Black Holes
When a collapsing stellar core is too massive even for neutron degeneracy pressure to save it, nothing in known physics can halt the collapse. The result is the most extreme object in the universe — one so dense that not even light can escape it.
When even neutron degeneracy pressure fails
The previous lesson showed that neutron degeneracy pressure can support a collapsing stellar core up to a certain mass — but, as with electron degeneracy pressure and the Chandrasekhar limit before it, this support also has an upper limit. If the collapsing core of a sufficiently massive star (typically the remnant of an original star considerably more massive than the Sun) exceeds this limit, neutron degeneracy pressure cannot halt the collapse either. There is no known form of pressure capable of resisting gravity beyond this point, so the collapse continues without limit. The result is a black hole.
Escape velocity exceeding the speed of light
The defining property of a black hole follows directly from the idea of escape velocity, covered in full in the Escape Velocity lesson later in this module: the minimum speed an object needs to completely escape a body's gravitational pull. As a collapsing core shrinks to a smaller and smaller radius while retaining its mass, the escape velocity from its surface increases without limit. Once the core has collapsed to a small enough radius, the escape velocity there reaches and exceeds $c$, the speed of light.
Since nothing can travel faster than $c$, this means nothing — not matter, not even light itself — can escape from within that radius. The object can no longer emit or reflect any light or other radiation, so it appears completely black to any outside observer, and cannot be observed directly. The boundary marking this point of no return is called the event horizon, and its radius is called the Schwarzschild radius.
This derivation uses the ordinary (non-relativistic) escape velocity formula, which is not strictly the correct framework for describing gravity this extreme — a full treatment requires general relativity. Remarkably, though, this simplified route happens to give exactly the same result as the full general-relativistic calculation, which is why it remains a useful and commonly used way to estimate the Schwarzschild radius at this level.
Estimate the Schwarzschild radius of a black hole with a mass of 10 solar masses ($M \approx 2.0\times10^{31}\,\text{kg}$).
$$R_s = \frac{2GM}{c^2} = \frac{2 \times (6.67\times10^{-11}) \times (2.0\times10^{31})}{(3.00\times10^{8})^2} \approx 3.0\times10^{4}\,\text{m}$$
This gives an event horizon radius of around 30 km — despite containing ten times the mass of the Sun, this black hole's event horizon would fit comfortably within a large city. This is the general scale of stellar-mass black holes: event horizons only a few to a few tens of kilometres across, illustrating just how extraordinarily compact these objects are.
Supermassive black holes and the connection to quasars
Not all black holes form from the direct collapse of a single massive star. At the centres of most large galaxies, including our own, astronomers have found strong evidence for supermassive black holes, with masses ranging from millions to billions of times the mass of the Sun. Their formation and growth are still active areas of research, but their existence is now well established observationally, largely from their gravitational influence on the orbits of nearby stars and gas.
The Doppler Effect and Red Shift lesson earlier in this module introduced quasars as extraordinarily luminous, highly redshifted objects found at enormous distances. Quasars are now understood to be powered by exactly the supermassive black holes described here: as matter (gas, dust, and disrupted stellar material) spirals inward and accretes onto an active supermassive black hole, it is accelerated to enormous speeds and heated to extreme temperatures by friction and compression within the accretion disc, releasing colossal amounts of electromagnetic radiation before it crosses the event horizon. A quasar's extraordinary luminosity is not radiation from inside the black hole itself — which is impossible — but from this violently heated infalling matter just outside the event horizon.
Key takeaways
- A black hole forms when a collapsing stellar core is too massive for even neutron degeneracy pressure to halt the collapse.
- A black hole's defining property is that its escape velocity exceeds the speed of light, $c$, so nothing, including light, can escape from within its event horizon.
- Setting the escape velocity formula $v_{esc}=\sqrt{2GM/r}$ equal to $c$ and solving for $r$ gives the Schwarzschild radius, $R_s = \dfrac{2GM}{c^2}$, the radius of the event horizon.
- This simplified, non-relativistic derivation happens to match the result of a full general-relativistic treatment.
- A stellar-mass black hole of around 10 solar masses has an event horizon only tens of kilometres across.
- Supermassive black holes, with masses of millions to billions of solar masses, are found at the centres of most large galaxies, including our own.
- Quasars are powered by matter accreting onto active supermassive black holes, releasing enormous energy as it heats up while spiralling inward, just outside the event horizon.