Black Holes and Revelations , A Scientific Paper By Jamie Donnelly

By: admin Category: Papers 2 Comments January 11, 2012

There is one human fear that transcends all eras, peoples and ways of thinking, and which can be generalized to any person’s human experience at one point or another in his or her life. That fear is the fear of the unknown, that feeling of trepidation and unease we get when we try to conceive and think of something we do not even know, yet we think can potentially harm us. It is the reason a child is afraid of the dark; it is not a fear of the blackness of night that fills the bedroom, but rather that which lies (or might lie) within it. We are secure as long as we know everything within our environment and that it is safe, but when the light by which we see is gone, our surroundings are no longer guaranteed to remain constant, and therefore, our security cannot either. That is what makes the unknown so exotic – what our senses cannot tell us, our minds fill in the gaps.

Our senses give us accurate information, but our minds can only estimate, meaning that the not-so-distant object that goes clunk in the night can take the form of anything, from our best expectation, to our worst nightmare. Since our mind has a tendency to assume the worst, it is normally the latter, which means that a normal bedroom covered in darkness becomes exponentially more frightening for every drop in light levels. Though fear of the dark is something we outgrow as we learn more and more about the real world, one thing remains: fear of the dark is simply a special case of fear of the unknown. The unknown, however, is infinite and is the name we give to that which is outside our reach, such as post-death and outer space. The list of things we may potentially fear, therefore, has only just begun.

When we consider how uniquely fear of the unknown and darkness coincide with each other, it is interesting that there remains one scientific phenomenon that never fails to attract wonder, intrigue and often a pang of scariness – it is the black hole. We have all wondered at some point what happens if you fall into a black hole and had dreams about the world ending by being sucked into a black hole that decides to pay a flying visit by our solar system and spew us out into a different universe, or worse, condemn us to an eternal fate of nonexistence. One American astronomer noted that when he gives talks at schools, the one topic about which he is always asked questions by curious kids is about black holes. It is worth noting that in recent years, the most asked question has shifted from black holes to whether the world will end or not in 2012, another thing people fear because it is in the future – an unknown in and of itself. There is one critical difference, however; claims that the world will end in 2012 are utterly false, and the reasons why are enough to form the basis of another paper. Black holes, on the other hand, are very real and an exploration of them hits two birds with one stone; the unknown and the fear thereof – and thus we have, Black Holes and Revelations.

The concept of a black hole goes all the way back to the 18th century, when the French Mathematician Pierre-Simon Laplace was studying escape velocities, ie. the speed at which an object has to travel to escape another object’s gravitational field. Escape velocity depends on two things; first, on how far you are from the object from which you want to escape and second how much mass that object has – the bigger the object and the closer you are to it, the faster you must travel to escape its gravitational field. The idea of an escape velocity, in turn, is based on Isaac Newton’s Law of Universal Gravitation, which states that two objects exert equal and opposite attractive forces on each other that are directly proportional to each of their masses and inversely proportional to the distance between them squared. The law is completed with the addition of the universal gravitational constant*. The escape velocity of an object at a set distance from a fixed mass can therefore be calculated using integration to find the speed at which an object must travel if it can only hypothetically come to a stop an infinite distance from the object.

The general escape velocity turns out to be the square root of twice the universal gravitational constant multiplied by the mass of the object from which one is escaping, divided by the distance one is from the object. For the Earth, this is 11.2km/s at the surface (though it decreases the further up you go), any slower and our rockets come falling back to Earth, any faster and it can keep travelling away from the Earth forever. On the moon, meanwhile, it is 2.4km/s, which is why the moon has no atmosphere; all the gases’ molecules are moving around fast enough to escape into space. On Jupiter, the escape velocity is 59.5km/ s, and on the surface of the sun, it is 617.5km/s. That’s over 1/500 the speed of light, the upper bound of all speeds in the universe. Meanwhile, Laplace found that if you kept the mass of the Earth the constant, but made its radius smaller, the escape velocity at the surface got bigger. As you quarter the Earth’s radius, you double the escape velocity. If you keep on making the Earth smaller and smaller, you reach a point beyond which the escape velocity exceeds the speed of light – Laplace calculated this distance to be 0.9mm. If the Earth were compressed that small, nothing, not even light would be able to escape its gravitational pull.

An extension of this idea is that a star could be compressed small enough for its light never to leave its surface, yet the thought of such a “dark star” was met with scepticism in a time when outer space was still considered to be relatively simple. The notion of a body so dense that its escape velocity at its surface exceeded the speed of light was so ahead of its time that it wasn’t truly revisited again until the 20th century, when the German physicist Karl Schwarzchild worked on the general case of this field in astrophysics. He found that the radius of a non-rotating sphere must have to achieve this speed-of-light escape velocity was equal to twice the sphere’s mass times the universal gravitational constant, divided by the speed of light squared. This radius is called the Schwarzchild radius. It is synonymous with the event horizon of a black hole, but since black holes, depending on their electric charge and how much they spin, tend to have an inner and an outer horizon, the term Schwarzchild radius will be used for simplicity.

It is also worth mentioning that what we have described so far is not actually a black hole, but a dark star; the distinction is that the latter is a product of classical mechanics, whereas a black hole takes into account breakthroughs made by Albert Einstein’s General Theory of Relativity. Indeed, the discovery of the Schwarzchild radius had a very new and fascinating relevance as early as 1905, after Einstein published his famous paper on Special Relativity, one in which he postulated that the speed of light is constant for all observers, moving or stationery. Bearing in mind the speed of light is also a maximum for all observers in the universe, any body that enters inside the Schwarzschild radius of a compressed mass will not come out again. There was another rather interesting implication of the Schwarzchild radius, which came from a prediction by Einstein in his paper on General Relativity that the path of light can be bent by gravity, a prediction that was verified during subsequent solar eclipses. Many physicists of the time believed that light was made up of massless particles, called “photons” that possessed wave properties, and hence, were unaffected by gravity. While there is some truth in this view, it wouldn’t be accurate to say that photons are massless. Although the idea that light was subject to gravity was counter-intuitive, experimental evidence backed it up and gave legitimacy to the concept of a body so condensed that it sucks in light and never lets it out again.

The existence of such an exotic scientific notion was doubted at first, until the launching of a rocket with X-ray detectors in 1962 to measure amounts of radiation coming from the Moon. While it detected small amounts of background radiation from all directions of space, when pointed in one particular direction, it encountered a rather strong stream of X-rays from the constellation of Scorpius, approximately 9,000 lightyears away. This was a rather profound and unexpected discovery, as X-rays require very high amounts of energy to generate in small quantities alone, never mind one so large its intensity is still quite strong 9,000 lightyears away. Thus, scientists had discovered the first ever extra-solar source of X-rays.

Eight years later, a satellite was launched, this time to map all such X-ray sources. This mostly consisted of highly radioactive supernovæ remnants, distant galaxies and binary stars, where matter from one star was accelerated to very high velocities by another nearby star (normally a neutron star) to generate the energy required to emit X-rays. There was one binary star pairing, however, whose X-ray source was so massive that it was estimated to be 5-8 solar masses. When a neutron star forms, its mass has an upper bound of three solar masses, which means if it is more than three times more massive than the sun, it will not be able to withstand the power of its own gravitational collapse. A later estimate showed the star’s mass to be closer to ten times the mass of the sun, which was so much over the top to be a neutron star that it could only be one thing – a black hole.

One of the things that make black holes an object of intrigue to people who are generally not interested in astrophysics is its very extreme nature – it is an object so strong that nothing can escape from it, sucking in all in its path. It is also this reason that black holes become a basis for fantasy and science fiction, becoming responsible for catastrophes and disasters which although they make for great cinema and fascinating end- of-world scenarios, simply aren’t true. In fact, we can gain a deeper insight into the nature of black holes by looking at what they are not and contrasting them with what they are.  Black holes are generally thought to be all-consuming and inescapable, affecting all within a vast radius around it, and while this is true, it does not tell the whole story. They are only all-consuming and inescapable if you enter its Schwarzschild radius, which is still very, very small in astronomical terms. It is also true that the force of its gravity has a massive range, in fact it’s infinite, but just as the gravitational force of any object only truly disappears at infinity, we must remember that Newton’s Law of Gravity is an inverse square law, which means it gets significantly weaker the more you move away from it. Every object in the universe exerts a gravitational pull on the Earth, but the Sun and the Moon are the only objects that exert in any way a significant pull; it is accepted by astronomers that nearly if not all galaxies contain a supermassive (more than one million times the mass of the sun) black hole at the centre, and the Milky way is no exception, but it’s so far away from us (~100 lightyears) that the force exerted upon us by the Sun is approximately a billion times the force exerted on us by the black hole. Since the solar system orbits the centre of the galaxy, we’re not going to be seeing that black hole anytime soon.

It is also thought that if a black hole passes near us (in astronomical terms) that we’re all going to die; while this is true, it’s because of the vast amount of radiation we’d experience due to the black hole’s accretion disk, a fine layer of debris that spins around it, if it has one. Unless a black hole actually passed through or just outside our solar system, there’s very little possibility we could get sucked in, though if it passed further out still, its respective pulls on the Earth and the Sun could still potentially mean we’ll slip out of the circumstellar habitable zone, a small imaginary spherical shell in which we must reside for the Earth’s temperature to support life. However, the chances of a black hole ever passing this close to us are next to zero. Even if our sun were to collapse into a black hole (which it can’t; it’s not massive enough), we would not be sucked into it, because though its size would be smaller, its gravitational force on us would remain the same, a fact we can verify by applying Friedrich Gauss’ Law of Gravitational Flux. The main difference would be that sunlight would never reach us, meaning that we would live a life of temporary darkness, before freezing to death.

There is another belief, and that is that if you get sucked into a black hole, you’ll either come out on the other side of the universe, be warped into another universe, or disappear completely. While these prospects very much appeal to our sense of fantasy, they simply don’t hold water. However, there is a small amount of truth in these notions; while a discussion of the role of black holes in General Relativity forms the basis for a more advanced paper, we may make minor allusions to Einsteinean Mechanics. We use a concept in General Relativity called spacetime, which refers to the combined state of space, time and gravity.

When we have a supermassive object with a very strong gravitational pull, we say it curves and distorts spacetime,  which in its briefest possible form means it warps the properties of the space around it. At the centre of a black hole, we have a very small space called a singularity, which is a spot where the gravitational pull is infinitely strong. This point in space, therefore, we call infinitely curved. Unfortunately, we do not know precisely what that means because this is where our understanding of high-energy physics breaks down.

We do, however, have a few ideas of what happens here. A gravitational well is a name we give to a trough in an object’s potential energy function. An interesting effect of a gravitational well is that when we place a clock inside it, it ticks slower. It is not an illusion, it is because two observers, one outside the well and one inside, record different amounts of time. If a clock is placed at the same spot as a singularity, it theoretically stops. Since spacetime is infinitely curved, one may hypothesise that you may actually be able to leave the gravitational well via a white hole, which is the opposite of a black hole. A white hole can never suck things in, it can only spit them out. When you end up with a black hole connected to a white hole we call this allegorical tunnel a wormhole, which leads to two possibilities. First, you may find yourself back at the spot you entered the black hole with the black hole gone and the rest of the universe advanced or even reversed millions of years. On the other hand, you may even end up in another part of our universe at the same time you arrived at the singularity, or even in another universe. These ideas, of course, are not facts, they are probably very wrong; one aspect in which they are very critically undermined is that the mysterious white hole is in reality a mythical beast, just like the massless rope or the frictionless plane. A white hole can hypothetically exist because like a black hole, it is a valid solution to Einstein’s equations of General Relativity. In nature, however, we have yet to conceive of a way it could possibly exist, and so, these exotic ideas we have just explored are relegated to the fantasies at the back of our minds.

Black holes have even occasionally created end-of-the-world scares, the most notable case being that involving CERN’s Large Hadron Collider. In Zurich in Switzerland, CERN have a research facility that contains a large, circular underground tunnel. Using this tunnel, they may accelerate tiny particles to huge velocities and by clashing them off each other, create new particles. When CERN decided to create tiny black holes for their Big Bang investigation, there was a proliferation of stories that CERN were playing with fire and that the world would get swallowed by one of these black holes. This certainly wasn’t the case, and very little has been heard of these black holes since. The reasons were that the black holes CERN were creating were way too small to accrete matter, which is essential for the survival of a black hole, and that the black holes evaporated in a fraction of a second due to Hawking Radiation, a theoretical concept we won’t explore here.

We shall conclude this paper with a journey through a black hole; an attempt to answer that scientific curiosity that intrigues so many children. Of all the various forms of spinning black holes, black holes with charge and black holes rotating in a binary star pairings, we shall only look at the case of a spherical black hole with zero angular momentum, ie. no spinning, no motion, a black hole that is completely still. As we arrive within eyesight of the black hole, we will be gaining speed, with a faster acceleration as we get closer and closer and the gravitational pull gets stronger and stronger. As we look directly at the black hole, we will be able to see all the stars around it, but not what’s inside, since no light can escape a black hole. We will see, however, a ring of lights around the black hole with small curved streaks, not dissimilar to the ring of light that appears around the edge of a bright, shiny doorknob on a sunny day. This is known as gravitational lensing, as the black hole pulls on the light coming from stars behind the black hole and curves its trajectory.

As we come to the verge of the black hole’s Schwarzschild radius, we encounter the photonsphere, the last possible distance from the centre of the black hole in which photons of light may remain in orbit without being sucked in by the massive force therein. Beyond the photonsphere, we see the massive, black, lightless space of the black hole in front of us, and behind us, the rest of space, which we now see one last time. Once we cross over into the black hole, there is no way we will ever exit again, no matter how strong we’re able to power ourselves. When we cross this verge, however, something very strange happens; all black holes have something called an event horizon, and for a simple black hole, this coincides with the  Schwarzschild radius. Once we enter the black hole, the event horizon splits into two, a horizon and an anti- horizon. The horizon is simply the one we just entered, and its speed relative to us is astronomical, enough that we can never catch it. The anti-horizon, meanwhile, is advancing away from us at a similar rate, such that even as the singularity attracts us rapidly towards the centre, we will see the anti-horizon, but we will never touch it. Contrary to popular belief, the inside of a black hole is not black; all those photons have to go somewhere, so they rush in towards the centre of the black hole. We would not be able to actually see anything inside the black hole for a variety of reasons, but we will ignore this in our journey.

Meanwhile, to an observer outside the black hole (the following is possible to know through other means), a clock held in our hand runs slower and slower as we get closer to the centre of the black hole, the mystical singularity. This is because of an earlier result mentioned, the slowing down of a clock as it rests deep in a gravitational well caused by a large object. There is something more of note to the observer, and that is that we are never actually seen crossing over into the black hole. This is because the observer requires light to see us, and the closer we get to the black hole, the more the gravitational pull holds back light. Another phenomenon occurs called the gravitational redshift, which causes the light to take on longer wavelengths and go towards the infrared part of the electromagnetic spectrum. Eventually, as we enter the black hole, the last photons carrying information of our transit become maximally affected by gravity, meaning that to our observer, we will be stuck forever on the edge of the black hole, dim and redshifted, unable to pass over. This is the reason black holes used to be called “frozen stars”; not because of a low temperature (quite the opposite, really!), but because its gravitational pull gives the illusion that time comes to a complete stop around its edges. The only way we’ll ever be seen passing over the Schwarzschild radius is if the black hole evaporates, releasing with it the gravitational force that held back that final photon.

Back inside the black hole, meanwhile, our journey inside is very real, although nobody will ever see it, except for ourselves. When we are a matter of a few lightseconds away from the singularity, the gravitational pull at the front of our body is stronger than that at the back, which means tidal forces begin to stretch and strain us. The larger gravitational pull at the front of our body means we become stretched while the tangential forces around the middle of our body compress us and stretch us like an elastic band. As this happens, the intense gravity of the black hole distorts our view, with what we see straight ahead bright and blueshifted, and what we see above and below dim and redshifted. A fraction of a second before we reach the singularity, the point of infinite gravity, the tidal forces are so big that the molecules that form our body are ripped apart violently and all that remains of us becomes part of the influential galactic sphere that is the black hole.

This concludes our journey into a black hole, and this paper as a whole. Luckily, we will never come into close contact with a black hole, but it is still interesting to explore how we would interact with it. As science advances and we are no longer afraid of the terrestrial phenomena that terrified our ancestors, such as lightning and comets, there still remains the most unknowable object of all, the black hole. There is nothing that creates a fear quite like it, because it is not a mythological curiosity, but a genuine article. Crossing its Schwarzschild radius is very much like death – in order to find out the secrets within, it is necessary to take a leap into the abyss, one from which it is impossible to return and tell these secrets to everyone that remains on the other side of the black hole; for these truly would be Black Holes and Revelations.

Jamie Donnelly