A Roller Coaster Ride through Relativity/Black Holes

Black Holes

In 1783, the Rector of Thornhill parish church in Yorkshire, a man called John Michell wrote to the Royal Society with an extraordinary idea. Newton had showed that for every planet there was a critical speed called the escape velocity which it was necessary to achieve if you were to throw a rock off the surface of the planet into space and it is easy to show that, for a planet (or star) of mass M and radius R, the escape velocity v is given by the formula:

$v_{escape}={\sqrt {2GM \over R}}$

Michell's idea was this. What if a star was either so massive, or alternatively, so small that the escape velocity was equal or greater than that of light? According to the corpuscular theory of light which was still popular at that time, at least in England, light would never be able to escape from such a star and even though it was burning brightly, it would look to us completely black. The radius of such a 'black star' would be equal to:

$R={2GM \over c^{2}}$

The idea did not catch on. Within a couple of decades, Thomas Young had shown pretty conclusively that light was in fact a wave and therefore probably immune to gravity. Nevertheless, we have now shown that light is indeed susceptible to gravity and that, as it climbs away from the surface of a star, it is not slowed down - it is red-shifted instead.

The modern equivalent of Michell's idea is therefore this. Is it possible for a star to be so massive that light is red-shifted all the way to infinite wavelength (or zero frequency)? In which case it would become completely invisible just like Michell's corpuscles. To answer this question we must use the accurate formula for gravitational time dilation:

$T={1 \over {\sqrt {1-2\Delta \phi /c^{2}}}}\,T_{0}$

and ask ourselves under what circumstances will T (the observed rate of flow of time on the surface of the star) to become infinite. The answer is of course when:

$1-2\Delta \phi /c^{2}=0$
$2\Delta \phi /c^{2}=1$

Now, the gravitational potential at the surface of a star is (see Appendix H)

$\phi =-{}GM \over R$

so putting Δφ = -φ, we get

${2GM \over Rc^{2}}=1$

and hence

$R_{s}={2GM \over c^{2}}$

which happens to be exactly the same formula that Michell proposed in the first place!

This is the radius of a Black Hole - the so called Schwarzschild radius. (As we have seen, the actual radius, ie the distance from the centre to the edge, is in general larger than this and for a black hole it is probably infinite. The radius used here is defined as being the circumference divided by 2π.)

For a black hole with the same mass as the Sun, R_s turns out to be almost exactly 3 km. It is clear that our own Sun is a long way from being a black hole but theoretical physicists and astronomers are now fairly convinced that most stars with a mass greater than about 5 solar masses will end their lives by turning into a black hole. Moreover the existence of supermassive black holes at the centres of galaxies is widely accepted and the existence of tiny mini-black holes is a possibility.

What would the mass of a black hole the size of an atom be? The answer is an incredible 10¹⁷ kg which is about the mass of an iron meteorite 32 km across. If the Earth were to encounter one of these, it would either cause a massive explosion big enough to obliterate all life on Earth, or, more likely, it would drill a neat hole through the Earth and come out the other side!

The gravitational field strength at the surface of a black hole (or the Event Horizon as it is more properly called) is equal to

$g_{s}={GM \over R_{s}^{2}}={c^{2} \over 2R_{s}}$

What this means is that really big black holes have very modest gravitational field strengths at their event horizons. For example, a black hole with a mass of 1.5 trillion solar masses (that is about a thousand galaxies) would have a Schwartzchild radius of about half a light year and a gravitational field strength at the surface of 10 ms⁻² – the same as Earth. If you wandered close to or even inside this black hole, you would not feel anything that you do not feel every day here on Earth – but, of course, you wouldn't be able to get out, however hard you tried!

It is instructive, when considering the possibility of black holes of varying sizes, to consider the density of the material inside the hole. (For our purposes here we shall assume that the hole is 'filled' with a material on uniform density up to its event horizon.) Putting

$M={\tfrac {4}{3}}\pi R_{s}^{3}\rho$

we obtain

$R_{s}={8\pi R_{s}\rho G \over 3c^{2}}$

so

$R_{s}={\sqrt {3c^{2} \over 8\pi \rho G}}={1.26\times 10^{1}3 \over {\sqrt {\rho }}}$

What this means is that the smallest black hole which can be made with ordinary matter (eg iron of density 7000 kg m⁻³) will have a radius of 1.5 x 10¹¹ m. By coincidence, this happens to be exactly the same as the orbital radius of the Earth. So imagine the whole of the solar system out to the Earth filled with iron and you have a simple black hole. It would have a mass of 51 million Suns and a gravitational field at the surface of 30,000 g!

Actually this scenario is quite academic because, as I have said earlier, it only needs about 5 solar masses of ordinary matter to make a black hole owing to the inability of ordinary matter to withstand the crushing effect of gravity. On the other hand, if we choose to make a black hole out of very rarefied matter, then the 'pressure' at the centre does not have to be very great at all. In fact, we could be living inside a black hole right at this very moment! If so, the last formula quoted above could be telling us something very important about the relation between the amount of matter in the universe and the density of material that it contains.

Current theories about the size of the universe suggest that it might have a 'radius' of about 15 billion light years or 1.4 x 10²⁶ m. Plugging this figure into the formula gives an average density for the universe of about 8 x 10⁻²⁷ kg m⁻³. Now the mass of a single hydrogen atom is 1.7 x 10⁻²⁷ kg, so this density corresponds to about 5 hydrogen atoms per cubic metre.

It is exceedingly difficult to estimate the average density of the universe but the best present day estimates of the average density of all the observable matter in the stars and galaxies is about 0.3 x 10⁻²⁷ kg m⁻³, that is one thirtieth of the required value. When you consider by what disparate routes the two figures were arrived at in the first place, it is surprising that it comes anywhere close at all. What it means is this. If the average density of the universe is less than the critical density, we are living in what is known as an open universe. The universe is expanding now and will continue to expand for ever. If, on the other hand, our estimates are wrong and/or we discover other sorts of matter which we have not included so far and the density of the universe is greater than the critical density, then we are effectively living inside a black hole; the universe will one day stop expanding and will fall back in on itself.

The most satisfying result would be that the universe has exactly the right critical density to make it into a perfect black hole - not an atom more, not an atom less.

If this is the case, it will not be by coincidence. Just as it is no coincidence that everyone agrees on the speed of light or that gravitational mass is exactly equal to inertial mass, there will have to be some over-arching principle of cosmology which makes the following conclusion inevitable:

Profound consequence of the (as yet unknown) Fundamental Principle of Cosmology
Every viable universe has a density exactly equal to its critical density

Perhaps some 21st century Einstein will come up with a brilliant new principle which will transform our understanding of the universe we live in and explain why things are as they are.

Are you that person?

You never know – perhaps you are!

Don't forget – even Einstein failed his exams on occasions too!

The Last encounter

. . . but just when we can stand it no more, the stretching and squashing begin to ease and a violet light appears ahead of us which appears to spread wider and wider turning bluer and bluer. The electric lights appear again but they are not like our lights. The aching blackness recedes and suddenly we tumble out into a graceful landscape with a blue sky, green grass and distant purple-headed hills.

What the **** happened then? you say. (You always were prone to colourful language.)

It is true that there is something strange and unfamiliar about everything. The grass has a strange feel and the sky is not quite the right colour. I pick up a stone and let it drop.

Well the laws of Physics seem to be much the same here.

As we look round we find that we are standing on the edge of a vertical shaft. Peering down we see the electric lights reddening in the distance and that aching void again.

We have fallen through a wormhole into another universe.

Turning suddenly, we see a strange figure standing nearby with an outstretched hand. "Welcome" he says, "Are you from Earth? Visitors from your planet do drop in from time to time . . ."

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