## Black holes

by Wm. Robert Johnston
last updated 14 October 2001

Introduction

In pre-Einstein physics, a black hole might have been defined as a object with an escape velocity exceeding the speed of light. This definition does not work, however, because a black hole involves such curvature of space-time that it can only be explained with the use of general relativity:

A black hole is the result of mass-energy in such a concentration that space-time is curved enough to cut off contact with the outside universe.
As early as 1795, Laplace used Newtonian (non-relativistic) physics to speculate about "dark stars" or objects meeting the pre-Einstein "definition" above. Robert Oppenheimer and H. Snyder in 1939 first predicted (based on general relativity) that a large enough collapsing star would curve space-time and cut itself off from the universe. From 1964 to 1975 considerable theoretical work was done on black hole physics.

Every feature of a black hole can be determined from only three characteristics: its mass, its rate of spin, and its electric charge. For practical matters, its only influence on the rest of the universe is its gravity. Astronomers can only detect black holes by observing their gravitational influence on nearby stars or other matter.

Nature of a black hole

We will start simple: consider a black hole with three times the mass of the Sun, not rotating, no nearby matter. This black hole would have a circumference of about 55 kilometers. (If one were to erroneously use pre-Einstein physics one could calculate an escape velocity equal to the speed of light for this mass and circumference.) Compare this to a circumference of 10,000,000 km for a normal star with the same mass.

Far from the black hole, its gravity is indistinguishable from that of a object like a normal star of the same mass. For example, suppose there were a planet in a circular orbit around the 3-solar mass star and another planet in a circular orbit at the same distance from the 3-solar mass black hole. The planets would have the same orbital velocity and same orbital period. Twin astronauts, one 2,000,000 km from the center of the star and one 2,000,000 km from the black hole would be subjected to the same gravitational forces and fall at the same rate of acceleration. Of course, one would be vaporized by light from the nearby star while the other would see no light from the black hole.

The difference is that the black hole's mass is far more concentrated, producing very different results as we approach. Recall that general relativity explains that concentrations of mass produce curvature of space-time. For a black hole, this curvature becomes infinite. Thus the environment near the black hole is quite different from elsewhere.

Suppose the astronaut above continues to fall towards the black hole. One thing which he has already noticed is the distortion of star light around the black hole. Light from the stars behind the black hole follows a curved path due to the curved space-time near the black hole. (This path is a geodesic, or the shortest possible path.) The result is that the astronaut sees star images distorted into arcs. This is gravitational lensing (which has been used to identify some suspected black holes in Hubble Space Telescope images).

The gravitational force due to an object decreases with distance. For a spacecraft orbiting the Earth, this means that the force of gravity is slightly greater on the side closer to the Earth than on the side further from the Earth. This effect is very small for spacecraft like the Shuttle Orbiter. (It is the reason that NASA now uses the term "microgravity" instead of "zero gravity" to describe the environment in Earth orbit.) It is more significant for larger objects: this is the source of tides on the Earth (caused by the Moon and Sun).

For our astronaut falling toward the black hole, the tidal forces will not remain very small. He will actually notice a stretching force along a direction towards the black hole and a squeezing force perpendicular to that. By the time he is only 5,500 km from the black hole the stretching force will be as great as the force of gravity on Earth, and shortly after that he will be pulled apart lengthwise (forcing us to continue our thought experiment without our astronaut!).

The 55-km circumference for the black hole describes the point of no return. From within this sphere no matter or radiation (no communication of any kind) can escape. This is called the event horizon, since in effect no event beyond that limit (horizon) can ever be seen by the rest of the universe.

The curvature of space-time affects our measurements of distance. We have referred to the circumference of the black hole but not its radius, and we have spoken of the distance to the black hole without specifying which part of the black hole. Space-time is stretched towards the center of the black hole such that if we could measure the distance to the center (the radius) it would be infinite.

This central point of infinite curvature is the singularity--at least general relativity as we understand it says there is a singularity. All matter and energy that formed the black hole or enters the black hole must unavoidably reach the singularity.

Outside the event horizon there are peculiar effects on time and light. Someone falling toward the black hole (ignoring the fact that they would be pulled apart) would find that it takes a limited amount of time to reach the event horizon. He would see light from all objects in his sky compressed into a smaller and smaller circle opposite the black hole. In contrast, those at a safe distance from the black hole would see his fall slow down as he approached the event horizon. At the same time light from him would become dimmer and redder, until he faded from view apparently "frozen" at the event horizon.

In practice black holes rotate, and this produces some differences. The singularity will be a ring rather than a point. One can produce a solution to the equations of relativity in which the black hole is connected to a white hole in this case. Outside the event horizon the rotation of the black hole drags space-time around the black hole. As a result objects cannot move near the black hole in the opposite direction of its rotation.

When gas near a black hole falls into it, it tends to form a disk around the black hole. This is for two reasons: any component of velocity the gas has which is not directly toward the black hole causes the gas to swing around the black hole; and most black holes rotate which means they wrap space-time around themselves, forcing the gas to spiral inward. Gas in the disk drags against gas in adjacent parts of the disk. This causes the gas to spiral inward and causes it to heat up, releasing high energy x-rays and gamma radiation before finally falling into the black hole.

Possible origins and types of black holes

• Stellar-mass black holes: Such black holes would be the end states of stars much more massive than the Sun.

The Sun, like other stars, has sufficient temperature and pressure in its core to sustain nuclear fusion reactions. The heat and radiation released balances the tendency of the Sun to collapse inward under its own gravity. It would take a star like the Sun about 5 billion years to exhaust the hydrogen fuel for fusion in its core. Fusion of other elements would then inflate the star into a supergiant, but once available sources for fusion are expended the star must collapse.

As a supergiant star, the outer layers will be blown off before the final collapse. Stars more massive than perhaps ten times the Sun's mass will, as collapse begins, undergo an explosion of fusion of heavy elements. This is a supernova.

If the amount of collapsing mass is up to about 1.4 times the mass of the Sun, then it will collapse to a white dwarf, about the size of the Earth or one-one hundreth the diameter of the Sun. If the amount is from 1.4 to 2-3 times the mass of the Sun, it will collapse further. Electrons and protons will combine under these pressures, leaving a neutron star about 24 kilometers in diameter. Above 2-3 times the mass of the Sun, collapse is unstoppable and the star collapses indefinitely to form a black hole.

Astronomers have identified a few dozen objects which appear to be stellar-mass black holes, ranging from a few times the Sun's mass to a few dozen times its mass. Such black holes in binary systems with a close nearby star can pull gas from the companion star. As mentioned above, this will form a disk of gas spiralling into the black hole giving off x-rays. The black hole and companion star orbit their mutual center of mass. Since x-rays will also be produced by gas spiralling onto a neutron star, the evidence from the orbital motion is important to show that the invisible object is too massive to be a neutron star--and must be a black hole.

More recently, gravitational lensing has indicated some possible stellar-mass black holes.

• Supermassive black holes: These black holes are found mostly in the cores of galaxies and are hundreds to billions of times the mass of the Sun.

Suppose a stellar-mass black hole is formed in a region where stars are close together, such as the core of a galaxy. Other stars straying too close to the black hole could fall in, adding to its mass. This would explain what appear to be supermassive black holes which have accumulated the mass of millions to billions of stars. Of course, far from the black hole a star can safely orbit it. However, stars are so dense in the cores of galaxies that they gravitationally perturb each other, "feeding" such black holes with more infalling stars.

By observing the motion of stars in the cores of galaxies, astronomers have found a few dozen cases where the stars are moving under the influence of a very large amount of mass in a small region. In many cases this is accompanied by x-ray emissions from these regions, which would result from gas being superheated as it falls into a black hole.

Recently a few suspected black holes have been found with intermediate masses, ranging from hundreds to thousands times the mass of the Sun.

• Primordial black holes: These are hypothetical black holes formed during the Big Bang.

No process at work in the current universe can form a black hole smaller than 2-3 times the Sun's mass. In the 1970s Stephen Hawking and other physicists suggested that smaller black holes could have been formed during the Big Bang. Hawking has also shown that black holes do radiate gamma radiation due to quantum mechanical effects. This effect is negligible for stellar or supermassive black holes. This evaporation rate increases as the size of the black hole becomes smaller. Thus a primordial black hole with the mass of a small mountain would soon evaporate in a explosion of gamma radiation.

Astronomers have not detected gamma radiation that can be identified as coming from the evaporation of small primordial black holes, so such black holes remain hypothetical.

© 2001 by Wm. Robert Johnston.