// Special & General Relativity · Gravitational Physics · Quantum Radiation
Relativistic Physics
Calculator Suite
Scientifically rigorous tools for exploring the extremes of space, time, and gravity — from orbital mechanics to black hole horizons.
Lorentz Factor & Relativistic Momentum
Special relativity kinematics — time dilation, length contraction, relativistic momentum & kinetic energy
Governing Equations
γ = 1 / √(1 − v²/c²) · Lorentz factor
t = γ·t₀ · Time dilation
L = L₀/γ · Length contraction
p = γ·m·v · Relativistic momentum
K = (γ−1)·m·c² · Relativistic kinetic energy
E_total = γ·m·c² · Total energy (mass-energy)
Enter the object's speed as a fraction of c (e.g. 0.99 for 99% of light speed), rest mass, proper time, and rest length. All relativistic effects scale with Lorentz factor γ ≥ 1.
Presets:
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Relativistic Velocity Addition
Einstein's velocity addition theorem — velocities cannot naïvely sum past c
Einstein Velocity Addition (1905)
w = (u + v) / (1 + u·v/c²)
For collinear motion. w always < c when |u|, |v| < c. Classic mechanics recovers at u,v ≪ c.
Enter two velocities as fractions of c. In classic Galilean mechanics, w = u + v, which violates special relativity. Einstein's formula ensures the result is always sub-luminal.
Presets:
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Gravitational Time Dilation (Schwarzschild Metric)
General Relativity — clocks run slower in deeper gravitational wells
Schwarzschild Time Dilation (Einstein 1915)
t_far / t_near = 1 / √(1 − r_s/r) · where r_s = 2GM/c²
Δf/f ≈ GM/(r·c²) · Gravitational redshift (weak field)
t_elapsed = t₀ · √(1 − r_s/r) · Proper time at radius r
At r = r_s (Schwarzschild radius): time stops. At r → ∞: t_elapsed → t₀. GPS satellites require +45.9 μs/day correction.
Enter the central body's mass and the observer's distance. The calculator computes proper time elapsed vs. coordinate time at infinity — the essence of gravitational time dilation as tested by Pound–Rebka (1959) and GPS satellites.
Body Presets:
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Combined Dilation: Orbital Velocity + Gravity (GPS Problem)
Real-world scenario — both SR (velocity) and GR (gravity) dilation act simultaneously
Total Time Rate (Schwarzschild + Kinematic)
dτ/dt = √( (1 − r_s/r) − v²/c² )
ΔT_GR = +t₀ · GM/(r·c²) · gravitational gain (clocks speed up aloft)
ΔT_SR = −t₀ · v²/(2c²) · kinematic loss (orbital velocity slows clock)
GPS clocks run fast by +45.9 μs/day (GR) but slow by −7.2 μs/day (SR), net +38.7 μs/day. Without correction, GPS drifts ~10 km/day.
Model a satellite (or any orbiting/moving body) with both orbital speed and altitude. Orbital velocity for a circular orbit is computed automatically if you check "circular orbit". Compare with GPS results.
Presets:
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Schwarzschild Black Hole — Complete Model
Horizon radius, photon sphere, ISCO, Hawking temperature, evaporation time, tidal forces
Key Black Hole Equations
r_s = 2GM/c² · Schwarzschild (event horizon) radius
r_ph = 1.5·r_s · Photon sphere radius
r_ISCO = 3·r_s · Innermost Stable Circular Orbit
T_H = ħc³/(8πGMk_B) · Hawking temperature
t_evap = 5120·π·G²·M³/(ħc⁴) · Evaporation lifetime
Δa_tidal ≈ 2GM·l/r³ · Tidal acceleration across body of length l
Enter the black hole's mass in solar masses (M☉ = 1.989 × 10³⁰ kg). Real astrophysical examples range from stellar-mass (~3–100 M☉) to supermassive (10⁶–10¹⁰ M☉). Hawking evaporation is negligible for astrophysical black holes.
Known BHs:
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Escape Velocity & Surface Gravity
Classical and relativistic escape conditions from compact objects
Escape Velocity & Compactness
v_esc = √(2GM/r) · Classical escape velocity
v_esc/c = √(r_s/r) · Relativistic compactness ratio
g_surface = GM/r² · Surface gravitational acceleration
Φ = −GM/r · Gravitational potential (Newtonian)
Bodies:
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Radiation Pressure & Photon Momentum
Electromagnetic radiation pressure — solar sails, stellar winds, Eddington luminosity
Radiation Pressure Equations
P_rad = I/c · Radiation pressure (absorbing surface)
P_rad = 2I/c · Radiation pressure (perfect reflector)
I = L/(4πr²) · Irradiance at distance r from isotropic source
F = P_rad · A · Force on surface of area A
p_photon = E/c = hν/c · Photon momentum
P_thermal = (4σ/3c)·T⁴ · Equilibrium radiation pressure (blackbody)
Enter the source luminosity, distance, and sail/surface parameters. The calculator gives radiation force, acceleration (if mass is provided), and compares to solar wind pressure. Choose "reflective" (sail) or "absorptive" surface mode.
Scenarios:
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Eddington Luminosity & Stellar Mass Limit
Equilibrium between radiation pressure and gravity — determines maximum stellar luminosity
Eddington Limit (Radiation–Gravity Balance)
L_Edd = 4πGMc/κ · κ ≈ 0.2(1+X) cm²/g (electron scattering opacity)
L_Edd ≈ 1.26 × 10³¹ · (M/M☉) · [W] for pure hydrogen plasma
M_crit = κ·L/(4πGc) · Minimum mass to remain gravitationally bound
Stars exceeding L_Edd drive mass loss via radiation. AGN accreting at super-Eddington rates form ULX/quasar winds.
Enter the object mass and hydrogen fraction X. For pure hydrogen, X = 1; for solar composition X ≈ 0.74. The calculator also shows where real sources fall relative to Eddington and their Eddington ratio η = L/L_Edd.
Objects:
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