Relativistic Escape Velocity Calculator
Calculates the escape velocity from a massive body using a relativistic correction for the Schwarzschild geometry. Useful for neutron stars, white dwarfs, and objects near black hole thresholds.
Last updated: September 2026
Formula below · 2 sources (nasa.gov, Wikipedia) · Updated Sep 2026
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About this calculator
Classical escape velocity is v_esc = √(2GM/r). In general relativity the Schwarzschild radius r_s = 2GM/c² sets the scale at which relativistic effects dominate, and a remarkable result holds: the escape speed measured by a stationary observer at radius r in Schwarzschild spacetime is exactly v_esc = √(2GM/r) = c·√(r_s/r) — the same expression as Newton's. What changes is its meaning: it is a locally measured speed, it reaches c exactly at the event horizon (r = r_s), and inside the horizon no escape is possible. This calculator uses G = 6.6743 × 10⁻¹¹ N·m²/kg² and c = 299,792,458 m/s, and reports when the radius is inside the Schwarzschild radius.
How to use
Consider a neutron star with mass M = 2 × 10³⁰ kg (about 1 solar mass) and radius r = 10,000 m (10 km). Step 1: r_s = 2 × 6.6743×10⁻¹¹ × 2×10³⁰ / (2.998×10⁸)² ≈ 2,970 m. Step 2: v_esc = c × √(r_s/r) = 2.998×10⁸ × √0.297 ≈ 1.634 × 10⁸ m/s ≈ 0.545c. Enter mass = 2×10³⁰ kg and radius = 10,000 m to reproduce this result. At r = r_s the escape speed would equal c.
Frequently asked questions
How does relativistic escape velocity differ from classical escape velocity?
Classical escape velocity assumes Newtonian gravity and ignores the curvature of spacetime, giving v_esc = √(2GM/r). In Schwarzschild spacetime the escape speed measured by a static observer turns out to be exactly the same expression, √(2GM/r) = c√(r_s/r). The difference is interpretation: in relativity it is a locally measured speed that reaches c exactly at the event horizon, and inside the horizon escape is impossible at any speed.
What is the Schwarzschild radius and why does it matter for escape velocity?
The Schwarzschild radius r_s = 2GM/c² is the radius at which an object's escape velocity equals the speed of light, forming a black hole event horizon if the object is compressed to that size. For Earth, r_s ≈ 9 mm; for the Sun, r_s ≈ 3 km. When an object's physical radius equals its Schwarzschild radius, no information or matter can escape — this is the definition of a black hole. In the escape velocity formula, the ratio r_s/r acts as a relativistic correction: the closer this ratio is to 1, the stronger the departure from Newtonian predictions.
What happens to escape velocity as the radius approaches the Schwarzschild radius?
Because v_esc = c√(r_s/r), the escape speed rises toward c as r approaches r_s and equals c exactly at the event horizon. Inside the horizon every future-directed path leads inward, so there is no escape speed at all; the calculator says so when you enter a radius below r_s.