Orbiting a Black Hole

If the Sun were to collapse into a black hole right now, would we survive?

Before we tackle this question, let’s get our hands dirty with a bit of the physics behind black holes. Firstly we have to find out what they actually are, and how they were proposed in the first place. On top of this we have to prove that they exist, which should be easy. Just point the Hubble telescope into the distance recesses of the universe until you see one, right? Well, not quite. Unfortunately no one has actually seen one before because light itself becomes trapped within it! All that we can see is a patch of darkness…. surrounded by hot gas and stars zipping around at close to the speed of light. Clearly something with a monstrous amount of gravity is lurking there!

— What are Black Holes? —

Despite Hawking’s massive contributions to our current understanding of Black holes, he was not the first to postulate their existence. Newton’s Laws had already mathematically shown that the escape velocity for a particular object was dependent only on its mass and size ^[1]. For a black hole, this means that not only does it have to have an incredible amount of mass but it must also be very small. How small you ask? Take the Earth and compress it all the way down to the size of a marble. At this point even the atoms themselves collapse. This is the density of a black hole. If you are interested in the escape velocity, it is found by equating the kinetic and potential energies of the object ^[2].

Back to the lecture at hand, the idea of making an object so compact that it’s escape velocity exceeds that of the speed of light was first thought up independently by John Mitchell and Pierre Laplace in the late 18th century ^[1]. Interestingly though, they believed that light emitted from this peculiar object would be emitted and eventually come to a stop, falling back to the surface (much like throwing a rock up into the air) ^[1].

Over a century later, Albert Einstein published his revolutionary ‘General Theory of Relativity’ in 1915 which described the space-time curvature that predicts how gravity works. It is important to note that not only does gravity affect mass – but it also affects energy! This means that light can actually be subject to gravitational attraction! But that is absurd. And luckily Einstein’s theory has a very good explanation for this. Light only knows straight lines. However the presence of mass causes the fabric of space-time to become warped around the object (a good analogy is placing a bowling ball on a trampoline) and this effectively means that light itself does not deviate from a straight line through space-time, but rather space-time itself is what is curved and light is just following that curved path ^[1][3].

Only one month after Einstein published his theory, Karl Schwarzchild calculated the radius of the event horizon* for an object that had an escape velocity greater than light, which came to be known as the Schwarzchild radius* ^[1]. Technically, the event horizon is defined as the boundary between the inside of a black hole and the universe that exists outside of it. This is an important definition as no information can travel faster than the speed of light – hence whatever crosses that boundary can never escape and similarly can never communicate with the rest of the universe ever again ^[1]. It is important to note however, that in 1958 it was David Finkelstein who identified the important connection between the event horizon and the radial sphere that Schwarzchild had calculated earlier ^[5].

In the image above we see the different orbits that the rays of light enter around the black hole. The outer most rays follow a hyperbolic path, where the light is merely deflected. However as we get closer towards the event horizon we see something very interesting happen. The second inner most path actually finds some sort of a stable orbit where the light rays can continuously orbit the black hole just like a satellite does around Earth. This orbit is often referred to as the ‘Photon Sphere’, something you may encounter if you find yourself falling into the black hole. If you were to somehow enter at the perfect angle (such that you shared this orbit with the light), then hypothetically you should be able to see the back of your head as the light bounces off of it and completes a full orbit around to your eyes! Finally, the inner most orbit ends up crossing the event horizon and eventually spirals inwards towards the singularity.

The name ‘Black Hole’ was coined by physicist John Wheeler in 1967, likely due to the increased attention given to the still mysterious hypothetical object after the Neutron Star (object with the greatest density at the time) was discovered by Jocelyn Bell at around the same time ^[4]. Then finally, in 1970, Roger Penrose and Stephen Hawking published their singularity* theorems that would solve the space and time-like singularity problems ^[6]. These solutions effectively provided a description on how singularities existed in our universe and satisfied certain physical conditions that allow gravitational collapse to occur ^[6].

At this singularity we have infinite density – that is, an extreme amount of mass all located at a single point. This is absurd. And the mathematics thinks so too. Our best physical laws all seem to break down inside a black hole, and quite frankly no-one can say with certainty that they know what exists beyond the event horizon. However we may still have a chance at figuring it out one day. The catch? Reconciling quantum mechanics with Einstein’s General Theory of Relativity.

Let us take see the attempts which have been made at doing this, as well as the progress which has flipped our understanding of black holes more than once.

— Hawking Radiation —

In 1974 Stephen Hawking published a very important paper which postulated how quantum effects interact with black holes. Basically it describes how some particles, even after they have crossed the event horizon, can still manage to escape a black hole. This is what Hawking Radiation is known as. Now I know that it flies in the face of what I’ve told you so far, but just hear me out.

The first approach I will take to explaining this process is through the famous particle-antiparticle analogy. Imagine that energy is ‘borrowed’ from the vacuum of space to create two ‘positive energy’ particles. Remember that energy must be conserved though, so in turn a small region of space must be created containing the same magnitude of ‘negative energy’ in order to conserve our total energy. Normally these particle-antiparticle pairs would recombine and annihilate each other on a time scale so small that it would be undetectable (this is a manifestation of Heisenberg’s Uncertainty Principle) ^[7]. This process is usually referred to as a quantum fluctuation. However think about this process occurring at the brink of an event horizon; one particle would fall into the black hole whereas the other would escape it. This escaping particle makes up our Hawking Radiation. Clearly we have broken conservation laws and have created something out of nothing ^[7]. Or have we?

The solution to this problem is answered by understanding what happens to the Black Hole as a result of the opposite particle falling into it. If we imagine that the particle is experiencing a gravitational force towards the singularity, then we can identically say that the gravitational field of the Black Hole exerts a potential energy on the particle ^[8]. This potential is converted into the kinetic energy of the in-falling particle. Therefore the gravitational field of the black hole has done work on the particle to provide it with the kinetic energy to fall towards the singularity. Thus to conserve energy it must have lost equally as much of its own potential energy. Ultimately this means that the Black Hole has lost energy itself by providing it to the particle. The difference in this loss of energy of the Black Hole is exactly equal to the gain in energy by the outside universe, and hence, energy is conserved! ^[8].

While this is a really good explanation for Hawking Radiation, a more technical approach deals with real particles actually being able to ‘tunnel’ out of the impossibly high potential. I’ll revisit this point in the future as it goes far too deep into the rabbit hole for now. Just for the curious reader out there, it has to do with the wave-particle duality of matter!

— Orbiting a Black hole —

Now that we know a little more about a black hole and how it behaves, we can finally answer the question: what would happen if the Sun suddenly turned into a black hole?

Well, not much actually… the Earth (along with all of us) would be fine. Besides the fact that before turning into a black hole the explosive supernova ensuing our Sun’s death would swallow the Earth up, whole, the gravitational attraction would actually remain the same. So if we could somehow shrink the Sun to less than 6 kilometres (down from its current 1 million kilometres), then the black hole it would become wouldn’t suck us in at all. This is because the mass has remained exactly the same, so there is no additional gravity.

Credit: ESA/Hubble (M. Kornmesser)

But not all would be dull. No, something very interesting indeed would happen. Something known as gravitational lensing. At the time Einstein did his calculation for this phenomena, he believed that this effect would never be observed. But it was, and its effect is extremely amplified in the case of a black hole. Here is an animation of a black hole lensing the light from a distant galaxy.

And if the Earth was in orbit? From a point outside, say a satellite watching us, then initially the Earth would look completely normal. But as it progressed its orbit and began to pass behind the black hole we would begin to observe the gravitational lensing. The space surround the black hole is extremely warped and distorted, so the light reflected off the Earth would also follow these wacky contours in space-time, producing something like this:

Definitions:

Event Horizon:  the boundary that separates the inside of a black hole and the universe that exists outside of it.

Schwarzchild radius: radius of the event horizon.

Singularity: a point where the curvature of space-time becomes infinite.

References:

[1] – Bennett, Donahue, Schneider, Voit,  2014, The Cosmic Perspective: Seventh Edition, Pearson, Chapter 18: The Bizarre Stellar Graveyard

[2] – Halliday, Resnick, Walker,  2014, Fundamentals of Physics: 10th edition, WILEY, Chapter 8: Potential Energy and Conservation of Energy

[3] – Einstein’s Relativity. Last updated September 3, 2012. Einstein’s Relativity. [ONLINE] Available at: http://www.astronomynotes.com/relativity/s4.htm. [Accessed 20 September 2014].

[4] – N.K. Gledenning, Neutron Stars and Pulsars, 2014.  [ONLINE] Available at: http://www2.lbl.gov/Science-Articles/Archive/sb/Nov-2004/03-Neutron-Stars.pdf. [Accessed 20 September 2014].

[5] – Bennett, Donahue, Schneider, Voit,  2014, The Cosmic Perspective: Seventh Edition, Pearson, Chapter S3: Spacetime and Gravity

[6] – Singularity Theorems. 2014. Untitled Document. [ONLINE] Available at: http://www.personal.soton.ac.uk/dij/GR-Explorer/singularities/singtheorems.htm. [Accessed 20 September 2014].

[7] – Stephen Hawking, A Brief History of Time, 1988. Bantam Dell Publishing.

[8] – More on Hawking Radiation. 2014. More on Hawking Radiation. [ONLINE] Available at: http://imagine.gsfc.nasa.gov/docs/ask_astro/answers/011125b.html. [Accessed 21 September 2014].