A very brief introduction to Black Holes by Gordon Dennis

1. Introduction

The existence of what we today call Black Holes was first predicted by the Englishman John Mitchel (1724-1793) in a 1783 paper presented to the Royal Society in London. He reasoned that a star’s gravitational pull might be so strong that the escape velocity would exceed the speed of light.

Albert Einstein (1879–1955) published his Special Theory of Relativity (SR), which formalised the concept of space-time, in 1905. Einstein followed this with the General Theory of Relativity (GR) in 1917, which built on SR and provided a new theory of gravity in space-time. GR predicts that gravitational collapse can lead to an object so dense that nothing - not even light - can escape.

Karl Schwarzschild (1873–1916) developed a solution to Einstein’s Field Equations and in that process discovered the radius of the event horizon of a Black Hole. The event horizon is the radius at which if an object were to fall below it, the object would no longer be visible to an external observer. Note that the event horizon is not the same as the Black Hole itself. At the centre of a sphere of radius equal to the Schwarzschild Radius, Rs, as predicted in GR, there is a singularity of r = 0 and of infinite density which is the Black Hole.

Today (2021), It is believed there are at least two different cases of origin for a Black Hole:

  • The final stage of the evolution of large stars (≥ 8M⊙) is gravitational collapse in a Type II supernova event, the final compression leading to the formation of a stellar mass Black Hole.
  • It is also believed that as a result of stellar coalescence, a super-massive Black Hole lies at the centre of most, if not all, galaxies in the universe. The Black Hole at the centre of our Galaxy, the Milky Way, is called Sagittarius A* (pronounced ’A star’).


2 The event horizon of a Black Hole

2.1 The event horizon and popular misconceptions
The event horizon of a Black Hole can be thought of as the boundary within which the black hole’s escape velocity is greater than the speed of light, i.e. Vesc > c. Since nothing can exceed c, this means that any object falling into the event horizon will never be able to escape from it.

A common misconception about Black holes is that they ”hoover up” any matter that approaches them. This is not the case. Black Holes can’t seek out material to consume any more than any other massive object such as a planet or a star.

Another common misconception is that matter can be observed falling into a black hole. This is also impossible. A distant observer will witness the object moving slower and slower, while any light the object emits will be further and further redshifted.

We can detect accretion disks around black holes, where material moves with such speed that friction creates high-energy radiation [4]. Some matter from these accretion disks is forced out at near relativistic velocity along the Black Hole’s spin axis. When this matter collides with the inter-stellar medium, visible jets are created which may be light years in length. An example of this is the active galaxy M87, which is emitting a relativistic jet at least 1,500 parsecs long.

2.2 The Schwarstschild metric
The Schwarzschild metric, sometimes called the Schwarzschild solution, is a solution of Einstein’s field equations in empty space. The solution is valid only outside the gravitating body; for a spherical body of radius R, the solution is valid for r > R. The Schwarzschild solution describes spacetime under the
influence of a massive, non-rotating, spherically symmetric object. Using the Virial Theorem, we may equate the kinetic energy of a small mass, m trying to escape the gravity of a much larger mass M:

The actual Solar radius, R⊙ = 6.96 ∗ 10⁵ km is evidently very much larger than the three km Schwarstschild radius for one solar mass (by way of comparison, the Schwarstschild radius for an object of mass equal to one Earth mass is about 9 millimeters).
 
Equation 4 provides us with a useful calibration tool, in that RS scales as 3kmM⊙−1.

Putting this in the context of the solar system, we know that the mean orbital distance of Mercury is 5.79 ∗ 10⁷km. Hence we can conclude that if the Sun was somehow replaced by the hypothetical one solar mass Black Hole, Mercury’s current orbital parameters would place it well outside the event horizon.

This in turn means that Mercury (or any of the other Solar system planets) would not be in danger of being ”sucked into the Black Hole” as is sometimes popularly suggested.

2.3 Other geometric representations
The Virial Theorem assumes a non-rotating system. Examples of these in astronomy include globular clusters. However, it is known that Black Holes, like stars and spiral galaxies, do rotate.

Whereas the Schwarstschild metric describes a non-rotating Black Hole, the Kerr metric, discovered in 1963 by New Zealand mathematician Roy Kerr (1934-), describes the geometry of empty space-time around a rotating, uncharged, axially-symmetric Black Hole with a quasi-spherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of GR.

Rotating Black Holes have two surfaces with the size and shape of these surfaces depending on the black hole’s mass and angular momentum. The outer surface bounds a region called the ergosphere and has a shape similar to an oblate spheroid. The inner surface marks the event horizon; objects passing into the interior of this horizon can never again communicate with the world outside that horizon. Note neither surface has a physical presence.

In summary, if Q represents the body’s electric charge and J represents its spin angular momentum:

• In the Schwarzschild metric, Q = 0 and J = 0.
• In the Kerr metric, Q = 0 and J ≠ 0.

The Kerr metric can be represented in Cartesian Coordinates as:

Kerr’s mathematical discovery was almost serendipitous. Prior to that time, Black Holes were commonly regarded as interesting theoretical objects which might or might not exist in nature. However, the first quasars (3C 48 and 3C 273) were discovered in the late 1950s, as radio sources in all-sky radio surveys. They appeared mysterious in that the emission lines of quasar spectra did not correspond to any known chemical elements. In 1963, the same year that Kerr published his result, Dutch-born astronomer and Caltech professor Maarten Schmidt (1929-) was studying 3C273 and realized that one emission line was that of hydrogen, albeit very highly redshifted. In turn, that indicated 3C273 was very distant (actually ≈ 3 ∗ 10⁹lyr distant). For the quasar to be so far away and still visible; Schmidt concluded it must be of very high luminosity. (3C273’s apparent V magnitude is 12.9, so within range of amateur telescopes. 3C273 has the highest apparent magnitude of any currently known quasar and is the closest quasar to the Solar system.)

Following these discoveries, Black Holes were no longer a theoretical curiosity and there has been much recent observational work done on quasars. An indication of how technology has advanced in recent years can be discerned from the fact that just one quasar was observable by the European Space Agency Hipparchus mission (launched 1989, ran for 4 years). The follow-on ESA GAIA mission is observing more than 500,000 quasars (Perriman, 2022).

In fact, quasars (which are distant active galaxies, and should not to be confused with pulsars, which are neutron stars) are the intensely powerful centres of distant, active galaxies, powered by an accretion disc of particles surrounding a supermassive Black Hole. 3C273 is now estimated to have luminosity 2∗10¹²L⊙. Yet 3C273 appears to be less than a light-year across (our Galaxy is thought to be                  ≈  100,000lyr in diameter).


3 Sagittarius A*

Sagittarius A* is the designation assigned to the Black Hole at the centre of our Galaxy. It was first discovered in 1951 by Australians Harry Minnett (1917-2003) and Jack Piddington (1910-1997) who used radio observations at 1200MHz and 3000MHz (Winnett and Piddington, 1951) .

This was confirmed by observations in 1994 by a team led by Reinhard Genzel (Genzel1, Hollenbach and Townes, 1994) of the Max Planck Inst. fur Extraterrestrische Physik. Genzel and Andrea Mia Ghez of the University of California, Los Angeles were jointly awarded the Nobel Prize in Physics in 2020 for their discovery that positively identified Sagittarius A* as a supermassive compact object, for which a Black Hole is the only currently known explanation.

Genzel’s team reported ten years of observations in 2002 of the motion of the star S2 orbiting Sagittarius A*. The observations of S2 used near-infrared (NIR) interferometry at λ = 2.2m. The VLBI radio observations of Sagittarius A* were aligned with the NIR images. The rapid motion of S2 (and other stars close to Sagittarius A*) were easily distinguishable from slower-moving stars along the line-of-sight so these could be subtracted from the images. They found that S2 is in a highly eccentric Keplerian orbit (ϵ = 0.8843, one focus of which was at Sagittarius A*. From this, the team deduced the mass of Sagittarius A* to be 4.1∗ 10⁶M⊙, and that the source radius is no more than 17 light-hours (120AU)

Using this result and Equation 4, we may deduce that the Schwarstschild radius of Sagittarius A* is:

i.e ≈ 3 orders of magnitude larger than the Schwarstschild radius of Sagittarius A*.

If you prefer. this can be equivalently expressed as ≈ 5.9 ∗ 10⁻⁴parsecs (recall that M87s jet extends at least 1,500parsecs). Alternatively this is also equivalent to 1.9 ∗ 10⁻³lyr or 120AU.

4 Hawking radiation and Black Hole evaporation

In 1974, Stephen Hawking (1942-2018), determined that a Black Hole should emit radiation with a perfect black body spectrum (Hawking, 1974). Assuming Hawking’s theory is correct, Black Holes are expected to shrink and evaporate over time as they lose mass by the emission of photons and other particles via this phenomenon.

The temperature of this Hawking radiation, the Hawking temperature, is directly proportional to the Black Hole’s surface gravity:
                                                                 TH ∝ gBH                                                        (10)

and in the case of a Schwarzschild Black Hole, it’s gravitational force is inversely proportional to the mass:
                                                              gBH ∝ 1/MBH                                                     (11)

Hence, large black holes emit less radiation than small black holes, and so large Black Holes will take longer to fully evaporate. A stellar black hole of M⊙ has a Hawking temperature of only 6.2 ∗ 10⁻⁸K, whereas the cosmic microwave background radiation has T ≈ 2.7K. Hence, stellar-mass Black holes receive more radiation (and thus, more mass) from the cosmic microwave background than they emit through Hawking radiation and will grow instead of shrinking.

Although the theoretical evidence for Hawking radiation is very strong, the onlyblack holes we have  seen in nature are the size of stars or galaxies, and their Hawking radiation is invisible. To be able to evaporate, a Black Hole must have a Hawking temperature larger than 2.7K and)mass less than the Moon. Using Equation 4, the event horizon of this Black Hole would have a RS < 10⁻⁴m.


5 Conclusions

Black Holes are objects so dense that nothing -not even light can escape their gravitational field. Black Holes are at the centre of a sphere the radius of which is called the event horizon. any object passing through the event horizon can never escape out of it.

The possible existence of Black Holes was first suggested in the eighteenth Century. In the second half of the twentieth Century, the discovery of Quasars prompted renewed interest in Black Holes as the only conceivable source of such vast energy in active galaxies such as M87.

It is now recognised that Black Holes are located at the centre of most galaxies in the Universe. Our own Galaxy, the Milky Way, has a Black hole at the Galactic centre named Sagittarius A*. It is estimated that Sagittarius A* has a mass ≈ 4.15M⊙. This has been confirmed by observations of stars orbiting the Black Hole.


6 References

R Genzel1, R Hollenbach, D and Townes, C (1994). The nucleus of our Galaxy Report on Progress in Physics, Volume 57, Number 5 https://iopscience.iop.org/article/10.1088/0034-4885/57/5/001/pdf Perriman, M. In lecture presented to AGM of Wells and Mendip Astronomers, January 23 2021.

Hawking, S (1974). Black hole explosions?. Nature 248, 30–31 (1974). https://doi.org/10.1038/248030a0

Wang, Q et al (2013). Dissecting X-ray-Emitting Gas Around the Center of Our Galaxy. https://ui.adsabs.harvard.edu/abs/2013Sci...341..981W/abstract.

Winnett, H and Piddington, J (1951). Observations of Galactic radiation at 1200 and 3000 Mc/s https://articles.adsabs.harvard.edu/pdf/1951AuSRA...4..459P