Tag Archives: General relativity

A meditation on physical units: Part 2

[Preface: This is the second part of my discussion of this paper by Craig Holt. It has a few more equations than usual, so strap a seat-belt onto your brain and get ready!]

“Alright brain. You don’t like me, and I don’t like you, but let’s get through this thing and then I can continue killing you with beer.”    — Homer Simpson

Imagine a whale. We like to say that the whale is big. What does that mean? Well, if we measure the length of the whale, say by comparing it to a meter-stick, we will count up a very large number of meters. However, this only tells us that the whale is big in comparison to a meter-stick. It doesn’t seem to tell us anything about the intrinsic, absolute length of the whale. But what is the meaning of `intrinsic, absolute’ length?

Imagine the whale is floating in space in an empty universe. There are no planets, people, fish or meter-sticks to compare the whale to. Maybe we could say that the whale has the property of length, even though we have no way of actually measuring its length. That’s what `absolute’ length means. We can imagine that it has some actual number, independently of any standard for comparison like a meter-stick.

"Not again!"
“Oh no, not again!”

In Craig’s Holt’s paper, this distinction — between measured and absolute properties — is very important. All absolute quantities have primes (also called apostrophes), so the absolute length of a whale would be written as whale-length’ and the absolute length of a meter-stick is written meter’. The length of the whale that we measure, in meters, can be written as the ratio whale-length’ / meter’ . This ratio is something we can directly measure, so it doesn’t need a prime, we can just call it whale-length: it is the number of meter sticks that equal a whale-length. It is clear that if we were to change all of the absolute lengths in the universe by the same factor, then the absolute properties whale-length’ and meter’ would both change, but the measurable property of whale-length would not change.

Ok, so, you’re probably thinking that it is weird to talk about absolute quantities if we can’t directly measure them — but who says that you can’t directly measure absolute quantities? I only gave you one example where, as it turned out, we couldn’t measure the absolute length. But one example is not a general proof. When you go around saying things like “absolute quantities are meaningless and therefore changes in absolute quantities can’t be detected”, you are making a pretty big assumption. This assumption has a name, it is called Bridgman’s Principle (see the last blog post).

Bridgman’s Principle is the reason why at school they teach you to balance the units on both sides of an equation. For example, `speed’ is measured in units of length per time (no, not milligrams — this isn’t Breaking Bad). If we imagine that light has some intrinsic absolute speed c’, then to measure it we would need to have (for example) some reference length L’ and some reference time duration T’ and then see how many lengths of L’ the light travels in time T’. We would write this equation as:

eq1

where C is the speed that we actually measure. Bridgman’s Principle says that a measured quantity like C cannot tell us the absolute speed of light c’, it only tells us what the value of c’ is compared to the values of our measuring apparatus, L’ and T’ (for example, in meters per second). If there were some way that we could directly measure the absolute value of c’ without comparing it to a measuring rod and a clock, then we could just write c’ = C without needing to specify the units of C. So, without Bridgman’s Principle, all of Dimensional Analysis basically becomes pointless.

So why should Bridgman’s Principle be true in general? Scientists are usually lazy and just assume it is true because it works in so many cases (this is called “proof by induction”). After all, it is hard to find a way of measuring the absolute length of something, without referring to some other reference object like a meter-stick. But being a good scientist is all about being really tight-assed, so we want to know if Bridgman’s Principle can be proven to be watertight.

A neat example of a watertight principle is the Second Law of Thermodynamics. This Law was also originally an inductive principle (it seemed to be true in pretty much all thermodynamic experiments) but then Boltzmann came along with his famous H-Theorem and proved that it has to be true if matter is made up of atomic particles. This is called a constructive justification of the principle [1].

The H Theorem makes it nice and easy to judge whether some crackpot’s idea for a perpetual motion machine will actually run forever. You can just ask them: “Is your machine made out of atoms?” And if the answer is `yes’ (which it probably is), then you can point out that the H-Theorem proves that machines made up of atoms must obey the Second Law, end of story.

Coming up with a constructive proof, like the H-Theorem, is pretty hard. In the case of Bridgman’s Principle, there are just too many different things to account for. Objects can have numerous properties, like mass, charge, density, and so on; also there are many ways to measure each property. It is hard to imagine how we could cover all of these different cases with just a single theorem about atoms. Without the H-Theorem, we would have to look over the design of every perpetual motion machine, to find out where the design is flawed. We could call this method “proof by elimination of counterexamples”. This is exactly the procedure that Craig uses to lend support to Bridgman’s Principle in his paper.

To get a flavor for how he does it, recall our measurement of the speed of light from equation (1). Notice that the measured speed C does not have to be the same as the absolute speed c’. In fact we can rewrite the equation as:

eq2

and this makes it clear that the number C that we measure is not itself an absolute quantity, but rather is a comparison between the absolute speed of light c’ and the absolute distance L’ per time T’. What would happen if we changed all of the absolute lengths in the universe? Would this change the value of the measured speed of light C? At first glance, you might think that it would, as long as the other absolute quantities on the left hand side of equation (2) are independent of length. But if that were true, then we would be able to measure changes in absolute length by observing changes in the measurable speed of light C, and this would contradict Bridgman’s Principle!

To get around this, Craig points out that the length L’ and time T’ are not fundamental properties of things, but are actually reducible to the atomic properties of physical rods and clocks that we use to make measurements. Therefore, we should express L’ and T’ in terms of the more fundamental properties of matter, such as the masses of elementary particles and the coupling constants of forces inside the rods and clocks. In particular, he argues that the absolute length of any physical rod is equal to some number times the “Bohr radius” of a typical atom inside the rod. This radius is in turn proportional to:

eq3

where h’, c’ are the absolute values of Planck’s constant and the speed of light, respectively, and m’e is the absolute electron mass. Similarly, the time duration measured by an atomic clock is proportional to:

eq4

As a result, both the absolute length L’ and time T’ actually depend on the absolute constants c’, h’ and the electron mass m’e. Substituting these into the expression for the measured speed of light, we get:

eq5

where X,Y are some proportionality constants. So, the factors of c’ cancel and we are left with C=X/Y. The numbers X and Y depend on how we construct our rods and clocks — for instance, they depend on how many atoms are inside the rod, and what kind of atom we use inside our atomic clock. In fact, the definition of a `meter’ and a `second’ are specially chosen so as to make this ratio exactly C=299,792,458 [2].

Now that we have included the fact that our measuring rods and clocks are made out of matter, we see that in fact the left hand side of equation (5) is independent of any absolute quantities. Therefore changing the absolute length, time, mass, speed etc. cannot have any effect on the measured speed of light C, and Bridgman’s principle is safe — at least in this example.

(Some readers might wonder why making a clock heavier should also make it run faster, as seems to be suggested by equation (4). It is important to remember that the usual kinds of clocks we use, like wristwatches, are quite complicated things containing trillions of atoms. To calculate how the behaviour of all these atoms would change the ticking of the overall clock mechanism would be, to put it lightly, a giant pain in the ass. That’s why Craig only considers very simple devices like atomic clocks, whose behaviour is well understood at the atomic level [3].)

image credit: xetobyte – A Break in Reality

Another simple model of a clock is the light clock: a beam of light bouncing between two mirrors separated by a fixed distance L’. Since light has no mass, you might think that the frequency of such a clock should not change if we were to increase all absolute masses in the universe. But we saw in equation (4) that the frequency of an atomic clock is proportional to the electron mass, and so it would increase. It then seems like we could measure this increase in atomic clock frequency by comparing it to a light clock, whose frequency does not change — and then we would know that the absolute masses had changed. Is this another threat to Bridgman’s Principle?

The catch is that, as Craig points out, the length L’ between the mirrors of the light clock is determined by a measuring rod, and the rod’s length is inversely proportional to the electron mass as we saw in equation (1). So if we magically increase all the absolute masses, we would also cause the absolute length L’ to get smaller, which means the light-clock frequency would increase. In fact, it would increase by exactly the same amount as the atomic clock frequency, so comparing them would not show us any difference! Bridgman’s Principle is saved again.

Let’s do one more example, this time a little bit more extreme. According to Einstein’s theory of general relativity, every lump of mass has a Schwarzschild radius, which is the radius of a sphere such that if you crammed all of the mass into this sphere, it would turn into a black hole. Given some absolute amount of mass M’, its Schwarzschild radius is given by the equation:

eq6

where c’ is the absolute speed of light from before, and G’ is the absolute gravitational constant, which determines how strong the gravitational force is. Now, glancing at the equation, you might think that if we keep increasing all of the absolute masses in the universe, planets will start turning into black holes. For instance, the radius of Earth is about 6370 km. This is the Schwarzschild radius for a mass of about a million times Earth’s mass. So if we magically increased all absolute masses by a factor of a million, shouldn’t Earth collapse into a black hole? Then, moments before we all die horribly, we would at least know that the absolute mass has changed, and Bridgman’s Principle was wrong.

Of course, that is only true if changing the absolute mass doesn’t affect the other absolute quantities in equation (6). But as we now know, increasing the absolute mass will cause our measuring rods to shrink, and our clocks to run faster. So the question is, if we scale the masses by some factor X, do all the X‘s cancel out in equation (6)?

Well, since our absolute lengths have to shrink, the Schwarzschild radius should shrink, so if we multiply M’ by X, then we should divide the radius R’ by X. This doesn’t balance! Hold on though — we haven’t dealt with the constants c’ and G’ yet. What happens to them? In the case of c’, we have c’ = C L’ / T’. Since L’ and T’ both decrease by a factor of X (lengths and time intervals get shorter) there is no overall effect on the absolute speed of light c’.

How do we measure the quantity G’? Well, G’ tells us how much two masses (measured relative to a reference mass m’) will accelerate towards each other due to their gravitational attraction. Newton’s law of gravitation says:

eq7

where N is some number that we can measure, and it depends on how big the two masses are compared to the reference mass m’, how large the distance between them is compared to the reference length L’, and so forth. If we measure the acceleration a’ using the same reference length and time L’,T’, then we can write:

eq8

where the A is just the measured acceleration in these units. Putting this all together, we can re-arrange equation (7) to get:

eq9

and we can define G = (A/N) as the actually measured gravitational constant in the chosen units. From equation (9), we see that increasing M’ by a factor of X, and hence dividing each instance of L’ and T’ by X, implies that the absolute constant G’ will actually change: it will be divided by a factor of X2.

What is the physics behind all this math? It goes something like this: suppose we are measuring the attraction between two masses separated by some distance. If we increase the masses, then our measuring rods shrink and our clocks get faster. This means that when we measure the accelerations, the objects seem to accelerate faster than before. This is what we expect, because two masses should become more attractive (at the same distance) when they become more massive. However, the absolute distance between the masses also has to shrink. The net effect is that, after increasing all the absolute masses, we find that the masses are producing the exact same attractive force as before, only at a closer distance. This means the absolute attraction at the original distance is weaker — so G’ has become weaker after the absolute masses in the universe have been increased (notice, however, that the actually measured value G does not change).

Diagram of a Cavendish experiment for measuring gravity.

Returning now to equation (6), and multiplying M’ by X, dividing R’ by X and dividing G’ by X2, we find that all the extra factors cancel out. We conclude that increasing all the absolute masses in the universe by a factor of a million will not, in fact, cause Earth to turn into a black hole, because the effect is balanced out by the contingent changes in the absolute lengths and times of our measuring instruments. Whew!

Craig’s paper is long and very thorough. He compares a whole zoo of physical clocks, including electric clocks, light-clocks, freely falling inertial clocks, different kinds of atomic clocks and even gravitational clocks made from two orbiting planets. Not only does he generalize his claim to Newtonian mechanics, he covers general relativity as well, and the Dirac equation of quantum theory, including a discussion of Compton scattering (a photon reflecting off an electron). Besides all of this, he takes pains to discuss the meaning of coupling constants, the Planck scale, and the related but distinct concept of scale invariance. All in all, Craig’s paper just might be the most comprehensive justification for Bridgman’s principle so far in existence!

Most scientists might shrug and say “who needs it?”. In the same way, not many scientists care to examine perpetual motion machines to find out where the flaw lies. In this respect, Craig is a craftsman of the first order — he cares deeply about the details. Unlike the Second Law of Thermodynamics, Bridgman’s Principle seems rarely to have been challenged. This only makes Craig’s defense of it all the more important. After all, it is especially those beliefs which we are disinclined to question that are most deserving of a critical examination.

math

Footnotes:

[1] Some physical principles, like the Relativity Principle, have never been given a constructive justification. For this reason, Einstein himself seems to have regarded the Relativity Principle with some suspicion. See this great discussion by Brown and Pooley.

[2] Why not just set it to N=1? Well, no reason why not! Then we would replace the meter by the `light second’, and the second by the `light-meter’. And we would say things like “Today I walked 0.3 millionths of a light second to buy an ice-cream, and it took me just 130 billion light-meters to eat it!” So, you know, that would be a bit weird. But theorists do it all the time.

[3] To be perfectly strict, we cannot assume that a wristwatch will behave in the same way as an atomic clock in response to changes in absolute properties; we would have to derive their behavior constructively from their atomic description. This is exactly why a general constructive proof of Bridgman’s Principle would be so hard, and why Craig is forced to stick with simple models of clocks and rulers.

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Bootstrapping to quantum gravity

Kepler

“If … there were no solid bodies in nature there would be no geometry.”
-Poincaré

A while ago, I discussed the mystery of why matter should be the source of gravity. To date, this remains simply an empirical fact. The deep insight of general relativity – that gravity is the geometry of space and time – only provides us with a modern twist: why should matter dictate the geometry of space-time?

There is a possible answer, but it requires us to understand space-time in a different way: as an abstraction that is derived from the properties of matter itself. Under this interpretation, it is perfectly natural that matter should affect space-time geometry, because space-time is not simply a stage against which matter dances, but is fundamentally dependent on matter for its existence. I will elaborate on this idea and explain how it leads to a new avenue of approach to quantum gravity.

First consider what we mean when we talk about space and time. We can judge how far away a train is by listening to the tracks, or gauge how deep a well is by dropping a stone in and waiting to hear the echo. We can tell a mountain is far away just by looking at it, and that the cat is nearby by tripping over it. In all these examples, an interaction is necessary between myself and the object, sometimes through an intermediary (the light reflected off the mountain into my eyes) and sometimes not (tripping over the cat). Things can also be far away in time. I obviously cannot interact with people who lived in the past (unless I have a time machine), or people who have yet to be born, even if they stood (or will stand) exactly where I am standing now. I cannot easily talk to my father when he was my age, but I can almost do it, just by talking to him now and asking him to remember his past self. When we say that something is far away in either space or time, what we really mean is that it is hard to interact with, and this difficulty of interaction has certain universal qualities that we give the names `distance’ and `time’.
It is worth mentioning here, as an aside, that in a certain sense, the properties of `time’ can be reduced to properties of `distance’ alone. Consider, for instance, that most of our interactions can be reduced to measurements of distances of things from us, at a given time. To know the time, I invariably look at the distance the minute hand has traversed along its cycle on the face of my watch. Our clocks are just systems with `internal’ distances, and it is the varying correspondence of these `clock distances’ with the distances of other things that we call the `time’. Indeed, Julian Barbour has developed this idea into a whole research program in which dynamics is fundamentally spatial, called Shape Dynamics.

Sigmund Freud Museum, Wien – Peter Kogler

So, if distance and time is just a way of describing certain properties of matter, what is the thing we call space-time?

We now arrive at a crucial point that has been stressed by philosopher Harvey Brown: the rigid rods and clocks with which we claim to measure space-time do not really measure it, in the traditional sense of the word `measure’. A measurement implies an interaction, and to measure space-time would be to grant space-time the same status as a physical body that can be interacted with. (To be sure, this is exactly how many people do wish to interpret space-time; see for instance space-time substantivalism and ontological structural realism).

Brown writes:
“One of Bell’s professed aims in his 1976 paper on `How to teach relativity’ was to fend off `premature philosophizing about space and time’. He hoped to achieve this by demonstrating with an appropriate model that a moving rod contracts, and a moving clock dilates, because of how it is made up and not because of the nature of its spatio-temporal environment. Bell was surely right. Indeed, if it is the structure of the background spacetime that accounts for the phenomenon, by what mechanism is the rod or clock informed as to what this structure is? How does this material object get to know which type of space-time — Galilean or Minkowskian, say — it is immersed in?” [1]

I claim that rods and clocks do not measure space-time, they embody space-time. Space-time is an idealized description of how material rods and clocks interact with other matter. This distinction is important because it has implications for quantum gravity. If we adopt the more popular view that space-time is an independently existing ontological construct, it stands to reason that, like other classical fields, we should attempt to directly quantise the space-time field. This is the approach adopted in Loop Quantum Gravity and extolled by Rovelli:

“Physical reality is now described as a complex interacting ensemble of entities (fields), the location of which is only meaningful with respect to one another. The relation among dynamical entities of being contiguous … is the foundation of the space-time structure. Among these various entities, there is one, the gravitational field, which interacts with every other one and thus determines the relative motion of the individual components of every object we want to use as rod or clock. Because of that, it admits a metrical interpretation.” [2]

One of the advantages of this point of view is that it dissolves some seemingly paradoxical features of general relativity, such as the fact that geometry can exist without (non-gravitational) matter, or the fact that geometry can carry energy and momentum. Since gravity is a field in its own right, it doesn’t depend on the other fields for its existence, nor is there any problem with it being able to carry energy. On the other hand, this point of view tempts us into framing quantum gravity as the mathematical problem of quantising the gravitational field. This, I think, is misguided.

I propose instead to return to a more Machian viewpoint, according to which space-time is contingent on (and not independent of) the existence of matter. Now the description of quantum space-time should follow, in principle, from an appropriate description of quantum matter, i.e. of quantum rods and clocks. From this perspective, the challenge of quantum gravity is to rebuild space-time from the ground up — to carry out Einstein’s revolution a second time over, but using quantum material as the building blocks.

Ernst Mach vs. Max Ernst. Get it right, folks.

My view about space-time can be seen as a kind of `pulling oneself up by one’s bootstraps’, or a Wittgenstein’s ladder (in which one climbs to the top of a ladder and then throws the ladder away). It works like this:
Step 1: define the properties of space-time according to the behaviour of rods and clocks.
Step 2: look for universal patterns or symmetries among these rods and clocks.
Step 3: take the ideal form of this symmetry and promote it to an independently existing object called `space-time’.
Step 4: Having liberated space-time from the material objects from which it was conceived, use it as the independent standard against which to compare rods and clocks.

Seen in this light, the idea of judging a rod or a clock by its ability to measure space or time is a convenient illusion: in fact we are testing real rods and clocks against what is essentially an embodiment of their own Platonic ideals, which are in turn conceived as the forms which give the laws of physics their most elegant expression. A pertinent example, much used by Julian Barbour, is Ephemeris time and the notion of a `good clock’. First, by using material bodies like pendulums and planets to serve as clocks, we find that the motions of material bodies approximately conform to Newton’s laws of mechanics and gravitation. We then make a metaphysical leap and declare the laws to be exactly true, and the inaccuracies to be due to imperfections in the clocks used to collect the data. This leads to the definition of the `Ephemeris time’, the time relative to which the planetary motions conform most closely to Newton’s laws, and a `good clock’ is then defined to be a clock whose time is closest to Ephemeris time.

The same thing happens in making the leap to special relativity. Einstein observed that, in light of Maxwell’s theory of electromagnetism, the empirical law of the relativity of motion seemed to have only a limited validity in nature. That is, assuming no changes to the behaviour of rods and clocks used to make measurements, it would not be possible to establish the law of the relativity of motion for electrodynamic bodies. Einstein made a metaphysical leap: he decided to upgrade this law to the universal Principle of Relativity, and to interpret its apparent inapplicability to electromagnetism as the failure of the rods and clocks used to test its validity. By constructing new rods and clocks that incorporated electromagnetism in the form of hypothetical light beams bouncing between mirrors, Einstein rebuilt space-time so as to give the laws of physics a more elegant form, in which the Relativity Principle is valid in the same regime as Maxwell’s equations.

Ladder for Booker T. Washington – Martin Puryear

By now, you can guess how I will interpret the step to general relativity. Empirical observations seem to suggest a (local) equivalence between a uniformly accelerated lab and a stationary lab in a gravitational field. However, as long as we consider `ideal’ clocks to conform to flat Minkowski space-time, we have to regard the time-dilated clocks of a gravitationally affected observer as being faulty. The empirical fact that observers stationary in a gravitational field cannot distinguish themselves (locally) from uniformly accelerated observers then seems accidental; there appears no reason why an observer could not locally detect the presence of gravity by comparing his normal clock to an `ideal clock’ that is somehow protected from gravity. On the other hand, if we raise this empirical indistinguishability to a matter of principle – the Einstein Equivalence Principle – we must conclude that time dilation should be incorporated into the very definition of an `ideal’ clock, and similarly with the gravitational effects on rods. Once the ideal rods and clocks are updated to include gravitational effects as part of their constitution (and not an interfering external force) they give rise to a geometry that is curved. Most magically of all, if we choose the simplest way to couple this geometry to matter (the Einstein Field Equations), we find that there is no need for a gravitational force at all: bodies follow the paths dictated by gravity simply because these are now the inertial paths followed by freely moving bodies in the curved space-time. Thus, gravity can be entirely replaced by geometry of space-time.

As we can see from the above examples, each revolution in our idea of space-time was achieved by reconsidering the nature of rods and clocks, so as to make the laws of physics take a more elegant form by incorporating some new physical principle (eg. the Relativity and Equivalence principles). What is remarkable is that this method does not require us to go all the way back to the fundamental properties of matter, prior to space-time, and derive everything again from scratch (the constructive theory approach). Instead, we can start from a previously existing conception of space-time and then upgrade it by modifying its primary elements (rods and clocks) to incorporate some new principle as part of physical law (the principle theory approach). The question is, will quantum gravity let us get away with the same trick?

I’m betting that it will. The challenge is to identify the empirical principle (or principles) that embody quantum mechanics, and upgrade them to universal principles by incorporating them into the very conception of the rods and clocks out of which general relativistic space-time is made. The result will be, hopefully, a picture of quantum geometry that retains a clear operational interpretation. Perhaps even Percy Bridgman, who dismissed the Planck length as being of “no significance whatever” [3] due to its empirical inaccessibility, would approve.

Boots with laces – Van Gogh

[1] Brown, Physical Relativity, p8.
[2] Rovelli, `Halfway through the woods: contemporary research on space and time’, in The Cosmos of Science, p194.
[3] Bridgman, Dimensional Analysis, p101.

Why does matter curve space and time?

This is one of those questions that has always bugged me.
black-hole
Suppose that, somewhere in the universe, there is a very large closed box made out of some kind of heavy, neutral matter. Inside this box a civilisation of intelligent creatures have evolved. They are made out of normal matter like you and me, except that for some reason they are very light — their bodies do not contain much matter at all. What’s more, there are no other heavy bodies or planets inside this large box aside from the population of aliens, whose total mass is too small to have any noticeable effect on the gravitational field. Thus, the only gravitational field that the aliens are aware of is the field created by the box itself (I’m assuming there are no other massive bodies near to the box).

Setting aside the obvious questions about how these aliens came to exist without an energy source like the sun, and where the heck the giant box came from, I want to examine the following question: in principle, is there any way that these aliens could figure out that matter is the source of gravitational fields?

Now, to make it interesting, let us assume the density of the box is not uniform, so there are some parts of its walls that have a stronger gravitational pull than others. Our aliens can walk around on these parts of the walls, and in some parts the aliens even become too heavy to support their own weight and get stuck until someone rescues them. Elsewhere, the walls of the box are low density and so the gravitational attraction to them is very weak. Here, the aliens can easily jump off and float away from the wall. Indeed, the aliens spend much of their time floating freely near the center of the box where the gravitational fields are weak. Apart from that, the composition of the box itself does not change with time and the box is not rotating, so the aliens are quickly able to map out the constant gravitational field that surrounds them inside the box, with its strong and weak points.

Like us, the aliens have developed technology to manipulate the electromagnetic field, and they know that it is the electromagnetic forces that keeps their bodies intact and stops matter from passing through itself. More importantly, they can accelerate objects of different masses by pushing on them, or applying an electric force to charged test bodies, so they quickly discover that matter has inertia, measured by its mass. In this way, they are able to discover Newton’s laws of mechanics. In addition, their experiments with electromagnetism and light eventually lead them to upgrade their picture of space-time, and their Newtonian mechanics is replaced by special relativistic mechanics and Maxwell’s equations for the electromagnetic field.

So far, so good! Except that, because they do not observe any orbiting planets or moving gravitating bodies (their own bodies being too light to produce any noticible attractive forces), they still have not reproduced Newtonian gravity. They know that there is a static field permeating space-time, called the gravitational field, that seems to be fixed to the frame of the box — but they have no reason to think that this gravitational force originates from matter. Indeed, there are two philosophical schools of thought on this. The first group holds that the gravitational field is to be thought of analogously to the electromagnetic field, and is therefore sourced by special “gravitational charges”. It was originally claimed that the material of the box itself carries gravitational charge, but scrapings of the material from the box revealed it to be the same kind of matter from which the aliens themselves were composed (let’s say Carbon) and the scrapings themselves seemed not to produce any gravitational fields, even when collected together in large amounts of several kilograms (a truly humungous weight to the minds of the aliens, whose entire population combined would only weigh ten kilograms). Some aliens pointed out that the gravitational charge of Carbon might be extremely weak, and since the mass of the entire box was likely to be many orders of magnitude larger than anything they had experienced before, it is possible that its cumulative charge would be enough to produce the field. However, these aliens were criticised for making ad-hoc modifications to their theory to avoid its obvious refutation by the kilograms-of-Carbon experiments. If gravity is analogous to the electromagnetic force — they were asked with a sneer — then why should it be so much weaker than electromagnetism? It seemed rather too convenient.

Some people suggested that the true gravitational charge was not Carbon, but some other material that coated the outside of the box. However, these people were derided even more severely than were the Carbon Gravitists (as they had become known). Instead, the popular scientific consensus shifted to a modern idea in which the gravitational force was considered to be a special kind of force field that simply had no source charges. It was a God-given field whose origin and patterns were not to be questioned but simply accepted, much like the very existence of the Great Box itself. This following gained great support when someone made a great discovery: the gravitational force could be regarded as the very geometry of spacetime itself.

The motivation for this was the peculiar observation, long known but never explained, that massive bodies always had the same acceleration in the gravitational field regardless of their different masses. A single alien falling towards one of the gravitating walls of the box would keep speed perfectly with a group of a hundred Aliens tied together, despite their clearly different masses. This dealt a crushing blow to the remnants of the Carbon Gravitists, for it implied that the gravitational charge of matter was exactly proportional to its inertial mass. This coincidence had no precedent in electromagnetism, where it was known that bodies of the same mass could have very different electric charges.

Under the new school of thought, the gravitational force was reinterpreted as the background geometry of space-time inside the box, which specified the inertial trajectories of all massive bodies. Hence, the gravitational force was not a force at all, so it was meaningless to ascribe a “gravitational charge” to matter. Tensor calculus was developed as a natural extension of special relativity, and the aliens derived the geodesic equation describing the motion of matter in a fixed curved space-time metric. The metric of the box was mapped out with high precision, and all questions about the universe seemed to have been settled.

Well, almost all. Some troublesome philosophers continued to insist that there should be some kind of connection between space-time geometry and matter. They wanted more than just the well-known description of how geometry caused matter to move: they tried to argue that matter should also tell space-time how to curve.

“Our entire population combined only weighs a fraction of the mass of the box. What would happen if there were more matter available to us? What if we did the Carbon-kilogram experiment again, but with 100 kilograms? Or a million? Surely the presence of such a large amount of matter would have an effect on space-time itself?”

But these philosophers were just laughed at. Why should any amount of matter affect the eternal and never-changing space-time geometry? Even if the Great Box itself were removed, the prevailing thought was that the gravitational field would remain, fixed as it was in space-time and not to any material source. So they all lived happily ever after, in blissful ignorance of the gravitational constant G, planetary orbits, and other such fantasies.

***

Did you find this fairytale disturbing? I did. It illustrates what I think is an under-appreciated uncomfortable feature of our best theories of gravity: they all take the fact that matter generates gravity as a premise, without justification apart from empirical observation. There’s nothing strictly wrong with this — we do essentially the same thing in special relativity when we take the speed of light to be constant regardless of the motion of its source, historically an empirically determined fact (and one that was found quite surprising).

However, there is a slight difference: one can in principle argue that the speed of light should be reference-frame independent from philosophical grounds, without appealing to empirical observations. Roughly, the relativity principle states that the laws of physics should be the same in all frames of motion, and from among the laws of physics we can include the non-relativistic equations of the electromagnetic field, from which the constant speed of light can be derived from the electric and magnetic constants of the vacuum. As far as I know, there is no similar philosophical grounding for the connection between matter and geometry as embodied by the gravitational constant, and hence no compelling reason for our hypothetical aliens to ever believe that matter is the source of space-time geometry.

Could it be that there is an essential piece missing from our accounts of the connection between matter and space-time? Or are our aliens are doomed by their unfortunately contrived situation, never to deduce the complete laws of the universe?

Skin Deep, by Xetobyte
Image Credit: Xetobyte

 

So can we time-travel or not?!

In a comment on my last post, elkement asked:

“What are exactly are the limits for having an object time-travel that is a bit larger than a single particle? Or what was the scope of your work? I am asking because papers as your thesis are very often hyped in popular media as `It has been proven that time-travel does work’ (Insert standard sci-fi picture of curved space here). As far as I can decode the underlying papers such models are mainly valid for single particles (?) but I have no feeling about numbers and dimensions, decoherence etc.”

Yep, that is pretty much THE question about time travel – can we do it with people or not? (Or even with rats, that would be good too). The bottom line is that we still don’t know, but I might as well give a longer answer, since it is just interesting enough to warrant its own blog post.

First of all, nobody has yet been able to prove that time travel is either possible or impossible according to the laws of physics. This is largely because we don’t yet know what laws govern time travel — for that we’d almost certainly need a theory of quantum gravity. In order for humans to time-travel, we would probably need to use a space-time wormhole, as proposed by Morris, Thorne and Yurtsever in the late eighties [1]. Their paper originated the classic wormhole graphic that everyone is so fond of:

(Used without permission from the publisher -- shh!)
(Used without permission from the publisher — shh!)

However, there are at least a couple of compelling arguments why it should be impossible to send people back in time, one of which is Stephen Hawking’s “Chronology Protection Conjecture”. This is commonly misrepresented as the argument “if time travel is possible, where are all the tourists from the future?”. While Stephen Hawking did make a comment along these lines, he was just joking around. Besides, there is a perfectly good reason why we might not have been visited by travellers from the future: according to the wormhole model, you can only go back in time as far as the moment when you first invented the time machine, or equivalently, the time at which the first wormhole mouth opens up. Since we haven’t found any wormhole entrances in space, nor have we created one artificially, it is no surprise that we haven’t received any visitors from the future.

via Saturday Morning Breakfast Cereal
via Saturday Morning Breakfast Cereal

The real Chronology Protection Conjecture involves a lot more mathematics and head-scratching. Basically, it says that matter and energy should accumulate near the wormhole entrance so quickly that the whole thing will collapse into a black hole before anybody has time to travel through it. The reason that it is still only a conjecture and has not been proven, is that it relies upon certain assumptions about quantum gravity that may or may not be true — we won’t know until we have such a theory. And then it might just turn out that the wormhole is somehow stable after all.

The other reason why time travel for large objects might be impossible is that, in order for the wormhole to be stable and not collapse in on itself Hawking-style, you need matter with certain quantum properties that can support the wormhole against collapse [2]. But it might turn out that it is just impossible to create enough of this special matter in the vicinity of a wormhole to keep it open. This is a question that one could hope to answer without needing a full theory of quantum gravity, because it depends only on the shape of the space-time and certain properties of quantum fields within that space-time. However, the task of answering this question is so ridiculously difficult mathematically that nobody has yet been able to do it. So the door is still open to the possibility of time-travelling humans, at least in theory.

To my mind, though, the biggest reason is not theoretical but practical: how the heck do you create a wormhole? We can’t even create a black hole of any decent size (if any had shown up at the LHC they would have been microscopic and very short-lived). So how can we hope to be able to manipulate the vast amounts of matter and energy required to bend space-time into a loop (and a stable loop no less), without annihilating ourselves in the process? Even if we were lucky to find a big enough, ready-made wormhole somewhere out in space, it will almost certainly be so far away as to make it nearly impossible to get there, due to sheer demands on technology. It’s a bit like asking, could humans ever build a friendly hotel in the centre of the sun? Well, it might be technically possible, but there is no way it would ever happen; even if we could raise humungous venture capital for the Centre-of-the-Sun Hotel, it would just be too damn hard.

The good news is that it might be more feasible to create a cute, miniature wormhole that only exists for a short time. This would require much smaller energies that might not destroy us in the process, and might be easier to manipulate and control (assuming quantum gravity allows it at all). So, while there is as yet no damning proof that time-travel is impossible, I still suspect that the best we can ever hope to do is to be able to send an electron back in time by a very short amount, probably not more than one millisecond — which would be exciting for science nerds, but perhaps not the headline that the newspapers would have wanted.

[1] Fun fact: while working on the movie  “Contact”, Carl Sagan consulted Kip Thorne about the physics of time-travel.

[2] For the nerds out there, you need matter that violates the averaged null energy condition (ANEC). You can look up what this means in any textbook on General Relativity — for example this one.

Why Quantum Gravity needs Operationalism: Part 2

(Update: My colleagues pointed out that Wittgenstein was one of the greatest philosophers of the 20th century and I should not make fun of him, and anyway he was only very loosely associated with the Vienna circle. All well and true — but he was at least partly responsible for the idea that got the Vienna Circle onto Verificationism, and all of you pedants can go look at the references if you don’t believe me.)

“Where neither confirmation nor refutation is possible, science is not concerned.”    — Mach

Some physicists give philosophy a bad rap. I like to remind them that all the great figures in physics had a keen interest in philosophy, and were strongly influenced by the work of philosophers. Einstein made contributions to philosophy as well as physics, as did Ernst Mach, whose philosophical work had a strong influence on Einstein in formulating his General Theory of Relativity. In his own attitude to philosophy, Einstein was a self-described “epistemological opportunist” [1]. (Epistemology is, broadly speaking, the philosophy of knowledge and how it is acquired.) But philosophy sometimes gets in the way of progress, as explained in the following story.

A physicist was skipping along one day when he came upon a philosopher, standing rigid in the forest. “Why standeth you thus?” he inquired.

“I am troubled by a paradox!” said the philosopher. “How is it that things can move from place to place?”

“What do you mean? I moved here by skipping, didn’t I?”

“Yes, sure. But I cannot logically explain why the world allows it to be so. You see, a philosopher named Zeno argued that in order to traverse any finite distance, one would have to first traverse an infinite number of partitions of that distance. But how can one make sense of completing an infinite number of tasks in a finite amount of time?”

“Well dang,” said the physicist “that’s an interesting question. But wait! Could it be that space and time are actually divided up into a finite number of tiny chunks that cannot be sub-divided further? What an idea!”

“Ah! Perhaps,” says the philosopher, “but what if the world is indeed a continuum? Then we are truly stuck.”

At that moment, a mathematician who had been dozing in a tree fell out and landed with a great commotion.

“Terribly sorry! Couldn’t help but overhear,” he said. “In fact I do believe it is conceptually possible for an infinite number of things to add up to a finite quantity. Why, this gives me a great idea for calculating the area under curves. Thank you so much, I’d better get to it!”

“Yes, yes we must dash at once! There’s work to do!” agreed the physicist.

“But wait!” cried the philosopher, “what if time is merely an illusion? And what is the connection of abstract mathematics to the physical world? We have to work that out first!”

But the other two had already disappeared, leaving the philosopher in his forest to ponder his way down deeper and ever more complex rabbit-holes of thought.

***

Philosophy is valuable for pointing us in the right direction and helping us to think clearly. Sometimes philosophy can reveal a problem where nobody thought there was one, and this can lead to a new insight. Sometimes philosophy can identify and cure fallacies in reasoning. In solving a problem, it can highlight alternative solutions that might not have been noticed otherwise. But ultimately, physicists only tend to turn to philosophy when they have run out of ideas, and most of the time the connection of philosophy to practical matters seems tenuous at best. If philosophers have a weakness, it is only that they tend to think too much, whereas a physicist only thinks as hard as he needs to in order to get results.

After that brief detour, we are ready to return to our hero — physicist Percy Bridgman — and witness his own personal fling and falling-out with philosophy. In a previous post, we introduced Bridgman’s idea of operationalism. Recall that Bridgman emphasized that a physical quantity such as `length’ or `temperature’ should always be attached to some clear notion of how to measure that quantity in an experiment. It is not much of a leap from there to say that a concept is only meaningful if it comes equipped with instructions of how to measure it physically.

Although Bridgman was a physicist, his idea quickly caught on amongst philosophers, who saw in it the potential for a more general theory of meaning. But Bridgman quickly became disillusioned with the direction the philosophers were taking as it became increasingly clear that operationalism could not stand up to the demanding expectations set by the philosophers.

The main culprits were a group of philosophers called the Vienna Circle [2]. Following an idea of Ludwig Wittgenstein, these philosophers attempted to define concepts as meaningful only if they could somehow be verified in principle, an approach that became known as Verificationism. Verificationism was a major theme of the school of thought called `logical empiricism’ (aka logical positivism), the variants of which are embodied in the combined work of philosophers in the Vienna Circle, notably Reichenbach, Carnap and Schlick, as well as members outside the group, like the Berlin Society.

At that time, Bridgman’s operationalism was closely paralleled by the ideas of the Verificationists. This was unfortunate because around the middle of the 20th century it became increasingly apparent that there were big philosophical problems with this idea. On the physics side of things, the philosophers realized that there could be meaningful concepts that could not be directly verified. Einstein pointed out that we cannot measure the electric field inside a solid body, yet it is still meaningful to define the field at all points in space:

“We find that such an electrical continuum is always applicable only for the representation of electrical states of affairs in the interior of ponderable bodies. Here too we define the vector of electric field strength as the vector of the mechanical force exerted on the unit of positive electric quantity inside a body. But the force so defined is no longer directly accessible to experiments. It is one part of a theoretical construction that can be correct or false, i.e., consistent or not consistent with experience, only as a whole.” [1]

Incidentally, Einstein got this point of view from a philosopher, Duhem, who argued that isolated parts of a theory are do not stand as meaningful on their own, but only when taken together as a whole can they be matched with empirical data. It therefore does not always make sense to isolate  some apparently metaphysical aspect of a theory and criticize it as not being verifiable. In a sense, the verifiability of an abstract quantity like the electric field hinges on its placement within a larger theoretical framework that extends to the devices used to measure the field.

In addition, the Verificationists began to fall apart over some rather technical philosophical points. It went something like this:

Wittgenstein: “A proposition is meaningful if and only if it is conceivable for the proposition to be completely verified!”

Others: “What about the statement `All dogs are brown’? I can’t very well check that all dogs are brown can I? Most of the dogs who ever lived are long dead, for a start.”

Wittgenstein: “Err…”

Others: “And what about this guy Karl Popper? He says nothing can ever be completely verified. Our theories are always wrong, they just get less wrong with time.”

Wittgenstein: *cough* *cough* I have to go now. (runs away).

Carnap: Look, we don’t have to take such a hard line. Statements like `All dogs are brown’ are still meaningful, even though they can’t be completely verified.

Schlick: No, no, you’ve got it wrong! Statements like `All dogs are brown’ are meaningless! They simply serve to guide us towards other statements that do have meaning.

Quine: No, you guys are missing a much worse problem with your definition: how do you determine which statements actually require verification (like `The cat sat on the mat’), and which ones are just true by definition (`All bachelors are unmarried’)? I can show that there is no consistent way to separate the two kinds of statement.

(Everybody’s head explodes)

So you can see how the philosophers tend to get carried away. And where was poor old Percy Bridgman during all this? He was backed into a corner, with people prodding his chest and shouting at him:

Gillies: “How do you tell if a measurement method is valid? If there is nothing more to a concept than its method of measurement, then every method of measurement is automatically valid!”

Bridgman: “Well, yes, I suppose…”

Positivists: “And isn’t it true that even if we all agree to use a single measurement of length, this does not come close to exhausting what we mean by the word length? How disappointing.”

Bridgman: “Now wait a minute –”

Margenau: “And just what the deuce do you mean by `operations’ anyhow?”

Bridgman: “Well, I … hey, aren’t you a physicist? You should be on my side!”

(Margenau discreetly melts into the crowd)

To cut a long story short, by the time Quine was stomping on the ashes of what once was logical empiricism, Bridgman’s operationalism had suffered a similar fate, leaving Bridgman battered and bloody on the sidelines wondering where he went wrong:

“To me now it seems incomprehensible that I should ever have thought it within my powers … to analyze so thoroughly the functioning of our thinking apparatus that I could confidently expect to exhaust the subject and eliminate the possibility of a bright new idea against which I would be defenseless.”

To console himself, Bridgman retreated to his laboratory where he at least knew what things were, and could spend hours hand-drilling holes in blocks of steel without having to waste his time arguing about it. Sometimes the positivists would prod him, saying:

“Bridgman! Hey Bridgman! If I measure the height of the Eiffel tower, does that count as an operation, or do you have to perform every experiment yourself?” to which Bridgman would narrow his eyes and mutter: “I don’t trust any experimental results except the ones I perform myself. Now leave me alone!”

Needless to say, Bridgman’s defiantly anti-social attitude to science did not help improve the standing of operationalism among philosophers or physicists; few people were prepared to agree that every experiment has to be verified by an individual for him or herself. Nevertheless, Bridgman remained a heroic figure and a defender of the scientific method as the best way to cope with an otherwise incomprehensible and overwhelming universe. Bridgman’s stubborn attitude of self-reliance was powerfully displayed in his final act: he committed suicide by gunshot wound after being diagnosed with metastatic cancer. In his suicide note, he wrote [3]:

“It isn’t decent for society to make a man do this thing himself. Probably this is the last day I will be able to do it myself.”

Bridgman’s original conception of operationalism continues to resonate with physicists to this very day. In the end he was forced to admit that it did not constitute a rigorous philosophical doctrine of meaning, and he retracted some of his initially over-optimistic statements. However, he never gave up the more pragmatic point of view that an operationalist attitude can be beneficial to the practicing scientist. Towards the end of his life, he maintained that:

“…[T]here is nothing absolute or final about an operational analysis […]. So far as any dogma is involved here at all, it is merely the conviction that it is better, because it takes us further, to analyze into doings or happenings rather than into objects or entities.”

Bridgman2

[1]  See the SEP entry on Einstein’s philosophy: http://plato.stanford.edu/entries/einstein-philscience/

[2] SEP entry on the Vienna Circle: http://plato.stanford.edu/entries/vienna-circle/

[3] Sherwin B Nuland, “How We Die: Reflections on Life’s Final Chapter” Random House 1995

Why quantum gravity needs operationalism: Part 1

This is the first of a series of posts in which I will argue that physicists can gain insight into the puzzles of quantum gravity if we adopt a philosophy I call operationalism. The traditional interpretation of operationalism by philosophers was found to be lacking in several important ways, so the concept will have to be updated to a modern context if we are to make use of it, and its new strengths and limitations will need to be clarified. The goal of this first post is to introduce you to operationalism as it was originally conceived and as I understand it. Later posts will explain the areas in which it failed as a philosophical doctrine, and why it might nevertheless succeed as a tool in theoretical physics, particularly in regard to quantum gravity [1].

Operationalism started with Percy Williams Bridgman. Bridgman was a physicist working in the early 20th century, at the time when the world of physics was being shaken by the twin revolutions of relativity and quantum mechanics. Einstein’s hand was behind both revolutions: first through the publication of his theory of General Relativity in 1916, and second for explaining the photoelectric effect using things called quanta, which earned him the Nobel prize in 1921. This upheaval was a formative time for Bridgman, who was especially struck by Einstein’s clever use of thought experiments to derive special relativity.

Einstein had realized that there was a problem with the concept of `simultaneity’. Until then, everybody had taken it for granted that if two events are simultaneous, then they occur at the same time no matter who is observing them. But Einstein asked the crucial question: how does a person know that two events happened at the same time? To answer it, he had to adopt an operational definition of simultaneity: an observer traveling at constant velocity will consider two equidistant events to be simultaneous if beams of light emitted from each event reach the location of the observer at the same time, as measured by the observer’s clock (this definition can be further generalised to apply to any pair of events as seen by an observer in arbitrary motion).

From this, one can deduce that the relativity principle implies the relativity of simultaneity: two events that are simultaneous for one observer may not be simultaneous for another observer in relative motion. This is one of the key observations of special relativity. Bridgman noticed that Einstein’s deep insight relied upon taking an abstract concept, in this case simultaneity, and grounding it in the physical world by asking `what sort of operations must be carried out in order to measure this thing’?

For his own part, Bridgman was a brilliant experimentalist who won the Nobel prize in 1946 for his pioneering work on creating extremely high pressures in his laboratory. Using state-of-the-art technology, he created pressures up to 100,000 atmospheres, nearly 100 times greater than anybody before him, and then did what any good scientist would do: he put various things into his pressure chamber to record what happened to them. Mostly, as you might expect, they got squished. At pressures beyond 25,000 atmospheres, steel can be molded like play-dough; at 50,000 atmospheres all normal liquids have frozen solid. (Of course, Bridgman’s vessel had to be very small to withstand such pressure, which limited the things he could put in it). But Bridgman faced a unique problem: the pressures that he created were so high that he couldn’t use any standard pressure gauge to measure the pressures in his lab because the gauge would basically get squished like everything else. The situation is the same as trying to measure the temperature of the sun using a regular thermometer: it would explode and vaporize before you could even take a proper reading. Consequently, Bridgman had no scientific way to tell between `really high pressure’ and `really freaking high pressure’, so he was forced to design completely new ways of measuring pressure in his laboratory, such as looking at the phase transition of the element Bismuth and the resistivity of the alloy Manganin [2]. This led him to wonder: what does a concept like `pressureor `temperature’ really mean in the absence of a measuring technique?

Bridgman proposed that quantities measured by different operations should always be regarded as being fundamentally different, even though they may coincide in certain situations. This led to a minor problem in the definitions of quantities. The temperature of a cup of water is measured by sticking a thermometer in it. The temperature of the sun is measured by looking at the spectrum of radiation emitted from it. If these quantities are measured by such different methods in different regimes, why do we call them both `temperature’? In what sense are our operations measuring the same thing? The solution, according to Bridgman, is that there is a regime in between the two in which both methods of measuring temperature are valid – and in this regime the two measurements must agree. The temperature of molten gold could potentially be measured by the right kind of thermometer, as well as by looking at its radiation spectrum, and both of these methods will give the same temperature. This allows us to connect the concept of temperature on the sun to temperature in your kitchen and call them by the same name.

This method of `patching together’ different ways of measuring the same quantity is reminiscent of placing co-ordinate patches on manifolds in mathematical physics. In general, there is no way to cover an entire manifold (representing space-time for example) with a single set of co-ordinates that are valid everywhere. But we can cover different parts of the manifold in patches, provided that the co-ordinates agree in the areas where they overlap. The key insight is that there is no observer who can see all of space-time at once – any physical observer has to travel from one part of the manifold to another by a continuous route. Hence it does not matter if the observer cannot describe the entire manifold by a single map, so long as they have a series of maps that smoothly translate into one another as they travel along their chosen path – even if the maps used much later in the journey have no connection or overlap with the maps used early in the journey. Similarly, as we extend our measuring devices into new regimes, we must gradually replace them with new devices as we go. The eye is replaced with the microscope, the microscope with the electron microscope and the electron microscope with the particle accelerator, which now bears no resemblance to the eye, although they both gaze upon the same world.

Curiously, there was another man named Bridgman active around the same time, who is likely to be more familiar to artists: that is George Bridgman, author of Bridgman’s Complete Guide to Drawing From Life. Although they were two completely different Bridgmans, working in different disciplines, both of them were concerned with essentially the same problem: how to connect our internal conception of the world with the devices by which we measure the world. In the case of Percy Bridgman, it was a matter of connecting abstract physical quantities to their measurement devices, while George Bridgman aimed to connect the figure in the mind to the functions of the hands and eyes. We close with a quote from the artist:

“Indeed, it is very far from accurate to say that we see with our eyes. The eye is blind but for the idea behind the eye.”

[1] Everything I have written comes from Hasok Chang’s entry in the Stanford Encyclopedia of Philosophy on operationalism, which is both clearer and more thorough than my own ramblings.

[2] Readers interested in the finer points of Percy Bridgman’s work should see his Nobel prize lecture.

Is physics in a crisis?

We live in very interesting times, especially if you are a theoretical physicist like me. To understand what kind of time we are living in physics-wise, it will be helpful to review some ideas of Thomas Kuhn, a famous philosopher of science. Kuhn described science as proceeding through a series of paradigms. A `paradigm’ is a sort of established framework in which scientists work to solve problems using an agreed-upon set of tools. The paradigm provides both the puzzles to be solved and the tools to solve them. Over time, scientists discover that the tools of the paradigm cannot solve every puzzle. The problems that lie beyond the reach of a paradigm are called anomalies. When enough serious anomalies are discovered, scientists begin to lose confidence in the existing paradigm and a crisis occurs. Historically, each crisis has been resolved by a subsequent scientific revolution, in which the old paradigm was replaced by a new paradigm that is capable of resolving the anomalies [1].

Interestingly, although the new paradigm solves more problems than the old paradigm, it also represents a complete change in perspective, so that even those problems that were solved by the old paradigm have to be `re-solved’ by the new paradigm, from a completely new point of view. As a result, there might be the odd puzzle that was solved by the old paradigm but suddenly cannot be solved by the new paradigm! This phenomenon is known as `Kuhn-loss’. The new paradigm is successful so long as it solves more important puzzles than the ones it loses through Kuhn-loss. I mention this only to illustrate how significant a change in paradigm is from Kuhn’s point of view: it is not merely a period of accelerated science, but a complete reworking of how scientists see the world.

We are currently in a period of crisis. Some physicists might disagree with me, but I think one can make a strong case that the paradigm that has taken us this far is showing cracks. In this post, I won’t directly compare current events to Kuhn’s description of a crisis, nor will I spend effort trying to define what the present paradigm is. For the moment I will content myself by pointing to some (just a few!) of the major puzzles that are facing us, and explain why they may represent `anomalies’ that require a new paradigm in order to solve them [2].

Dark matter / energy: One of the best-known puzzles of our time is the mystery of dark matter and dark energy in cosmology. Briefly, the matter that we can see in the universe (galaxies, nebulae and so on) is moving around as though it is being pushed and pulled by gravitational forces that have no visible source. In fact, there seems to be 95% more `stuff’ in the universe that we can’t actually see directly – we can only deduce its presence by its gravitational interactions with visible matter. The fact that we don’t know what this stuff is has been called the most embarrassing problem in physics for good reason: if somebody asks me what kind of matter and energy there is in the universe, I have to admit that, for the most part, I have no freaking idea.

Quantum gravity: Going by Kuhn’s picture of science, the key tool of the present paradigm is the Standard Model (SM) of particle physics. This model is impressively accurate down to really tiny scales and has been spectacularly confirmed time and time again in the world’s big particle accelerators, right up to the recent discovery of the Higgs Boson at the Large Hadron Collider (LHC). However, a major limitation of the Standard Model is that it does not tell us how gravity fits into the picture. While we have brought electromagnetism and the nuclear forces up to date with quantum mechanics, our theory of gravity is still straggling behind by over a hundred years. All the other forces have been given a quantum makeover, but gravity remains the shy stepsister, cloaked in a classical veil. Despite some pioneering attempts to get behind that veil, most notably String Theory and Loop Quantum Gravity, there is still no agreement among the community about which approach is correct or whether we have to try something else entirely [3].

Quantum foundations: It is often said that nobody understands quantum mechanics. This would be very worrying if it were true, since much of today’s technology is based on it! So what is the situation really? Well, obviously we understand the theory well enough to use it in practical applications. The trouble is more on the philosophical side: physicists can’t agree on why quantum mechanics works so well. In fact, we still can’t agree on why the universe should be quantum mechanical in the first place! John Wheeler’s famous question `why the quantum?’ still keeps many of us awake at night. There is an ongoing body of research on quantum foundations, whose goal is to improve our understanding of quantum mechanics to the point where most of us can agree on a single interpretation. This interpretation (it is hoped) would reveal quantum mechanics in such a way that nearly every physicist will reflexively slap their forehead and declare `of course! It had to be that way’! The interpretation should be so compelling that classical physics will look absurd by comparison and quantum mechanics will be the most natural way to describe the world.

As an example, since Einstein, the gravitational force is now widely interpreted as the curvature of space and time. However, technically it is possible to explain gravity in terms of fields operating in flat spacetime, in a way that agrees with current experimental data – yet if you ask any physicist what gravity is, nearly all of them will say `the curvature of space-time due to matter’. By contrast, if you ask them what the wave function of quantum mechanics is, you will get all kinds of different answers, and probably an invitation to a conference on foundations where such matters are still being hotly debated. Whereas curved space-time seems like an elegant, simple and compelling way of visualizing gravity, we have no similarly compelling paradigm for visualizing quantum mechanics.

***

One of the tasks a physicist faces during a crisis is to identify which anomalies deserve our attention and which ones are less important. This decision is guided by one’s intuitions and one’s chosen philosophy, hence a physicist must embrace some philosophy in order to make progress. For my part, I am most interested in the latter two anomalies: quantum gravity and quantum foundations. I think that the two are deeply connected. Since the regime of quantum gravity is still far from being accessible to experiments, the success of a theory of quantum gravity will be decided by the intuitive appeal of the physical principles on which it is based, as well as its elegance and explanatory power. We cannot hope to meet these demands all the way down at the level of quantum gravity (the Planck scale) if we still can’t do it up here on our home turf for quantum mechanics. Indeed, it is embarrassing that we cannot claim to have such a compelling picture of quantum mechanics, given that we have so much experimental data to guide us!

In upcoming blog posts I intend to elaborate on quantum gravity and quantum foundations and their possible connection to one another. I will also present my own ideas about how we should try to resolve the connected anomalies, using a philosophy based on a modern revival of operationalism and ideas from the exciting new field of quantum information. Stay tuned!

[1] This is a very rough version of Kuhn’s picture of scientific progress. The reader is encouraged to read the entry on Thomas Kuhn in the Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/thomas-kuhn/ . The less lazy reader is referred to Kuhn’s seminal work The Structure of Scientific Revolutions, University of Chicago Press, 2nd ed. (1970).

[2] There are of course far more anomalies in physics than the three listed here, although many of them can be linked to the same broad categories. For a more thorough list, see John Baez’s `Open Questions in Physics’: http://math.ucr.edu/home/baez/physics/General/open_questions.html .

[3] Some people have gone as far as to argue that String Theory is a failure. As an ignoramus, my own stance on this is more cautious, but that is a topic for another blog post.