Stephen Hawking’s Scientific Legacy

Stephen Hawking died Wednesday morning, age 76. Plenty of memories and tributes have been written, including these by me:

I can also point to my Story Collider story from a few years ago, about how I turned down a job offer from Hawking, and eventually took lessons from his way of dealing with the world.

Of course Hawking has been mentioned on this blog many times.

When I started writing the above pieces (mostly yesterday, in a bit of a rush), I stumbled across this article I had written several years ago about Hawking’s scientific legacy. It was solicited by a magazine at a time when Hawking was very ill and people thought he would die relatively quickly — it wasn’t the only time people thought that, only to be proven wrong. I’m pretty sure the article was never printed, and I never got paid for it; so here it is!

(If you’re interested in a much better description of Hawking’s scientific legacy by someone who should know, see this article in The Guardian by Roger Penrose.)

Stephen Hawking’s Scientific Legacy

Stephen Hawking is the rare scientist who is also a celebrity and cultural phenomenon. But he is also the rare cultural phenomenon whose celebrity is entirely deserved. His contributions can be characterized very simply: Hawking contributed more to our understanding of gravity than any physicist since Albert Einstein.

“Gravity” is an important word here. For much of Hawking’s career, theoretical physicists as a community were more interested in particle physics and the other forces of nature — electromagnetism and the strong and weak nuclear forces. “Classical” gravity (ignoring the complications of quantum mechanics) had been figured out by Einstein in his theory of general relativity, and “quantum” gravity (creating a quantum version of general relativity) seemed too hard. By applying his prodigious intellect to the most well-known force of nature, Hawking was able to come up with several results that took the wider community completely by surprise.

By acclimation, Hawking’s most important result is the realization that black holes are not completely black — they give off radiation, just like ordinary objects. Before that famous paper, he proved important theorems about black holes and singularities, and afterward studied the universe as a whole. In each phase of his career, his contributions were central.

The Classical Period

While working on his Ph.D. thesis in Cambridge in the mid-1960’s, Hawking became interested in the question of the origin and ultimate fate of the universe. The right tool for investigating this problem is general relativity, Einstein’s theory of space, time, and gravity. According to general relativity, what we perceive as “gravity” is a reflection of the curvature of spacetime. By understanding how that curvature is created by matter and energy, we can predict how the universe evolves. This may be thought of as Hawking’s “classical” period, to contrast classical general relativity with his later investigations in quantum field theory and quantum gravity.

Around the same time, Roger Penrose at Oxford had proven a remarkable result: that according to general relativity, under very broad circumstances, space and time would crash in on themselves to form a singularity. If gravity is the curvature of spacetime, a singularity is a moment in time when that curvature becomes infinitely big. This theorem showed that singularities weren’t just curiosities; they are an important feature of general relativity.

Penrose’s result applied to black holes — regions of spacetime where the gravitational field is so strong that even light cannot escape. Inside a black hole, the singularity lurks in the future. Hawking took Penrose’s idea and turned it around, aiming at the past of our universe. He showed that, under similarly general circumstances, space must have come into existence at a singularity: the Big Bang. Modern cosmologists talk (confusingly) about both the Big Bang “model,” which is the very successful theory that describes the evolution of an expanding universe over billions of years, and also the Big Bang “singularity,” which we still don’t claim to understand.

Hawking then turned his own attention to black holes. Another interesting result by Penrose had shown that it’s possible to extract energy from a rotating black hole, essentially by bleeding off its spin until it’s no longer rotating. Hawking was able to demonstrate that, although you can extract energy, the area of the event horizon surrounding the black hole will always increase in any physical process. This “area theorem” was both important in its own right, and also evocative of a completely separate area of physics: thermodynamics, the study of heat.

Thermodynamics obeys a set of famous laws. For example, the first law tells us that energy is conserved, while the second law tells us that entropy — a measure of the disorderliness of the universe — never decreases for an isolated system. Working with James Bardeen and Brandon Carter, Hawking proposed a set of laws for “black hole mechanics,” in close analogy with thermodynamics. Just as in thermodynamics, the first law of black hole mechanics ensures that energy is conserved. The second law is Hawking’s area theorem, that the area of the event horizon never decreases. In other words, the area of the event horizon of a black hole is very analogous to the entropy of a thermodynamic system — they both tend to increase over time.

Black Hole Evaporation

Hawking and his collaborators were justly proud of the laws of black hole mechanics, but they viewed them as simply a formal analogy, not a literal connection between gravity and thermodynamics. In 1972, a graduate student at Princeton University named Jacob Bekenstein suggested that there was more to it than that. Bekenstein, on the basis of some ingenious thought experiments, suggested that the behavior of black holes isn’t simply like thermodynamics, it actually is thermodynamics. In particular, black holes have entropy.

Like many bold ideas, this one was met with resistance from experts — and at this point, Stephen Hawking was the world’s expert on black holes. Hawking was certainly skeptical, and for good reason. If black hole mechanics is really just a form of thermodynamics, that means black holes have a temperature. And objects that have a temperature emit radiation — the famous “black body radiation” that played a central role in the development of quantum mechanics. So if Bekenstein were right, it would seemingly imply that black holes weren’t really black (although Bekenstein himself didn’t quite go that far).

To address this problem seriously, you need to look beyond general relativity itself, since Einstein’s theory is purely “classical” — it doesn’t incorporate the insights of quantum mechanics. Hawking knew that Russian physicists Alexander Starobinsky and Yakov Zel’dovich had investigated quantum effects in the vicinity of black holes, and had predicted a phenomenon called “superradiance.” Just as Penrose had showed that you could extract energy from a spinning black hole, Starobinsky and Zel’dovich showed that rotating black holes could emit radiation spontaneously via quantum mechanics. Hawking himself was not an expert in the techniques of quantum field theory, which at the time were the province of particle physicists rather than general relativists. But he was a quick study, and threw himself into the difficult task of understanding the quantum aspects of black holes, so that he could find Bekenstein’s mistake.

Instead, he surprised himself, and in the process turned theoretical physics on its head. What Hawking eventually discovered was that Bekenstein was right — black holes do have entropy — and that the extraordinary implications of this idea were actually true — black holes are not completely black. These days we refer to the “Bekenstein-Hawking entropy” of black holes, which emit “Hawking radiation” at their “Hawking temperature.”

There is a nice hand-waving way of understanding Hawking radiation. Quantum mechanics says (among other things) that you can’t pin a system down to a definite classical state; there is always some intrinsic uncertainty in what you will see when you look at it. This is even true for empty space itself — when you look closely enough, what you thought was empty space is really alive with “virtual particles,” constantly popping in and out of existence. Hawking showed that, in the vicinity of a black hole, a pair of virtual particles can be split apart, one falling into the hole and the other escaping as radiation. Amazingly, the infalling particle has a negative energy as measured by an observer outside. The result is that the radiation gradually takes mass away from the black hole — it evaporates.

Hawking’s result had obvious and profound implications for how we think about black holes. Instead of being a cosmic dead end, where matter and energy disappear forever, they are dynamical objects that will eventually evaporate completely. But more importantly for theoretical physics, this discovery raised a question to which we still don’t know the answer: when matter falls into a black hole, and then the black hole radiates away, where does the information go?

If you take an encyclopedia and toss it into a fire, you might think the information contained inside is lost forever. But according to the laws of quantum mechanics, it isn’t really lost at all; if you were able to capture every bit of light and ash that emerged from the fire, in principle you could exactly reconstruct everything that went into it, even the print on the book pages. But black holes, if Hawking’s result is taken at face value, seem to destroy information, at least from the perspective of the outside world. This conundrum is the “black hole information loss puzzle,” and has been nagging at physicists for decades.

In recent years, progress in understanding quantum gravity (at a purely thought-experiment level) has convinced more people that the information really is preserved. In 1997 Hawking made a bet with American physicists Kip Thorne and John Preskill; Hawking and Thorne said that information was destroyed, Preskill said that somehow it was preserved. In 2007 Hawking conceded his end of the bet, admitting that black holes don’t destroy information. However, Thorne has not conceded for his part, and Preskill himself thinks the concession was premature. Black hole radiation and entropy continue to be central guiding principles in our search for a better understanding of quantum gravity.

Quantum Cosmology

Hawking’s work on black hole radiation relied on a mixture of quantum and classical ideas. In his model, the black hole itself was treated classically, according to the rules of general relativity; meanwhile, the virtual particles near the black hole were treated using the rules of quantum mechanics. The ultimate goal of many theoretical physicists is to construct a true theory of quantum gravity, in which spacetime itself would be part of the quantum system.

If there is one place where quantum mechanics and gravity both play a central role, it’s at the origin of the universe itself. And it’s to this question, unsurprisingly, that Hawking devoted the latter part of his career. In doing so, he established the agenda for physicists’ ambitious project of understanding where our universe came from.

In quantum mechanics, a system doesn’t have a position or velocity; its state is described by a “wave function,” which tells us the probability that we would measure a particular position or velocity if we were to observe the system. In 1983, Hawking and James Hartle published a paper entitled simply “Wave Function of the Universe.” They proposed a simple procedure from which — in principle! — the state of the entire universe could be calculated. We don’t know whether the Hartle-Hawking wave function is actually the correct description of the universe. Indeed, because we don’t actually have a full theory of quantum gravity, we don’t even know whether their procedure is sensible. But their paper showed that we could talk about the very beginning of the universe in a scientific way.

Studying the origin of the universe offers the prospect of connecting quantum gravity to observable features of the universe. Cosmologists believe that tiny variations in the density of matter from very early times gradually grew into the distribution of stars and galaxies we observe today. A complete theory of the origin of the universe might be able to predict these variations, and carrying out this program is a major occupation of physicists today. Hawking made a number of contributions to this program, both from his wave function of the universe and in the context of the “inflationary universe” model proposed by Alan Guth.

Simply talking about the origin of the universe is a provocative step. It raises the prospect that science might be able to provide a complete and self-contained description of reality — a prospect that stretches beyond science, into the realms of philosophy and theology. Hawking, always provocative, never shied away from these implications. He was fond of recalling a cosmology conference hosted by the Vatican, at which Pope John Paul II allegedly told the assembled scientists not to inquire into the origin of the universe, “because that was the moment of creation and therefore the work of God.” Admonitions of this sort didn’t slow Hawking down; he lived his life in a tireless pursuit of the most fundamental questions science could tackle.

 

31 Comments

31 thoughts on “Stephen Hawking’s Scientific Legacy”

  1. Very enjoyable articles — many thanks. We must all be very grateful to the UK’s National Health Service, which gave our species an extra 50+ years of Stephen Hawking’s genius.

    As an irrelevant aside, near the foot of your Atlantic piece, if they’re Scotch they’re whiskies, not whiskeys.

  2. Question from a layman: If Bekenstein discovered “Hawking” radiation, why is Hawking more associated with that result?

  3. He didn’t. Bekenstein suggested that black holes have entropy; Hawking calculated the exact amount of entropy, and showed that there would be radiation. Thus, it’s the “Bekenstein-Hawking entropy,” and “Hawking radiation.”

  4. Thank you for the article. It was a great read. I see that you’ve mentioned the virtual particle-antiparticle way of understanding Hawking radiation is hand-wavey. Can you give us the non-hand-wavey explanation please?

  5. Loved the WHAT WOULD STEPHEN HAWKING DO? podcast… next time I am frustrated, I will just ask WWSH. We need to learn to stop complaining about what we can’t control and just do our thing.

  6. I really enjoyed reading Stephen Hawking’s books. He did much to inspire interest in the fields of physics and cosmology by making important theories understandable to the general public. His gifted insight will be greatly missed.

  7. I asked him once if he could have the answer to one question, what would it be?
    He said, he would just like to know there WAS an answer.

  8. So, in physics, a singularity is a point in time or space or both having some infinite values (curvature, energy, density). But neither our minds nor our mathematics can handle infinities. In addition, there are practical limits to how close we can approach the singular state in experiments because of the very high energies involved. And where experimentation is limited, theories are little more than possibilities. So perhaps Pope John Paul II and science are in agreement in that there is a limit to what we can possibly know, with any real confidence, about how the universe began.

  9. Dr Carroll,

    An off-topic but slightly related question – I heard that Raychaudhuri (RC) equation extends black hole singularity to BB singularity. How far is it true? If it is not so, what does RC equation exactly do (and infer) and what is Hawking’s critical insight that built on RC equation?

    Thank you

  10. ER=EPR for black holes that has his roots in hawkings study of black holes;he made a giant step in provin gravity and space -time quantum nature;i am curios was hawking an everettian or he believed in copenhagen or other model of qm?

  11. “But according to the laws of quantum mechanics, it isn’t really lost at all; if you were able to capture every bit of light and ash that emerged from the fire, in principle you could exactly reconstruct everything that went into it, even the print on the book pages.”

    Actually this is the part I can never understand. If you could, even in principle, take a bunch of particles, reverse the trajectory and reconstruct any earlier state, wouldn’t that imply that you could measure the position and momentum of each particle to an arbitrarily high precision? Which of course you can’t.

  12. The state isn’t given by “a bunch of particles”; it’s given by a single wave function. The evolution of that is completely reversible (if we ignore collapse due to observation).

  13. But doesn’t the reversal of the wave function just take you back to a superposition of possible encyclopaedias? Any specific encyclopaedia would be lost.

  14. @Robin:

    I think you are getting confused because you are taking the analogy of “reconstructing backwards” too literally. AFAIK, the physics works like this:

    1. In classical physics, position (x) and momentum (p) are the necessary aspects you need to know to define the system completely. You give me these two, I have what I need to know about every physically meaningful characteristic of the system. E = p^2/2m and so on.

    This is another way of saying that the position (x) and momentum (p) give all the necessary information about the system. Information is abstract in this sense. Momentum and position are not the information in themselves, they give the information about the system.

    2. In quantum mechanics, we have a “wave function” which does the job of “position and momentum” in classical mechanics. It means that given a wave function, I have all the information needed to know about the system.

    3. There are three interlinked foundational principles or assumptions about the wave function

    a) the amplitude squared of the wavefunction gives the probability of finding a particle at given point.

    b) the sum of probabilities at all points where the particle can possibly be, should be 1. If not, the particle is disappearing or the wavefunction is not correct, which is not acceptable.

    c) Following from (b), the sum of probabilities is 1 not just now, but it was also 1 in the past, and it should be 1 in the future. This is the crucial part. This is the other way of saying “information is NOT lost”.

    When we say that information is NOT lost, what we mean is that the condition of “sum of probabilities is 1” is satisfied at all times.

    Note that the form of wavefunction can change in future, but we are concerned only about the sum of probabilities (amplitude squared). We are fine as long as the sum is 1.

    In case if you want to go back in time, all you need is the probabilities as they were in the past, which will have to add up to 1. Unlike position and momentum where there are uncertainty restrictions on measuring them precisely, there is no such precision restriction on probability.

    4. What’s happening in the Hawking radiation is that – as wavefunctions (particles) fall into the black hole, you are putting information inside it. Once the black hole evaporates, all the wave functions (information) thrown inside is gone. It means that after the black hole evaporates, the sum of probabilities would NOT be 1 for all those wavefunctions thrown in. Hence, we say that information is lost, which is a violation of the foundational principle of quantum mechanics, 3 (c) discussed above.

    In other words, we are not cobbling the book (thrown into a black hole) together going back in time, in its literal sense, as you think. We are just wanting to ensure that the sum of probabilities of the wavefunction to be 1 at all times.

    Note that we are not using “position” and “momentum” here. The high precision is required only in such case, which is not possible due to the uncertainty principle. If we are using the probabilities of the wave function, all you need to do is to ensure the probability at a given point, which can be done precisely.

    @Others: Please correct me if I am wrong. 🙂

  15. Torbjörn Larsson

    Lovely article, and nice illustration!

    @Bruce: “But neither our minds nor our mathematics can handle infinities. In addition, there are practical limits to how close we can approach the singular state in experiments because of the very high energies involved. And where experimentation is limited, theories are little more than possibilities. So perhaps Pope John Paul II and science are in agreement in that there is a limit to what we can possibly know, with any real confidence, about how the universe began.”

    Who says mathematics cannot handle infinities? There are mathematical theorems on infinities, and one that implies we can always derive one bit of information (order of infinity) from them. Limited theorems, but certainly not information free.

    Physics on the other hand tend to avoid infinities, which is one reason quantum field theory extend classical field theory. Sean has remarked more than once that Penrose-Hawking singularity theorems only apply classically and breaks down at Planck energies. But mostly, generic inflation avoids singularities (and the extraordinary and evidence-free religiously hallowed idea of ‘nothing instead of something’) altogether. It predicts a local universe as a ghost of (eventual exit from a series of) quantum fluctuations, and more tangibly laws as a result of selection bias of observers and structures as a result of previous quantum fluctuations.

    I simply do not understand why people confuse myth with knowledge. How hard can it be to understand the difference!?

  16. Concerning Hawking radiation and black hole evaporation. A black hole of 1 solar mass will take 2 * 10**67 years to evaporate. That’s approximately 10**58 times the current age of the universe. I also read that a black hole of 1 solar mass absorbs much more energy from cosmic background radiation than it emits by Hawking radiation.
    My question is: as the universe continues to expand is there a point where the cosmic background radiation “dilutes” to the point where Hawking radiation leads to a net loss of mass? I suppose there must be but I haven’t seen an explicit statement on this.

  17. Hi K

    In other words, we are not cobbling the book (thrown into a black hole) together going back in time, in its literal sense, as you think.

    I was referring to the following:

    If you take an encyclopedia and toss it into a fire, you might think the information contained inside is lost forever. But according to the laws of quantum mechanics, it isn’t really lost at all; if you were able to capture every bit of light and ash that emerged from the fire, in principle you could exactly reconstruct everything that went into it, even the print on the book pages.

    which is not referring to a black hole, but tossing an encyclopedia into a fire.

    If you could, in principle, reconstruct the burned up encyclopedia from capturing every bit of light and ash then how, in principle, would you do it? In what sense could it be reconstructed? What properties of that light and ash would you use to reconstruct the encyclopedia and how would you use them to reconstruct it?

    As far as I can see that is, even in principle, impossible.

  18. @Robin

    Falling into black holes or catching fire all are metaphors. Don’t take them literally. All that the metaphor is trying to convey is that – even after the “form of information changes”, it is conserved. If not, the literal meaning of reconstruction would also mean that you can do backwards time travel and so on.

    Also, all that the metaphor is saying is that “information is conserved”, not that you can do it in practice. It’s same as saying that energy is conserved. A metaphorical explanation of energy conservation would be that, if you go back in time, you would get the precise energy in the way it was in the past. It neither means that we can do backwards time travel nor does it mean that we can “practically” reconstruct the precise form of energy. It’s just a conservation principle.

    If you still have trouble wrapping your head around this, I recommend doing an undergraduate course in Quantum Mechanics. Solving QM problems will help you get this intuition, without getting confused with metaphors.

    I recommend Allan Adams https://www.youtube.com/watch?v=lZ3bPUKo5zc&list=PLUl4u3cNGP61-9PEhRognw5vryrSEVLPr Just around 30 hours of video lectures.

    Cheers

  19. Sad to hear about Stephen’s passing.

    While as a creationist I do aspire to be Sean’s unofficial metaphysics nemesis, I must admit I enjoyed the ‘What Would Stephen Hawking Do?’ video immensely.

  20. I suppose I could frame my question better.

    In the case of the burning encyclopedia, the answer the the question “where did the information go?” is answered by “You could, in principle, reconstruct that information from currently existing information”.

    Couldn’t that also go for black hole evaporation? Could you, in principle, take the information following the evaporation (ie the particles which escaped), roll it backward and find that it could only have resulted from there having been a black hole there, ie to ‘unevaporate’ the black hole? Then couldn’t you, in principle, roll it back to the point where a particle approaches the black hole, another comes out and they annihilate each other, so we get back to the reverse of the original pair production which was described as the origin of the problem?

    Or would the evaporation leave insufficient information to reverse the physics?

    Would that mean that there couldn’t be a wave function of the Universe? Or at least, not a reversible one?

  21. Robin – I believe the issue is that the particles which caused the evaporation are unrelated to the particles/objects which were consumed by the black hole previously, and because of that, the evaporation-causing virtual particle pairs don’t contain any information about the previous consumed stuff that contributed to the actual mass of the black hole.
     
    Whereas, if we burn an encyclopedia, the particles which are coming off of that chemical reaction were created directly from it: heat + book + oxygen -> ash + smoke + water vapor + carbon dioxide + light + heat (heat is on both sides of the equation because the initial heat just gets us over the activation energy of the burning reaction; after that point, the heat produced in the reaction is enough to drive it forward), and you would need every possible bit of information about the products to reconstruct: velocity and energy of individual particles, not just the bulk information that we can actually collect. If you could truly examine every photon and particle and follow its path backwards, you would eventually “rewind” far enough to reconstruct the encyclopedia.

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