Invisible stars

To begin our story of black holes we must go back two hundred
years to the end of the eighteenth century, since that is when
scientists first realized that black holes might exist. Back then
they were known as invisible stars and their existence followed
logically and reasonably from a combination of Newton’s law of
gravity and his particle theory of light.
Until fairly recently it was thought that the first person to
predict the existence of black holes was the world famous French
mathematician and astronomer Pierre Laplace in 1795. It is now
clear that he was beaten to it by an English geologist named John
Michell, who was rector of Thornhill Church in Yorkshire.
Michell is considered to be the father of the field of seismology
and was the first to explain, in the aftermath of the Lisbon
earthquake of 1755, that quakes started as a result of the buildup
of gas pressure from boiling water due to volcanic heat. He
also pointed out that earthquakes could start underneath the ocean
bed and that the Lisbon one was an example of this. His ideas on
the formation of black holes in space were presented to the Royal
Society of London in 1783. Both Michell and Laplace had based
their quite similar arguments on the idea of escape velocity. I recently watched a documentary on television about
amateur rocket builders. These guys take their hobby very
seriously and have competitions to see which rocket can reach the
highest altitude before Earth’s gravity reclaims it. The problem is,
of course, that rockets need to achieve an escape velocity before
they can break away fromEarth’s pull and make it into outer space.
Surprisingly, space is not really that far away—about a two hour
drive by car if we were able to head straight up. It is just that
whatever speed an object starts off at, gravity will immediately
begin to slow it down, and so it has to start off fast enough to
allow for this slowing down. Remember that the force of gravity
becomes weaker with distance and so a rocket does not need to be
travelling very fast once it has reached a certain height. In practice,
rockets only get into orbit gradually, by firing engines in successive
stages.
The escape velocity on the surface of the Earth is eleven
kilometres per second (or forty thousand kilometres per hour). On
the moon it is a little over two kilometres per second, which is
why the Apollo missions’ Lunar Modules did not need such large
rocket engines to leave the moon and return to Earth.
The escape velocity on the Sun is 620 kilometres per second.
This is a number which Michell had worked out, based on the
size and density of the Sun. He also knew with some accuracy
another figure: the speed of light, which had been measured a
century earlier, and which is 500 times bigger than the Sun’s escape
velocity. Michell therefore calculated that a star 500 times bigger
than the Sun, but with the same density, would have an escape
velocity equal to the speed of light.1
Michell was following the conventional wisdom of the time;
that of Isaac Newton, and believed that light was composed of
particles. It therefore followed that light should be affected by
gravity like any other object. But a star with the same density as
the Sun, but more than 500 times as large, would have an escape velocity that exceeded the speed of light, and so the particles of
light would not be fast enough to escape its gravitational pull.
Such a star must therefore look black to the outside world. In fact,
it would be invisible!
Michell had explained the ‘black’ part but we have to fast
forward to the twentieth century to understand what the ‘hole’
bit means. After all, an invisible star, as explained by Michell
and Laplace, is not really very interesting. In fact, using the idea
of escape velocity to explain black holes is a bit like saying that
the Big Bang was just a very big explosion of matter and energy,
and leave it at that. As I explained in the previous chapter, the Big
Bang didn’t happen somewhere in space and at some point in time
but rather encompassed space and time within it in a way that is
extremely hard for us to grasp. In the same way, black holes are
much more than large dense clumps of dead star whose gravity
is too strong to let light escape. In fact they differ from Michell’s
invisible stars in some astonishing ways. For a start, Michell’s
black stars are solid objects of some definite size. Black holes, as
we understand them today, comprise almost entirely empty space!
In fact they are literally holes in space, inside which the properties
of space and time are completely altered. And although we have
never come face to face with a black hole, we have a rough idea
what it would be like to fall into one (not very nice). The reason for
this confidence is our trust in Einstein’s general theory of relativity.
For if general relativity is correct, and we have no reason to doubt
it so far, then it suggests that black holes not only exist in our
Universe but are an inevitable consequence of Einstein’s version
of gravity. One of the world’s leading experts on general relativity,
Kip Thorne, goes so far as to state that “the laws of modern physics
virtually demand that black holes exist”.
We broke off at the end of Chapter 2 after briefly describing
what happens to a star much more massive than the Sun when it
runs out of its nuclear fuel. Having completed our bumpy—but I
hope exhilarating—ride through some of the ideas in cosmology in
the last chapter we are now ready to look in more detail at exactly
how and why a black hole forms. Remember that Einstein’s view
of gravity is to do with the curvature of space, and the stronger the gravitational field of a massive body, the more curved and
distorted the space will be around it.
When a large star explodes as a supernova, it will often shed
most of its mass into space leaving behind a neutron star core
which no longer has enough mass to collapse any further. Inside
such a dense object, matter is packed so tightly that even atoms
cannot retain their original identity. In normal matter, such as the
stuff that makes up everything around us, including ourselves,
atoms are mostly empty space themselves despite being so small.
They comprise of a tiny core known as the atomic nucleus which
is surrounded by even tinier electrons. The laws of quantum
mechanics govern how these electrons behave within atoms and
explain why they keep their distance from the nucleus. Inside
a neutron star, gravity is so strong that the atoms get squashed
together and the electrons are squeezed into the nuclei. The laws
of quantum mechanics state that there will now be an outward
pressure that prevents the neutron star fromcollapsing any further
under its own weight.
What if, after a star has shed part of its mass in a supernova,
its remaining core is still above some critical mass (roughly three
times the mass of the Sun)? Now even a highly compact object
with the density of a neutron star is not ‘solid’ enough. Its matter
does not have sufficient internal pressure to withstand further
gravitational contraction. In fact the star has no choice but to keep
collapsing. Rather than slowing down, the gravitational collapse
actually speeds up. It is rather like a ball rolling over the crest of a
hill. Once it gets past the highest point and starts to roll down the
other side it will just get faster and faster. The question is, what
happens next? Surely the collapse must stop somewhere? The
star is being squeezed smaller and smaller with the matter inside
it being packed more and more closely together.
We now see that the escape velocity from the surface of a star
depends both on its mass and its size. So we do not need a star that
has the same density as the Sun and which is five hundred times
bigger for it to have an escape velocity exceeding the speed of light.
We can achieve the same result if the Sun itself could be squeezed
down to a size just a few kilometres across since then, despite it having the same mass (the same original amount of material) as it
did before being squeezed, it is now much more densely packed.
Such a density would be considerably greater even than that of
a neutron star (which would typically have an escape velocity of
about half the speed of light) and will continue to collapse further
to form a black hole.
Thus Michell and Laplace’s argument that a collapsing star
would eventually disappear from view should also hold for
collapsing stars that have a density that is greater than the critical
value for a neutron star. But this does not even begin to describe
the exotic nature of black holes. After all, we would like to know
what, if anything, can halt the apparent runaway gravitational
collapse of such an object, even if we can no longer ‘see’ what is
happening.
The clue is in the fact that Michell and Laplace were using
Newton’s version of gravity and not Einstein’s. In Chapter 2 it
seemed as though the main difference between Newtonian and
Einsteinian gravity was in the way it was interpreted. Newton
described it simply as an attractive force between any two objects,
while Einstein said that it was a curving, or warping, of space
around an object due to its mass, which causes other objects
close by to roll into the dent in space around it and thus move
closer to it. But surely the final result is the same however we
choose to interpret it? It turns out that this is not the case.
Once gravity becomes very strong (such as in the vicinity of a
collapsing massive star) Einstein’s version of gravity begins to
depart radically from Newton’s. In fact, Newtonian gravity is
said to be only approximately correct. It works well in the weak
gravitational field of the Earth, but to understand black holes we
must ditch it completely.
As soon as Einstein completed his general theory of relativity
in 1915 he began trying to solve his field equations. These equations
were the complicated (yet mathematically beautiful) embodiment
of his ideas on the connection between matter, space and time.
But being able to write down the equations is only half the
battle. They then have to be applied to particular situations
and scenarios which involves much more than simply ‘putting' numbers into a formula, but involves page after page of tedious
and complicated algebra. This is in sharp contrast with the
mathematics of Newtonian gravity, which is so simple it is taught
at school.
The first exact solution of the equations of general relativity
was obtained by a German astronomer named Karl Schwarzschild.
He completed his calculations while on his death-bed, having
contracted an incurable skin disease during the First World War,
and only a few months after Einstein had himself published his
work. The Schwarzschild solution, as it is now known, described
the properties of space and time due to the gravitational field
around any spherical concentration of mass. It was only later
realized that Schwarzschild’s result contained a description of a
black hole in space. In fact, it was not until 1967 that the American
physicist John Wheeler first coined the phrase ‘black hole’ which
has since captured the public’s imagination so spectacularly