Hawking Radiation

by Nicholas Mee on March 19, 2018

What is the connection between a steam engine and a collapsed star?

Not much, you might think. There is, however, a very deep and subtle connection that is still not completely understood.

James Prescott Joule (1818-1889).

Brewing Up New Theories of Physics

James Prescott Joule, the son of a wealthy Manchester brewer, was taught physics by John Dalton, famous for introducing the idea of atoms into chemistry. When Joule inherited the brewery he began experimenting with ways to make the brewing process more efficient. These investigations led to one of the most important principles in physics—conservation of energy or the First Law of Thermodynamics—which he established in the 1840s.

Although energy may be converted from one form into another, if all types of energy are taken into account, including heat, then the total amount of energy never changes. We may burn coal to heat water and produce steam that drives a turbine to generate electricity that powers our computers, but at every step the total amount of energy remains the same.

Revolutionary Ideas

In the early decades of the 19th century the development of steam power was rapidly increasing Britain’s industrial and military might. Decades before Joule’s experiments, Sadi Carnot realised that France needed more efficient steam engines if she was going to maintain her military strength.

Sadi Carnot (1796-1832) aged 17 painted by Louis-Léopold Boilly.

Sadi had trained as a military engineer and was the son of Lazare Carnot who, as Minister for War, had established France’s Revolutionary Army. Sadi Carnot is responsible for the second pillar of thermodynamics.

Many processes conserve energy, but are never seen. For instance, heat never flows from a cold object, such as a lump of ice, into a warm object, such as a cup of tea. If this were possible, we could imagine adding ice to the tea in such a way that the ice becomes colder and the tea gets warmer. This might conserve energy, yet we know that it never happens—we cannot boil a cup of tea by adding any amount of ice to it.

Carnot invented the concept of entropy as a book keeping device to account for this prohibition. He defined entropy as a quantity of heat divided by the temperature. For a given amount of heat, the entropy is smaller at a high temperature than at a low temperature, so when heat is transferred from a hot object to a cold object, entropy increases, whereas if heat were transferred from a cold object to a hot object, entropy would decrease. We are very familiar with the first of these processes, but our teatime experiments tell us the second process never happens. Carnot captured this observation in the statement that in any allowed process the total entropy of the universe can never decrease. This is the Second Law of Thermodynamics. It is now known to be due to the statistical behaviour of large numbers of interacting particles.

Battersea Power Station on the banks of the Thames in London. (Famous for its flying pigs.) Credit: David Samuel.

The Second Law is less familiar than the First, but it is equally important. One consequence is that when energy is converted from one form to another in a power station, a portion of the energy is always emitted as heat. It is, therefore, impossible to convert all the energy in a lump of coal into electricity, as some of the energy will always escape as heat. However, another consequence of the Second Law is that by increasing the temperature difference between the turbines and the surrounding environment, the proportion of the energy that is lost as heat can be reduced, thereby improving the efficiency of the power station. So there are very important real world applications. (As well as some rather otherworldly applications.)

Black Hole Dynamics

Jacob Bekenstein (1947-2015).

Israeli physicist Jacob Bekenstein worried that black holes might offer the universe the ultimate waste disposal system by which it could lose some of its entropy. Black holes are featureless and seemingly have no entropy, so any material falling into a black hole would be lost along with the entropy that it contains. This appeared to reduce the entropy of the universe in violation of the Second Law.

After some contemplation, Bekenstein realised there might be a solution. When matter falls into a black hole, the black hole’s mass increases and it therefore increases in size, in particular, the surface area of the black hole’s event horizon increases. Perhaps the increase in the surface area of the event horizon might compensate for the loss of entropy contained in the material falling into the black hole. Hawking had recently proved that in all physical processes the total area of black hole event horizons can never decrease, as described in Hawking Crosses the Event Horizon. Bekenstein was struck by the parallel between Hawking’s result and the Second Law of Thermodynamics. He proposed a generalization of the law by combining:

Hawking’s Area Theorem: The total surface area of all the black holes in the universe can never decrease.

And the Second Law of Thermodynamics: The total entropy of the universe can never decrease.

To produce the generalized Second Law: The total entropy of the universe outside black hole event horizons plus the total area of all the black hole event horizons can never decrease.

This is rather a mouthful, but it can be made much more succinct by simply identifying the area of a black hole’s event horizon as the black hole’s entropy. This is what  Bekenstein tentatively proposed. But how could this be? The area theorem of black holes is a result about gravity and geometry, whereas the Second Law of Thermodynamics is a statistical law about heat.

A Restless Night

Now this was all very strange. How could there be a link between gravity and a theory devised to explain steam engines? Furthermore, black holes were believed to be characterized simply by their mass and the rate at which they spin plus possibly their electric charge. How could they have any statistical properties?

Hawking’s reaction when he heard of Bekenstein’s proposal was that it was ridiculous. It could not possibly be true. He went to bed that night convinced there must be a fundamental flaw in the argument. But he couldn’t sleep. He lay awake looking for a counterargument that would demolish this absurd idea. It was clear to Hawking that if a black hole had entropy, then it must also have a temperature and that was nonsense. All objects with a temperature greater than absolute zero must emit radiation, and the hotter the object, the greater the intensity of the radiation. But black holes are inherently black—they cannot emit radiation because nothing, not even light can escape a black hole.

An artist’s conception of the black hole known as Cygnus X-1. Credit: NASA/CXC/M.Weiss.

Then Hawking had a revelation. He realised there was a fatal flaw in his own argument and that Bekenstein’s was right—the surface area of a black hole really does measure the black hole’s entropy. Indeed, contrary to what everyone believed a black hole must behave like a body with a well-defined temperature and it will therefore emit radiation. Despite this counter-intuitive conclusion, everything fitted together perfectly. There was no chance that Hawking would sleep now. Hawking’s illness had progressed to the point where he could no longer get out of bed by himself, so he had to wait several hours for his nurse to arrive before he could put his thoughts down on paper.

Hawking Radiation

General relativity remains the best theory of gravity that we have. But general relativity is a classical theory not a quantum theory and this was the crux of Hawking’s breakthrough. It is true that when a black hole is modeled using general relativity its temperature is zero, because, as everybody knew, radiation may fall into a black hole, but nothing ever comes out, so black holes do not give off heat. However, although general relativity is a fantastically accurate theory, the world plays by the rules of quantum mechanics, so any description of black holes in the real world must incorporate the effects of quantum mechanics. Hawking realised that when quantum effects are included black holes have a non-zero temperature and consequently black holes are not completely black they emit radiation. Hawking announced his startling conclusion at a meeting of his colleagues in February 1974 and was met with utter disbelief.

This was the first link between gravity and quantum mechanics that anyone had devised, so it is no surprise that it took physicists a while to realise that Hawking’s bold idea must be true. It is very surprising that a temperature can be assigned to a black hole and contrary to what everyone expected. The radiation emitted by a black hole is now known as Hawking radiation. 

If you fell into a black hole you would be crushed and stretched and it would get pretty warm. In fact, you would be vaporised well before reaching the centre of the black hole. But the temperature of the black hole referred to by Hawking is the temperature measured by someone outside the black hole. This is determined by the radiation, or in other words heat, that the black hole is emitting. So, despite whatever incredible violence might be going on inside the black hole its temperature can be very low. A black hole with the mass of a star has a temperature that is unmeasurably low. Hawking’s calculations show that the temperature of a black hole of ten solar masses is less than one ten millionth of a degree above absolute zero. Any radiation emitted by such a black hole would be completely swamped by background radiation, so unfortunately it would be completely undetectable.

Mini Black Holes

Hawking speculated that if the early universe was quite lumpy, then some of the denser regions might have undergone the ultimate gravitational collapse to from primordial mini black holes in the immediate aftermath of the Big Bang. These hypothetical mini black holes might have a mass equal to that of an asteroid packed into a region smaller than an atom. Their low mass would make them much hotter than stellar mass black holes and, being hot, they would emit large amounts of radiation, thereby losing some of their mass. This decrease in mass would raise their temperature further, and this would further increase the radiation they emit in a runaway process that can only end one way. The temperature of a mini black hole would rise dramatically in its final moments until it exploded and disappeared in a huge blast of radiation.

Stephen Hawking (1942-2018). Credit: NASA

If primordial black holes really exist, then it should be possible for astronomers to detect them going bang as they disappear in a puff of gamma rays. The smallest such primordial black holes would already have exploded. Black holes with a mountain-sized mass of around 100 billion kilograms should currently be on the verge of detonation. Slightly larger and the mini black holes will continue emitting X-rays and gamma rays for many aeons to come. To date none have ever been seen.

Whether mini black holes exist or not, there is absolutely no doubt the general principles of Hawking’s theory are correct and that black holes do emit Hawking radiation. The combination of general relativity, quantum mechanics and thermodynamics mesh together so well that these ideas must play an important role in the fundamental structure of the universe. This remains the most important result linking quantum mechanics to gravity. It is among the most profound ideas in the history of physics.

Further Reading

A video version of this post is now available here: Black Holes and Hawking Radiation on The Cosmic Mystery Tour YouTube Channel. Please don’t forget to click the subscribe button.

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

The Physical World: An Inspirational Tour of Fundamental Physics, that I co-authored with Nick Manton, contains a mathematical introduction to thermodynamics, including further details of Hawking’s ideas about black holes.

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