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Neutron stars

The more massive a body, the greater its gravity. This means that in general suns have the highest gravities around, and the gravitational forces associated with their huge mass attempt to collapse them.

Suns generate energy by fusing the nuclei of small atoms together. Energy can be obtained by fusing nuclei up to the atomic number of iron. Above that, it takes energy (rather than provides energy) to fuse these heavier nuclei (energy is in fact released when these larger nuclei are fissioned or spilt as in the fission of uranium in a nuclear power plant). The smaller the nuclei that are fused the greater the energy released, so suns get most of their energy by fusing hydrogen to make helium.

The temperature and pressure provided by hydrogen fusion hold back the natural gravitational collapse. After a few billion years the hydrogen runs low and the temperature drops a little; what happens then depends on the mass of the star.

For stars over five times the mass of our Sun the interior of the star collapses a little, creating higher temperatures at the core in which higher and higher-numbered nuclei will fuse, for a while generating enough energy to hold the collapse at bay. The outer shell of the star is puffed out into a red giant by the extreme temperature of the core. Eventually the star’s core will have been fused to iron and no more energy is available. The collapse then continues.

In the early life of a star hydrogen is fused into helium, converting 0.0072 kg of mass into energy for every kg of hydrogen - an efficiency of only 0.72%. Later as heavier and heavier elements are fused together to make iron, every kg of the original hydrogen eventually converts a total of 0.009 kg into energy - a total efficiency of 0.9%. The initial conversion of hydrogen to helium is the most efficient part of the process. But in gravitational collapse in a Neutron star 0.05kg is converted out of every kilogram of matter - at 5% conversion efficiency this is the highest energy mechanism to large stars.

Now it is only the mass contained within a radius that contributes to the gravitational gradient at that radius (so there is no gradient inside a hollow sphere). During the collapse more and more mass gets crowded into a smaller radius, so the gradient at any radius inside the star increases dramatically although the external field at a planetary radius is almost unchanged. This increases the pressure at internal solar radii and boosts the collapse. In under a second the core collapses. In this collapse it releases more energy from gravitational sources than it released through fusion in the whole of its life prior to the collapse. The core reaches reaches temperatures beyond 100 billion degrees, at incredible pressures under which the iron atoms themselves collapse.

Ordinary atoms have a minute nucleus from 1.0E^-15 meters to 7.0E^-15 meters in radius, surrounded by an atomic space of 0.5E^-10 to 1.5E^-10 meters in radius. When an atom collapses to nuclear size the tiny electrons are merged into the nucleus and it compacts to a density of the order of 1.0E⁺¹⁴ greater than normal matter. In the collapse of the core the atoms are compressed into the space of their nuclei.

During the collapse matter reaches near-light-speeds with massive inertia. As the core compresses massive amounts of energy are released from the inner regions of the star at the heart of the compression. This energy is so massive it reverses the collapse for all but the innermost  regions of the star, and blows a huge part of the star away into space; the compressive pulse of new energy fuses iron into heavier elements. This immense blast of energy lights up the whole of that region of space for just a few weeks; remember that a Sun will burn for 10 billion years fusing hydrogen to helium, yet for only a few weeks for a process that is more efficient - the output is orders brighter than a galaxy for this brief period. Nearby solar systems are sterilized by the flood of gamma and Xrays.

This is a Supernova. The Crab Nebula is the best-known example of such a supernova remnant - the original supernova was seen on Earth in 1054 A.D.

At the heart of that remnant lie the final remains of the star, now a neutron star. The electrons and protons are compressed into neutrons, and the star’s diameter is only about 10km. It has lost 80% of its mass, mostly blasted out into the universe, but also by the gravitational contraction of its mass. Its surface gravity means that every kilogram weighs about 1.0E⁺¹² Newtons.

If you were near the surface of a neutron star and not compressed your head would be about 1.9 meters from your feet. Assume your feet were nearer to the star than your head. Each kilogram in your head would weigh 1.000760E⁺¹² Newtons and each kilogram in your feet would weigh 1.0E⁺¹² Newtons, a difference of 760 MegaNewtons. You could get the same effect on Earth by fixing your head to a solid support and attaching a 100,000-tonne mass to your feet! The effect exists on Earth, but is really tiny because of the lower gravity field and the larger radius of the Earth. However, it is responsible for something much more visible - the oceans on the side of the Earth nearer the moon are closer than those on the opposite side by the diameter of the Earth, and the difference in the pull of the moon’s gravity between points is strong enough to create our tides.

Another danger is the star’s magnetic field. It can be 60MTesla, with an energy density of 1.4E+21 Joules/meter³, with a mass equivalent of 1.6E+04 kg/meter³ - 16 times the density of (uncompressed) water. At the surface of the star it sweeps round with a velocity of a million meters/second. If you were near the star and in caught in the path of this field you would evaporate in a microsecond - it would hit you like an immense white-hot lead wall. The energy density is so high it contributes to the angular momentum of the star and inertial forces tend to push the magnetic field to lie on the equator. (There is another similar stellar phenomenon called a magnetar, where the magnetic fields apparently approach 1E⁺¹¹ Tesla; here the energy density becomes 4E⁺²⁷ Joules/meter³, the mass-density equivalent being 4.4E⁺¹⁰ kg/meter³, 44 million times the density of water.)

We can detect neutron stars because of three things...

1. Much of the star’s angular momentum remains, but with the small diameter this means that the star rotates at high speed - perhaps 30 times/second.
2. Neutron stars have high magnetic fields that are not aligned with their axis of rotation. This means their magnetic fields sweep through their local space as the star rotates.
3. They collect charged particles.

The high magnetic field sweeping past the charged particle streams causes synchrotron radiation at X-ray wavelengths. So if the magnetic axis sweeps through the direction in which the Earth lies we will detect a radiative source pulsing once per revolution at perhaps 30 beats a second - a very distinctive stellar signature!

Black Holes

So what happens when the distortion becomes really massive? One idea is that we get a black hole. First I will cover the general idea of a black hole, then discuss some of the issues involved.

In the collapse of a star more than 15 times the size of our sun, the theory is that the gravitational collapse cannot ever stop. First the atoms collapse to nuclear size, then the nuclei themselves collapse without limit under the extreme gravity, compressing the star to a point. Unlike neutron stars there may be no explosion that blasts part of the star out into space after the initial collapse - the collapse is one-way only, especially if the angular momentum is low.

Take two observers - ‘A’ who stays far from the black hole and ‘B’ who travels from ‘A’’s position into the black hole; both carry clocks that the other can see.

As ‘B’ leaves ‘A’ he travels in free-fall towards the black hole. As he does so ‘A’ perceives ’B’’s clock to be slowing down drastically, and his rate of acceleration into the block hole seems to be less than it should be.  For ‘B’, however, his time is unaffected and he falls as he would expect. This time dilation increases as ‘B’ gets closer to the black hole.

Behind him ‘B’ sees a picture of the universe behind him that distorts more and more as he falls. At first he sees simply what is behind him, but the gravitational lensing causes him to see more and more of the universe round the black hole until at a radius 3.G.M/c² the whole of the universe around the black hole is visible behind him (where G is the gravitational constant and M is the mass of the black hole). This is the photon sphere, where photons can orbit around the black hole. Beyond this point the view behind him closes up into a cone surrounded by light that has come up from below him and been lensed back down. The view is similar to that of an underwater swimmer looking up at the surface - he sees a cone that compresses everything visible above the surface, surrounded by reflections of what is under the water.

 He continues until at a certain point the time dilation as seen by ‘A’ becomes infinite. At this point ‘A’ sees that ‘B’ has stopped dead - he hangs forever at this point, stuck in frozen time till the end of the universe. ‘B’’s perception is that he is still accelerating into the black hole and there is nothing abnormal about his fall.

This point, where ‘B’’s time is infinitely dilated, is an event horizon termed the Schwartzchild radius. It exists at this point only for a remote observer such as ‘A’. As you approach the black hole the event horizon moves ahead of you, so that for ‘B’ there is a different event horizon ahead of him, where someone further into the black hole than he will experience infinite time dilation relative to ‘B’’s time - or the “square of infinite” time dilation relative to ‘A’’s time. There is however only one Schwartzchild radius - the outermost event horizon - and this occurs at 2.G.M/c². For a star with just two solar masses left after collapse the Schwartzchild radius is about 6km and the photon sphere radius is about 9km.

Let us continue ‘B’’s journey to the centre of the black hole. He falls normally, according to his own perception, even though with respect to the outside world of ‘A’ he ultimately has many powers of infinity of time dilation. Finally reaching the centre the gravitational field is so high that nothing stops the collapse into a single point, and ‘B’ is swept into it. At this point the mass density reaches infinity, causing the gravitational field equations to have a mathematical singularity at this point - that is, they have no solution. So the laws of space and time do not apply.

As ‘B’ falls into the black hole his mass and charge are simply added to the black hole. Mass and charge are simply added to the black hole. So if the sun suddenly collapsed into a black hole the gravitational field at Earth’s orbit would be unaffected and Earth would continue in its normal orbit without perturbation.

Anything wrong with this picture? Well, there have been a great deal of massive stars in the lifetime of the universe, and so far we have not detected one despite looking hard. We should see them by radiation from infalling cosmic gas and dust. So maybe black holes do not exist.

What arguments are there against the existence of black holes? The first and most obvious is that spatial dilation has not been allowed for in the above argument, and it thereby violates energy conservation laws. Let us review the above fall of ‘B’ into the region of the black hole.

As ‘B’ falls into the black hole ‘A’ perceives that he shrinks in size. ‘B’ on the other hand sees ‘A’ grow in size together with the outer universe. As ‘B’ free-falls into the gravitational field he exchanges mass for kinetic energy, so that at the Schwartzchild radius, where spatial dilation is infinite, he has become of negligible size and virtually all his mass has been exchanged for kinetic energy. ‘A’ perceives him to have zero mass and his kinetic energy is m₀.c², where ‘m₀’ was his starting mass in the greater universe, so his ratio of kinetic energy to mass is infinite. ‘B’ perceives himself to have the same mass as he started with, but he has picked up an infinite kinetic energy, so again his ratio of kinetic energy to mass is infinite. With this ratio he can travel at only one speed - light speed. He travels with the photons across the Schwartzchild radius.

His kinetic energy does not contribute to the gravitational field, since that is reference-frame dependent. Only his mass can. As far as ‘A’ is concerned he contributes zero mass to the black hole. By extension, nothing inside the Schwartzchild radius can contribute any mass or charge to the black hole - all its energy is kinetic - and hence the event horizon cannot form - it is self-limiting by the spatial dilation. Hence there is no singularity.

Another factor that affects the formation of the singularity is that at the putative event horizon ‘B’ is infinitely dilated, but the event horizon is finite. Hence ‘B’ perceives that the space he is entering has expanded dramatically - black holes are infinite on the inside. At the event horizon he perceives the radius to have become infinite, although in reality it is he whose size has become negligible.

Truly  massive stars may therefore collapse like neutron stars, with the piledriving pulse of the collapse converting mass into kinetic energy in a violently-dilated space. The resulting rebound explosion blasts all but a small remnant out into space. The gravitational field turns much of the remnant’s mass into kinetic energy which fuels the explosion, and so is lost to the star forever. And so it can be seen that massive suns create energy in the following sequence:-

1. Fuse hydrogen into helium - 0.72% efficient
2. Fuse helium and hydrogen into heavier elements up to iron - 0.2% efficient
3. Convert mass into kinetic energy in gravitational collapse - 5.0 % or more

For massive suns the last is the most efficient, give by far the highest ratio of energy output to total mass involved. Since the collapse is so fast, taking less than a second, the resulting release of energy is incredible. This energy output is greater than all the energy produced in the previous life of the star.

There are other arguments against singularities:-

Relativity Theory uses the simplifying assumption that all masses are points. Since a point mass is itself a singularity the black-hole singularity is inevitable. But Quantum Theory shows that the position of a particle is associated with a probability distribution. Clearly if the position of a particle is indeterminate the singularity cannot form since it requires a point of zero dimensions with no indeterminacy. “Quantum Relativity” attempts to address this issue. An associated argument is that for the same reason a particle’s positional probability distribution can lie across the event horizon and so a black hole will slowly “leak” particles and thereby evaporate. The counter argument is that if a black hole can form the particle becomes embedded in an infinite space inside an event horizon, and its probability distribution cannot extend beyond infinity.

 The time taken to form the singularity (again according to ‘A’) would be infinity to a high power so cannot happen in a real limited universe. We could of course take the role of ‘B’ or some deeper observer to make it more possible, but that would be in a purely theoretical daughter universe that is forever closed to us by infinite time.