The Schwarzschild metric and black holes

Solving Einstein’s equations in the simplest possible case — the vacuum outside a static spherical source of mass $M$ — Karl Schwarzschild found in 1916, just weeks before his death at the front, the so-called Schwarzschild metric. Without writing the metric tensor in full tensorial form, the effect on the spacetime “distance” (the Minkowski invariant) between two nearby events can be summarised as:

$(\Delta\tau)^2 = \left(1 - \frac{r_s}{r}\right)(\Delta t)^2 - \frac{1}{c^2\left(1 - r_s/r\right)}(\Delta r)^2 - \frac{r^2}{c^2}\Delta\Omega^2$

where $r$ is the distance from the centre of the mass, $\Delta\Omega$ encodes the angular variations, and $r_s$ is the Schwarzschild radius.

Schwarzschild radius

For a mass $M$ the Schwarzschild radius is $\ev{r_s = \frac{2\,G\,M}{c^2}}$ It is the characteristic scale of the gravitational metric produced by $M$. For the Sun it is about $3$ km; for the Earth about $9$ mm; for the mass of a human being ($\sim 70$ kg) about $10^{-25}$ m, infinitesimal compared with the diameter of an atom.

For $r \gg r_s$ the metric returns to flat, indistinguishable from that of Minkowski: the corrections of general relativity become noticeable only near very massive and compact objects. At large distances the formula reproduces gravitational time dilation: a clock at rest at distance $r$ measures a proper time $\Delta\tau \approx \sqrt{1 - r_s/r}\,\Delta t \approx \left(1 - \frac{r_s}{2r}\right)\Delta t.$ The closer one gets to the mass, the more $\sqrt{1-r_s/r}$ decreases, and the more $\Delta\tau$ shortens relative to $\Delta t$ (the time read by an observer at infinity): the clock slows down.

Schwarzschild radius of the Sun and of the Earth

With $M_\odot = 1.99\cdot 10^{30}$ kg and $M_T = 5.97\cdot 10^{24}$ kg: $r_s^\odot = \frac{2\cdot 6.67\cdot 10^{-11}\cdot 1.99\cdot 10^{30}}{(3\cdot 10^8)^2} \approx 2.95\cdot 10^3\;\text{m} \approx 2.95\;\text{km}.$ $r_s^T = \frac{2\cdot 6.67\cdot 10^{-11}\cdot 5.97\cdot 10^{24}}{(3\cdot 10^8)^2} \approx 8.87\cdot 10^{-3}\;\text{m} \approx 8.87\;\text{mm}.$ To become a black hole, the Sun would have to collapse into a sphere of about $3$ km radius; the Earth into one of about $9$ mm.

Black holes

What happens to an object so compact that its effective radius $r$ is smaller than $r_s$? The metric becomes singular: the factor $(1 - r_s/r)$ changes sign, and time and space “swap” roles. The surface $r = r_s$ is called the event horizon: anyone crossing it while descending towards the centre can no longer turn back, because all light cones now point towards $r = 0$. This is what we call a black hole.

Black hole (operational definition)

A black hole is an object whose mass $M$ is concentrated within a region of radius smaller than the Schwarzschild radius $r_s = 2GM/c^2$. The surface $r = r_s$ is called the event horizon: nothing, not even light, can escape from it.

To an external observer, an object falling into a black hole appears to slow down as it approaches $r_s$ (its clock runs ever more slowly), and at the same time the light reaching us from it shifts ever further into the red, until it fades out entirely. For the infalling object, however, crossing $r_s$ is a perfectly ordinary event: locally, at the horizon, nothing special happens. This is one of the most peculiar asymmetries in general relativity.

Real black holes

In 2019 the international Event Horizon Telescope collaboration published the first “shadow” image of a black hole, the one at the centre of the galaxy M87 (Event Horizon Telescope Collaboration 2019): mass $\sim 6.5\cdot 10^9$ solar masses, $r_s\approx 2\cdot 10^{13}$ m. In 2022 a similar image arrived of Sagittarius A*, at the centre of the Milky Way (mass $\sim 4.3\cdot 10^6$ solar masses, $r_s \approx 1.3\cdot 10^{10}$ m). Stellar black holes, formed from the collapse of a star of more than about $20$ solar masses, have $r_s \approx 30$ km; the supermassive ones at galactic centres range from $10^6$ to $10^{10}$ solar masses.

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Topics: Relatività ristretta Concepts: Invariante spazio-temporale · Legge di gravitazione universale

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