Wednesday, September 02, 2015

GR, QG: horizons, gravitational waves, Hawking radiation etc. are more than beliefs

We haven't visited a black hole – no one will ever return alive from the black hole interior. We haven't directly and/or clearly observed the Hawking radiation, the Unruh radiation, a singularity in general relativity, gravitational waves let alone gravitons, excited strings, additional compactified spatial dimensions, superpartners, quanta of the inflaton field, regions with a different value of the inflaton field, regions with differently compactified extra dimensions than our way, and many other things.

Nevertheless, theoretical physicists generally assume that all those things – or at least most of them – exist. They do so most of the time. They say that they "believe" that those things are needed. Is the word "believe" another proof that their activity has evolved into a religion?

Not at all. The words "I believe X" simply means "my opinion is that X is probably right". Both religious and irreligious people have the right to "believe". The churches have no monopoly over the word. And religious and irreligious people may become convinced about something equally staunchly and feel the same psychological certainty about something. Where the churches differ from the scientists is in the methods to arrive to a "belief".

The methods matter. And because they're so different, science – including modern physics – is something totally different than religion, despite some similarities that may exist as well.

The religious people generally decide that "X is true" if someone, especially some religious authorities, say or write that "X is true" or that "X is positively correlated with God" in any sense. Or they are convinced that God told them the truth directly. The scientific people don't believe the holy texts or religious authorities. They should ultimately decide that "X is true" if X is a simple enough hypothesis that implies something about Nature which seems to agree with the experience (ideally lots of data and patterns we observe) even though there exist lots of alternatives to X that would disagree.

The scientific method starts by "guessing" a hypothesis. The hypothesis may be falsified by the observations. Or it may survive. When it survives many tests, it becomes important in the scientific thinking. When such a successful theory is falsified in the future, newer replacements are being guessed as "variations" of the old theory of some kind because the scientists know that the replacement theory will have to pass the same tests that the old theory did.

It's important that the hypotheses are not being "directly deduced or read" from Nature. All the work of "guessing" viable hypotheses is a creative work of induction that relies on some special talents of the people. David Hume (1711-1776) was the philosopher who famously understood this point – that the "creation" of viable hypotheses doesn't and can't proceed by strict deductive reasoning but it has to be about induction. Science just can't work without induction. The scientific process always depends on some kind of "reverse engineering" and there may exist no universal recipe how to successfully "reverse engineer" the knowledge in every situation. Search for "David Hume" and "induction".

Interestingly enough (1642-1726), Isaac Newton didn't appreciate that point yet. In a 1713 essay, he presented his famous slogan hypotheses non fingo, "I feign no hypothesis". He was clearly convinced that he was deducing the laws of physics directly from the observed phenomena. They were no "hypotheses" for him. They were the "truth" that he directly saw. That's what he believed and it was clearly wrong. After all, Newton's laws were wrong, when presented as the complete truth about Nature, for several independent reasons.

So what Newton actually did was to apply his creative genius and invent a great guess – along with the new discipline of mathematics that was needed to make it work. His framework of physics remained the basis for all of physics in the following 250 years before it was found insufficient. A part of his "hypotheses non fingo" may have been due to his genuine misunderstanding. A part of it may have been (probably pretended) modesty: Everyone else could have noticed Newton's laws written all over Nature, too.

No, they couldn't.

Physics differs from some less structured disciplines by its building a tall pyramid or skyscraper where the higher floors depend on the lower ones. In the analogy, the older or approximate or effective theories may be either be placed at the bottom or the top – there are different ways to adjust the details of this metaphor. But in all cases, the stability of the skyscraper depends on the robustness of the floors. You can't build a good skyscraper out of sand.

So physicists are really working – are forced to be working – with some concrete blocks. It's actually at least iron-reinforced concrete. What they want to understand simply has many levels or layers or floors and resilient building blocks are needed for any structure that can sustain those floors. Mathematics, often nearly rigorous and boasting very accurate numbers and propositions everywhere, is what makes these blocks so sound.

This robustness guarantees that physicists may confidently predict the results of many experiments they haven't observed. The physical laws and especially the metalaws – the laws about laws – that the physicists first guess and then validate are so powerful that physicists may trust (again, without any link to religion) interpolations and even extrapolations that would be pure speculations in less structured and less rigorous scientific disciplines.

A zoologist may observe a spider that has six legs. A larger rabbit has four. A human has two. Does it follow that an even greater elephant has one leg? You may see that almost none of these extrapolations would ever work in zoology. But in physics, those extrapolations typically do work. They work because some quantities are described as basically smooth functions of others. The form of the function may be guessed and constrained by observations and linearized in some regions and lots of other things may be done with them. With these tools, you may predict infinitely many "similar" situations, situations that wouldn't be similar for a zoologist.

And the extrapolations are not just extrapolations of some values of parameters. Physicists may deduce the "natural" answers to qualitatively different, more complex questions out of their validated answers to different, simpler, or more elementary questions. Physicists are doing such things all the time. They wouldn't have gotten anywhere if they were not doing so. If they were imitating the zoologist, pretty much every situation and every question about it would be completely independent from the previous ones, from all the observations that have already been made. The "structure" of physics would evaporate. "Physics" would degenerate into a telephone directory of independent observations. It wouldn't deserve to be called "physics", of course. There would be no "laws". The absence of any "laws" is obviously a key reason why e.g. Feynman has said that pseudosciences such as social sciences are not really sciences.

But physics has lots of "structure". Physics is an iron-reinforced concrete that runs through Nature and connects objects and situations that are apparently so different and so distant that no non-physicist would dare to connect them (except for the religious people who connect them through God, of course). In effect, physicists may have and do have lots of (irreligious) faith about numerous questions whose answers simply couldn't have been extracted from direct observations – and most likely, they won't be extracted directly from the experience in the coming years, decades, centuries, and more.

I claim that those who don't understand why physicists feel so confident while answering questions without "directly observing" the answer simply do not understand the very character of physics, what kind of science it is and what it allows us to do and why.

A "linearly polarized" basis of the 2-dimensional space of polarizations of a gravitational wave.

Gravitational waves

The gravitational waves are analogous to the electromagnetic waves. In \(d=3+1\), they also have two independent polarizations for each direction+frequency. But the very existence of the gravitational waves as a consequence of Einstein's equations remained controversial for an unreasonably long time. The linearized calculation is straightforward.

But some people had irrational problems with the linearizations and they preferred to (semi-religiously) believe their prejudices that general relativity should ban the gravitational waves. These prejudices were remnants of Mach's principle which basically wants you to believe that the empty spacetime without "objects" is always the same. But GR says otherwise. The gravitational waves may be present or absent and the two situations demonstrably differ (well, if the spacetime allows the presence of some measurement apparatuses).

We haven't directly observed the gravitational waves. LIGO was running between 2002 and 2010 and found none. A new gadget, Advanced LIGO, is in the construction on the same site, I think. Italy's VIRGO started in 2007 and has found nothing so far, either. LISA, the space antenna to measure those things in space, should be launched in 2034. Even the plans are very far – and they may still turn out to be overly optimistic.

A person who is viscerally hostile towards physics may call the opinion that gravitational waves exist "religion". A physicist knows that the person from the previous sentence is an ignoramus. After all, a Nobel prize in physics has already been given primarily for the indirect detection of gravitational waves. In 1993, Hulse and Taylor were rewarded for having seen a pulsar, a pair of neutron stars orbiting one another, whose orbital period is getting shorter every revolution. The rate of "speeding" exactly agrees with the calculation in general relativity that implies that these two objects emit gravitational waves, lose energy, and therefore are falling closer to each other. The time from the zeroth revolution to the \(N\)-th revolution is a nice quadratic function of \(N\) with a negative quadratic coefficient.

Can it be coincidence? Could a theory without gravitational waves imply the same "acceleration" of the orbits of this object? Miracles – something that has nothing to do with any explanations we are aware of now – may in principle occur. But scientists generally assume that they are unlikely.

If we admit that we're not 100% certain that gravitational waves exist, what probability should we assign to them? In his discussion with David Simmons-Duffin, Richard Muller suggested 80% – well, maybe to a different question but the spirit was the same. Well, if science would imply a conclusion to be true with the 80% probability, scientists would have a reason to build on this assumption and it is deeply misleading to call this type of work "religion", David stressed. It's science. 80% certainty is something we must get used to.

On the other hand, I think that the figure 80% for the existence of the gravitational waves is just insanely low. My figure is about 99.99999%, some six-sigma. Einstein's equations have been successfully tested – every piece of them, if you wish, but the pieces co-exist with each other in a pretty much unique way (if you only write down the two-derivative terms). A theory that deviates from GR so substantially that it predicts no gravitational waves would almost certainly have made some other predictions that would have been falsified by now. And the pulsar is one of them. A theory without gravitational waves seems to be 99% certain to have no explanation of the acceleration of the pulsar. And even if it had some different explanation, the value of the quadratic coefficient it predicts would almost certainly – 99% – be different than the observed one. If you combine these lines of evidence and appreciate that they're largely independent of each other, you may get something like 99.99999% certainty that the the gravitational waves exist.

There may exist other ways to think about the phenomena as well. But the right theory must confirm that the existence of gravitational waves is a valid way to explain what is going on.

While David has said that 80% would still make it fair to call it science, the science couldn't be extended too far. Once you were depending on 3 or 4 independent assumptions whose likelihood is just 80%, you would already deal with an axiomatic system that is less likely to be true than 50%. I am absolutely confident that this is not the case for modern fundamental physics as we know it. The near-certainty extends through many additional floors of claims about Nature that we haven't directly observed – that we have observed even "much less" than the gravitational waves.

OK, I obviously consider those who disagree with the existence of gravitational waves to be close to the "deniers of high school science" who also reject the existence of ice ages or heliocentrism. If you're one of them, please try to adjust your comments to the fact that I basically consider you a wild animal, a skunk of a sort. This is not meant to be insulting. It's meant to be as accurate an appraisal of the intellectual depth of two mammals as I am capable of producing.


Now, another layer. The gravitational waves are analogous to the electromagnetic waves. And electromagnetic waves are now interpreted as streams of many photons. Similarly, gravitational waves have to be composed of many gravitons. All successful theorists obviously agree with that statement; all others are considered cranks even though most well-known physicists are pathologically polite and avoid this accurate description.

Why do we know that gravitational waves are composed of gravitons?

The theoretical reasons work just like in the case of the electromagnetic field. Consider a somewhat messy gravitational wave packet moving in a certain direction that still has a pretty well-defined frequency \(f\). The fields at a point are approximately periodic with the period \(1/f\). But because the wave function of an energy-state depends on time as \[


\] and a general superposition of such terms isn't periodic, not even "up to phase", we have to demand that the allowed values of \(E\) that are included in the superposition differ by integer multiples of \(E=\hbar \omega = 2\pi\hbar f = hf\), the energy of one graviton. The argument and the energy-frequency relationship is identical for gravitons and photons. It actually holds for all particles in the Universe (although only bosons are able to team up in macroscopically strong "classical" waves).

So the energy carried by electromagnetic or gravitational waves has to be quantized in the units of \(E=hf\).

If you don't like it, there is another argument that has already worked in the electromagnetic case: the ultraviolet catastrophe. If you agree that general relativity implies the existence of gravitational waves, they may exist for any value of the vector \(\vec k\). Even in a finite box, there are infinitely many allowed values of \(\vec k\) – belonging to a lattice. Each of them behaves as a mode and in a thermal oven, it will carry the average energy \(kT/2\), as any degree of freedom in thermodynamics. (Well, it's two times \(kT/2\) for the kinetic and potential part of the energy, and times another two for the number of polarizations per \(\vec k\).)

Most of this infinite collection of contributions scaling like \(kT/2\) has to be suppressed if space is supposed to carry a finite energy density at a nonzero temperature. It is suppressed because the modes with too high \(|\vec k|\) contribute much less than \(kT/2\). It's because the single photon is too energetic for them and they become exponentially unlikely to harbor more than zero photons. So these high-momentum degrees of freedom are basically frozen and only the modes with low \(|\vec k|\) substantially contribute to the energy of the space.

It's no coincidence that the "finite and low heat capacities" have also been used to disprove all the realistic interpretations of quantum mechanics because "realistic interpretations" and "non-quantum theories" are synonymous. The ultraviolet catastrophe is just a field-theoretical version of the diseases that the classical and realistic theories suffer from when they predict insanely high heat capacities of the atoms, empty space, and everything else.

Again, this argument works identically to the case of the electromagnetic field. If there weren't gravitons, the high-frequency modes of the gravitational field wouldn't be suppressed, they would carry the same thermal energy as well, and the heat capacity of a cubic meter of empty space would be infinite.

At any rate, I think it is utterly unreasonable from a scientific viewpoint to attribute the probability lower than 99.99995% to the existence of gravitons.

Hawking radiation

Jacob Bekenstein made some ingenious guesses about the entropy of the black hole – which is proportional to the area of the event horizon. Bekenstein's claims were indeed guesses or speculations, if you wish, which were supported by analogies and visions. Building upon them would resemble religion to some extent – but still an extremely scientific yet beautiful religion.

However, Hawking has changed the status of those things. He derived the Hawking radiation – the thermal radiation whose temperature is basically the gravitational acceleration at the event horizon in the Planck units – out of the framework whose all assumptions are clearly formulated and have basically been validated.

His framework is the quantum field theory on curved spaces. Or semiclassical gravity. The matter (non-gravitational) fields are treated in the usual quantum field theory way, with creation and annihilation operators etc. The metric tensor field (spacetime geometry) is treated as a classical field which isn't possible in a complete and consistent theory but it's OK in an approximation. It's an analogy of the Born-Oppenheimer approximation in which the positions of the nuclei are first treated classically, and only the electrons are quantized, and only at the end, one may treat the nuclei quantum mechanically as well, with the full eigenvalues of the electrons' problem playing the role of "potential energy" between the nuclei. One may justify the applicability of the semiclassical gravity calculation in a similar way as in the Born-Oppenheimer approximation.

But at the end, Hawking's is just a technical calculation. The very fact that some particles are created may already be seen in the toy model, the Unruh effect (which was ironically found after Hawking's complex calculation), and the Unruh effect just boils down to the Bogoliubov transformation.

If you have a harmonic oscillator and you write Hamiltonians such as\[

H(\alpha,\beta) = \alpha x^2 + \beta p^2,

\] then classically, \(x=0\) and \(p=0\) would be "the ground state" of zero energy, regardless of the values of \(\alpha,\beta\in \RR^+\). However, quantum mechanically, you can't have a well-defined \(x\) and \(p\) at the same moment, thanks to the uncertainty principle. The ground state is a Gaussian and its width depends on \(\alpha/ \beta\). So the ground state for one value of \(\alpha / \beta\) is a linear superposition of the ground state and excited states of a different Hamiltonian with a different \(\alpha/ \beta\).

The Unruh effect, and with some purely technical complications, the Hawking effect as well are just an infinite-dimensional version of the harmonic oscillator argument above. Observers who slice the spacetime differently – like differently accelerating observers – have to use different Hamiltonians (to switch from a slice to the infinitesimally following one) and different Hamiltonians correspond to different quantum mechanical ground states. So what seems as the "empty vacuum" to the inertial observer becomes a "heat bath with lots of particle" from the accelerating observer's viewpoint.

In the case of the Hawking radiation, the radiation becomes "real" even for the ordinary distant observer because thanks to the curvature created by the black hole, this late distant observer's frame is basically connected with a frame that was accelerating when the black hole was young.

Quantum field theory on curved spaces is a highly trustworthy tool for a modern physicist. But it isn't the only piece of evidence that the Hawking radiation has to exist. The radiation may be derived "microscopically" in specific models of string theory, too. I think that it is unreasonable to assign a probability lower than 99.9999% to the existence of the Hawking radiation.

Event horizons

I should have started with that. Why do we believe that black holes actually exist? Well, the black hole at the galactic center is an important example. We observe that it is devouring highly energetic objects that are shining and do all kinds of things and once they're devoured, the radiation disappears.

This can't happen without event horizons because if there were any ordinary object over there, the "dinner" would have added the energy to the object, and the object would have an increasing temperature and would be increasingly radiating. The more it would eat, the more it would radiate. We observe exactly the opposite. So the apparent temperature of the object (the big black hole) we observe through its radiation must be vastly lower than the local temperature of the matter that falls in, and the basically unlimited red shift is needed for that.

Needless to say, the main reason why I (irreligiously) believe in the existence of black holes and event horizons is that they clearly follow from general relativity – in a regime where the curvature is as mild as in all the situations where GR has succeeded. I don't think it's too plausible for black holes not to exist. At least 99.999999% certainty.

Extra dimensions

The probability will obviously drop here because we're approaching more abstract layers of physics. But the probability that extra dimensions are relevant for some more accurate description of Nature than the Standard Model is still above 99.9%, even if I claimed that I am not sure which kind of string/M-theory compactification if any is valid.

Even if it turned out to be irrelevant for Nature for some reasons that I can't imagine, string/M-theory has already taught us that some assumptions we may have had are just irrational, unlikely, provincial, rationally unjustifiable. The assumption has been that \(d=3+1\) is the only value we need and care about. String theory not only "forced us" to study theories in different dimensions. But it also showed that the higher number may be consistent with all the basic observations we can do. In fact, it taught us that the extra dimensions are extremely helpful in explaining the spectrum of the elementary particles (including the three/many generations) and the gauge groups. But string/M-theory has also told us that consistency can actually dictate the dimensionality to be a particular number different than \(d=3+1\).

The critical dimension of strings is \(d=9+1\) and M-theory has \(d=10+1\) and there may exist mutually dual descriptions so it makes no sense to ask which of them is more right. All of them are equally right.

I view the critical dimensions as extremely important but I am only 95% sure that the right form of extra dimensions that will be relevant in the future phenomenology will copy the 10- or 11-dimensional spacetimes of supersymmetric string/M-theory. I can imagine that something like "free fermionic heterotic models" will be superior and won't allow any geometrization of these degrees of freedom. Or, less likely, something like sub- or supercritical string theory may be needed.

But the general lesson that "the dimensions we easily see don't have to be the only ones" is a lesson that physicists are unlikely to unlearn again. The Earth is much more easily visible than other planets which doesn't mean that there aren't other planets. It would really be contrived, unnatural, if all the planets were equally visible from a hospitable perspective. The dimensions are analogous. There's no reason why all the dimensions that act as degrees of freedom on the world sheet or anything that generalizes it must be the dimensions that we know from long-distance physics.

It's hard to convey all this reasoning to someone who doesn't understand the whole string-theoretical predictive framework. But I view these lessons of string theory to be analogous to the apple that Adam and Eve ate in the paradise (and in this sense, extra dimensions are anti-religious). Once they did so, they were able to notice that they had non-isomorphic sexual organs that can be played with, sometimes in complementary ways. It's hard to unlearn that lesson once you actually start to play with those things. ;-) The people who are hoping in a future of physics that will forget about the extra dimensions again are analogous to the people who want to forget that they had any sexuality, people who want to un-eat the apple. I just don't think that anything of the sort is realistic.


String theory was born as a temporarily failed theory of the strong interaction (that in the atomic nuclei). A year later, it was shown that it actually described the gravitational force consistently. Ten more years later, it was seen that it actually predicts the exact general types of matter particles and forces that are needed to explain everything we know in Nature.

String theory is consistent where all other conceivable competing theories look hopelessly inconsistent. It confirms – from totally different viewpoints – previous insights including the Bekenstein-Hawking entropy of black holes and the Hawking radiation and lots of other things. It allows us to geometrize many things that seemed to have nothing to do with geometry.

But I actually think that we have much more "direct" and speculation-free ways to see that string theory has to be taken seriously when one studies quantum gravity beyond the approximations above.

Take AdS/CFT. Imagine that you have any consistent theory of quantum gravity and it allows you to be formulated on an anti de Sitter background. That background has the isometry that is isomorphic to the conformal symmetry on the boundary. Because most of the volume of the AdS space lies "very close to the boundary", there should be a way to describe the quantum gravitational physics on the AdS space that works with the degrees of freedom that are localized on the boundary.

Use this argument or some 't Hooft-Susskind-like general arguments about holography. But you will decide that there should exist an equivalent description using a boundary CFT (conformal field theory). So try to construct the theory by considering conformal field theories. The supersymmetric ones are the easiest ones to be guaranteed to be exactly conformal. You will find a few simple examples, especially the \(\NNN=4\) superconformal Yang-Mills theory in \(d=4\), and you may analyze what the physics in the bulk looks like if the AdS/CFT holds.

You will find out that at low energies, the bulk is the 10D supergravity. But if you study the physics at a higher accuracy, you will see that the exact theory in the bulk isn't just supergravity – which is inconsistent – but it has to be the type IIB string theory. The symmetries are those predicted by string theory, not SUGRA, and you will also find all the excited and interacting strings and branes that all other descriptions of string theory imply.

There are lots of consistency reasons why the "improvements" that string/M-theory actually makes to supergravity are absolutely needed for consistency. You may either see those facts on the examples – there are lots of CFTs, in assorted dimensions, that produce the AdS bulk dual matching a vacuum of 10- or 11-dimensional string/M-theory. Or you may check these arguments by careful reasoning. See e.g. Two roads from \(\NNN=8\) SUGRA to string/M-theory. All these "improvements" – characteristic features of string theory that distinguish it from its low-energy limit – may be seen to be necessary in many different contexts, using very different and (if you reject that string theory is a framework that should be thought of as connecting all of physics) logically independent arguments. This increases the probability that these "improvements" are actually needed in general, whether or not you have already embraced the need for string theory as the all-encompassing theory of physics.

I think that the probability one should assign to the statement that "quantum gravity has to be extended to string theory to remain well-defined" is at least 99.5%.

String theory is several floors "above" the first layers of scientific assumptions that some people already question. But I am still confident about those floors. String theorists are actually confident about roughly 5 additional "detailed floors" built within string theory. String theorists don't just write papers saying "string theory of course". They make detailed advances within string theory – making lots of specific statements about particular situations or more special questions. To a sloppy thinker, those things could look hopelessly speculative. But they are not. Due to the iron-reinforced concrete of the mathematical formalism, machinery, and argumentation that the physical reason is based upon, theoretical physicists in general and string theorists in particular can get much much further from the direct observations.

Just think about a schoolkid who is asked to calculate a problem in mechanics – the motion of some contrived triple pendulum. No one has constructed it before but you should be able to understand that a good schoolkid can make valid predictions what will happen, anyway. String theorists can do analogous things – not just with a pendulum but with the Universe and everything important it contains. It's possible. 350 years ago, physicists may have been understanding things like pendulum only but these days, we're much more certain about many more things, almost all things we know.

At the end, I do think that people are probably making some mistake – or missing something simple, something that may change some opinions in the future and make the replacement statements look much more obvious than we can imagine today. But what's important is that the critics of theoretical physics have nothing whatever to do with these future discoverers and geniuses. The contemporary critics actually don't have any valuable idea, any viable alternative, or any valuable anything, for that matter. These individuals are just piles of stinky crap and it is deeply dishonest for them (e.g. Richard Muller) to pretend that they are something different.

The future realizations in physics may bring – and, hopefully, will bring – new Eureka moments and paradigm shifts. But the right recipe to get them isn't to criticize science, compare it with religion, or to focus on ideas that don't look promising. One – and the physics community – simply has to enforce some meritocratic criteria and focus on ideas that do look promising for one reason or another. So if the physicists are missing something important, it's all of them who is missing it.

And even if some of the assumptions we are making today will be found strictly speaking invalid in the future, this invalidity may be (and probably is) totally unhelpful for the next 100 years of the progress in science. If the truth is "totally different" than the (irreligious) beliefs of the best contemporary theoretical physicists but you don't know exactly what it is and why it is what it is, you should better shut your mouth because vague enough propositions about a "different future" are always "true in some sense" but their value for the scientific progress is usually zero or negative. Their looking valid from some direction(s) is a largely inevitable coincidence or noise.

Talented theoretical physicists may only divide their time to ideas that are already known or that have been "glimpsed" by themselves or someone else. The totally unknown types of ideas and theories aren't eligible for the competition yet. You may discover them and present the evidence why they're as promising as (or more promising than) the ideas on the market today. But if you have nothing of the sort, you have zero rational or moral justification to pretend that you're a peer (or even better than) the top physicists today.

The top physicists have found something that seems to make sense according to themselves and other good physicists. They have already produced some products that science is supposed to produce. One doesn't need a direct observation to verify that they have created a valuable product. Mathematical, logical, and indirect empirical tests are enough to quantify the value, too. Who doesn't want to see the value of these things is an anti-science person because basically everything that science ever finds is qualitatively analogous to these discoveries.

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