Monthly Archives: November 2012

Firewalls

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There’s been a huge kerfuffle in the quantum gravity community since this summer, when some people here at UCSB published a paper arguing that (old enough) black holes may actually be surrounded by a wall of fire which burns people up when they cross the event horizon.  This is huge, because if it were true it would upset everything we thought we knew about black holes.

General relativity is our best theory of gravity to date, discovered by Einstein.  This is a  classical theory.  (In the secret code that we physicists use, classical is our code-word for “doesn’t take into account quantum mechanics”.  Don’t tell anyone I told you.)

In my other posts on physics, I’ve been trying to explain the fundamentals of physics in the minimum number of blog posts.  This post is out of sequence, since I haven’t described general relativity yet!  But I wanted to say something about exciting current events.

In classical general relativity, a black hole is a region of space where the gravity is so strong that not even light can escape.  They tend to form at the center of galaxies, and from the collapse of sufficiently large stars when they run out of fuel to hold them up.  A black hole has an event horizon, which is the surface beyond which if you fall in, you can’t ever escape without travelling faster than light. The information of anything falling into the black hole is lost forever, at least in classical physics.

In the case of a non-rotating black hole, without anything falling into it, the event horizon is a perfect sphere.  (If the black hole is rotating, it bulges out at the equator.)  If you fall past the event horizon, you will inevitably fall towards the center, just as in ordinary places you inevitably move towards the future.

At the center is the singularity.  As you approach the singularity, you get stretched out infinitely in one direction of space, and squashed to zero size in the other two directions of space, and then at the singularity time comes to an end!  Actually, just before time comes to an end, we know that the theory is wrong, since things get compressed to such tiny distances that we really ought to take quantum mechanics into account.  Since we don’t have a satisfactory theory of quantum gravity yet, we don’t really know for sure what happens.

Now it’s important to realize that the event horizon is not a physical object.  Nothing strange happens there.  It’s just an imaginary line between the place where you can get out by accelerating really hard, and the place where you can never get out.  Someone falling into the black hole just sees a vacuum.  If the black hole was formed from the collapse of a star, the matter from the star quickly falls into the singularity and disappears.  The black hole is empty inside, except for the gravitational field itself.

We don’t know how to describe full-blown quantum gravity, but we have something called semiclassical gravity which is supposed to work well when the gravitational effects of the quantum fields are small.  In semiclassical gravity, one finds that black holes slowly lose energy from thermal “Hawking” radiation.  This radiation looks exactly like the random “blackbody radiation” coming from an ordinary object when you heat it up. Here’s the important fact: You can prove that the radiation is thermal (i.e. random) just using the fact that someone falling across the horizon sees a vacuum (i.e. empty space) there.

The Hawking radiation comes from just outside the event horizon.  It does not come from inside the black hole, so in Hawking’s original calculation it doesn’t carry any information out from the inside.   Nevertheless, for various reasons I can’t go into right now, most black hole physicists have convinced themselves that the information eventually does come out.

As the black hole radiates into space, it slowly evaporates, and eventually probably disappears entirely (although knowing what happens at the very end requires full-blown quantum gravity).  If the outgoing Hawking radiation carries all the radiation out, then for a black hole at a late enough stage in its evaporation, the radiation must not be completely random, because it actually encodes all the information about what fell in.

The gist of what Almheiri, Marolf, Polchinski, and Sully argued, is that if we take both of these statements in bold seriously, then it follows that the black holes are NOT in the vacuum state from the perspective of someone who falls in.  Instead you would get incinerated by a “firewall” as you cross the horizon.  (It’s not clear yet whether this is only for really old black holes, or if it applies to younger ones too.)  That’s if we still believe there is an “inside” at all.  The argument shows that semiclassical gravity is completely wrong in situations where we would have expected it to work great.

If this is right, then it’s devastating to the ideas of many of us who have been thinking about black holes for a long time.  As a reluctant convert to the idea that information is not lost, I’m wondering if I should reconsider.  At the end of this month, I’m going to Stanford for a weekend, since Lenny Susskind has invited a bunch of us to try to get this worked out.  Exciting times!

 

Pillar of Science IV: Precise Descriptions

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Science involves Precise Description.

To be capable of being confirmed or ruled out at the high levels of reliability associated with Science, a hypothesis must be stated in a way which is precise enough to do definitive tests.  Even if a scientific hypothesis may not be experimentally testable at the present time, a precise formulation helps indicate ways that it could be tested in the future.  If experiment is capable of making everyone eventually agree on whether the idea worked or not, then the words it is expressed in shouldn’t mean different things to different people.

Mathematical models of the physical world are the most precise form of description available, because they can describe complicated systems with perfect exactitude.  In theoretical physics, this kind of quantitative description is the usual way to make things precise.  We like to think about systems that are simple enough to describe mathematically (of course, this requires first making certain approximations).  In fact, fundamental physics is so mathematical that, even when there are no or few experiments, one can often make progress just by demanding that the model be logically consistent, and that it conform to known physical principles.  (Known, because they have been tested in other situations where we can do experiments.)  Mathematical consistency is nearly our only guide in speculative fields like my own (quantum gravity); however, it cannot completely substitute for observations, since no matter how consistent or beautiful your model is, Nature could always do something else when you finally are able to take a look.

So Math is great when you can get it.  Nevertheless, systems which are less regular, more complex, or less well-understood (such as biological life) cannot always be described mathematically, but may still be described through technical vocabulary that minimizes imprecision, without removing it altogether.  I’m not a biologist, so I’m probably not the best person to ask how this usually works, but I didn’t want to give the impression that math is the only way to make ideas precise.  Even in physics, at one time it was possible to describe everything in words.  The great experimenter St. Faraday (whose work helped established the concept of the electromagnetic field), once wrote a letter to St. Maxwell (who wrote down the equations for electromagnetism) expressing surprise at Maxwell’s need to translate everything into mathematical equations.  Yet no one could accuse Faraday’s journals of being imprecise.

But not all concepts will do.  Ideas that are apprehended in words or images rich with heavy associations or mottled with variegated meanings—in short, using the common language of humanity—such ideas are excluded from Science.  Not because it is impossible to discuss and test these ideas; if that were true, then it would be impossible to think accurately at all about most matters of ordinary human concern.  Rather, it is because they involve elements of human and holistic judgement which are unsuitable for scientific inquiry.  The question “Is xenophobia a frequent cause of war?” could be given an informed and accurate answer by a historian, but it does not become a scientific question until the terms “xenophobia”, “frequent”, and “war” are given technical meanings sufficiently precise that a social scientist can do a statistical analysis.

Medieval Bashing

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Recently I ran across a pretty good explanation of the Higgs mechanism (hat tip Siris) by a certain Rob Knopp, which I thought I’d link to because of its connection with my previous post on fields.  When I first looked at his blog, it seemed like maybe I’d found a kindred spirit: someone who blogs on science while identifying as Christian.  Unfortunately, it turns out he actually denies almost all traditional Christian beliefs.  On this blog, religion won’t mean something watered down until it makes few if any factual claims.  Instead I adhere to the red-hot supernaturalist “original brand” of Christianity that includes real miracles, a divine Incarnation, Atonement, inspired Scripture (including the Old Testament), the Second Coming, etc.  Just in case you were wondering.

However, what really stood out to me from the last link was this throwaway line about medieval culture:

[Jesus] lived an errant life, eschewing the temporal power that monarchs would later claim was justified by the “divine right of kings,” even eschewing extreme temporal influence.  He preached the opposite of a lot of medieval social philosophy: the poor are not lesser humans and thus worthy of their lot, but if anything exalted. (Blessed are the poor, and all of that.)

Assuming that this was meant as a criticism of Medieval Europe (and not, say, the caste system of Medieval India), this is almost as far off base as if he said that Medievals had landed on the Moon.  The Medievals did not believe that the poor are “lesser humans and thus worthy of their lot”.  It would be much more accurate to say that they believed that the poor were superior humans and that poverty is, not indeed strictly necessary for salvation, but highly desirable for anyone wanting to live more spiritually.  Which explains all those people who swore vows of poverty in order to live in monasteries.  It’s almost as though they were familiar with the teachings of Jesus on the subject of poverty!

It’s true that Medieval Europeans believed that the rule of earthly Kings was ordained by God.  But the claim that so-and-so was the rightful King of England originally had nothing whatsoever to do with spiritual superiority, any more than your claim to own your car, or the President’s claim to be legitimately elected, implies any notion of being spiritually superior to other people.  Medieval Christians (like Ancient and Modern Christians) believed that God has ordained the existence of human governments to enforce justice, and that therefore (barring exceptional circumstances) it is our duty to obey whatever government one happens to live under.  There was indeed a much more extreme theory of the “divine right of kings” that basically said that the monarch could not be resisted under any circumstances whatsoever.  However, this theory was popular, not in Medieval times, but rather the Early Modern era (roughly the 16th-18th centuries).

As St. Chesterton pointed out, people are happy to accuse almost anything that seems old and bad as “Medieval” without checking to see what Medieval people actually thought and did.  In reality, prior to the Renaissance, the Medievals:

  • invented Academia as we know it, and founded the first Universities,
  • strongly believed that one could discover the nature of the Universe using logical reasoning based on appropriate authorities,
  • held in high regard ancient pagan learning and culture,
  • like all educated people after Aristotle, knew the Earth is spherical and that the universe is huge compared to the Earth,
  • prohibited military attacks on civilians, and tried to restrict war to certain days,
  • ended nearly all chattel slavery in Europe (but see below),
  • developed the notion that government must respect certain human rights (in feudalism, serfs were tied to the land but had customary rights which the lord was required to respect),
  • officially taught that witchcraft was impossible, and that the popular belief in witches was a superstition to be discouraged.

Yes, you heard that right.  They didn’t burn witches.  They did burn heretics, but those were real whereas witches were a figment of the peasants’ imagination.  Witch trials didn’t become popular until the supposedly more enlightened Renaissance and Early Modern Era (mainly 1484-1750).  Even then they didn’t burn witches, they hung them.  (What about all that stuff about dumping the witch in water to see if she floats?  You know, either she floats and is convicted, or she drowns and is posthumously acquitted.  Of course, no one would actually be so stupid as to devise a trial system that kills the innocent on purpose.  In reality, the witch-hunters would pull those who sank out of the water before they drowned.)

The early Medievals did occasionally use trial by ordeal, when the evidence of guilt was doubtful—for other crimes than witchcraft, which they didn’t believe in, remember!  However, the trials actually appear to have been rigged to produce acquittals.  In any case, these mostly ended soon after 1215 when the Church refused to allow priests to cooperate.  In England this method was replaced with trial by jury.

The Modern Period also brought the racist version of the slave-trade into the world.  Unfortunately, because of failure to turn the other cheek when pagans or Muslims would capture Christians as slaves, some slavery of non-Christians was permitted.  Unfortunately, this meant that slave-traders were in existence when the New World was opened up…

The idea that the Medieval scholars believed the earth was flat is a lie invented in the 19th century by rationalists eager to find a historical precedent useful for mocking creationists, as documented by the historian St. Jeffrey Burton Russell here and here (I haven’t read his book but I’ve seen him talk).  For more information about the actual Medieval worldview, you can’t do better than reading St. Lewis’ wonderful book The Discarded Image.  I should have put a gazillion more links in this post, but you all know how to use Wikipedia.

The idea of inevitable moral progress with time is much easier to believe if you only have a superficial notion of history.  I do think that we’ve made important progress in justice over the past 2 or 3 centuries, but notice that a lot of these involve undoing the moral mistakes of the Early Modern era.  Like all eras, the Medievals had many moral blind spots.  But then again, so do we.

 

Fields

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What is the world made out of?  In the most usual formulations of our current best theories of physics, the answer is fields.  What are those?

Well, if you know what a function is, you’re already most of the way there.  A function, you will recall, is a gadget where, for any number you input, you can get a number out as an output.  We can write \(f(x)\) where \(x\) is the number you input, and \(f(x)\) is the number you output.  The function \(f\) itself is the rule for going from one to the other, e.g.  For example \(f(x) = \sqrt{\sin x^2 + 1}\).

Now, nothing stops you from having a function that depends on multiple numbers as input; for example the function \(f(x,\,y) = xy^2 + x^3y\) depends on two input variables, \(x\), and \(y\).  If there are \(D\) input numbers, then the \(D\)-dimensional space of possible combinations of input numbers is called the domain of the function.

Also nothing stops you from having the output be a set of several numbers.  In this case we would need some sort of subscript \(i\) to refer to the different possible output numbers.  For example, if we had a function with one input number \(x\) and three output numbers \(y\), then we could write \(f_i(x)\), where \(i\) takes the values 1, 2, or 3.  Then \(f_i(x)\) would really be just a package of three different functions: \(f_1(x)\), \(f_2(x)\), and \(f_3(x)\).  So if you specify the input \(x\), you get three output numbers \((f_1, f_2, f_3)\).  If there are \(T\) different output numbers, the \(T\) dimensional space of possible outputs is called the target space.

Now a field is just a function whose domain is the points of spacetime.  For example, the air temperature in a room may vary from place to place, and it may also change with time.  So if you imagine checking all possible points of space in the room at all possible times, you could describe this with a temperature field \(T(t, x, y, z)\).  However, the temperature field isn’t a fundamental entity that exists on its own.  It subsists in a medium (air) and describes its motion.  When the air molecules are moving around quickly in a random way, we say it’s hot, and when they start to move around slower, we say it’s getting chilly.  An example of a field which actually is fundamental (as far as we know) would be the electromagnetic field.  This has 6 output numbers, since the electric field can point in any of the 3 spatial directions, and the magnetic field also has 3 numbers.

For a while in the 19th century, scientists were confused about this.  They thought that electromagnetic waves had to be some sort of excitation of some sort of stuff, which they called the aether.  That’s because they were assuming (based on physical intuitions filtered through Newtonian mechanics) that matter is something solid and massy, which interacts by striking or making contact with other things.  The 20th century scientific advances partly came from realizing that its okay to describe things with abstract math.  Any kind of mathematical object you write down satisfying logically consistent equations is OK, as long as it matches experiment.  So electromagnetic waves don’t have to be made out of anything.  They just are, and other things are (partly) made out of them.

In our current best theory of particle physics, the Standard Model, there are a few dozen different kinds of fields, and all matter is explained as configurations of these fields.  I can’t tell you exactly how many fields there are, because it depends on how you count them.  Not counting the gravitational field, there are 52 different output numbers corresponding to bosons, and 192 different output numbers corresponding to fermions (Don’t worry about what these terms mean yet).  So you could say that there are 244 different fields in Nature, each with one output number.

That sounds awfully complicated.  But there’s also a lot of symmetries in the Standard Model which relate these output numbers to each other.  This includes not only the Poincaré group of spacetime symmetries, but also various internal symmetries related to the dynamics of the strong, weak, and electromagnetic forces.  They are called internal because they don’t move the points of spacetime around.  Instead they just mutate the different kinds of output numbers into each other.

So normally, particle physicists just package the output numbers into sets, such that the numbers in each set are related by the various kinds of symmetry.  (For example, the 6 different numbers of the electromagnetic field are related by rotations and Lorentz boosts.)  Each of these sets is called a field.  In future posts I’ll give more details about the different kinds of fields.  As always, questions are welcome.

UPDATE: I forgot to include the 4 vector components of the spin-1 gauge bosons, so the numbers of degrees of freedom of the bosons were wrong before.  Note to Experts: These are the “off-shell” degrees of freedom before taking into consideration constraints or gauge symmetry.  Note to Non-Experts: the numbers in this post are just for flavor, in order to give you the sense that there are a LOT of different fields in Nature.  You won’t need to understand how I got these numbers in order to enjoy future posts!

Pillar of Science III: Approximate Models

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Science requires Approximations.

Every kind of professional activity changes the way you think.  It rewires your brain so that even when you’re off the job, things start looking a certain way.  For example, to a computer programmer everything looks like an algorithm.  To a teacher, everything is pedagogical.  As a physicist, what goes through my head every day is approximations.

Every time I think about a situation involving black holes, or prove a theorem, or do a calculation, I always have to keep in the back of my mind what kinds of physical effects I’m ignoring, not taking into account.  This habit has leaked out into my thinking about life in general.  Ideas don’t have to be just true or false, instead they can be good approximations in some contexts, and bad approximations in other contexts.

(Partly related: before I started doing physics seriously, I think I had the idea in the back of my head that when I went to grad school, I’d learn how to calculate the really hard problems.  But it turns out there is no way to calculate the answers to the hard math problems.  There are only clever tricks for simplifying hard problems so that they become easy problems.  Frequently, the clever trick is finding some parameter that can be taken to be small, in order to justify some approximation.  The way this works is: first you figure out what happens if the parameter is zero, and then you calculate the tiny effects of it not quite being zero.)

It is impossible for any model of the universe to capture every feature of reality, or else it would be too complicated for human beings to analyze, or to compare to experiment, at the high level of precision demanded by science.  Consequently every scientific theory is applicable only to some limited range of phenomena.  In other words, it isolates some feature of reality which is as free as possible from contaminating influences, and which is simple enough to be either measured experimentally or calculated theoretically (ideally, one can do both, to compare the theory to the experiment).

Therefore, Science consists of a bunch of partly overlapping models which cover different patches of reality. Some of these patches are smaller, and cover very specific situations (e.g. Bernoulli’s principle for fluid dynamics) and others cover a very broad range of situations (e.g. Quantum Electrodynamics, which covers everything related to electricity, chemistry, and light). None of these patches covers everything, and the two broadest patches, Quantum Field Theory and General Relativity, cannot yet be fully reconciled with one another.

One of the implications of this principle is that scientific revolutions seldom result in the complete discrediting of the old well-established theory. The reason is that if the first theory explained a significant patch of data, the new theory can only supercede the old one if it explains all the things the first one explained, and more. Usually this means that the old theory is a limiting situation or special case of the new theory. Thus the old theory is still valid, just in a smaller patch than the new theory. For example, Einstein’s theory of General Relativity superceded Newton’s theory of gravity, but it predicts nearly the same results as Newton in the special case that the objects being considered are travelling much slower than light, and their gravitational fields are not too strong.

Thus the empirical predictions of Newton’s theory are still correct when applied in the proper domain.  However, the philosophical implications regarding the nature of space and time could hardly be more different in the two theories, because the Newtonian theory regards space and time as fixed, immutable, separate entities, while Einsteinian theory regards spacetime as a single contingent field capable of being affected by the flow of energy and momentum through the spacetime.

Philosophers who reason from scientific discoveries should take warning from this: although the empirical predictions of a theory usually survive revolution, the philosophical implications often do not.  Thus our current scientific views on such matters should be taken as somewhat provisional.  On the other hand, it would be even more foolhardy to try to discuss the philosophical nature of space, time, causality, etc. without taking into account the radical changes which Science has made to our naïve intuitions about these concepts.  Some improvement of our thinking is better than none.