Relativistic Escape Velocity

Relativistic Escape Velocity

In Newtonian gravity, the escape velocity from radius $r$ is $v_{\text{esc}} = \sqrt{2GM/r}$. General Relativity modifies this slightly, but the deeper insight is that the Schwarzschild radius marks the point where even light — moving at $c$ — cannot escape.

The GR Escape Velocity

From the Schwarzschild metric, the escape velocity from radius $r$ is:

$v_{\text{esc}} = c \sqrt{\frac{r_s}{r}} = c \sqrt{\frac{2GM}{c^2 r}}$

As a fraction of $c$:

$\frac{v_{\text{esc}}}{c} = \sqrt{\frac{2GM}{c^2 r}}$

At $r = r_s$, this equals 1 — the escape velocity equals the speed of light, which is why nothing escapes from inside the event horizon.

Event Horizon Check

A radius $r$ is inside (or at) the event horizon if $r \leq r_s = 2GM/c^2$.

Physical Values

Your Task

Implement these functions with all constants defined inside each function:

• escape_velocity(M, r) — returns $v_{\text{esc}} = c\sqrt{2GM/(c^2 r)}$ in m/s
• escape_velocity_fraction(M, r) — returns $v_{\text{esc}}/c = \sqrt{2GM/(c^2 r)}$
• event_horizon_check(M, r) — returns True if $r \leq 2GM/c^2$, else False