How We Came to Know the Cosmos:
Space & Time

by Dr Helen Klus

# Chapter 14. Black Holes

## 14.1 Stellar black holes

After a supernova, the cores of the most massive stars can’t even be held together by neutron degeneracy and collapse in on themselves, becoming black holes. A black hole is defined as an object with a radius from which light cannot escape. This radius is known as the event horizon, which occurs at the Schwarzschild radius.

Black holes can be observed by their effect on the matter that surrounds them. In X-ray binaries, an accretion disc can form. Matter within the disc travels at different speeds, creating friction. This heats up the material, and X-rays are emitted close to the black hole, where it’s hottest. Jets of matter are also sometimes emitted along the black hole’s rotation axis.[1]

In 1973, Israeli-American physicist Jacob Bekenstein and American physicist John Archibald Wheeler devised the no-hair theorem, which states that only three properties can be attributed to black holes: mass, charge, and angular momentum.[2]

In 1974, British physicist Stephen Hawking showed that the only thing that appears to escape a black hole is Hawking radiation, which is a consequence of quantum mechanics[3] (discussed in Book II).

### 14.1.1 Escape velocity

To escape from a gravitational field, you need to achieve a kinetic energy equal to the gravitational potential energy (discussed in Chapter 5):

 1/2m1v2 = Gm1m2/r (14.1)

Here, G is the gravitational constant (G = 6.674×10-11 Nm2kg-2), m1 is your mass, v is your velocity, m2 is the mass of the object you are trying to escape from, and r is the distance between you and the centre of mass of the object you are trying to escape from.

Ignoring air resistance, the velocity you need to escape is therefore:

 vesc = √2Gm1/r (14.2)

The escape velocity of the Sun is:

 vesc of the Sun = √2G × (1.99×1030)/696,000,000 (14.3) = 617,623 m/s

The speed of light (c) is 299,792,458 m/s. In order for the gravitational field to be so strong that not even light can escape, the escape velocity must be higher than this:

 vesc = √2Gm1/r and so if vesc ≥ c then √2Gm1/r ≥ c and so 2Gm1/c2 ≥ r (14.4)

This is known as the Schwarzschild radius (rSch), after German physicist Karl Schwarzschild:

 2Gm1/c2 ≥ rSch (14.5)

The Sun has a Schwarzschild radius of:

 2G × (1.99×1030)/c2 ≥ 2,955 m (14.6)

This is well within its 696,000 km radius.

If a person were to approach the event horizon of a black hole, then what would happen next is a matter of perspective. For those watching from beyond the event horizon, gravitational time dilation (discussed in Chapter 8) means that an object falling into a black hole appears to slow down as it approaches the event horizon, taking an infinite time to reach it. Light also appears redder near the black hole, and so before an object reaches the event horizon it becomes too dim to be seen.[4]

From the perspective of a person falling into a black hole, the rest of the universe would appear to speed up, with time moving much faster. They would cross the event horizon in a finite time, although they would not be aware of when this was, as they would not be able to determine its exact location. In most cases, they would be drawn to the centre where they would probably be torn apart in a process known as spaghettification.[4]

One day, it may be possible for people to use black holes to travel to a different space or time, although this idea is still highly speculative because no one knows what’s at the centre of a black hole. Classical relativity predicts a singularity, a region of zero volume containing all of the star’s mass, but this is probably not correct because a theory of quantum gravity (discussed in Book II) is needed to explain the behaviour of objects so small and heavy, and these are still being developed.

### 14.1.3 Mass and density

When objects fall into a black hole, they add to its mass. A non-rotating black hole with the mass of the Sun will have a Schwarzschild radius of about 3 km, and this diameter will increase by about 3 km for every solar mass that falls into it:

 2G × (1 ×1.99×1030)/c2 ≥ 2,955 m (14.7) 2G × (2 ×1.99×1030)/c2 ≥ 5,911 m (14.8) 2G × (3 ×1.99×1030)/c2 ≥ 8,866 m (14.9) 2G × (10 ×1.99×1030)/c2 ≥ 29,555 m (14.10)

Black holes become less dense as they increase in mass. If a black hole’s mass doubles, its Schwarzschild radius doubles, and its volume increases by a factor of 23 = 8 because:

 Volume of a sphere = 4/3 πr3 (14.11)

If the mass doubles and the volume increases by a factor of 8, then the average density decreases by a factor of 4:

 Average density = 2 × Mass/23 × Volume = 1/4 Mass/Volume (14.12)

A black hole twice the mass of the Sun will have a Schwarzschild radius of 5,908 m, and an average density of:

 2 × 1.99×1030/4/3 × π × 5,9083 = 4.6 billion billion kg/m3 (14.13)

This density is an average, and so it’s still possible that the entire mass is contained in a region of zero volume and infinite density in the centre of the sphere created by the Schwarzschild radius. However, a theory of quantum gravity (discussed in Book II) is needed to explain what is at the centre of a black hole.

## 14.2 Supermassive black holes

The most massive black holes are known as supermassive black holes,[5] and these are thought to exist within the centre of most galaxies, including our own. The supermassive black hole at the centre of the Milky Way is known as Sagittarius A* and it is about 4 million times as massive as the Sun.[6]

Sagittarius A* has a Schwarzschild radius of:

 2G × (4×106 × 1.99×1030)/c2 ≥ 11.8 billion m (14.14)

and an average density of:

 4×106 × 1.99×1030/4/3 × π × (1.2×1010)3 = 1,099,716 kg/m3 (14.15)

H1821+643, one of the most massive supermassive black holes that we know of, is about 30 billion times as massive as the Sun. H1821+643 has a Schwarzschild radius of:

 2G × (3×1010 × 1.99×1030)/c2 ≥ 88,620 billion m (14.16)

and an average density of:

 3×1010 × 1.99×1030/4/3 × π × (8.9×1013)3 = 0.02 kg/m3 (14.17)

This is less dense than hydrogen gas.

While all the stars in the Galaxy orbit the Galactic centre, which contains Sagittarius A*, Sagittarius A* is also directly orbited by a number of stars from well beyond the Schwarzschild radius.[7] It’s likely that most other supermassive black holes are also directly orbited by stars.

 Figure 14.1Image credit A jet emanating from active galaxy M87, optical image taken by the Hubble Space Telescope in 1998.
 Figure 14.2Image credit Artist’s impression of a supermassive black hole with an accretion disc.

Supermassive black holes may also have accretion discs, which emit X-rays, just like the accretion discs around stellar black holes. Supermassive black holes like this are known as active galactic nuclei (AGN), and they were first detected in the 1960s.[8]

At the rotation axis of the supermassive black hole, matter from the accretion disc can be pushed away at the speed of light, creating jets that can extend for thousands of light-years. As these jets run out of energy, they flare out, creating radio lobes, which mostly emit lower energy radio waves.[9] Jets can sometimes emit light of other wavelengths, and the first optical jets were observed by American astronomer Heber Curtis in 1918, coming from the galaxy M87.[10] X-ray observations by the Einstein observatory later showed that M87 contains a supermassive black hole.[11]

AGN can be identified in many different ways depending on the angle they are viewed from. They appear most powerful when viewed along a region close to the jet. AGN that are viewed from this angle are known as blazars and quasars. When viewed at a 90° angle to this, AGN appear less luminous, and are known as radio galaxies.[12,13]

 Figure 14.3Image credit AGN are labelled blazars, quasars, or radio galaxies, depending on the angle at which we view them.

Jets can also be produced from dormant black holes if stars get too close, and are pulled within the Schwarzschild radius. Evidence that stars can fall into supermassive black holes came in 2005, when hyper-velocity stars were discovered.[7] These are thought to have been part of a binary star system that broke apart as it approached a supermassive black hole. As one star was captured, and the other was pushed away at a velocity exceeding the escape velocity of the Galaxy.

In 2011, a jet was produced from an otherwise dormant supermassive black hole in a galaxy 3.9 billion light-years away.[14,15] This was attributed to it capturing a star, in an event known as Swift J1644+57.[16] This supermassive black hole is thought to be about 8 million times the mass of the Sun, and twice the mass of Sagittarius A*.

 Figure 14.4Image credit Artist’s impression showing how supermassive black hole Swift J1644+57 produced a jet after destroying a star.

## 14.3 References

1. University of Cambridge, Black Holes and X-ray binaries, Institute of Astronomy X-Ray Group.

2. Misner, C. W., Thorne, K. S., Wheeler, J. A., Gravitation, W. H. Freeman, 1973.

3. Hawking, S. W., Communications in mathematical physics 1975, 43, 199–220.

4. University of California, Santa Barbara, What happens beyond an event horizon?, UCSB Science Line.

5. Luminet, J. P., Black Holes, Cambridge University Press, 1992.

6. Schödel, R., Ott, T., Genzel, R., Hofmann, R., Lehnert, M., Eckart, A., Mouawad, N., Alexander, T., Reid, M. J., Lenzen, R., Hartung, M., Nature 2002, 419, 694–696.

7. Brown, W. R., Geller, M. J., Kenyon, S. J., Kurtz, M. J., The Astrophysical Journal Letters 2005, 622, 33–36.

8. Lynden-Bell, D., Nature 1969, 223, 690–694.

9. Hardcastle, M. J., Philosophical Transactions of the Royal Society of London A: Mathematical Physical and Engineering Sciences 2005, 363, 2711–2727.

10. Curtis, H. D., Publications of Lick Observatory 1918, 13, 9–42.

11. Lea, S. M., Mushotzky, R., Holt, S. S., The Astrophysical Journal 1982, 262, 24–32.

12. NASA, Blazars and Active Galaxies, NASA.

13. Burrows, D. N., Kennea, J. A., Ghisellini, G., Mangano, V., Zhang, B., Page, K. L., Eracleous, M., Romano, P., Sakamoto, T., Falcone, A. D., Osborne, J. P., Nature 2011, 476, 421–424.

14. Zauderer, B. A., Berger, E., Soderberg, A. M., Loeb, A., Narayan, R., Frail, D. A., Petitpas, G. R., Brunthaler, A., Chornock, R., Carpenter, J. M., Pooley, G. G., Nature 2011, 476, 425–428.

15. NASA, Researchers Detail How a Distant Black Hole Devoured a Star, NASA, 2011.