Hawking Crosses the Event Horizon

by Nicholas Mee on March 22, 2018

Stephen Hawking died on 14 March 2018. As a student in the 1960s he was diagnosed with motor neurone disease and given just two years to live. Confounding these predictions, he lived to become the world’s most famous scientist since Einstein. His incredible determination to succeed in the face of any obstacle and his drive to live life to the full should be an inspiration to everyone. We should also remember his incredible communication skills and his lifelong commitment to those less able and less fortunate than himself.

A computer generated image showing what the Event Horizon Telescope is expected to reveal.

Hawking’s research was critical in establishing our modern understanding of black holes. When he began investigating their properties these weird objects lay at the outer fringes of respectable science. Although compelling theoretical arguments supported the idea of black holes, observational confirmation was unavailable. Hawking lived to see the arrival of overwhelming evidence for their existence and their acceptance within mainstream established physics.

We now know that a supermassive black hole of four million solar masses lies at the heart of our galaxy. The Event Horizon Telescope is currently attempting to image this beast. A monster black hole is now believed to lurk at the centre of all galaxies.

Equally marvellous are the results from the LIGO gravitational wave detector in the United States that felt the first tremor of a gravitational wave signal in 2015 originating from the collision and merger of two black holes in some far flung region of the universe. There is more about this event here: Cosmic Ripples. Several more gravitational wave signals have now been detected.

Artist’s impression of a close binary black hole system. Credit: LIGO/Caltech/MIT/Sonoma State (Aurore Simonnet).

Much can be deduced from gravitational wave signals. We know the black holes whose collision produced the first signal had 29 and 36 solar masses and they coalesced into a rapidly spinning black hole of 62 solar masses. This might look rather odd as 29 + 36 = 65, so three times the mass of the Sun seems to have disappeared in the process. In fact, this mass was converted into pure energy in the form of gravitational waves in a remarkable instance of Einstein’s famous formula E = mc^2 . Here E is energy, m is mass and c is a constant—the speed of light. Einstein’s formula implies that mass is just another form of energy, and as energy is conserved in all processes, any change in mass has to be taken into account when we balance the books.

Still, you might say: Black holes are supposed to be black. How could this mass escape from the black holes?

It is perfectly true that a black hole is surrounded by a one-way boundary known as its event horizon and anything within the event horizon is on a one-way trip into the abyss. Once inside nothing comes out again. The solution of this conundrum is that the gravitational waves did not come out of the black holes, they formed outside their event horizons. The gravitational waves carry away the binding energy of the black holes that is released as they approach and merge. To see what is going on it helps to consider analogies from other areas of physics.

Binding Energy

Schematic diagram of hydrogen fusion.

Stars are powered by converting hydrogen nuclei into helium nuclei. Four protons are transformed into a helium nucleus which consists of two protons and two neutrons. But the mass of the helium nucleus is about 4% less than the mass of the four protons. This difference in mass is the binding energy of the helium nucleus, which is released in the form of various particles during the nuclear fusion process, as shown schematically on the right. This nuclear binding energy is what lights up the Sun and stars.

Another example, possibly more familiar, but involving much less energy, is the formation of a hydrogen atom from a proton and an electron. The combined mass of a well separated proton and an electron is slightly greater than the mass of the hydrogen atom they form when bound together. There is an electromagnetic attraction between the positively charged proton and the negatively charged electron. When a proton and an electron come together to form a hydrogen atom one or more photons are emitted and these pulses of electromagnetic wave carry away the binding energy of the atom. The energy of the photons equals the mass that is lost when the hydrogen atom forms. (To separate the proton and the electron the same amount of energy must be put back in, which is possible in various ways, such as by a photon interacting with the hydrogen atom, in a chemical reaction or by heating hydrogen to a few thousand degrees.)

Schematic illustration of gravitational wave emission by a binary black hole system. Credit: LIGO Scientific Collaboration (LSC)/NASA.

Now back to the colliding black holes. There is a very strong gravitational attraction between them. The closer they approach each other the greater the amount of binding energy that is released, and this energy is radiated away in the form of gravitational waves. Consequently the mass of the bound pair of black holes is less than the total mass of the two black holes when they were well separated.

Although the length scales are vastly different, there is a clear parallel between the binary black hole and the hydrogen atom. The hydrogen atom is bound by the electromagnetic force and the binding energy of the proton and electron is carried away as electromagnetic waves. The binary black hole system is bound by gravity and the binding energy is carried away as gravitational waves.

How Big is a Black Hole?

The size of the event horizon of a black hole is determined by its mass and the rate at which it spins – the greater the mass, the bigger the black hole. If the Sun collapsed and formed a black hole, its diameter would be about six kilometres. As material falls into a black hole, its mass increases and therefore its size will also increase. The diameter of the supermassive black hole at the centre of our galaxy is about 24 million kilometres, so it would fit well within Mercury’s orbit around the Sun. Other galaxies harbour much bigger black holes. Nothing can get out of the black hole so as time passes and the black hole feeds on its surroundings it will inevitably grow in size. One way of putting this is to say that the area of the event horizon of the black hole will inevitably increase with time. This turns out to be a very general feature of black holes.

Hawking was speculating about what happens when black holes collide well before any firm evidence for their existence had been discovered. In 1971, he proved a remarkable result about what happens in such an event. When two black holes collide and merge, no matter how fast they are spinning and whatever the details of their encounter, the area of the event horizon of the final black hole must be greater than the total area of the event horizons of the two black holes that collided. Indeed, Hawking showed that in any process whatsoever, according to general relativity, the total area of black hole event horizons can never decrease. This is Hawking’s Area Theorem, a very abstract result proved by Hawking using very sophisticated mathematical techniques. Although it might sound like a quirky fact without much physical significance, it opened the door to a profound re-evaluation of the role of gravity in the universe. It is the key to an astonishing connection between collapsed stars and steam engines! But that is a story for another day. (See Hawking Radiation.)

Stephen Hawking helping to launch Breakthrough Starshot in 2016.

Stephen Hawking was Lucasian Professor of Mathematics at the University of Cambridge from 1978 until his retirement in 2009. This was the post held by Isaac Newton 300 years earlier. Hawking will be buried in Westminster Abbey close to the tomb of Sir Isaac Newton.

Further Information

There is more about Stephen Hawking and black holes in my book Gravity: Cracking the Cosmic Code.

{ 2 comments… read them below or add one }

Julius Mazzarella March 24, 2018 at 3:26 pm

Thank you for this memorial article.

The analogy you outlined about binding energy is interesting. During fission I assume the process happen in reverse for the binding from electromagnetic forces so there is symmetry. So now I am wondering about the binding energy itself from the strong interaction. I wonder if it must also be the same when quarks are bound together or separated or is this different than gravity?

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Nicholas Mee March 26, 2018 at 3:31 pm

Dear Julius
In answer to your question. The neutrons and protons in the nucleus are attracted to their nearest neighbours by the strong force. The protons also feel the electromagnetic repulsion of all the other protons. As the number of protons rises this long range electromagnetic repulsion becomes increasingly important and this is why heavy nuclei such as uranium are unstable. The net effect of the attractive strong force and repulsive electromagnetic force is that the binding energy per nucleon decreases for very heavy nuclei as the number of protons increases. This means that medium heavy nuclei are bound more tightly than uranium, so energy is released when a uranium nucleus splits into two lighter nuclei.

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