Confusions about the relationships of special relativity and general relativity

Sabine Hossenfelder wrote about the confusions surrounding the relationship of Einstein's 1905 special theory of relativity and Einstein's 1915 general theory of relativity. Edward Pig Measure is one of the laymen who are somewhat confused; many others are vastly more confused.



First of all, I find it very important that all the discussions on the two blogs above are about physics topics that have been settled for 100 years, about the high-school understanding of relativity. I think it is desirable to emphasize this point because much of the confusion arises when complete crackpots such as Lee Smolin say or write totally wrong things about relativity and they sell these totally wrong things as a cutting-edge research.




Special relativity is a 108 years old or new theory of space and time that correctly accounts for new phenomena that are known to occur when the observers' speeds approach the speed of light. It is a principled theory that constraints what particular constructive theories of individual phenomena and their classes may say and what they mustn't say.




All other theories must be made compatible with the two postulates of special relativity:

• Relativity postulate: the laws of physics have the same form in the coordinate systems of all observers moving by constant speeds in a constant direction (inertial frames)
• Constancy of the speed of light: the speed of light is constant, \(c\), regardless of the speed of the source and the speed of the observer

Maxwell's theory of electromagnetism was actually compatible with those principles before relativity was found; that's why Einstein's good understanding of electromagnetism helped him to discover special relativity. However, ordinary mechanics was only compatible with the first postulate (which is referred to as the Galilean invariance in non-relativistic mechanics); it didn't respect the constancy of the speed of light because the speed of light was supposed to become \(c\pm v\) if an observer was moving relatively to the aether – a preferred environment in which the speed of light is \(c\) (independently of the speed of the source!) – by the speed \(v\). The 1887 Morley-Michelson experiments made it clear that the speed was always \(c\), regardless of the speed of the observer.

So Einstein's special relativity primarily modified mechanics – mechanics was forced to change. For example, the relative speed between two bodies on a collision course on a line whose speeds are \(u,v\) is no longer \(u+v\) but it is \((u+v)/(1+uv/c^2)\). But any other kind of phenomena if there were one aside from mechanics and electrodynamics – hydrodynamics, aerodynamics, thermodynamics etc. (although they're really derived theories from mechanics and perhaps electrodynamics, not really fundamentally new) – had to be adjusted to agree with the two postulates. Particle physics only accepts theories that agree with relativity, too. For particle physics, this is so automatic – quantum field theory and string theory are the frameworks of choice and all of them are relativistic – that we don't even realize how much the possible "theories of particles" have been constrained by relativity.

The postulates imply – and Einstein was able to prove from them – that the length of objects shrinks in the direction of motion; the rate at which (any) clocks are "ticking" is slowing down if the clocks are speeding up; the total relativistic mass is increasing with the speed; its conservation law is merged with the momentum conservation law to the 4-momentum conservation law; this also implies that the mass and energy conservation laws become one identical part of the 4-momentum law and (what we used to call) energy may be converted to (what we used to call) mass and vice versa via \(E=mc^2\), the most well-known equation of relativity among the laymen.

While mechanics (I really mean kinematics, the description of motion influenced by forces that are given and whose origin isn't analyzed) was adjusted to relativity in 1905 – it was the main point of it – the physics of gravity (the description of a particular force that causes the motion – and such descriptions belong to "dynamics", not "kinematics") remained mysterious because (among related problems), Newton's gravity seems to operate instantaneously which violates the speed limit, \(c\), that relativity imposes on the speed of propagation of any usable information.

Einstein spent the decade after the discovery of special relativity, 1905-1915, by attempts to reconcile the laws of gravity with the principles of special relativity. The result of this long but successful work, the general theory of relativity, pretty much inevitably and uniquely follows from special relativity (that is required to hold whenever the gravitational fields are negligible) and the equivalence principle (the statement that all bodies accelerate in gravitational fields by the same acceleration which means that freely falling frames are indistinguishable from the life outside gravitational fields, and must therefore locally preserve special relativity).

I will discuss GR as an unavoidable extension of SR momentarily. But let me first address a more trivial question:

Is SR applicable to phenomena in which objects accelerate?

The answer is, of course, Yes. Special relativity would be useless if it were requiring all objects to move without any acceleration; after all, almost everything in the real world accelerates, otherwise the world would be useless. The correct claim similar to the proposition above is that special relativity has the same, simpler form in coordinate systems associated with non-accelerating observers. But that doesn't mean that we can't translate the predictions of a special relativistic theory to an accelerating frame. Yes, we can. It's as straightforward as a coordinate transformation. Fictitious forces will appear in the description. All of them are fully calculable.

We should point out that if it were impossible to consider accelerating observers, special relativity couldn't tell us anything about the twin "paradox". At least one of the twins, the astronaut, has to intensely accelerate during his life. But the total time measured by his clock – and by the aging of his organs, which is just another type of clocks (not too accurate one) -´is clearly composed of the proper times of tiny line intervals into which his world line may be divided. The infinitesimal pieces of his world line are straight so special relativity simply has to hold. When we compute i.e. integrate the total proper time along the world line, of course that we will find out that the twin-astronaut will be younger than his brother who spent decades on Earth.

We don't need general relativity because the presence of acceleration doesn't mean that there's a gravitational field. The curvature of the spacetime is still zero. Acceleration is locally equivalent to gravity by the equivalence principle but the clever way to use it isn't to envision unnecessary gravitational fields but, on the contrary, to undo the gravity whenever we can by replacing it with acceleration combined with no gravity – and for this combination, special relativity is sufficient.

Not being able to produce this right answer to the twin "paradox" means not to understand special relativity at the high-school level (at least we did learn basics of special relativity at the high school, a pretty ordinary high school). It's not wise, deep, clever, or sophisticated to be doubtful about the usual resolution to the twin "paradox". It is nothing more than a sign of brutal ignorance. (Christine Dantas is among those who believe that special relativity doesn't imply that the astronaut-twin will be younger because acceleration makes it impossible to use the theory. Holy cow. This lady has had a full big mouth about quantum gravity while high school physics is apparently way too hard for her.)

Now, let me switch to general relativity again. Sabine promotes a particular definition of special relativity:

Ask some theoretical physicist what special relativity is and they’ll say something like “It’s the dynamics in Minkowski space” or “It’s the special case of general relativity in flat space”. (Representative survey taken among our household members, p=0.0003). But open a pop science book and they’ll try to tell you special relativity applies only to inertial frames, only to observers moving with constant velocities.

I don't think that it is downright incorrect to describe special relativity in Sabine's way. But I don't think it's the deepest or most natural way, either. More importantly, I do agree with the criticized books that at least something in special relativity does apply to observers moving with constant velocities only – the Lorentz symmetry only mixes the viewpoints of these observers and, consequently, the laws of physics only have the usual simple form in the coordinate systems connected with these observers. The difference between inertial and non-inertial systems is essential in special relativity and if that's the claim that Sabine criticizes, she is completely wrong.

Moreover, her "definition" of special relativity is useless. A definition is meant to be helpful to someone whose knowledge is at a lower level than the level at which the defined object is "obvious". If someone doesn't know special relativity, you won't help him much if your explanation will assume the knowledge of general relativity because, you know, general relativity is harder than special relativity.

But there's another, more conceptual reason why I consider Sabine's definition to be a sign of her (and her spouse's, as we were told) shallow knowledge of the subject. What is the reason? Her definition implicitly says that general relativity is the fundamental set of insights, rules, and principles and special relativity is just a minor corollary of it. While it's true that special relativity is a limit of general relativity obtained for gravitational fields going to zero, the actual "hierarchy of power" is the opposite: general relativity is just one application of special relativity – the incorporation of the gravitational field in a special-relativity-invariant way. While general relativity is arguably the prettiest (and geometrically most non-trivial) classical application of the rules of special relativity, in principle it is on par with Yang-Mills theory or any other (special) relativistic field theory.

This claim of mine may be interpreted as a modern interpretation of the philosophy underlying relativity – and widely appreciated by most of the competent modern theoretical/phenomenological particle physicists (people who were clearly not included in Sabine's low-brow survey). But there's a sense in which it's ancient, too. What's the sense? Well, the insight is ancient because Einstein simply didn't have a choice when he was searching for a relativistic theory of gravity between 1905 and 1915. General relativity is the unique theory obeying the postulates of special relativity that describes the gravitational force – by which I mean a force (and we can prove that it's the force because such a force must be unique for a physical system) that respects the equivalence principle.

The gravitational field must be given by some components of the mass/energy/momentum-encoding stress-energy tensor. Because the strength of the field around a physical system as measured at infinity cannot change (in analogy with the field around a charge in electrostatics), it must be conserved quantities that source the gravitational field/influence. Because our goal is a gravitational force that depends on the mass, it's clearly the whole stress-energy tensor \(T_{\mu\nu}\) that must be involved in sourcing the gravitational field (\(T_{00}\) which must surely influence the gravitational field isn't a Lorentz-invariant quantity and the Lorentz transformations of this quantity involve all other components of the tensor). The corresponding "potentials" of the gravitational field must be organized as a symmetric tensor with two indices, too. It's \(h_{\mu\nu}\).

However, the derivatives of the field \(h_{\mu\nu}\) contribute to the energy as well, like the derivatives of any matter field. We are led to the question how the field sources itself. We're brutally constrained by the equivalence principle because physics in the \(h_{\mu\nu}\) field that linearly depends on the coordinates must be indistinguishable from physics outside any nonzero fields: a freely falling observer (in the linear \(h\)-field) mustn't be able to figure out that he's in a gravitational field at all.

This is only possible if there is a rather large symmetry that is able to identify configurations with different profiles of \(h_{\mu\nu}\) – identify some configurations where this field is nonzero (and even non-constant) with the configuration where it's zero. So this symmetry must be mixing the gravitational field \(h_{\mu\nu}\) with something that was nonzero to start with. It must have the same tensor structure and we conclude that it must be the pre-existing metric tensor \(\eta_{\mu\nu}\). The only symmetry that is able to produce the right number of symmetries acting on these metric tensors is the diffeomorphism symmetry under which the "total metric"\[

g_{\mu\nu}=\eta_{\mu\nu} + h_{\mu\nu}

\] transforms as the tensor field. So we're led to general relativity as the only possible (special) relativistic description of gravity that uses fields.

This was a sequence of arguments that tried to be as classical as possible. Modern particle physicists would present a similar but quantum-field-theory-based version of the ideas. Because it's locally sourced by the stress-energy tensor, gravity must involve spin-two fields. In the covariant, manifestly Lorentz-invariant description, spin-two fields have some positively definite components \(h_{ij}\) and perhaps \(h_{00}\) (which will also be mostly killed, despite its good sign) and some negative-normed components \(h_{0i}\), ghosts. The latter is unacceptable because it leads to the prediction of negative probabilities for some processes. So there must exist a symmetry that decouples all the ghosts. The symmetry has to be local and have a whole "vector" of parameters at each spacetime point. Coordinate redefinitions \(\delta x^\mu\) are the only solution. For gravity in terms of quantum fields, you need spin-two fields and the diffeomorphism invariance is necessary to get rid of their pathological, negative-normed components. The rest of the GR follows; the Ricci scalar is the lowest-order (in the number of derivatives) coupling compatible with the required symmetry but there may also be higher-order corrections (whose effect becomes negligible at long distances).

Some people would declare all the derivations above to be heresies because they think it is a blasphemy to ever write the metric tensor as a sum of two or several pieces because such a blasphemy contradicts the holy beauty of general relativity as written in an unwritten commandment somewhere. ;-) The price they pay for this medieval, unjustifiable, irrational, stupid taboo (the commandment really says "you shall never make your hands dirty by any science that actually applies to a situation in the real world or answers some questions beyond the questions whose answers you have been given to start with, by science that requires you to write anything else than the most beautiful form of the basic equations") is very high: They can't understand some key facts about modern physics, e.g. that and why the general theory of relativity is unavoidable given the validity of special relativity and the existence of gravity sourced by the energy-and-momentum density and their fluxes/currents.

Many people in the Backreaction discussion are confused about many other things.

For example, is a charged object sitting somewhere on the Earth's surface emitting electromagnetic and/or Unruh and/or gravitational radiation?

The answer is, of course, No. If it were radiating in any of the three ways (to be precise, by radiation I mean sending physical photons, gravitons, or other particles to infinity), it would have to lose energy to avoid the violation of the energy conservation law. But the charged object is already sitting at a place where the energy is minimized so there's no way to extract more energy out of the particle.

Relatively to a freely falling frame, the charged object sitting on the Earth's surface is accelerating so it should emit all three kinds of radiation, some people could argue. If it emits no radiation, doesn't it violate the equivalence principle?

No, it doesn't. First of all, the equivalence principle is only guaranteed locally. But in the previous paragraphs, we were asking whether particles are emitted to infinity. This requires us to connect the vicinity of the Earth with infinity, to compare them. But such a global connection turns the existence of the Earth's gravitational field into an objective fact. There exists no flat-space-based equivalent description of a region that would include both Earth's vicinity as well as the asymptotic region at infinity. So the equivalence principle isn't really applicable. There's no justifiable way to argue that the charged sitting object should emit radiation.

There are other ways to argue and reach the same conclusion.

For example, the equivalence principle identifies the experience of a freely falling observer with those of an inertial observer in the flat spacetime. But the identification only holds if "all other factors are equal". The freely falling observer who is going to hit the Earth's surface soon doesn't have "all other factors equal". In particular, there may be some extra radiation coming from the rest of the Universe. It just happens that the radiation is such that it perfectly cancels the would-be electromagnetic/Unruh/gravitational radiation of the charged object sitting on the Earth.

To make this discussion really complete, I would have to describe a formalism that has something to cancel at all and distinguish the different amounts of radiation as seen by a nearby static, nearby accelerating, or infinitely distant detector. The discussion could get unnecessarily messy and repetitive. But my point that shouldn't get lost in this technical material is that only the black holes emit the Hawking radiation. One actually needs the horizon for that. If there's no horizon, there's no energy loss by the Hawking or another acceleration-based radiation. (And this 1999 paper is just wrong. It's not the only one.)

Why does the horizon matter? If there's the horizon, one simple fact holds: the black hole interior can't possibly send any radiation (positive-energy one or a "compensating one") in the outward direction; nothing gets out of the black hole. That's why the frame of an observer who is freely falling into a black hole (with a horizon) is as equivalent to an inertial observer in an empty space as you can get. He could have been freely falling throughout his life which explains that no radiation was going in his direction.

On the other hand, there's no radiation going from the black hole interior, against him, either. It's forbidden by the blackness of the black hole. It's this latter property that doesn't hold for the Earth. The Earth imposes different boundary conditions on the surface than the black hole enforces on the event horizon. If the Earth were a conductor, the electrostatic potential would vanish on the surface. The relevant modes of waves would be standing waves above the Earth's surface. While the condition "killing" one-half of the modes in the black hole case says that "nothing is coming in the outward direction", the conditions are different for the Earth: "no waves are inside the conducting Earth". The latter condition is past-future-symmetric, unlike the condition for the black hole.

The vacuum is Unruh and electromagnetic radiation-free in the "most natural frames". For black holes, it's the freely falling frame because you can just freely fall and you will never notice that something is unnatural about that frame (the singularity kills you before you realize that). That's why there's no radiation in this frame while the frame of an observer keeping himself above the horizon by jets experiences Unruh radiation that penetrates through the black hole's gravitational field and becomes real, physical Hawking radiation at infinity.

For the Earth, the most "vacuum-like" frame is one associated with the surface because the freely falling observer will hit the Earth's surface and the headache will convince him it's not the frame most similar to the empty space. ;-) So the Earth stabilizes all the surrounding fields relatively to its static surface and relatively to this frame, there's no radiation – and frames accelerating relatively to the surface's frame will see some radiation. Of course, a semiclassical analysis of GR coupled to electromagnetism offers you a more reliable but less funny derivation of the same conclusion.

One should emphasize that the Unruh/Hawking radiation for the Earth, even if there were one, would be ludicrously weak. The typical wavelength of the emitted photons would be comparable to \(c^2/g\) which is about \(10^{16}\,{\rm meters}\), not far from a light year. It's clearly just an academic debate for the Earth as the very weak radiation would be totally unobservable – dozens of orders of magnitude weaker than the observable one. But it would still be an inconsistency if stable objects and particles like that would radiate because of some incorrectly applied equivalence principle.



Off-topic: Mr Ilja Hurník (*1922) died. He was a serious Czech composer of highly non-classical music for classical instruments and a piano virtuoso but people like your humble correspondent know him as the author of small pieces such as the "Merry Postman" (yes, he's ringing the bell and knocking the door) and "Little Soldier" above which I liked to play when I was 8 or so. ;-)

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A universal derivation of Bekenstein-Hawking entropy from topology change, ER-EPR

I have been intrigued by topology change in quantum gravity, especially its Euclidean version, for 15 years or so. Since the beginning, I liked a sketch of a derivation (that I invented) of the Bekenstein-Hawking entropy of a black hole that was based on a wormhole connecting two tips of the Euclidean black hole in the \(rt_E\) plane.



Ignore the wormhole-related captions.

Before the ER-EPR correspondence, I would interpret the two planes on the picture above (lower, upper) as spacetimes in the ket vector and the bra vector, respectively, and this need to double and complex conjugate the whole spacetime made the details of the argument confusing because the thermal calculation (which is inevitably connected with the cigar-like Euclidean black hole pictures) inevitably involves a trace over ket vectors (or bra vectors but not both).

Fortunately, one may now present the whole argument without any bra vectors. Thanks to Maldacena and Susskind, the doubling of the spacetime (note that there is the upper and lower plane on the picture above) may be interpreted as the presence of two distinct spacetimes or two faraway regions of one spacetime – or two faraway regions of the same spacetime; it won't really make a difference. With this reinterpretation of the pictures, I am more satisfied with the argument.




Try to calculate a thermal correlation function in a spacetime (or a pair of spacetimes if you really view the two planes as disconnected) of temperature \(1/\beta\) which will be chosen to agree with the black hole temperatures below. The operators in the correlation functions don't matter; assume that they are low-energy operators far away from all the celestial bodies we will consider.

We want to know how much the states with two black holes at places \(A,B\) (in arbitrary microstates) contribute to the correlator; and how much the states with two neutron stars at the same places \(A,B\) contribute. The ratio of the two contributions should be \(\exp(S_A+S_B)\) where the terms in the exponent are black hole entropies times some subleading corrections (all neutron stars' entropies will be negligible). Just to be sure, the contribution from two black holes should be exponentially larger. I will take the two celestial objects to be macroscopically the same so the ratio should be \(\exp(2S)\) where \(S=S_A=S_B\).




To confirm the Bekenstein-Hawking formula means to prove that the contribution from the two black holes is \(\exp(2A/4G) = \exp(A/2G)\) times greater than contribution from the two neutron stars.

By the neutron stars (a nickname chosen for the sake of simplicity), I really mean a celestial body that is on the verge of collapsing to a black hole. I want \(g_{00}\) to be very small right above the surface of this body. Because \(|g_{00}|\) wants to be even smaller in the stellar interior which makes it impossible for \(|g_{00}|\) to be near zero above the surface, I really need to consider a hollow star – a shell that is protected against the collapse by some skeleton or light gas inside or whatever. I hope that these awkward technicalities don't really matter and can be replaced by a less problematic treatment. Maybe it's enough to compare the two-black-hole contribution with the contribution having no objects at those places at all.

For the sake of clarity, let's assume that the black hole radii are equal to a few miles (a solar-mass black hole). The thermal correlators may be calculated from the path integrals\[

\langle \cdots \rangle = \int {\mathcal D}\,{\rm fields}(x,y,z,t_E)\,\exp(-S_E)\, (\cdots )

\] over the Euclidean geometries with Euclidean field configurations in a spacetime whose Euclidean time coordinate \(t_E\) has the periodicity \(\beta\).

Now, we don't want to study the detailed microscopic physics of the neutron stars. Their entropy (and any non-black-hole celestial object's entropy) is negligible in comparison with the black hole entropy. We don't even want to specify what exact short-distance degrees of freedom are responsible for the black hole entropy. Indeed, the goal is to derive the Bekenstein-Hawking formula "universally", for every quantum theory that resembles quantized general relativity in a limit.

But yes, in this geometrized picture of the degrees of freedom, all the entropy is carried by some degrees of freedom – field modes and their generalizations – that may be attached to the stretched horizon, a Planckian vicinity of the region that will host a throat in a minute.

To neglect the short-distance physics, why don't we integrate out all the field modes with wavelengths shorter than 1 millimeter (to be specific again)? When you do so, the two-neutron-star contribution looks like two disconnected pieces, essentially two planes (the upper and lower plane) not connected by the throat shown on the picture at the top. Even if there has been some entanglement between the stars, it was way too weak to produce the smooth throat. Instead, the thin tunnels disappeared as we integrated the high-energy degrees of freedom out. The stellar interior isn't clearly shown on the picture – the picture only shows the stellar exterior – but it's somewhere and the Ricci scalar \(R\) is essentially zero everywhere. Again, maybe I should replace the neutron stars by empty regions of space throughout this argument; I wanted the two compared situations (with and without black holes) to be as similar as possible, however, so that the difference may be blamed on the throat, as we will see momentarily.

What about the two-black-hole contribution?

Maldacena and Susskind taught us that the Hilbert space of 2 similar black holes – essentially \(\HH_{2BH}=\HH_{1BH}\otimes \HH_{1BH}\) – is isomorphic to (really the same as) the Hilbert space of an Einstein-Rosen bridge geometry that connects them. Despite the apparently different topologies of the two descriptions, they're the same Hilbert spaces. The bridge-based description is better for highly entangled states in the Hilbert space; the 2 isolated black hole description is better for the nearly unentangled states of the two black holes. (Note that "highly entangled states" and "almost unentangled states" don't form linear spaces because the properties "entangled" and "unentangled" aren't closed under addition.) The two-black-hole states that strongly entangle the two black holes look like smooth bridges; however, there are highly excited, unsmooth bridges that must describe all the other two-black-hole microstates as well.

In general, the two planes – see the picture at the top – are connected by "some" throat. When you integrate all the field modes shorter than one millimeter out, you also do it for the gravitational modes so the geometry can't be too thin or curved. In effect, the gradual integration out thickens the throat in the black-hole case while it cuts the throat(s) in the stellar case. When you're finished, the throat itself is about one millimeter thick. It was a randomly chosen distance scale that is much longer than the Planck scale but much shorter than the black hole radius.

Looking at the two-neutron-star and two-black-hole Euclidean geometries, they look very similar. The only difference is the throat near the event horizon (or near event horizon in the case of the stars). In that region, the \((d-2)\)-dimensional area of the angular variables is constant, \(A\), which simply enters as an overall factor to the difference of the actions, and the major components of the curvature tensor only exist in the two-black-hole case and in the Riemann components \(R_{rtrt}\) and its three copies dictated by the Riemann tensor's symmetries (\(t\) really denotes \(t_E\) as an index).

(The throat in the black-hole case isn't Ricci-flat; the nonzero Ricci tensor must be blamed on the high-energy matter that resides in the stretched horizon(s).)

So the two contributions to the path integral – from the two neutron stars; and from the two black holes – only differ by the extra "wormhole" in the two-black-hole case. This wormhole is a "handle" of a Riemann surface and the exponent of the Euclidean path integral is more negative in the black-hole-case (I hope) relatively to the neutron-star case by the factor\[

\exp[-(S_E^{\rm BH}-S_E^{\rm neut})] =
\exp\left(\!-\frac{A\int d^2 x\sqrt{|g|}R_{(2)}}{16\pi G}\right)=\dots

\] over the handle (wormhole). But the two-dimensional integral – the Einstein-Hilbert action above – is proportional to the Euler characteristic\[

\chi = \frac{1}{4\pi}\int d^2 x\,\sqrt{|g|}R_{(2)}.

\] Note that a sphere of radius \(a\) has \(R_{(2)}=2/a^2\) and \(\chi=2\). Each added handle (which has a negative curvature \(R_{(2)}\) in average) reduces the Euler character by two and (therefore) the integral of \(\sqrt{|g|}R_{(2)}\) by \(8\pi\). When you substitute this \(8\pi\) decrease above, it becomes an increase of the exponent due to the extra minus sign in the exponent and you will see that the two-black-hole contribution is greater by the factor of \[

\exp\zav{ \frac{A\cdot 8\pi}{16\pi G} } = \exp\zav{ \frac{A}{2G} },

\] exactly as expected from the Bekenstein-Hawking entropy of two black holes. This multiplicative increase implies that there are \(\exp(A/4G)\) black hole microstates per black hole whose precise identity doesn't significantly affect the correlator we agreed to compute. So if we trace over them (and we do so in a thermal calculation), they just influence the result by the simple multiplicative factor (the number of these microstates).

You may have some doubts about the sign of the Euclidean Einstein-Hilbert action used above. I have some doubts as well. I can enumerate about 6 things one must be careful about that may lead you to a wrong sign but I am not sure whether I am not missing some other sign flips. The probability that I keep on committing a sign error here is too close to 50 percent at the end ;-) which is why I must add that a more careful scrutiny is needed.

This argument may arguably be generalized to derive Wald's entropy formula for a more general action including higher-derivative terms. In these cases, one still has \(R_{rtrt}=2\pi \delta^{(2)}(r,t_E)\) per black hole located at the horizon and if we treat this modification of the Riemann tensor perturbatively, the change of the gravitational action produces Wald's entropy formula instead of the Bekenstein-Hawking formula above.

Incidentally, I think that quite generally, the black hole entropy must also be interpretable as the total order/volume of an approximate symmetry group of a given spacetime because a black hole may be interpreted as a codimension-2 "cosmic string" in the Euclidean spacetime (which is analogous to 7-branes in F-theory and requires us to study the monodromies). But why this gives the right results in weakly coupled string theory (where you have a \(U(1)\) for each free field-theory mode produced by the string theory); pure \(AdS_3\) with the monster symmetry group; and in BTZ-black-hole-based \(AdS/CFT\) calculations will be reserved for other blog entries, much like the connections of the ideas above with the representation of microstates as Mathur's fuzzballs.

String/M-theory gave us numerous pictures of the microscopic structure of the black holes. Those usually make it hard to see the locality in the bulk (and even hard to see into the black hole interior) and difficult to assign the degrees of freedom to the locations in the bulk. While unitarity etc. is manifest in these string/M-theoretical pictures, various geometric properties are less clear. Realizations such as the text above are meant to clarify all the remaining secrets of the black holes that are "universal" and independent of the microscopic description of the black holes.

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Nathaniel Craig's State of the SUSY Union address

I have known Nathaniel Craig since he was a brilliant Harvard undergraduate who was attending graduate courses – at least my string theory course (I believe he was the best student in the room). This young Gentleman has written 37 papers or preprints (if you subtract some namesakes) and the last one among them is sufficiently pedagogic for you to be interested in it:

The State of Supersymmetry after Run I of the LHC

These 71 pages are based on his talk at a June 2013 workshop.




The first section is an introduction. At the end of it, on page 5, Nathaniel summarizes 5 positive reasons why the LHC has strengthened our belief that SUSY is right and relevant and 1 way in which it has weakened the belief.

In the following section, he discusses the expectations – naturalness and parsimony (essentially minimality) of the right supersymmetric models. The section is summarized by an expected ordering of the superpartners' masses and reasons for this ordering.




The third section is about our knowledge, especially various limits. In this section, you start to encounter lots of handwritten yet colorful cartoons that reproduce various graphs that you might think that only computers can draw well. ;-) Colorful, electroweak, third-generation, Higgs-related superpartners are given special attention.

The fourth section is about indirect limits, mainly ones from various rare decays.

Implications of the Higgs and its suddenly known (SUSY-compatible) mass as well as Standard-Model-like couplings are discussed in Section 5.

There have been no signals proving SUSY reported by the LHC yet. This disfavors the minimal naive models and Nature reconciles SUSY with the observations in at least one of the two ways: by breaking the signal relatively to the most visible naive models or by breaking the spectrum.

Section 6 is dedicated to breaking of the signal. It's harder to see SUSY if the spectrum is compressed or SUSY is stealth or SUSY is R-parity-violating. Compressed spectrum means that the LSP isn't much lighter than the colored superpartners. If that's so, not many particles may be produced when the colored superpartners decay to the LSP and something else. Moreover, the missing transverse energy tends to cancel as it's copied from the oppositely moving colored superpartners.

Stealth supersymmetry has a light LSP (usually outside the MSSM) which decays into something truly "almost invisible", like a light gravitino, and its R-even superpartner whose mass is just a bit lighter than the LSP mass. This R-even superpartner consequently decays into well-known SM particles so almost nothing new – and, more importantly, almost no missing energy – is produced in the reaction.

R-parity violation makes it harder to economically explain dark matter and may worsen problems with the proton decay. For the latter reason, RPV operators should still preserve either lepton or baryon number. SUSY becomes less visible because the (new) superpartners may completely decay up to SM particles again.



Section 7 is about breaking of the spectrum. Natural SUSY became a newly recycled term for SUSY models where only particles that are "really needed" for the lightness of Higgs' being are light – especially the third-generation quarks (primarily the stops). Light stops have been discussed on TRF many times, of course. This lightness of the third generation should ideally be connected with the heaviness of the third generation of SM fermions. Such models are OK with the LHC data because the data still allow light third-generation sleptons and squarks; and there's nothing unnatural about the heavy (and safely LHC-compatible) first two generations of sfermions. Nathaniel discusses various strategies to obtain natural SUSY models by choices in the mediation.

By supersoft SUSY, he means a different way of breaking the spectrum. The squarks of all generations are comparably light but the gluino is much heavier which is enough to suppress the production of superpartners at the LHC (which is mostly performing gluon-gluon collisions, using a microscopic perspective). This would be unnatural in the minimal models but it's OK if the gluino is a Dirac particle, something that I like so it's been repeatedly discussed on this blog.

Nathaniel discusses one more unusual way of breaking the spectrum, folded or colorless SUSY, in which the relevant superpartners don't carry any color, unlike their known SM partners. I don't understand how this could be possible and will study this tonight. (I see, they're just some non-SUSY models that also cancel quadratic divergences but in a more general way. This looks contrived to me and the only way way how string theory could endorse such things is via some non-supersymmetric orbifolds.)

Focus point SUSY – another way to break the spectrum, a way that is considered rubbish by Nima Arkani-Hamed, by the way – is also dedicated a special subsection.

The final subsection of Section 7 is about minisplit SUSY – going in the direction of split SUSY by Arkani-Hamed et al. but not that extreme. In this approach, one sacrifices naturalness but tries to respect all the other attractive conditions.

The final Section 8 is dedicated to thoughts about the future and Nathaniel's recommendations how people should approach the 2015- LHC run at 13 or 14 TeV. Acknowledgements and 93 references are the only other thing expecting you after that section.

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An apologia for ideas from Hawking's BH bet concession

In Summer 2004, Stephen Hawking conceded his and Kip Thorne's bet against John Preskill: Preskill was the only one among the three who said in 1974 that the information about the initial state was preserved after a black hole evaporates. This blog is relatively ancient according to the blogospherical standards so there was already a story about the concession in late 2004.

Off-topic: Israel Gelfand would celebrate his 100th birthday today.

Kip Thorne hasn't conceded yet; I think that his position has become indefensible over the years.

In July 2005, Hawking wrote a paper (TRF comments) in which he presented his own arguments why the qualitative outcome was different than he used to think. Even though Hawking would admit that the developments in string theory and especially AdS/CFT were the main advances that made him change his mind, his 2005 ideas were presented as great new insights by Hawking and some of the journalists. Your humble correspondent and most experts in the field were skeptical. I wasn't hiding my skepticism either but it seems clear that I was more sympathetic to those ideas than most others.




The recent discussions about block non-diagonalizability of the black hole evolution operator in a classically accessible basis as well as the impossibility to identify local operators in a background-independent way have strengthened my feeling that Hawking was ahead of time when he pointed out an important feature of the evolution:

For a proper understanding of the black hole information puzzle, it's important to properly (quantum mechanically) treat superpositions of classically distinct black hole microstates; and, which is related, to include the interference between the histories with different intermediate states (black hole is not there; black hole in one location/shape/decoration is present).

This important observation wasn't emphasized just by your humble correspondent. I would say that Papadodimas and Raju; Nomura, Varela (and sometimes Weinberg); and Hsu consider the revelation above to be an important part of their knowledge that makes it clear that the arguments that black hole firewalls have to exist are flawed.




A key paragraph from Hawking's 2004 concession speech said:

Information is lost in topologically non-trivial metrics like black holes. This corresponds to dissipation in which one loses sight of the exact state. On the other hand, information about the exact state is preserved in topologically trivial metrics. The confusion and paradox arose because people thought classically in terms of a single topology for spacetime. It was either \(\RR^4\) or a black hole. But the Feynman sum over histories allows it to be both at once. One can not tell which topology contributed to the observation, any more than one can tell which slit the electron went through in the two slits experiment. All that observation at infinity can determine is that there is a unitary mapping from initial states to final and that information is not lost.

For Hawking, the interference between the black-hole-containing and black-hole-free intermediate states is what restores the purity of the final state. Most of us had the same feeling as the feelings recently shared by Scott Aaronson:

I should confess that I don’t understand this argument (and apparently I’m not alone — even Preskill, to whom Hawking conceded, said he didn’t understand it!). But Hawking does seem to be clearly asserting that the solution to information loss involves there being a nonzero amplitude for the black hole never forming in the first place. (Though an obvious issue is that he doesn’t say how large the amplitude is: if it were nonzero but exponentially small, that wouldn’t seem to help much.)

This complaint against Hawking's "key role" of the black-hole-free intermediate state sounds very natural. After all, it seems intuitively "obvious" that the contribution is either tiny in which case it can't have the potency to convert the near-maximally mixed thermal final state into a pure one; or the black-hole-free intermediate states are dominant but then we don't have any explanation why the evolution looks like any events in a black-hole-containing spacetime at all.



Off-topic: A dancing 3D model of a Calabi-Yau manifold. A girl with a 3D printer may print them for you. Via tweeting Maria Spiropulu.

I have repeatedly written the same objection against Hawking's thoughts in the past although my formulations were never as clear as they are today. However, with some newer realizations, I believe that the complaint is at least morally wrong. The basic weapon that challenges the intuitive explanation from the previous paragraph was articulated in the following blog entry and the paper mentioned therein:

Hawking radiation: pure and thermal mixed states are a micron away

The text argues that in a "truly generic" basis of the \(\exp(S)\)-dimensional Hilbert space relevant for the CV of a black hole, the pure and (maximally or near maximally) mixed density matrices may only differ by exponentially tiny matrix elements of order \(\O(\exp(-S))\).

There is something that I find a bit demagogic about this December 2012 text of mine today: the mixed density matrix has (diagonal) entries of order \(\O(\exp(-S))\), too. So while it was "small" in an absolute sense, the correction needed to perturb the approximate mixed final state to a pure state has to possess matrix elements that are of order \(\O(100\%)\). There was no wrong claim in my blog entry but I was sort of hiding this fact.

But this update doesn't really invalidate the point of the "micron" essay qualitatively. The point is that the off-diagonal entries that are comparable to the diagonal ones may still be invisible to the semiclassical calculations – in fact, they may be invisible at all finite orders of perturbation theory.

Path integral for mechanics

Let me begin with a physical system that has been understood for quite some time: non-relativistic quantum mechanics. In Feynman's path integral approach, the evolution amplitudes are computed as the functional integral\[

{\mathcal A}_{f\leftarrow i} = \int {\mathcal D}x(t)\,\exp(iS/\hbar)

\] over all trajectories that begin and end at the right places. Note that all trajectories, however weird ones, contribute equally (as far as the absolute value goes). If you made an error and treated the observable \(x(t)\) classically, you would expect that the integrand is only nonzero for the correct classical trajectory but it vanishes everywhere else.

Quantum mechanics says something different. The classical trajectory is "highlighted" in the classical limit because all the trajectories that sufficiently differ from the classical solution tend to have a "random", quickly variable phase as the integrand. These random phases tend to cancel and only the phases \(\exp(iS/\hbar)\) near the extremum of \(S\), i.e. near the classical solution, contribute "coherently" because the phase (the exponent) isn't changing much near the extremum (or extrema).

Back to black hole density matrices

Consider a black hole formed by a collapsed of a star in a pure state \(\ket\psi\). The black hole gets formed and then it evaporates. Hawking's approximate 1974 calculation of the final state reveals that the final state is a thermal one (with increasing Hawking temperature as the black hole shrinks), one given by a near maximally mixed density matrix. This result is likely to hold to all orders in perturbation theory.

We know from the AdS/CFT, Matrix theory, and other explicit constructions that the final state is actually pure. So if you describe it by a density matrix, it must be a density matrix of the form\[

\rho_{\rm final} = \ket{\psi}_{\rm final} \bra{\psi}_{\rm final}

\] which must still be rather close to the approximate density matrix \(\rho_{\rm approx}\) that we claimed to be near maximally mixed. They look "qualitatively different" but this type of "qualitative difference" is one that may actually result from tiny or (in practice) hardly observable "quantitative differences".

Let's sensibly assume that in a "classically natural" basis for the Hawking radiation, the final pure state is "generic". It means that in the relevant \(\exp(S)\)-dimensional Hilbert space, all the amplitudes are of the same order i.e. of order \(\O(\exp(-S/2))\) – which is needed for the normalization condition \[

\sum_{i=1}^{\exp(S)} |c_i|^2 = 1

\] to hold. The relative phases between \(c_i,c_j\) are important although the laymen are often led to believe (by sloppy presentations of the Schrödinger cat thought experiment and other things) that only the absolute values matter. The pure density matrix has matrix elements\[

\rho^{\rm final}_{ij} = c_i c^*_j

\] What would you think about the value of the density matrix if you committed a similar error we discussed in the "non-relativistic quantum mechanics" section above? Well, you would do exactly what the laymen usually do when they think that only the absolute values of the amplitudes in a basis matter: you would just keep the diagonal entries but incorrectly set the off-diagonal entries to zero:\[

\rho^{\rm final}_{ij} = c_i c^*_j \cdot \delta_{ij}

\] Apologies, the Kronecker delta must be interpreted "literally" and the usual checks for indices (repeated indices only occur if they're summed via the Einstein sum rule) don't hold here.

Now, my point is that it is perfectly compatible with everything we know – and, ultimately, inevitable – that the semiclassical approximate calculation ends up with a similarly castrated final density matrix as the density matrix with an extra Kronecker delta factor. Why?

Think about two mutually orthogonal microstates of the black hole (or the black hole radiation that results from them), \(\ket i\) and \(\ket j\), which are very similar to one another in some operational classical way of looking at things. For example, they are two black hole microstates that describe the black hole located at positions that differ by a sub-Planckian distance (which still allows the states \(\ket i\) and \(\ket j\) to be orthogonal if the black hole mass is much greater than the Planck mass, and it should be for the black hole interpretation to be OK); or \(\ket j\) is obtained by a creation of a soft photon or another quantum on top of the structure given by \(\ket i\).

What I want to emphasize is that the difference between \(\ket i\) and \(\ket j\) will be inevitably invisible in a semiclassical approximation to any calculation of the evolution of the black hole. If you think about the spinning Earth, you have no chance to distinguish the states of the Earth with the \(z\)-component of the spin equal to \(J_z\) and \(J_z+\hbar\) because \(J_z\gg \hbar\). So all such things are invisible in a calculation that treats \(J_z\) "classically".

In some cases, you may argue that the quantum evolution operator must be diagonal in such "classically indistinguishable microstates", anyway. This diagonal form may follow from the conservation laws (of the angular momentum, for example). The point is that this is not true for the differences between black hole microstates \(\ket i\) and \(\ket j\) described two paragraphs above.

For example, when a black hole is emitting the Hawking quanta, there is no reason for its center-of-mass location to be exactly conserved. In fact, we know for sure that it is not conserved. The black hole is recoiled once it shoots a Hawking particle in a specific direction. Referring to the black hole's large mass, such recoils have been largely neglected in all the (semiclassical – and sometimes "more ambitious") calculations of the Hawking radiation. But the black hole is actually moving because of these recoils and the motion resembles the Brownian motion at (very long) timescales comparable to the Hawking evaporation lifetime. It can get very far.

While the changes of some internal properties of the black hole such as the precise sub-Planckian location of the center-of-mass or the \(\O(1)\) changes to the number of soft quanta around it (which may be large) may be neglected for some purposes, they surely cannot be neglected if you want to calculate the final matrix element \(\rho_{ij}\) where \(\ket i\) and \(\ket j\) are two classically "nearby" microstates.

The actual behavior of \(\rho_{ij}\) for a pure initial state should be clear to you: in a "classically natural" basis for the radiation, all matrix elements \(\rho_{ij}\) are of the same order, whether they are diagonal ones or off-diagonal ones. This is clearly implied by the purity and genericity of the microstate. All approximate calculations tend to assume that the final density matrix – and/or the evolution operator – is diagonal or off-diagonal in some basis of microstates that look natural or easily accessible for classical measurements in the final spacetime (e.g. Fock space occupation number eigenstates of the radiation). But this assumption is completely wrong and the full, exact calculation shows that the off-diagonal elements are actually of the same order. One may only rightfully conclude that the off-diagonal elements (of the final density matrix or the evolution operator) are "almost zero" if we average them over many classically similar yet mutually orthogonal microstates but if we really treat them accurately, the off-diagonal elements in a basis of our choice are never negligible relatively to the diagonal ones. In fact, I would stress that the off-diagonal elements between "pretty much any two" classically natural states are comparable to the geometric average of the two diagonal states.

I believe that the mistake described in the previous sentences and many paragraphs above them is one of the most widespread and crucial mistakes made by Joe Polchinski and many others who end up with incorrect and seemingly paradoxical conclusions such as the existence of a "black hole firewall". They just treat the black hole's own properties – including the metric tensor around it – classically and they believe that the unitarity should hold in each "superselection" sector (with some classical properties; effectively, in each "exact" background spacetime) separately.

But this can't be the case. To guarantee unitarity, it is essential for quantum gravity to have interference – and nonzero off-diagonal matrix elements – between microstates of a black hole that look "similar in the classical approximation" but whose details differ (location of the black hole center mass measured with a sub-Planckian accuracy and/or infinitely many occupation numbers changing by much smaller additions than their rough classical value, to mention two major examples). Only with this full connectedness of the black hole microstates – nonzero off-diagonal entries through which you can connect (assuming many \(ij,jk,kl,lm\) jumps) a black hole state with any other state (including a black-hole-free state) – quantum gravity is capable of preserving all the principles simultaneously (unitarity, equivalence principle wherever it should hold, and locality in the appropriate approximation).



Hawking repeated and and rebranded (as an anti-firewall argument) his 2004-2005 thoughts on the recent Fuzz-Or-Fire workshop in Santa Barbara. Ironically enough (if you think about the bet), Hawking of 2013 is more in favor "unitarity is true and consistent with other principles" i.e. "anti-firewall" (like LM) than even Preskill. Use the hyperlinks if you don't have a VLC player plugin.

What does it have to do with Hawking's 2004-2005 bet concession? Well, he was talking about the need to consider the interference between intermediate states with a black hole and those without a black hole. While the intermediate states with a "totally eradicated" black hole are probably not enough to completely purify his mixed approximate answer, a generalization of his 2004-2005 thesis is true and very important: If we want to understand how the unitarity in the Hawking radiation is compatible with other cherished principles, it is totally essential to acknowledge the interference between a black hole intermediate states and an exponentially large number of intermediate states that aren't quite the same, even when it comes to a classical description of their appearance.

When studied with the full precision, semiclassical gravity just isn't consistent and treating black holes with slightly different classical properties as "superselection sectors" that can't interfere with each other does mean to make the assumption that the internal black hole observables do behave classically, at least in some respects, which they don't.

And that's the memo.

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One can't background-independently localize field operators in QG

...because the "basis" of coherent states is overcomplete...

Let me begin with something simple. John Preskill asked you "What's inside a black hole?" and offered you four options:

1. An unlimited amount of stuff.
2. Nothing at all.
3. A huge but finite amount of stuff, which is also outside the black hole.
4. None of the above.

Well, the option (D) may have been at the beginning and an obvious suboption of (D), "The black hole interior is a region just like any other region and independent from others", should have been offered as a special choice (E). A surprising result is that (E) is almost certainly wrong. Instead, (C) is right – at least if we omit the very highly curved region near the singularity that could justify (A) in a complicated way and if we allow the definition of a black hole to cover its rare microstates – if we only allowed the most generic black hole microstates, the answer would be (B): the interior has to be empty.

Well, (B) may also be interpreted as a claim allowing a firewall, in which case it's wrong in general (the firewall isn't necessary or generic) but of course that there are rare black hole microstates that contain something that burns you near the horizon much like there are rare black hole microstates with a bunny in the interior.

This point is simple but often misunderstood. A black hole is defined by its event horizon but it doesn't follow that the interior has to be empty. There can be a bunny in it. However, among microstates of localized matter, a black hole with a bunny is an exponentially rarer class of microstates. Most of the mass \(M\pm \delta M/2\) black hole microstates look empty – that's why the entropy-increasing evolution converges towards these states as the black hole keeps on devouring the surrounding matter to clean its interior (and vicinity). But don't make a mistake about it: a bunny in a black hole (or a nonzero occupation number of freely falling field operator modes) is unlikely yet possible.

But let me switch to a more complicated question.




During Suvrat Raju's talk at a recent Fuzz-Or-Fire workshop in Santa Barbara, a core of the pro-firewall/anti-firewall conflict became rather visible. On one hand, the Papadodimas-Raju "state-dependence" of the definition of the black hole interior field operators seems unacceptable to the firewall champions although it looks pretty much inevitable to many of us.

On the other hand, this disagreement may be described as a criticism of Polchinski's and pals' alternative: They believe that the bulk field operators and especially their location in a quantum theory with a dynamical geometry may be defined by a recipe described operationally in a "background-independent way". For example, start at the AdS boundary that must be close to the empty AdS geometry, pick a direction as if it were an empty AdS, and go in this direction for a certain proper time or proper length. Then you turn in the direction of the greatest curvature (defined in some other way) and walk for 5 meters of proper distance or 2 microseconds of proper time, and so on. You get to a point and there is a scalar field at that point that you may call \(\phi(x,y,z,t)\) and ask about its eigenvalues or what it does when it acts on a state \(\ket\chi\) etc.




The classical counterpart of such a prescription sounds totally OK in classical general relativity. You may imagine that one particular spacetime geometry is the "right one" and whatever the spacetime geometry is, the operational procedure involving the proper times, proper distances, and angles may be followed and the right value of the field at the point we just found becomes a well-defined \(c\)-number (let's talk about scalar functions of local tensor fields and their derivatives only).

Joe Polchinski and others believe that the same background-independent operational definition of field operators at various points may be used in quantum gravity, too. This belief is incorrect. There are several ways to see why. They seem very different but ultimately they are rooted in the same general properties or at least "spirit" of quantum gravity that is imposed on us by consistency.

To maintain their belief that the background-independent localization of field operators (and therefore state-independence) is possible, the firewall advocates must assume that

• the metric tensor is a good and precisely and uniquely defined degree of freedom (quantum observable) at arbitrarily short distances
• every ket vector in a quantum gravity theory may be uniquely rewritten as a sum of ket vectors each of which comes with a well-defined classical geometry

Both of these assumptions are incorrect, however. In some sense, the second problem is more damning for the firewall advocates' plans than the first one.

The metric tensor isn't any good at (sub)Planckian distances

The first point has to be true because they want to determine proper distances. You need a metric tensor for that. Because the definition must work even in rather general, potentially extreme environments near collapsing and other black holes where we often need an exponential precision to locate the events (note the coordinate singularity at the horizon etc.) while we have to resist high matter densities etc., the definition of the metric tensor has to be really exact for Joe's and pals' background-independent operational definitions of the points in a general spacetime to make any sense.

However, quantum gravity doesn't allow you things like that. The metric tensor is only good and well-defined in an effective description of quantum gravity. At shorter distances, it just ceases to be a good observable. Well-defined observables in quantum gravity are different; the gauge fields in the \(\NNN=4\) Yang-Mills theory involved in the most famous example of the AdS/CFT correspondence are an example. The matrices \(X,P,\Theta\) in Matrix Theory are another example.

Even if you had something like a "closed string field theory" that would apparently contain the metric tensor "everywhere", you would have to solve the problems of the mixing of its field modes with some other modes of fields arising from heavy excited string states (with the same charges and spin). To make the procedure well-defined, you would have to overcome the problem that there are many ways (related by field definitions involving all the massive string fields) how to define the metric tensor. They may be thought of as different "renormalization schemes". You may imagine that a different "renormalization scheme" amounts to switching the metric from something like the string frame to something like the Einstein frame but the rescaling depends on the massive scalar fields \(h\) in string/M-theory rather than the dilaton \(\phi_D\). Classically, \(h\) is constant so this rescaling doesn't change much. However, quantum mechanically, \(h\) is a dynamical, fluctuating field so an \(h\)-dependent redefinition of the metric tensor does matter.

But even if your procedure directing someone to walk over some proper distances in a general spacetime etc. did specify a particular "renormalization scheme", it would still be no good because at very short, near-Planckian distances, the geometry becomes brutally fluctuating and the proper distances and times, when accurately measured over the violent landscape of the quantum foam, are probably divergent and/or ill-defined. So Joe's prescription would break down.

My point is that whatever "renormalization scheme" you pick, \(g_{\mu\nu}(x,y,z,t)\) is a fluctuating degree of freedom that has nonzero probability amplitudes to be nonzero and substantial even in the vacuum state of the spacetime. By dimensional analysis, the magnitude of the contribution \(\delta L\) of these fluctuations to a proper distance \(L\) comparable to the Planck length is comparable to the Planck length i.e. 100 percent; I believe that this dimensional analysis, assuming \(g_s=O(1)\), is OK even in string theory despite its ability to "calm down" the quantum foam. You simply shouldn't assume that the flat and peaceful spacetime offers you good expectations about the behavior of proper distances, times, and angles near/below the Planck scale. Try to follow Joe's algorithms on the quantum foam (the picture at the bottom):



It's pretty obvious that you get caught in the weird tunnels and valleys of this quantum foam whatever recipe you choose. What you actually need is a geometric prescription that is allowed to use the smooth, nearly flat spacetime similar to the upper part of the figure. But using the proper distances and proper times calculated from the dynamic metric tensor just don't give you anything like a flat space even in the vacuum-like ket vectors. The quantum foam picture at the bottom of the picture above is an eigenstate of \(g_{\mu\nu}(x,y,z,0)\) and even the Minkowski-like vacuum state in quantum gravity is a superposition of states whose geometry looks like this. You won't really get anywhere with the background-independent protocols to isolate a location in the spacetime.

Non-uniqueness of a "geometry" associated with a ket vector in QG

But it's the second complaint against Joe's paradigm, if you allow me to call it in this way, that seems more damning and conceptual. You could imagine that for some unknown reasons, string theory calms down the quantum foam so nicely that the sub-Planckian terrain may still be imagined as a smooth space rather than the quantum foam and the procedure could get through with a potentially natural choice of the "renormalization scheme".

However, the procedure will still fail due to some facts that don't depend on the short-distance, Planckian physics. What are these general problems with the background-independent approach to the location of points in a dynamically curved quantum spacetime?

For the sake of simplicity, let's assume that the procedure "go here for 5 meters, turn left etc." is only used to move through a slice of the spacetime at a fixed value of the coordinate \(t\), whatever it is. If we considered trajectories deviating from the slice, we would open yet another can of worms because the metric tensor doesn't commute with its time derivatives (the uncertainty principle!) so it's just downright impossible to imagine that these behave classically in any ket vector (this assumption is as wrong as the assumption that arbitrarily sharp trajectories in the quantum phase space make sense).

Fine. Polchinski's procedure is meant to tell you what is the action of an operator \(\phi(P)\) on a general quantum gravity ket vector \(\ket\psi\). The point \(P\) is specified by an operational, background-independent procedure of the type "go for 5 meters, turn left, do this and that". Now, Joe believes that the action\[

\phi(P)\ket\psi

\] is another well-defined ket vector. We can see it can't be the case. Why? Well, the vector \(\ket\psi\) isn't an eigenstate of the metric tensor operators \(g_{\mu\nu}(Q)\) at the relevant points \(Q\) that may appear along the trajectory. To avoid the immediate ill-definedness of a recipe based on proper distances, we must decompose \(\ket\psi\) into eigenstates of the metric tensor variables \(g_{\mu\nu}(Q)\):\[

\ket\psi = \sum_j \ket{\gamma_j}

\] Well, the sum could actually be an integral and the normal people would tend to normalize \(\ket{\gamma_j}\) to unity and write the normalization factor as a special coefficient, and so on, but the equation above is good enough. In the previous section, I discussed the problems resulting from the violent character of the geometry in the \(g_{\mu\nu}\)-eigenstate. But even if you forget about these short-distance troubles and ambiguities and you assume that the proper distances through the apparent quantum foam behave just like your long-distance intuition suggests (up to a universal renormalization coefficient for the distances), you face insurmountable problems, even at long distances. They're related to the short-distance problems discussed previously but the arguments below hopefully make their independence on the UV physics more obvious.

Imagine that we want to apply the procedure to the most peaceful yet nontrivial state we can imagine, a smooth macroscopic gravitational wave in an otherwise empty spacetime. This state containing a gravitational wave may be written as a coherent state\[

\ket\psi = \exp\left[\int d^d k\,\alpha(k) c^\dagger(k)\right] \ket 0.

\] It's the exponential of a superposition of creation operators for some graviton states. As a homework exercise ;-), add sums over the polarizations and other indices and everything else you like or need. Now, additional particles may be created on top of the state \(\ket\psi\) and I think that Polchinski would say that the right way to apply his procedure is for the distances in the states that contain a few particles on top of the curved spacetime \(\ket\psi\) to use the geometry of this curved spacetime when we try to follow the procedure to "find the location in a general spacetime".

You should already feel uncomfortable at this point because the state \(\ket\psi\) is an excitation of the Minkowski vacuum state, too. Rewrite the exponential as a Taylor expansion if you want to make the point more suggestive. Gravitons are particles, too. You might say that it couldn't be a hopeless idea to use the flat spacetime's metric when you try to locate points in the spacetime except that it would also be obvious why the relationship between the local operators on top of the excited coherent, curved space \(\ket\psi\) with the local operators on top of the Minkowski space \(\ket 0\) is extremely convoluted.

So let me assume that Polchinski et al. really want to use the curved geometry from the coherent state \(\ket\psi\) when they follow their background-independent procedure. It means that to find the action of a local operator \(\phi(P)\) on \(\ket\chi\), they need to decompose \(\ket\chi\) into "matter-like" (and therefore geometry unchanging) excitations of coherent states of the type \(\ket\psi\) above for which the metric tensor is known.

The trouble with this background-independent physics is that the "basis" of the harmonic oscillator Hilbert space consisting of the coherent states is overcomplete.

See basic introductions to coherent states if you have any doubt about the statement. So even if you restrict your calculations to ket vectors \(\ket\chi\) that only contain purely gravitational excitations, you will need "the" decomposition of such states to coherent vectors to identify \(\phi(P)\) but "the" decomposition actually isn't unique.

This is a problem that makes your background-independent procedure break down even for states \(\ket\chi\) that are as simple as a low-energy, single-graviton excitation of the Minkowski vacuum state. On one hand, you could consider this excitation to only change the background geometry infinitesimally and use the Minkowski geometry to follow the procedure. The first excited state of a harmonic oscillator is proportional to a superposition of coherent operators weighted by \(\delta'(a)\) all of which are infinitesimally close to the origin of the phase space (interpreted as a flat space in the Fock space of gravitons). On the other hand, you may rewrite this first excitation of the harmonic oscillator as some linear superposition of coherent states centered elsewhere, even very far from the center at zero (effectively a linear superposition of highly curved spacetimes). It's clear that the point \(P\) where you get by following these spacetimes will depend on the way how you decompose your states to the coherent states. This way isn't unique and the infinitely many choices differ by differences that are unbounded from above.

If the procedure doesn't work for single-graviton states, you may be sure that the problems become exponentially worse if you try to apply the procedure to a black hole spacetime with a significant density of mass, coordinate singularities, and many other things. It's completely hopeless.

Incidentally, if you tried to replace the decomposition into coherent states by a decomposition into \(g_{\mu\nu}\)-eigenstates – in the harmonic oscillator analogy, \(x\)-eigenstates – discussed at the beginning, you could cure the overcompleteness problem of the basis but you would also totally delocalize the vectors in the values of \(\partial_t g_{\mu\nu}\) which means that the time-like geodesics of the recipe would probably become infinitely singular (the coherent states naturally balance the needs of the metric in the spatial and temporal directions); you wouldn't be guaranteed that the proper distances are well-behaved and finite at short distances. At the end, any attempt to define the recipe will fail because what all of them actually contradict is the equivalence principle: they are assuming that the spacetime geometry is classical enough so that the proper length/time of some generic trajectories going in many directions may be accurately measured which isn't so.

An alternative for the background-independent operational localization protocols

Once I have shown that the background-independent way of identifying locations of operators isn't possible, it may seem polite for me to tell you what's a legitimate replacement of it. We could be saying that no calculations based on strictly local operators attached to "points" are possible in quantum gravity. Except that I think that they are possible. However, you have to assume (manually and, whenever possible, cleverly choose) a background – a particular "curved space" vacuum-like state of the quantum gravitational theory which may also be obtained as a coherent state built from other vacuum-like states – and construct many other microstates out of this vacuum-like state by the action of a "finite" (not scaling with various parameters called \(N\) that would be increasing functions of the curvature radius etc.) number of field operators where these field operators are behaving much like they are behaving in the flat space, at least locally in regions where the curvature may be neglected. Papadodimas and Raju explain these conditions more quantitatively. In some sense, I believe that the ER-EPR correspondence with its ER bridges is a special visualizable "Ansatz" for solutions of such constraints.

Here I must say that people like Lee Smolin have been saying totally idiotic things about "background independence" for years. They would even criticize string theory for being able to write the Hilbert space of quantum gravity as a de facto Fock space built upon a particular background. Remember all the silly demagogy that no backgrounds can ever be talked about because GR imposes a democracy between all of them, and all this rubbish.

Feel free to impose a ban on talking about backgrounds but then you will be unable to make any calculations that may be compared with the experiments, too. The adjective "background-independent" may be given many meanings and some of them are respectable, at least in some contexts, but be sure that if your interpretation is that "we can't use any backgrounds in calculations at all", then you are throwing the baby out with the bath water.

Because I properly learned many of the computational techniques that existentially depend on the choice of a background (in the spacetime or the world sheet) from Joe Polchinski, I wouldn't have believed 14 years ago that he would ever be saying things "remotely similar" to the Smolinian rubbish on the background independence.

If we want to organize a Hilbert space (or, more typically, its subspace) as some collection of states with a spatial interpretation (states that tell us what is being observed here or there), then we simply need to associate the microstates with a background. We also need to gauge-fix the diffeomorphism gauge symmetry or redundancy, if you wish. Only when it's done, it's possible to define how local field operators act in between the states in this subspace of the Hilbert space. It's clear that if you create too many things in your background, or if you deform the geometry by too many gravitons, to be more specific, the added gravitons or the backreaction to the added matter make the original background's geometry an unnatural (or perhaps more accurately, practically not too useful) way to measure distances and times. You should better pick a different background to parameterize the relevant portion of the Hilbert space if you consider states whose geometry is too different from the original background. But you must choose a background because trying to leave the "job to measure the geometry" on the microstates without a choice of background requires a decomposition of the gravitons' Fock space states to coherent states which isn't unique.

ER-EPR's definitions of operators are clearly background-dependent, too

The state-dependence – well, really background-dependence – of the definitions of the black hole interior (and perhaps all other) local field operators is something most tightly associated with the insights by Papadodimas and Raju. But I believe that the Maldacena-Susskind ER-EPR correspondence makes this inevitable background dependence equally if not more self-evident.

Why?

It's simple. They say that the Hilbert space of one Einstein-Rosen bridge (a pair of black holes geometrically connected by a non-traversable wormhole) is the same Hilbert space as the Hilbert space of two faraway black holes (that are allowed to be entangled). Clearly, these two pictures of the same Hilbert space envision completely different background spacetimes – the spacetimes have different topologies, in fact. So the definitions of field operators in the black hole interior(s) are clearly different in these two pictures. In other words, the definition of local field operators depends on whether you describe the same Hilbert space as two black holes that can get entangled later (but you're "expanding" around the microstates for which the entanglement is low and the black holes are assumed to be independent to start with) or the Hilbert space of a single Einstein-Rosen bridge with just "one interior" (you're expanding around a particular microstate for which the entanglement entropy is maximized; note that there can be many such maximally entangled microstates for which the bridge is correspondingly "twisted"). In other words, the definition of the local field operators is background-dependent, i.e. dependent on the choice of the spacetime background you have to make manually and subjectively before you start your calculations. It's clear because the local operators depend even the topology which is totally different in the two choices. The two black holes have two interiors while the Einstein-Rosen bridge only has one component of the interior. For various situations or classes of microstates, one of the two descriptions is more convenient or practical than the other description, but there can't be a universal law that would make one description more correct than the other one a priori. You must predecide how many components the interior(s) has (have) before you start to talk about the field operators in the interior(s).

Finally, I must say that I believe that most of what I wrote above aren't my exclusive original insights but just a reinterpretation of some insights made by Papadodimas and Raju which uses different words. If this is not a legit way to describe what they concluded, they will tell me and I will inform you, too.

I like to think about the ER-EPR correspondence but again, I believe it is just a more specific, visualizable "Ansatz" how to write the field operators at different places and the general, non-visualized principles for the operators were already found by Raju and Papadodimas (and perhaps others whom I may have slightly overlooked). The Raju-Papadodimas conditions for the mutual relations between the field operators start to break down once you arrive to short enough distances where the Einstein-Rosen bridges with the Hawking radiation become visible.

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Imagine that the Universe is not expanding

Wetterich's cosmon claimed to be an alternative to the Big Bang singularity, inflation, and the recent apparent expansion



Image: NASA/JPL–Caltech...

Most papers trying to replace the usual cosmological concepts such as dark matter and dark energy by something entirely different may be shown to be wrong within minutes. As I learned from a Czech server called osel.cz ("osel" is a horse-like animal known as an ass: I don't know of a shorter way to explain that it's not the other ass), a rather achieved cosmologist Christof Wetterich posted an unusual clever yet apparently equally provoking preprint to the astro-ph arXiv at the beginning of this month:

Variable gravity Universe

Be ready for a wild ride: the proposed model claims to explain all the known observations, eliminate the Big Bang singularity, account for the patterns we attribute to inflation, the radiation-dominated era, and the matter-dominated era. And Wetterich also wants to boast that his construction "produces" the arrow of time – as if cosmology were needed for that (but that didn't make me stop reading). A single scalar field – the cosmon – may do all these wonderful things, the gospel say.

It's weird if not exciting, isn't it? ;-)




The idea of a time-dependent Newton's constant (variable strength of gravity) goes back to Jordan and Dirac. The latter man tried to use it to explain the existence of vast and tiny parameters in the Universe. The explanation doesn't really work, especially because dimensionless constants of physics are measured to be really constant.




This is a field – a minefield, to be more precise – that is full of failed and dead bodies. You don't want to go through all these failures because there are too many. This Wetterich guy wants to avoid the basic traps by assuming that the Planck mass is changing with time but the masses of all objects are changing at the same rate so the ratios remain fixed.

Such a claim is already a bit provoking to me because one always has the freedom to define the masses in the Planck units so with the prescription described in the previous paragraph, we may say that nothing is changing in the Planck units. Well, on page 4, he's a bit more specific about the role of his cosmon field \(\chi\) ("chi"). The effective action is\[

\Gamma = \int d^4 x \sqrt{g} \left\{
-\!\frac 12\! F(\chi) R + \frac 12 \! K(\chi) \partial^\mu\chi\partial_\mu\chi +V(\chi)
\right\}

\] To be authentic, I retyped the expression as he wrote it although the sign of the determinant of \(g\) seems problematic and so do other things. He considers two basic models, (A) and (B), which make the following fixed choices:\[

\eq{
(A):& F(\chi) = \chi^2, \,\,V(\chi)=\mu^2\chi^2\\
(B):& F(\chi) = \chi^2+m^2, \,\,V(\chi) = \bar\lambda_c.
}

\] The coefficients in the kinetic terms \(K(\chi)\) are allowed to vary throughout the paper to adjust the models.

Well, you see that Newton's constant depends on the cosmon in some way. The cosmon has some field-dependent kinetic term and some potential. The first thing that comes to my mind is that one could rescale the metric and nonlinearly redefine the cosmon field so that he would effectively eliminate up to two of the three functions above. So unless there are some global constraints or inequalities, wouldn't it become just an ordinary GR coupled to an ordinary scalar field with some potential?

I am confused by this basic point but it's probably because I have only been reading the paper for a few minutes so far. If and when I spend an hour with it – or if a more experienced reader offers his or her thoughts and observations – chances are that all the uncertainty will go away and the ambitious claims by Wetterich will turn out to be either strictly viable or demonstrably wrong.

Which way it is? ;-) I am unlikely to learn the answer tonight because I want to watch the second soccer match Maribor [SI] vs FC Viktoria Pilsen [CZ]. "Our" Pilsner team is likely to win in the aggregate match after the 3-to-1 victory at home last week which would mean that it will earn over $10 million and penetrate to the standard group of the UEFA Champion League for the second time.

Off-topic: graphene \(\heartsuit\) metals and makes them 100+ times stronger. Via Bahamas

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Light dark matter in NMSSM and non-diagonalization of BH evolution matrices

I want to mention two new papers today.

First, Jonathan Kozaczuk and Stefano Profumo of Santa Cruz discuss the possibility to embed the very light, sub-\(10\GeV\) dark matter particle (indicated by some of the direct search experiments) to the Next to Minimal Supersymmetric Standard Model (NMSSM: it's the MSSM in which the Higgs bilinear coefficient \(\mu\) is promoted a chiral superfield \(S\) which is, according to many criteria and physicists, more natural than the MSSM itself):

Light NMSSM Neutralino Dark Matter in the Wake of CDMS II and a \(126\GeV\) Higgs

They find out that there are regions in the parameter space of NMSSM that are able to produce this very light higgsino-singlino-mixed LSP dark matter candidate with a huge, spin-dependent cross section coupling it to the nucleons. The Higgs mass may be achieved sort of naturally, other "negative" constraints may also be satisfied, and the scenario produces some automatic predictions, e.g. a large invisible branching fraction of the Higgs decays.




Yesterday, I watched some black-hole-information talks in Santa Barbara, especially the whole talk by Suvrat Raju about his paper with Papadodimas. The talk was rather impressive, Suvrat (my former TA, I am tempted to proudly say, although his brilliance has nothing to do with your humble correspondent) was responding to a stream of questions and they addressed pretty much all the criticisms.




There is a new related paper by Steve Hsu today in which he reiterates and perhaps updates his February 2013 paper,

Factorization of unitarity and black hole firewalls (arXiv)

---

Promotion of the paper on Steve's blog

I've left the following comment on Steve's blog:

Nice paper, I completely agree with you: all the potential final microstates are being mixed in the most general way so it's certainly incorrect to assume that the evaporating black hole Hilbert space may be rewritten as a direct sum of many "superselection sectors" that evolve unitarily and independently so that the evolution matrix would block-diagonalize in that splitting. This is one of the wrong assumptions constantly made by AMPS and followers - one that effectively amounts to believing that the geometry is purely classical and the gross features of a system like BH are perfectly predictable which they're not. Just to be sure, the evolution matrix may not only be block-diagonalized but diagonalized but its eigenstates are not states with a simple classical representation, e.g. a sharp center-of-mass location, they're not eigenstates of some natural local operators etc. They have no reason to be.

Also, just to sure, you're not the first one who pointed out pretty much the same thing but it's nice that you cite Nomura, Varela, Weinberg, and friends for whom this was a key point to point out. In fact, I think that the main "controversial" point of the Raju-Papadodimas paper (at least according to the structure of questions during Raju's Monday talk in Santa Barbara) - that the definition of the interior BH field operators in terms of the CFT operators is microstate-dependent - is pretty much equivalent to this your or Nomura gang's claim that the superselection sectors aren't separated (the evolution matrix isn't block-diagonalized in those subsets). The relevant subset of the Hilbert space to which one may "plausibly" evolve by a simple action of a few operators etc. depends on the ket vector we start with and the spacetime has no good reason to be a good "description of the background" if one deviates too much (by too many creation operators etc.), because of the back-reaction.

So in a "neighborhood" (in the sense of measuring the number of simple actions of natural operators) of a microstate, the local operators are sort of well-defined, but they become increasingly inadequate for more general, "faraway" microstates. This Papadodimas-Raju statement implicitly says that the definition of the local operators must gradually change as you change the microstates but there are no sharp borders between the neighborhoods, so no block-diagonal decomposition is possible. Instead, it's essential for the unitarity that all the microstates from the superselection sectors may be transformed to each other. The exponentially small matrix entries (including those between what AMPS and others would consider different superselection sectors) can't be neglected because they're essential even for the difference between pure density matrices and the maximally mixed one, see Papadodimas-Raju or

Hawking radiation: pure and mixed states are microns away

The microstate-dependence or the non-decoupling to the classical superselection sectors seems like a totally obvious point when looked at from a proper direction: it just means that the "Hilbert space of plausible pure states" and their organization that one needs to consider is allowed to depend on the rough or gross or "classical" evolution of the system: the Hilbert space of finely grained microstates is "fibered" over the space of coarse degrees of freedom and the character of the fiber may change. This is obvious - that's why we consider things like "the mass M black hole Hilbert space" at all (the spaces for different M are different; after all, they have different dimensions, even though both belong to a grander space of string theory) and why we can separate these states from others in the Hilbert space.

But the point is that the organization of the subspaces of the Hilbert space by local operators in the black hole interior is dependent not just on the presence of the black hole and its mass but "most" of its microstate details if a black hole is present. To me, this sounds sort of inevitable because one needs a "more than infinite" Schwarzschild time to penetrate inside the event horizon which means that in a certain Schwarzschild-time-slicing-based basis, a huge amount of scrambling that can mix really everything that waits to be mixed is performed on the local degrees of freedom right when an observer is crossing the horizon. There's absolutely no reason why the evolution operator should be block-diagonalized in the "superselection sectors" that look like classical patches. Superpositions of all of them may occur and therefore will occur.

I think that many of you are saying the same thing - or at least a big portion of what I consider the right answer to most of the questions here - but you don't fully appreciate that you use different words for the same insights.

Incidentally, Scott Aaronson who attended the KITP Santa Barbara fuzz-or-fire workshop posted a blog entries with some links and quotes about this topic.

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