Black holes and the Schwarzschild radius

What happens if we compress a mass into an ever-smaller sphere? The escape velocity $v_\text{fuga} = \sqrt{2GM/R}$ grows as $R$ decreases. Taking this reasoning to the extreme, there is a critical radius beyond which not even light can escape: this gives a black hole.

Schwarzschild radius

Imposing that the escape velocity equals the speed of light, $v_\text{fuga} = c$: $\sqrt{\frac{2GM}{R}} = c \;\Rightarrow\; \ev{R_S = \frac{2GM}{c^2}}$

The calculation presented here is semiclassical — a rigorous description requires general relativity — but it captures the correct order of magnitude and the essential idea: below $R_S$ gravity is so intense that nothing, light included, escapes.

Orders of magnitude

• Earth: $R_S \approx 9\;\text{mm}$. The entire mass of the Earth would need to be compressed into a sphere less than a centimetre across.
• Sun: $R_S \approx 3\;\text{km}$.
• Sagittarius A*, the black hole at the centre of our galaxy: a mass of about $4\cdot 10^6$ solar masses and $R_S \approx 12\cdot 10^6\;\text{km}$.

The Schwarzschild radius is not the physical size of the object, but that of its event horizon: the ideal surface separating the region from which escape is still possible from the one where it is impossible. The actual black hole is concentrated in an enormously smaller and denser volume.

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Topics: Gravitazione Concepts: Velocità di fuga Skills: Analisi di casi limite e fantafisica

Related exercises: Velocità di fuga dalla Terra (energia) · Vero o falso su energia e orbite · Dimostrazione della velocità di fuga