contains $c \cdot c$: one $c$ from velocity $v = c\sin\theta$, one $c$ from momentum $p = m_0 c \tan\theta$. The factor $c^2$ is a computational consequence, not a postulate.

The total energy is $E = E_{kin} + m_0 c^2 = m_0 c^2 / \cos\theta$ (equation 7). At rest ($\theta = 0$) this reduces to $E_0 = m_0 c^2$.

Equations (16)–(19) – The Being-vector#

The Being-vector $\vec{S}$ has two components:

\[S_{space} = p = m_0 c \tan\theta \quad \text{(varies)} \tag{16}\]

\[S_{time} = m_0 c \quad \text{(invariant — already defined in equation 12)} \tag{17}\]

Being is time-momentum. The structure is symmetric:

	Through space	Through time
Velocity	$v = c\sin\theta$	$d\tau/dt = \cos\theta$
“Momentum”	$p = m_{rel} \cdot v$	$m_{rel} \cdot d\tau/dt = m_0$

With dimensionless time-velocity $d\tau/dt = \cos\theta$, Invariance of Being reduces to: $m_{rel} \cdot d\tau/dt = (m_0/\cos\theta) \cdot \cos\theta = m_0$. The conservation law simply says that rest mass is invariant.

Why $S_{time} = m_0 c$ and not $m_0$? The factor $c$ is not a unit choice. It follows directly from Principle 1: an object at rest moves at $v_{time} = c$ through time. Invariance of Being then gives $m_0 \cdot c = m_0 c$. The table above uses the dimensionless time-velocity $d\tau/dt = \cos\theta$ (which absorbs the $c$), while $S_{time}$ uses the full $v_{time} = c\cos\theta$.

The magnitude of the Being-vector:

\[|\vec{S}| = \frac{m_0 c}{\cos\theta} = \frac{E}{c} \quad \text{(derived, §4.4)} \tag{18}\]

Pythagoras on the Being-vector gives the energy-momentum relation:

\[E^2 = (pc)^2 + (m_0 c^2)^2 \tag{19}\]

Connection with the four-momentum (SRT)#

The Being-vector contains the same physics as the four-momentum $p^\mu$ from special relativity theory (SRT), but the presentation differs fundamentally:

Component	Four-momentum $p^\mu$ (SRT)	Being-vector $\vec{S}$ (ORT)
Space	$p$	$S_{space} = p$
Time	$p^0 = E/c$	$S_{time} = m_0 c$
Magnitude	$m_0 c$ (as norm)	$\lvert\vec{S}\rvert = E/c$

The four-momentum hides $m_0$ as a norm — you need to compute $\sqrt{p^\mu p_\mu}$ to recover the rest mass. The Being-vector makes the invariance of $m_0$ directly visible as a component: $S_{time} = m_0 c$ is right there.

The derivation chain#

\[\boxed{\text{Invariance of Being through time}} \xrightarrow{m \cdot v_t = m_0 c} m_{rel} = \frac{m_0}{\cos\theta} \xrightarrow{\int v \, dp} E = \frac{m_0 c^2}{\cos\theta} \xrightarrow{|\vec{S}| = E/c} E^2 = (pc)^2 + (m_0 c^2)^2\]

# Being-vector diagram
zijn_vector_diagram(lang='en')
pass

_images/aeac7c1a44bddb0078ec69e81b6a5f07ec83fc0c4aed95d95a6f19abfb66a289.png

# Verification: Invariance of Being for various velocities
m0 = 1.0  # kg (unit mass)
print(f"Invariance of Being: S_m = m_rel * v_time = m0 * c = {m0 * C:.6e} kg*m/s")
print()
print(f"{'beta':>6}  {'m_rel (gamma*m0)':>16}  {'v_time (m/s)':>14}  {'S_m':>14}  {'= m0*c?':>10}")
print("-" * 70)
for beta in [0.0, 0.3, 0.5, 0.8, 0.9, 0.99]:
    stv = ORT.from_beta(beta)
    m_rel = stv.relativistic_mass(m0)
    v_t = stv.v_time
    s_m = m_rel * v_t
    match = abs(s_m - m0 * C) < 1e-6
    print(f"{beta:6.2f}  {m_rel:16.6f}  {v_t:14.2f}  {s_m:14.6e}  {'YES' if match else 'NO':>10}")

print()
print("The Being value is constant -- independent of velocity!")

Invariance of Being: S_m = m_rel * v_time = m0 * c = 2.997925e+08 kg*m/s

  beta  m_rel (gamma*m0)    v_time (m/s)             S_m     = m0*c?
----------------------------------------------------------------------
  0.00          1.000000    299792458.00    2.997925e+08         YES
  0.30          1.048285    285983777.98    2.997925e+08         YES
  0.50          1.154701    259627884.49    2.997925e+08         YES
  0.80          1.666667    179875474.80    2.997925e+08         YES
  0.90          2.294157    130676502.85    2.997925e+08         YES
  0.99          7.088812     42290930.54    2.997925e+08         YES

The Being value is constant -- independent of velocity!

# Verification: energy-momentum relation E^2 = (pc)^2 + (m0*c^2)^2
m0 = 1.0  # kg
print("Energy-momentum relation (equation 19):")
print(f"  m0*c^2 = {m0 * C**2:.6e} J\n")

for beta in [0.0, 0.5, 0.8, 0.99]:
    stv = ORT.from_beta(beta)
    E = stv.total_energy(m0)
    p = stv.momentum(m0)
    lhs = E**2
    rhs = (p * C)**2 + (m0 * C**2)**2
    print(f"  beta={beta:.2f}: E={E:.6e} J, p={p:.6e} kg*m/s")
    print(f"    E^2              = {lhs:.6e}")
    print(f"    (pc)^2+(m0c^2)^2 = {rhs:.6e}")
    print(f"    Match: {abs(lhs - rhs) / lhs < 1e-12}\n")

Energy-momentum relation (equation 19):
  m0*c^2 = 8.987552e+16 J

  beta=0.00: E=8.987552e+16 J, p=0.000000e+00 kg*m/s
    E^2              = 8.077609e+33
    (pc)^2+(m0c^2)^2 = 8.077609e+33
    Match: True

  beta=0.50: E=1.037793e+17 J, p=1.730853e+08 kg*m/s
    E^2              = 1.077014e+34
    (pc)^2+(m0c^2)^2 = 1.077014e+34
    Match: True

  beta=0.80: E=1.497925e+17 J, p=3.997233e+08 kg*m/s
    E^2              = 2.243780e+34
    (pc)^2+(m0c^2)^2 = 2.243780e+34
    Match: True

  beta=0.99: E=6.371107e+17 J, p=2.103921e+09 kg*m/s
    E^2              = 4.059100e+35
    (pc)^2+(m0c^2)^2 = 4.059100e+35
    Match: True

# Numerical verification: work integral E_kin = integral(v dp) = (gamma - 1) m0 c^2
# This is the NON-CIRCULAR derivation of E = mc^2 (§4.4)
m0 = 1.0  # kg

print("Work integral: E_kin = ∫ v dp = m₀c² (1/cos(θ) - 1)")
print(f"{'':>58} {'Numerical':>14} {'Analytical':>14} {'Difference':>14}")
print("-" * 100)

for beta in [0.1, 0.3, 0.5, 0.8, 0.9, 0.95, 0.99]:
    stv = ORT.from_beta(beta)

    # Numerical integral via the new method
    E_numerical = stv.kinetic_energy_integral(m0, n_steps=1_000_000)

    # Analytical result: (gamma - 1) m0 c^2
    E_analytical = stv.kinetic_energy(m0)

    rel_diff = abs(E_numerical - E_analytical) / E_analytical if E_analytical > 0 else 0

    print(f"  β = {beta:.2f}  θ = {math.degrees(stv.theta):6.2f}°  γ = {stv.gamma:8.4f}  "
          f"{E_numerical:14.6e} {E_analytical:14.6e} {rel_diff:14.2e}")

print()
print("The numerical work integral matches the analytical result.")
print("c² arises as c · c: one c from v = c·sin(θ), one c from p = m₀c·tan(θ).")

Work integral: E_kin = ∫ v dp = m₀c² (1/cos(θ) - 1)
                                                                Numerical     Analytical     Difference
----------------------------------------------------------------------------------------------------
  β = 0.10  θ =   5.74°  γ =   1.0050    4.527763e+14   4.527763e+14       3.06e-14

  β = 0.30  θ =  17.46°  γ =   1.0483    4.339625e+15   4.339625e+15       2.56e-14
  β = 0.50  θ =  30.00°  γ =   1.1547    1.390379e+16   1.390379e+16       6.47e-14

  β = 0.80  θ =  53.13°  γ =   1.6667    5.991701e+16   5.991701e+16       3.59e-13
  β = 0.90  θ =  64.16°  γ =   2.2942    1.163131e+17   1.163131e+17       8.31e-13

  β = 0.95  θ =  71.81°  γ =   3.2026    1.979565e+17   1.979565e+17       1.83e-12
  β = 0.99  θ =  81.89°  γ =   7.0888    5.472351e+17   5.472351e+17       9.84e-12

The numerical work integral matches the analytical result.
c² arises as c · c: one c from v = c·sin(θ), one c from p = m₀c·tan(θ).

Chapter 5 – Simultaneity and Co-locality#

ORT contains an important duality principle:

Simultaneity and co-locality are dual concepts, symmetric around the light line (45 degrees).

Equation (20) – Slope of simultaneity line#

\[\alpha = \arctan(\sin\theta) \tag{20}\]

Equation (21) – Relativity of simultaneity#

\[\Delta t' = \frac{\sin\theta}{c \cos\theta} \, \Delta x = \frac{\tan\theta}{c} \, \Delta x \tag{21}\]

Two events that are simultaneous in the rest frame ($\Delta t = 0$) but at different positions, are not simultaneous in the moving frame.

In SRT this equation reads $\Delta t' = \gamma v \Delta x / c^2$, which is identical.

Equations (22a) and (22b) – The full Lorentz transformation#

\[x' = \frac{x - c \sin\theta \cdot t}{\cos\theta} \tag{22a}\]

\[t' = \frac{t - \frac{\sin\theta}{c} \, x}{\cos\theta} \tag{22b}\]

In SRT this is written as $x' = \gamma(x - vt)$ and $t' = \gamma(t - vx/c^2)$.

ORT derives the full Lorentz transformation: the spatial component (22a) follows from length contraction, the temporal component (22b) follows from the simultaneity-colocality duality.

# Relativity of simultaneity
simultaneity_diagram(beta=0.6, lang='en')
pass

_images/a7920e6f6fc22cd52c6239614a5532589762840d1c3d784b2be15a2d90e222a9.png

# Lorentz transformation: how the spacetime grid deforms
lorentz_transform_grid(beta=0.5, lang='en')
pass

_images/306363bc9db66a2798af2e92001a7f262ef0725d0750ebde02b6423b13fdca68.png

# Numerical verification of the Lorentz transformation
beta = 0.6
stv = ORT.from_beta(beta)
v = stv.v_space
gamma = stv.gamma

print(f"Lorentz transformation for beta = {beta}, gamma = {gamma:.6f}")
print(f"v = {v:.2f} m/s\n")

# Test events (x in light-meters, t in seconds)
events = [
    (0, 0, "Origin"),
    (C, 0, "x=c, t=0 (simultaneous but not co-located)"),
    (0, 1, "x=0, t=1 (co-located but not simultaneous)"),
    (C, 1, "x=c, t=1"),
]

print(f"{'Event':<50} {'x':>12} {'t':>12} {'x_prime':>14} {'t_prime':>14}")
print("-" * 105)
for x, t, label in events:
    x_p, t_p = stv.lorentz_transform(x, t)
    # Verification with Einstein's formulas
    x_e = EinsteinModel.lorentz_x(x, t, v)
    t_e = EinsteinModel.lorentz_t(x, t, v)
    print(f"{label:<50} {x/C:12.4f}c {t:12.4f}s {x_p/C:14.4f}c {t_p:14.4f}s")
    assert abs(x_p - x_e) < 1e-6, "Mismatch!"
    assert abs(t_p - t_e) < 1e-6, "Mismatch!"

print("\nAll values match Einstein's Lorentz transformation exactly.")

Lorentz transformation for beta = 0.6, gamma = 1.250000
v = 179875474.80 m/s

Event                                                         x            t        x_prime        t_prime
---------------------------------------------------------------------------------------------------------
Origin                                                   0.0000c       0.0000s         0.0000c         0.0000s
x=c, t=0 (simultaneous but not co-located)               1.0000c       0.0000s         1.2500c        -0.7500s
x=0, t=1 (co-located but not simultaneous)               0.0000c       1.0000s        -0.7500c         1.2500s
x=c, t=1                                                 1.0000c       1.0000s         0.5000c         0.5000s

All values match Einstein's Lorentz transformation exactly.

Chapter 6 – Relativistic Effects#

ORT describes all known relativistic effects in terms of the angle $\theta$:

Effect	ORT ($\theta$-notation)	Equivalent ($\gamma$-notation)
Time dilation	$\tau = t\cos\theta$	$= t/\gamma$
Length contraction	$L = L_0\cos\theta$	$= L_0/\gamma$
Relativistic mass	$m = m_0/\cos\theta$	$= \gamma m_0$
Momentum	$p = m_0 c\tan\theta$	$= \gamma m_0 v$

# Relativistic effects at v = 0.8c -- Newton vs SRT vs ORT
beta = 0.8
v = beta * C
stv = ORT.from_beta(beta)
m0 = 1.0  # kg
L0 = 1.0  # m (rest length)
t_coord = 1.0  # s (coordinate time)
theta_deg = math.degrees(stv.theta)

print(f"=== Relativistic effects at beta = {beta} ===")
print(f"  theta = {theta_deg:.2f} degrees,  gamma = {stv.gamma:.6f}")
print()

# --- Time dilation ---
td_newton = NewtonModel.proper_time(t_coord, v)
td_srt = EinsteinModel.proper_time(t_coord, v)
td_ort = stv.proper_time(t_coord)

print(f"Time dilation (tau for t = {t_coord} s):")
print(f"  {'Newton':>8}: {td_newton:.6f} s  (no dilation)")
print(f"  {'SRT':>8}: {td_srt:.6f} s  (= t/gamma)")
print(f"  {'ORT':>8}: {td_ort:.6f} s  (= t*cos(theta))")
print()

# --- Length contraction ---
lc_newton = L0  # Newton: no contraction
lc_srt = EinsteinModel.length(L0, v)
lc_ort = stv.length_contraction(L0)

print(f"Length contraction (L0 = {L0} m):")
print(f"  {'Newton':>8}: {lc_newton:.6f} m  (no contraction)")
print(f"  {'SRT':>8}: {lc_srt:.6f} m  (= L0/gamma)")
print(f"  {'ORT':>8}: {lc_ort:.6f} m  (= L0*cos(theta))")
print()

# --- Momentum ---
p_newton = NewtonModel.momentum(m0, v)
p_srt = EinsteinModel.momentum(m0, v)
p_ort = stv.momentum(m0)

print(f"Momentum (m0 = {m0} kg):")
print(f"  {'Newton':>8}: {p_newton:.6e} kg*m/s  (= m0*v)")
print(f"  {'SRT':>8}: {p_srt:.6e} kg*m/s  (= gamma*m0*v)")
print(f"  {'ORT':>8}: {p_ort:.6e} kg*m/s  (= m0*c*tan(theta))")
print()

# --- Kinetic energy ---
ek_newton = NewtonModel.kinetic_energy(m0, v)
ek_srt = EinsteinModel.kinetic_energy(m0, v)
ek_ort = stv.kinetic_energy(m0)

print(f"Kinetic energy:")
print(f"  {'Newton':>8}: {ek_newton:.6e} J  (= 0.5*m0*v^2)")
print(f"  {'SRT':>8}: {ek_srt:.6e} J  (= (gamma-1)*m0*c^2)")
print(f"  {'ORT':>8}: {ek_ort:.6e} J  (= (1/cos(theta)-1)*m0*c^2)")
print()

# --- Verification SRT = ORT ---
print("Verification SRT = ORT:")
print(f"  Time dilation:      diff = {abs(td_srt - td_ort):.2e}")
print(f"  Length contraction:  diff = {abs(lc_srt - lc_ort):.2e}")
print(f"  Momentum:            diff = {abs(p_srt - p_ort):.2e}")
print(f"  Kinetic energy:      diff = {abs(ek_srt - ek_ort):.2e}")
print("  SRT and ORT give identical results!")

=== Relativistic effects at beta = 0.8 ===
  theta = 53.13 degrees,  gamma = 1.666667

Time dilation (tau for t = 1.0 s):
    Newton: 1.000000 s  (no dilation)
       SRT: 0.600000 s  (= t/gamma)
       ORT: 0.600000 s  (= t*cos(theta))

Length contraction (L0 = 1.0 m):
    Newton: 1.000000 m  (no contraction)
       SRT: 0.600000 m  (= L0/gamma)
       ORT: 0.600000 m  (= L0*cos(theta))

Momentum (m0 = 1.0 kg):
    Newton: 2.398340e+08 kg*m/s  (= m0*v)
       SRT: 3.997233e+08 kg*m/s  (= gamma*m0*v)
       ORT: 3.997233e+08 kg*m/s  (= m0*c*tan(theta))

Kinetic energy:
    Newton: 2.876017e+16 J  (= 0.5*m0*v^2)
       SRT: 5.991701e+16 J  (= (gamma-1)*m0*c^2)
       ORT: 5.991701e+16 J  (= (1/cos(theta)-1)*m0*c^2)

Verification SRT = ORT:
  Time dilation:      diff = 1.11e-16
  Length contraction:  diff = 1.11e-16
  Momentum:            diff = 5.96e-08
  Kinetic energy:      diff = 1.60e+01
  SRT and ORT give identical results!

Chapter 7 & 8 – Model Comparison: Newton vs Einstein (SRT) vs ORT#

A comparison of different aspects across these three models:

Aspect	Newton	Einstein (SRT)	ORT
Time dilation	None	$ au = t/gamma$	$ au = tcos heta$
Length contraction	None	= L_0/gamma$	= L_0 cos heta$
Velocity addition	+ v_2$	Relativistic (rapidities additive)	Idem
Mass	Constant $	$gamma m_0$ (relativistic)	/cos heta$ from Being
= mc^2$	–	Derived (momentum conservation + Lorentz)	Derived (Invariance of Being)

Derivation chain SRT vs ORT#

Step	SRT	ORT
Starting point	Relativity principle + $ = constant	Everything moves at $ + Invariance of Being
Gives	Lorentz transformations	Velocity sphere
Plus	Momentum conservation invariant under Lorentz	cdot v_t = m_0 c$
Gives	= gamma m_0 v$	{rel} = m_0/cos heta$
Via	Work integral	Work integral
Gives	= gamma m_0 c^2$	= m_0 c^2/cos heta$

Einstein (SRT) and ORT give identical numerical results — they describe the same physics. The difference lies in interpretation: ORT makes the geometric structure visible through the angle $ heta$, and derives /cos heta$ and = mc^2$ from Invariance of Being. Newton diverges fundamentally at high velocities.

# Model comparison: time dilation at various velocities
betas_compare = [0.1, 0.3, 0.5, 0.8, 0.9, 0.99]
labels = [f'{b}c' for b in betas_compare]

newton_td = [1.0 for _ in betas_compare]  # Newton: no dilation
einstein_td = []
spacetime_td = []

for beta in betas_compare:
    v = beta * C
    stv = ORT.from_beta(beta)
    einstein_td.append(EinsteinModel.proper_time(1.0, v))
    spacetime_td.append(stv.proper_time(1.0))

values = {
    'labels': labels,
    'newton': newton_td,
    'einstein': einstein_td,
    'spacetime': spacetime_td
}
model_comparison_bar('Time Dilation (tau/t)', values, lang='en')
pass

_images/ab8b2721e4c8d60b9548cb1060baf640a8c08879533cff34667fc0bdb9775c36.png

# Detailed comparison table
m0 = 1.0  # kg
print(f"{'beta':>6} | {'Newton tau':>12} {'Einstein tau':>14} {'ORT tau':>14} | "
      f"{'Newton p':>14} {'Einstein p':>14} {'ORT p':>14}")
print("-" * 100)

for beta in [0.1, 0.3, 0.5, 0.8, 0.9, 0.95, 0.99]:
    v = beta * C
    stv = ORT.from_beta(beta)
    
    # Proper time for 1 second of coordinate time
    tau_n = NewtonModel.proper_time(1.0, v)
    tau_e = EinsteinModel.proper_time(1.0, v)
    tau_st = stv.proper_time(1.0)
    
    # Momentum
    p_n = NewtonModel.momentum(m0, v)
    p_e = EinsteinModel.momentum(m0, v)
    p_st = stv.momentum(m0)
    
    print(f"{beta:6.2f} | {tau_n:12.6f} {tau_e:14.6f} {tau_st:14.6f} | "
          f"{p_n:14.6e} {p_e:14.6e} {p_st:14.6e}")

print("\nEinstein (SRT) and ORT give identical results.")
print("Newton deviates increasingly at higher velocities.")

  beta |   Newton tau   Einstein tau        ORT tau |       Newton p     Einstein p          ORT p
----------------------------------------------------------------------------------------------------
  0.10 |     1.000000       0.994987       0.994987 |   2.997925e+07   3.013028e+07   3.013028e+07
  0.30 |     1.000000       0.953939       0.953939 |   8.993774e+07   9.428037e+07   9.428037e+07
  0.50 |     1.000000       0.866025       0.866025 |   1.498962e+08   1.730853e+08   1.730853e+08
  0.80 |     1.000000       0.600000       0.600000 |   2.398340e+08   3.997233e+08   3.997233e+08
  0.90 |     1.000000       0.435890       0.435890 |   2.698132e+08   6.189940e+08   6.189940e+08
  0.95 |     1.000000       0.312250       0.312250 |   2.848028e+08   9.120990e+08   9.120990e+08
  0.99 |     1.000000       0.141067       0.141067 |   2.967945e+08   2.103921e+09   2.103921e+09

Einstein (SRT) and ORT give identical results.
Newton deviates increasingly at higher velocities.

Chapter 9 – Thought Experiments#

9.1 The Twin Paradox#

A pair of twins: Alice stays on Earth, Bob travels at $v = 0.8c$ to a star 10 light-years away and back.

9.2 Muon Decay#

Muons are created in the upper atmosphere (about 10 km altitude) and move at $v \approx 0.998c$. Without relativity, they would decay in ~2.2 $\mu$s and travel only ~660 m.

9.3 GPS SRT Correction#

GPS satellites move at ~3.87 km/s. Special relativity causes a time delay.

# 9.1 Twin Paradox
print("=== Twin Paradox ===")
beta_bob = 0.8
distance_ly = 10  # light-years one way
stv_bob = ORT.from_beta(beta_bob)
theta_bob = math.degrees(stv_bob.theta)

t_alice_one_way = distance_ly / beta_bob
t_alice_total = 2 * t_alice_one_way

# ORT: proper time Bob = t * cos(theta)
t_bob_ort = t_alice_total * math.cos(stv_bob.theta)

# SRT: proper time Bob = t / gamma
t_bob_srt = t_alice_total / stv_bob.gamma

print(f"  Bob travels at {beta_bob}c to a star {distance_ly} light-years away")
print(f"  theta = {theta_bob:.2f} degrees")
print()
print(f"  Alice (Earth): {t_alice_total:.2f} years")
print(f"  Bob (ORT):  tau = t * cos(theta) = {t_alice_total:.2f} * cos({theta_bob:.2f}) = {t_bob_ort:.2f} years")
print(f"  Bob (SRT):  tau = t / gamma     = {t_alice_total:.2f} / {stv_bob.gamma:.4f}    = {t_bob_srt:.2f} years")
print(f"  Difference ORT vs SRT: {abs(t_bob_ort - t_bob_srt):.2e} years (identical)")
print()
print(f"  Bob is {t_alice_total - t_bob_ort:.2f} years younger upon return")

=== Twin Paradox ===
  Bob travels at 0.8c to a star 10 light-years away
  theta = 53.13 degrees

  Alice (Earth): 25.00 years
  Bob (ORT):  tau = t * cos(theta) = 25.00 * cos(53.13) = 15.00 years
  Bob (SRT):  tau = t / gamma     = 25.00 / 1.6667    = 15.00 years
  Difference ORT vs SRT: 0.00e+00 years (identical)

  Bob is 10.00 years younger upon return

# 9.2 Muon Decay
print("=== Muon Decay ===")
beta_muon = 0.998
tau_muon = 2.2e-6  # rest lifetime in seconds
h_atmosphere = 10000  # height in meters

stv_muon = ORT.from_beta(beta_muon)
theta_muon = math.degrees(stv_muon.theta)

d_classical = beta_muon * C * tau_muon

# ORT: lifetime = tau / cos(theta)
t_ort = tau_muon / math.cos(stv_muon.theta)
d_ort = beta_muon * C * t_ort

# SRT: lifetime = tau * gamma
t_srt = tau_muon * stv_muon.gamma
d_srt = beta_muon * C * t_srt

print(f"  Muon speed: {beta_muon}c, theta = {theta_muon:.2f} degrees")
print(f"  Rest lifetime: {tau_muon*1e6:.1f} microseconds")
print()
print(f"  Without relativity:")
print(f"    Distance = {d_classical:.0f} m ({d_classical/1000:.2f} km) -- does NOT reach ground")
print()
print(f"  ORT: lifetime = tau / cos(theta) = {t_ort*1e6:.1f} us")
ort_reaches = "YES" if d_ort > h_atmosphere else "NO"
print(f"    Distance = {d_ort/1000:.1f} km -- reaches ground: {ort_reaches}")
print()
print(f"  SRT: lifetime = tau * gamma      = {t_srt*1e6:.1f} us")
srt_reaches = "YES" if d_srt > h_atmosphere else "NO"
print(f"    Distance = {d_srt/1000:.1f} km -- reaches ground: {srt_reaches}")
print()
print(f"  Difference ORT vs SRT: {abs(d_ort - d_srt):.2e} m (identical)")

=== Muon Decay ===
  Muon speed: 0.998c, theta = 86.38 degrees
  Rest lifetime: 2.2 microseconds

  Without relativity:
    Distance = 658 m (0.66 km) -- does NOT reach ground

  ORT: lifetime = tau / cos(theta) = 34.8 us
    Distance = 10.4 km -- reaches ground: YES

  SRT: lifetime = tau * gamma      = 34.8 us
    Distance = 10.4 km -- reaches ground: YES

  Difference ORT vs SRT: 0.00e+00 m (identical)

# 9.3 GPS SRT correction
print("=== GPS Special Relativity Correction ===")
v_gps = 3870  # m/s (GPS satellite orbital speed)
stv_gps = ORT.from_v(v_gps)
theta_gps = stv_gps.theta
seconds_per_day = 86400

# ORT: proper time = t * cos(theta)
tau_ort = seconds_per_day * math.cos(theta_gps)
delta_ort = seconds_per_day - tau_ort

# SRT: proper time = t / gamma
tau_srt = seconds_per_day / stv_gps.gamma
delta_srt = seconds_per_day - tau_srt

print(f"  Orbital speed: {v_gps} m/s")
print(f"  theta = {math.degrees(theta_gps):.6f} degrees")
print()
print(f"  ORT: tau = t * cos(theta)")
print(f"    Time delay per day: {delta_ort*1e9:.2f} nanoseconds")
print()
print(f"  SRT: tau = t / gamma")
print(f"    Time delay per day: {delta_srt*1e9:.2f} nanoseconds")
print()
print(f"  Difference ORT vs SRT: {abs(delta_ort - delta_srt)*1e9:.4f} ns (identical)")
print()
print(f"  Satellite clock runs SLOWER due to motion (SRT).")
print(f"  (Note: GRT causes a SPEEDUP due to less gravity, which is larger.)")

=== GPS Special Relativity Correction ===
  Orbital speed: 3870 m/s
  theta = 0.000740 degrees

  ORT: tau = t * cos(theta)
    Time delay per day: 7198.86 nanoseconds

  SRT: tau = t / gamma
    Time delay per day: 7198.86 nanoseconds

  Difference ORT vs SRT: 0.0000 ns (identical)

  Satellite clock runs SLOWER due to motion (SRT).
  (Note: GRT causes a SPEEDUP due to less gravity, which is larger.)

Chapter 10 – Summary and Limitations#

What the model describes#

ORT reproduces complete special relativity from two principles: (1) everything moves at $c$ through 4D spacetime — time is a direction of motion just like space, and (2) Invariance of Being ($m \cdot v_t = m_0 c$).

Formula overview#

Nr	Name	ORT ($\theta$)	SRT ($\gamma, \beta$)	Description
(1)	Velocity norm	$v_r^2 + v_t^2 = c^2$	—	Velocity law (ORT principle 1)
(2)	Parameterization	$v_r = c\sin\theta$, $v_t = c\cos\theta$	$v_r = \beta c$	Velocity components
(5)	Time dilation	$d\tau/dt = \cos\theta$	$= 1/\gamma$	Proper vs coordinate time
(6)	Lorentz factor	$1/\cos\theta$	$\gamma = 1/\sqrt{1-\beta^2}$	Definition of $\gamma$
(7)	Total energy	$m_0 c^2/\cos\theta$	$\gamma m_0 c^2$	Energy of a moving object
(8)	Kinetic energy	$(1/\cos\theta - 1)\,m_0 c^2$	$(\gamma-1)\,m_0 c^2$	Difference from rest energy
(9)	Momentum	$m_0 c\tan\theta$	$\gamma m_0 v$	Spatial momentum
(10)	Velocity addition	$(v_1+v_2)/(1+v_1 v_2/c^2)$	idem	Addition of spatial velocities
(11)	Rapidity	$\text{artanh}(\sin\theta)$	$\text{artanh}(\beta)$	Additive velocity parameter
(12)	Being through time	$S_{time} = m_0 c$	—	Invariance of Being through time (ORT principle 2)
(13)	Energy corollary	$E \cdot v_t = S_{time} \cdot c^2$	—	Derived from (12) + (7)
(14)	Rel. mass	$m_0/\cos\theta$	$\gamma m_0$	Follows from Mass-Being
(15)	Work integral	$\int v\,dp = m_0 c^2(1/\cos\theta - 1)$	$(\gamma-1)\,m_0 c^2$	Derivation of $E = mc^2$ (from Being)
(16)	Being through space	$S_r = m_0 c\tan\theta$	$= p$	Space component Being-vector (varies)
(17)	Being through time	$S_t = m_0 c$	—	Time component Being-vector (invariant)
(18)	Being-magnitude	$m_0 c/\cos\theta$	$= E/c$	Magnitude Being-vector
(19)	Energy-momentum	$E^2 = (pc)^2 + (m_0 c^2)^2$	idem	Pythagoras on Being-vector
(20)	Simultaneity	$\alpha = \arctan(\sin\theta)$	$\arctan(\beta)$	Slope angle
(21)	Rel. simultaneity	$(\tan\theta/c)\,\Delta x$	$\gamma v \Delta x/c^2$	Time difference
(22)	Lorentz transf.	$(x - c\sin\theta\cdot t)/\cos\theta$	$\gamma(x - vt)$	Coordinate transformation

The derivation chain#

\[\boxed{\text{Invariance of Being through time}} \xrightarrow{m \cdot v_t = m_0 c} m_{rel} = \frac{m_0}{\cos\theta} \xrightarrow{\int v \, dp} E = \frac{m_0 c^2}{\cos\theta} \xrightarrow{\theta = 0} E_0 = m_0 c^2\]

Limitations#

This notebook only covers special relativity (spacetime without gravitational sources)
Gravity (curved spacetime) is covered in Notebook 02
The model is descriptive: it gives the same predictions as SRT, but offers an alternative, geometrically intuitive interpretation

Note on $v_{time}$: The time-velocity is just as “real” as spatial velocity — both are components of the same velocity vector through spacetime. An observer at rest sees others (also at rest) moving at one second per second through time: that is $v_{time} = c$, directly observable.

Conclusion#

ORT offers an elegant and intuitive framework for special relativity. All formulas follow logically from the two principles. The model is complete for SRT.

Component	Four-momentum \(p^\mu\) (SRT)	Being-vector \(\vec{S}\) (ORT)
Space	\(p\)	\(S_{space} = p\)
Time	\(p^0 = E/c\)	\(S_{time} = m_0 c\)
Magnitude	\(m_0 c\) (as norm)	\(\lvert\vec{S}\rvert = E/c\)

Effect	ORT (\(\theta\)-notation)	Equivalent (\(\gamma\)-notation)
Time dilation	\(\tau = t\cos\theta\)	\(= t/\gamma\)
Length contraction	\(L = L_0\cos\theta\)	\(= L_0/\gamma\)
Relativistic mass	\(m = m_0/\cos\theta\)	\(= \gamma m_0\)
Momentum	\(p = m_0 c\tan\theta\)	\(= \gamma m_0 v\)

ORT: Special Relativity

Contents

ORT: Special Relativity#

Chapter 1 – Introduction#

Chapter 2 – The Core Principle: The Velocity Sphere#

Equation (1) – The velocity norm#

Equations (2a) and (2b) – Parameterization with \(\theta\)#

Chapter 3 – Mathematical Formulation#

Equation (5) – Time dilation#

Equation (6) – The Lorentz factor#

Equations (7) and (8) – Energy#

Equation (9) – Momentum#

Equation (10) – Relativistic velocity addition#

Equation (11) – Rapidity#

Chapter 4 – Invariance of Being#

Equation (12) – Invariance of Being through time#

Equation (13) – Energy corollary (derived)#

Equation (14) – Relativistic mass#

Equation (15) – E = mc² via the work integral#

Equations (16)–(19) – The Being-vector#

Connection with the four-momentum (SRT)#

The derivation chain#

Chapter 5 – Simultaneity and Co-locality#

Equation (20) – Slope of simultaneity line#

Equation (21) – Relativity of simultaneity#

Equations (22a) and (22b) – The full Lorentz transformation#

Chapter 6 – Relativistic Effects#

Chapter 7 & 8 – Model Comparison: Newton vs Einstein (SRT) vs ORT#

Derivation chain SRT vs ORT#

Chapter 9 – Thought Experiments#

9.1 The Twin Paradox#

9.2 Muon Decay#

9.3 GPS SRT Correction#

Chapter 10 – Summary and Limitations#

What the model describes#

Formula overview#

The derivation chain#

Limitations#

Conclusion#

	Through space	Through time
Velocity	\(v = c\sin\theta\)	\(d\tau/dt = \cos\theta\)
“Momentum”	\(p = m_{rel} \cdot v\)	\(m_{rel} \cdot d\tau/dt = m_0\)

Aspect	Newton	Einstein (SRT)	ORT
Time dilation	None	\( au = t/gamma\)	\( au = tcos heta\)
Length contraction	None	= L_0/gamma$	= L_0 cos heta$
Velocity addition	+ v_2$	Relativistic (rapidities additive)	Idem
Mass	Constant $	\(gamma m_0\) (relativistic)	/cos heta$ from Being
= mc^2$	–	Derived (momentum conservation + Lorentz)	Derived (Invariance of Being)

Nr	Name	ORT (\(\theta\))	SRT (\(\gamma, \beta\))	Description
(1)	Velocity norm	\(v_r^2 + v_t^2 = c^2\)	—	Velocity law (ORT principle 1)
(2)	Parameterization	\(v_r = c\sin\theta\), \(v_t = c\cos\theta\)	\(v_r = \beta c\)	Velocity components
(5)	Time dilation	\(d\tau/dt = \cos\theta\)	\(= 1/\gamma\)	Proper vs coordinate time
(6)	Lorentz factor	\(1/\cos\theta\)	\(\gamma = 1/\sqrt{1-\beta^2}\)	Definition of \(\gamma\)
(7)	Total energy	\(m_0 c^2/\cos\theta\)	\(\gamma m_0 c^2\)	Energy of a moving object
(8)	Kinetic energy	\((1/\cos\theta - 1)\,m_0 c^2\)	\((\gamma-1)\,m_0 c^2\)	Difference from rest energy
(9)	Momentum	\(m_0 c\tan\theta\)	\(\gamma m_0 v\)	Spatial momentum
(10)	Velocity addition	\((v_1+v_2)/(1+v_1 v_2/c^2)\)	idem	Addition of spatial velocities
(11)	Rapidity	\(\text{artanh}(\sin\theta)\)	\(\text{artanh}(\beta)\)	Additive velocity parameter
(12)	Being through time	\(S_{time} = m_0 c\)	—	Invariance of Being through time (ORT principle 2)
(13)	Energy corollary	\(E \cdot v_t = S_{time} \cdot c^2\)	—	Derived from (12) + (7)
(14)	Rel. mass	\(m_0/\cos\theta\)	\(\gamma m_0\)	Follows from Mass-Being
(15)	Work integral	\(\int v\,dp = m_0 c^2(1/\cos\theta - 1)\)	\((\gamma-1)\,m_0 c^2\)	Derivation of \(E = mc^2\) (from Being)
(16)	Being through space	\(S_r = m_0 c\tan\theta\)	\(= p\)	Space component Being-vector (varies)
(17)	Being through time	\(S_t = m_0 c\)	—	Time component Being-vector (invariant)
(18)	Being-magnitude	\(m_0 c/\cos\theta\)	\(= E/c\)	Magnitude Being-vector
(19)	Energy-momentum	\(E^2 = (pc)^2 + (m_0 c^2)^2\)	idem	Pythagoras on Being-vector
(20)	Simultaneity	\(\alpha = \arctan(\sin\theta)\)	\(\arctan(\beta)\)	Slope angle
(21)	Rel. simultaneity	\((\tan\theta/c)\,\Delta x\)	\(\gamma v \Delta x/c^2\)	Time difference
(22)	Lorentz transf.	\((x - c\sin\theta\cdot t)/\cos\theta\)	\(\gamma(x - vt)\)	Coordinate transformation