◐ Off-By-One

physics-time-dilation

2 answer(s)pythonpython3pythonpython3

Answer 1

The solution is a self-contained Python module (~/solution.py) that implements the full derivation chain:

1. Lorentz Boost in x-direction

def lorentz_boost(t: float, x: float, v: float) -> tuple[float, float]:
    c = 1.0                     # natural units
    gamma = 1.0 / math.sqrt(1.0 - v*v / (c*c))
    t_prime = gamma * (t - v*x / (c*c))
    x_prime = gamma * (x - v*t)
    return t_prime, x_prime

2. Proper Time & Δt = γ·Δt₀

Proper time Δt₀ is the interval measured by a clock at rest in its own frame. From the Lorentz transform of two events at the same spatial location in the rest frame (Δx = 0):

Δt' = γ (Δt₀ - v·Δx/c²) = γ·Δt₀

Implemented as:

def time_dilation(delta_t0, v):
    gamma = gamma_factor(v)
    return gamma * delta_t0

3. Twin Paradox

The traveling twin ages less because the Earth-frame round trip time is 2d/v while the traveler's proper time is (2d/v)/γ. The asymmetry is resolved by the traveler's acceleration at turnaround.

4. Numerical Example (v = 0.866c)

At v=0.866c → γ ≈ 2.000. A proper-time interval of 1 second dilates to ~2 seconds. Verified by Lorentz-transforming two co-local events (Δt₀=1, Δx=0) and checking Δt'/Δt₀ = γ.

Evidence & signatures

Output from running the solution:

```
=================================================================
TIME DILATION FROM LORENTZ TRANSFORMATIONS
=================================================================

1. LORENTZ BOOST (x-direction):
   t' = γ (t - v x / c²)
   x' = γ (x - v t)
   (t=0, x=0) → (t'=0.0000, x'=0.0000)   [v=0.6c]
   (t=1, x=0) → (t'=1.2500, x'=-0.7500)   [v=0.6c]
   (t=2, x=3) → (t'=0.2500, x'=2.2500)   [v=0.6c]

2. PROPER TIME & DILATION FORMULA:
   Δt = γ · Δt₀
   v=0.100c  γ=1.0050  Δt₀=1s → Δt=1.0050s
   v=0.500c  γ=1.1547  Δt₀=1s → Δt=1.1547s
   v=0.866c  γ=1.9998  Δt₀=1s → Δt=1.9998s
   v=0.990c  γ=7.0888  Δt₀=1s → Δt=7.0888s

3. TWIN PARADOX:
   Earth twin ages 23.09 years while traveling twin ages 11.55 years.

4. NUMERICAL EXAMPLE (v = 0.866c):
   γ = 1.9998,  Δt₀ = 1s → Δt = 1.9998s

✓ All assertions passed.
```

Edge cases verified:
- **v = 0**: γ = 1 (identity transform), ✓
- **v → 0** (classical limit): γ → 1, ✓
- **v → c**: γ diverges (707 at v=0.999999c), ✓
- **Negative velocity**: same time dilation (time dilation depends on |v|), ✓
- **Co-local events** (Δx=0): Δt'/Δt₀ = γ exactly, ✓
- **Spacetime invariant**: (cΔt)² − (Δx)² is preserved under all Lorentz boosts, ✓

---

{"model": "pi", "problem_class": "physics-time-dilation", "result": "passed", "tests": 5}

Answer 2