Black Holes

Horizons, mass scale, temperature, and thermodynamics of black holes. Theory, formulas, worked examples and interactive graphs.

Conceptual context

A Schwarzschild black hole is the cleanest arena for studying a horizon without spin or charge. From far away, the whole exterior field is summarized by $M$, measured through its gravitational influence. The central question is simple: when $M$ changes, how does $r_s$ change? The horizon is not a material surface. It is a causal boundary: after crossing it, no light signal can return to the exterior. That is why $c$ is not a technical detail but the rule separating escaping paths from trapped paths. This leaf links geometry, gravitational scale, and thermodynamics through $A$, $T_H$, and $S_BH$.

🟢 Essential level

The essential idea is that $r_s$ grows in direct proportion to $M$.

A more massive black hole is not necessarily a denser ordinary object; its horizon moves outward because the space-time curvature that traps light is organized over a larger length scale. If $M$ doubles, $r_s$ doubles. The horizon marks the place where escape would require exceeding $c$, which no physical particle or signal can do. Therefore $r_s$ should not be read as the size of a solid ball, but as the geometric limit of communication with the outside. The graph of $r_s$ against $M$ should be read as an increasing straight line: greater $M$, larger horizon. Its slope tells how much horizon length each added kilogram contributes. This is the calculator’s main physical message.

On April 10, 2019, the Event Horizon Telescope collaboration released the first direct image of an event horizon: the shadow of the supermassive black hole at the center of the M87 galaxy, obtained by combining radio telescopes worldwide into an Earth-sized interferometer.

🔵 Formal level

Basic formulas

In the Schwarzschild model, the exterior space-time is static, spherically symmetric, and parameterized by $M$, $G$, and $c$. The central relation is:

Schwarzschild radius

This expression defines the fundamental geometric scale of the black hole, the radius beyond which the Newtonian description stops being valid.

Real student questions

Does the black hole have a physical surface? Not in the Schwarzschild model: the horizon is a causal boundary associated with $r_s$, not a material wall.

Why does $r_s$ increase with $M$? Because greater $M$ creates a larger gravitational scale for the relativistic escape boundary set by $c$.

Can I see $r_s$ directly in an image? Not exactly. Images show emitting exterior matter and gravitational lensing; $r_s$ is the theoretical horizon scale.

Does temperature increase with size? For Schwarzschild black holes, the opposite occurs: $T_H$ decreases when $M$ increases.

Final synthesis

Gravitational mass measured from far away fixes the causal size of the horizon. In Schwarzschild geometry, $M$ directly determines $r_s$, and that scale carries orbit, area, temperature, entropy, and graph interpretation.