Black Holes

299-311




Gravity is both midwife and undertaker for the stars. Stars are born through the gravity-driven collapse of large interstellar gas and dust clouds. Throughout their evolution stars wage a continual struggle against the constant, inward crush of gravity. Finally all stars sucuumb, in one way or another to gravity. For low mass stars such as our sun the end comes as a dead and slowly cooling white dwarf. Large stars will die violently as supernovae and become neutron stars. For other massive stars their is one more possible end - the black hole.

Although the detailed theory of black holes is less than fifty years old the idea does extend back to the late 18th century when two very different astronomers mused over the possible existence of objects that we, today call black holes. As early as 1784 John Michell (clergyman and amateur astronomer) wrote of objects so massive that their escape velocity equalled or exceeded the velocity of light. French physicist Pierre Simon de Laplace echoed this in early editions of Mechanique Celeste. The conclusion reached was that the most massive stars in the heavens may, in fact, not be visible at all!

Our modern understanding of black holes has only emerged since the development of Einstein's theory of General Relativity and today the black hole plays a vital role in many porcesses that occur in the universe.

 

Escape Velocity and "Newtonian Black Holes"

Surprisingly Newton's law of gravitation "predicts" the existence of BH's. Even though the physics is a bit "flakey" and the final result agrees (probably coincidentally) with Einstein's, it is instructive to go through the reasoning that leads to the idea of a black hole:

  1. As you saw earlier, if you toss a baseball straight up from the surace of the earth it will climb to a point were all of the initial kinetic energy that you gave the ball by throwing has been converted into gravitational potential energy. Once the ball has lost its kinetic energy it will begin to fall back down.
  2. The greater the initial kinetic the higher the object climbs before beginning its descent
  3. If you provide an object with enough kinetic energy so that even "at infinity" it still has some kinetic energy left over then it will never fall back down.
  4. The escape velocity is the initial velocity that you must give an object to enable it to continue to move away from another body and never "fall back down" in that bodie's gravitational field.
  5. The more massive an object the greater its escape velocity and the denser an object the greater its escape velocity

Newton's theory of gravity gives a simple formula for calculating escape velocity and it was this that led both Mitchell and Laplace to conclude that an object could be massive enough for its escape velocity to be greater than the speed of light. Therefore, they reasoned, light could not move fast enough to escape from the star and we would never see it!

Another way to understand this is to realize that of you compress an object without changing its mass then the escape velocity will also increase. If you compress an object enough its escape velocity will eventually equal the speed of light - again - we won't see the star. We can express this with the following "simple" formula:

In this expression "RS"is called the Schwarzschild Radius named after Karl Schwarzschild who derived this expression from Einstein's equations in 1917.

The Schwarzschild Radius represents the size at which the escape velocity from and object equals the speed of light. For the earth this is a very tiny 8.9 millimeters or about the size of a grape! The following interactive graph shows the Schwarzschild radii for a number of objects:

 
Figure 11.7 Interactive graph showing the Schwarzschild radius for a given mass. The radius is expressed in km while the mass is in solar units.

Example 11.4 What is the size of the Schwarzschild radius for our sun? What does this mean?

Solution: Either by using the formula given above or the graph in Figure 11.7 you can determine that the Schwarzschild radius for the sun is 2.95 km. If a star the mass of the sun was compressed to this size or smaller it would become a black hole.

The Schwarzschild Radius and Einstein's Black Holes

Recall from an earlier discussion that Einstein's theory of gravity interprets gravity, not as a force but as a "warping" of space. The phrase:

Space tells matter how to move: Matter tells space how to curve

sums this idea up. Any mass causes a warping or bending of the structure of space. This is often depicted using the "rubber sheet" analogy which is illustrated in Figure 11.8.

Figure 11.8 The rubber sheet analogy compares the warping of a rubber sheet when a mass is placed on it to the effect that mass has on the geometry of space.

The emergence of Black Holes in Einstein's theory is a consequence of the warping effect that matter has on space. In Figure 11.8 the analogy shows the "mild" warp created by an object that is still much larger than its Schwarzschild radius. As the density of an object grows so does the intensity of its gravitational affect on its surroundings which causes an even greater warping of space. Eventually the space "engulfs" the object - it is pinched off from the rest of the universe. The Schwarzschild Radius represents the boundary separating the Black Hole from the universe. Once you have crossed the Schwarzschild Radius you are gone forever!

Event Horizons, Singularities and Spacetime

Mass affects not only space but time as well. In fact it is more correct to talk about spacetime since an essential point in Einstein's theory is that space and time are not independent concepts. Just as the curvature of space increases in the presence of an intense gravitational field, clock rates decrease. If you moved a clock toward a Black Hole the clock rate would slow until, when you reached the Schwarzschild radius the clock would appear to stop altogether. Physicists call this the event horizon for the black hole.

The event horizon is the barrier that separate the black hole from our universe. Once a star (or anything) crosses the event horizon it continues to fall inward to a central point called a singularity at which point the object occupies zero volume and has infinite density! The singularity is a mathematical idea - we cannot see singularity - it is not in our universe!

Figure 11.9 illustrates these ideas. The strong warping of space around the event horizon causes light from background objects to be deflected with the result that background objects are distorted in exactly the same way that a strong magnifying lens can distort images. This effect is called gravitational lensing and has been observed in many places in the universe.

We will return to this idea in a later chapter. Figure 11.9 Roll-over image showing the event horizon and singularity for a black hole with a starfield in the basckground.

How Black Holes Form - Uncontrolled Collapse

The final stages of stellar evolution involve the collapse and possible new equilibrium position for a stellar core. A few key numbers emerge:
  • White Dwarfs form if mass is less than 1.4 solar masses. The star's weight is supported by a degenerate electron gas in a predominantly carbon-oxygen core.
  • Neutron Stars form if mass less than approximately 3 solar masses. The star now consists of a degenerate neutron gas which is able to produce enough pressure to support the star as long as the mass does not exceed
  • Black Hole forms if there is too much mass present - neutron degeneracy pressure cannot support the star and if comresses to within its Schwarzschild radius.
So, unless a massive star is able to shed most of its mass (less than 3 solar masses) it will likely produce a black hole.

Looking for Black Holes

An important question to ask is how do we know black holes exist? If, by definition, nothing "escapes" from a black hole then it would seem impossible to verify their existence. However, it is important to be clear on what you mean when you say "oberve a black hole". The concensus is that we will never observe the the singularity - it is forever hidden on the other side of the event horizon. We can, under the right circumstances, however detect the event horizon.

X-ray Binaries

If a black hole is part of a close binary system then the same accretion processes that produce x-ray emission in white dwarf and neutron star binaries should be active here as well. So - one signature could be the emission of x-rays from a compact region. This in itself would not be distinguisable from a neutron star binary. However, if we can determine the orbital characteristics of the x-ray binary then we can use the measurement of mass to tell whether the system contains a neutron star or a black hole. Table 11.1 shows a partial list of black hole candidates. These are all binary systems for which accurate masses of "unseen" companions have been measured. For example, Cygnus X-1 was the first candidate black hole to be discovered. The unseen companion should have a mass of 16 solar masses - more than enough to become a black hole.

Candidate
Distance (ly)
Companion star Spectral Type
Period (h)
Mass of
black hole (Mo)
GRO J0422+32
Nova Per 1992
6,500
M2 V
5.0918
4.30.7
LMC X-3
180,000
B3 Ve
40.9150
7.61.6
LMC X-1
R148
180,000
O7 III-V
101.491
73
A 0620-00
Nova Mon 1975
2,700
K4 V
7.7524
10.82.1
GRS 1009-45
Nova Vela 1993
10,000
K8 V
6.8449
4.20.6?
XTE J1118+480
KV Uma
5500
K7-M0 V
4.0783
6.50.4
GRS 1124-683
Nova Muscae 1991
3,000
K5 V
10.3825
7.30.9
4U 1543-47
IL Lup
13,000
A2 V
26.7938
9.41.0
XTE J1550-564
8,500
K3 III
37.044
9.61.2
GRO J1655-40
Nova Sco 1994
10,000
F4IV
62.926
6.30.3
GRS 1659-487
GX 339-4
13,000
F8-G2 III
14.8
5.80.5
H 1705-250
Nova Oph 1977
33,000
K5 V
12.5
7.01.3
XTE J1819-254
V4641 Sgr
32,000
B9 III
67.6152
7.10.3
Cygnus X-1
HDE 226868
7,000
09.7 Iab
134.396
165
GS 2000+250
Nova Vul 1988
6,500
K5 V
8.2582
7.50.3
GS 2023+338
V404 Cygni
8,000
K0 III
155.31
11.71.7
Table 11.1 Partial list of black hole candidates

Example 11.5 Both Neutron stars and Black Holes accrete matter. What kind of radiation would you expect to be given off when matter accretes onto a neutron star or black hole? What would be very different about the way in accretion onto a black hole would appear?

Solution: In either case the gravitational field is very intense and an enormous amount of energy will be released as matter accretes onto either object. The energy comes from the conversion of gravitational potential energy into kinetic energy. As the accreting matter approaches either the nutron star or event horizon of the black hole it reaches very high speeds and any collisions between particles will heat the gas to very high temperatures measured in the millions of degrees Kelvin. Hence you should expect x-rays to be emitted by the very hot gas. What would be different, however, is what happens when the matter hits the neutron star or cross the event horizon of the black hole. If the object is a neutron star you would expect one last and intense burst of x-ray energy. On the other hand matter crossing the event horizon would disappear from our universe and we would not expect to see a final burst of energy.

Black Holes in the Cores of Galaxies

In a future chapter we will look at the important role that astronomers think supermassive black holes play in the centers of many galaxies. Figure 11.10 shows Hubble space telescope image of the centre of the distant galaxy NGC 4261. Measurements reveal that as much as 1 billion solar masses of material is contained within a region smaller than our solar system - this would be smaller than the Schwarzschild radius. Although not all astrophysicists accept the existence of black holes, a large majority do and it is evidence such as the discovery of x-ray binaries and compact objects in the cores of galaxies that provide compelling evidence for this bizarre but apparently true phenomenon.
  Figure 11.10 Possible black hole is contained in the inner most region of the galaxy NGC 4261.

 

Features of Black Holes: What they are and What they aren't

There are a number of misconceptions about black holes. The following summarizes a number of key points that we have discussed:

  • black holes are regions of zero volume, infinite spacetime curvature (singularities) that are hidden by an event horizon (Cosmic Censorship Theorem). The event horizon is situated at the Schwarzschild Radius.
  • clock rates are affected by the intense gravitational field of the black hole. A clock falling in toward the event horizon would run slower and slower and appear to stop ("frozen in time") on the event horizon. From an outside observer, it would take forever for a clock (or unfortunate astronaut) to cross the event horizon.
  • black holes would behave like any other gravitating body until you got very close to the event horizon. If the sun suddenly became a black hole the earth and other planets would continue to orbit much as they do now.
  • black holes are not cosmic vacuum cleaners scouring space for unwary stars! They are passive acceptors of space debris.
  • It is possible to extract energy from black holes! Here are two ways in which we think a black hole can return energy:

 

Polar Jets

Just as we have seen in the case of neutron stars acting as pulsars, accreting matter forms a swirling disk around the black hole. The infalling matter heats and, especially if magnetic fields are present, intense jets (particle beams) can be produced flowing outward at right angles to the accretion disk. Jets are a very efficient mechanism to convert gravitational energy into kinetic energy and eventually heat a gas. In this case the black hole acts as a "catalyst" by turning some of the gravitatinal energy of accreting matter into energy that is propelled back out into space. Figure 11.11 illustrates this.

  Figure 11.11 Artist's rendering of what a jet formed by a black hole and accretion disk might look like. (Credit: NASA/CXC/M.Weiss)

The Hawking Process

Quantum theory makes a very strange prediction. It says that at any moment of time, at any place in space and for a fleeting instant a particle-anti-particle pair can "pop" into existence. These are called virtual particle pairs. Just as quicky the two particles "re-combine" and sink back into nonexistence. Should a virtual pair be created on the event horizon, however, one half could fall into the black hole and the other would be stranded - it would appear as a new particle in the universe. The only way to conserve mass is to have the black hole lose an equivalent amount of mass. In other words this is the same as saying that the black hole has emitted the particle. As bizarre as this sounds it works!

Figure 11.12 Animation depicting the Hawking process. Yellow dots represent virtual particle pairs "popping" in and out of existence. When this occurs at the event horizon particle emission can occur.

 

Advice For Space Travelers:

  • never put your toe in a black hole! As you cross the event horizon the prospects are bleak. If the Schwarzschild radius is small enough the tidal force between your toes and head will be enormous and you will be stretched into spaghetti! Even if the black hole is larger you are just delaying the inevitable. As you approach the singularity the tidal forces will become enormous. You're pasta - pal!
  • don't bring a black hole home. Fine - did you ever stop to think how you would contain the black hole!!? What happens if the black hole gets out?
  • don't throw any thing in a black hole. Just like the case of neutron stars, as matter funnels into a black hole it heats up and emits x-ray and gamma ray radiation back at you!.

 
   

Practice

  1. How does the sun's present size compare to its Schwarzschild radius? Express your answer quantitatively.
  2. Explain how it is possible for a black hole to emit x-rays.
  3. Suppose that you are investigating an x-ray binary in which a B giant star is orbitting an unseen companion of mass 10 Mo. The x-ray emission from this system suggest that the unseen companion must be very small - less than 1000 km across. Eventhough neutron stars can also emit x-rays if they are part of an x-ray binary system why is it more likely that the unseen companion in this system is a black hole?
  4. Could our sun ever become a black hole?

 

 

To understand the formation and features of black holes

Chp 14.2,3