While I was buying and installing my new fridge, I kept on burning my brain by analyzing various refinements and implications of the paper by Maldacena and Susskind.
Some business with the M2-brane topology change and entanglement looked too obvious to me so I returned to the question how do the black hole exterior and interior really interact. The burning of the brain is composed of various steps that combine and recombine analytic continuation, diverse choices of coordinates, unusual ways to redefine the connectedness of the spacetime, and connections between previously disconnected regions through the complexified spacetime.
At each step, I try to ask not only HOW does it work but also WHY a particular trick that looks clever at a given moment should be picked and WHETHER it is inevitable or unique. Some of the partial conclusions are more convincing than others but I don't want to waste your time with an incomplete picture.
Instead, let me mention that we're probably victims of some bad habits and ultimately invalid lore related to the way how we think about certain issues in general relativity.
What do I mean?
General relativity is composed of the equations that are locally valid, Einstein's equations, and one may derive them from the equivalence principle and/or the diffemorphism symmetry that is needed for a consistent, ghost-free theory of spin-2 fields. We know that Einstein's original equations include the leading terms in a derivative expansion but there may be higher-order operators, too. All this stuff seems obvious and undeformable.
But we're also making some conceptual and "global" assumptions which may be wrong – and this wrongness may be especially harmful when one considers the black hole information puzzle and related topics.
General relativity is capable of producing spacetimes of nontrivial topology and the topology is a classical property of a spacetime. Therefore, it's been implicitly assumed that the discrete data defining the spacetime or space topology are observables that are represented by linear operators in the quantum theory.
Maldacena and Susskind have pretty much completely convinced me that this can't be the case, however. A spin-up electron here and a spin-down electron there, \(\ket{\uparrow\downarrow}\), that propagate on a flat spacetime seem to be eigenstates of the "topology" operator with the "trivial topology" eigenvalue. The same seems to hold for \(\ket{\downarrow\uparrow}\). If the "space topology" operator were linear, it would also obey\[
(\hat{\text{space topology}})\cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}) = (\text{trivial topology}) \cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}) \text{ NOT!}
\] However, the correct eigenvalue is different, Maldacena and Susskind argue:\[
(\hat{\text{space topology}})\cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}) = (\text{topology with an ER bridge}) \cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}).
\] (Please, open the mobile version of this page to see the whole equations; I don't want to change them or divide them to several lines.)
After all, this singlet state is the simplest quantum state that lives on the background with a single Einstein-Rosen bridge. So there's no linear operator that would count the Einstein-Rosen bridges. It's not a good observable. You can't use the number of ER bridges as a quantity that defines coarse-grained histories, e.g. in the Consistent Histories approach to quantum mechanics. In other words, the state \(\ket{\uparrow\downarrow}\) isn't orthogonal to the singlet state above: they're not mutually excluding although the former seems to be living on a bridge-free spacetime while the latter is living in a spacetime with a bridge.
Note that there's nothing wrong about the non-existence of an operator – observable – that would count the Einstein-Rosen bridges in a simple way. The bridges may be very thin and thin bridges are examples of those that are strongly affected by the "Planckian" details of a theory of quantum gravity. More generally, there isn't any clear operational method to determine the spacetime topology. A strategy to "see" a wormhole is to look for clumps of information and patterns that are repeated in two regions – but that's the same thing that defines correlations or entanglement. Entanglement of a state isn't associated with an observable, either.
You should notice that all the childish proposals to construct a theory of quantum gravity such as loop quantum gravity, causal dynamical triangulations, and so on are incompatible with the Maldacena+Susskind insight (not that it is the first lethal disease that has killed these stupidities, far from that). Why? Because they construct a spacetime (dreaming about its being large and nearly smooth) that clearly has some combinatorial properties and some topological invariants may be clearly extracted from their preferred basis of spin networks and other silly animals they use. So they have linear operators of "space topology".
It's easy to see why these approaches to quantum gravity have this property – which we now believe to be a pathological one. They have it because they belong among the naive models of quantum mechanics that start with a preferred basis and observables for which all the basis vectors are eigenvectors. But such theories are extremely unnatural as quantum theories because quantum theories don't come with any preferred bases or preferred operators. Viable quantum theories always offer us many different operators that don't commute with each other – which also implies that bases constructed from the eigenvectors of one set or another set are different bases.
We have to jettison this excess baggage. In quantum gravity, there aren't any linear operators that would honestly count the number of Einstein-Rosen bridges in the spacetime or other things. We must be careful about this incorrect assumption. It's a trap.
Bridges that invite you to count them with a linear operator are probably not the only trap. I became convinced that there is something wrong about the way how we think of black hole singularities.
The smart contemporary relativist – smart according to the "mainstream" – knows that black holes are defined by their event horizons. The event horizons is what separates the black hole interior from which nothing can escape – and that's why the black holes are black. And it's true. But...
The singularity plays a secondary role. That's where the infalling observer is finally killed. Things get sick around the singularity, the low-energy or low-curvature effective theory breaks down. One needs the full theory of quantum gravity but it's generally believed that it can't help us to eliminate the singularity as a violent and irregular end of the spacetime, anyway. The existence of the singularities is a robust conclusion of the low-curvature effective equations of motion that more accurate dynamical laws can't change because the singularity is too localized, after all. And it's true as well. But...
Let me say what is the problem. Singularities are hard to be avoided but we generally assume that they don't spoil the dynamics elsewhere in the spacetime. After all, all singularities are covered by event horizons so they can't spread the illness outside these black holes etc. Penrose strengthened this ideology – and yes, it's an ideology, not robust science supported by evidence – by the Cosmic Censorship Conjecture which we now kind of know to be wrong even in the most weakened versions you may think of. Classical general relativity can't save us from the singularities whose impact has the potential to be harmful. So we have to deal with them, after all.
My impression is that we're doing something wrong with the singularities and this error may be shown to be responsible – at least in one scheme of attribution – for some of the confusion we're still seeing when it comes to the black hole information puzzle.
So I was comparing how we imagine a well-defined problem in semiclassical (or more accurate) general relativity with quantized fields on a background with a computational framework that has much higher standards of physics rigor, namely with calculations on the string theory world sheet. The semiclassical gravity on the background of an evaporating black hole has a singularity and fields become crazy at that point but we don't care. Because of causality – something we believe to be "inherited" from the special relativistic core of the classical general relativity – the Armageddon near the Schwarzschild singularity won't spoil anything else.
Except that we have sort of known for quite some time that this strict version of locality doesn't exactly hold in quantum gravity. Can't the required non-local behavior be blamed upon the singularity? I guess it can.
Beginners who learn string theory are also told about the possible boundary conditions at the stringy endpoints. The requirement is that the boundary terms in \(\delta S\), the variation of the action, have to cancel. Otherwise we would just not get a consistent theory. What the boundary conditions are always matters. So we learn that there may be open strings – with Neumann and/or (as Polchinski and others taught us later) Dirichlet boundary conditions – or closed strings with some kind of periodic boundary conditions (the periodic conditions may also be twisted because closed string states may exist in the twisted sectors, too).
But we never ignore this problem in string theory. We don't sweep it under the rug.
Our habits in general relativity are different. Replace the world sheet with a spacetime – a background spacetime with an evaporating black hole. It has some boundaries, the null infinities (SCRIs), the \(r=0\) "line" (an artifact of the spherical coordinates), and the black hole singularity. If these boundaries of the Penrose diagram were boundaries of a world sheet, we would insist that consistent boundary conditions cancelling the boundary terms of \(\delta S\) are picked. In general relativity, we seem to be more sloppy. After all, we're not afraid of any singularity because of the causal protection, are we?
So I began to think that we're simply obliged to solve these problems. We're solving them sort of correctly – analogously to the open strings – in the case of the null infinities and the \(r=0\) line (well, lines – before and after the black hole evaporates). The problematic boundary of the Penrose diagram (essentially the spacetime) is the singularity.
In their 2003 paper called Black Hole Final State, Maldacena and Horowitz proposed some "final boundary conditions" at the Schwarzschild singularity. Effectively, the singularity would behave as yet another open-string-like boundary, a mirror of a kind. It led to lots of puzzling threats for causality and at least some complicated transformation had to be inserted next to this "mirror" for the behavior of the object not to be self-evidently acausal or teleological.
Today, I started to think that the right boundary conditions are different than those in the black hole final state: one must define sort of closed-string-like boundary conditions for the spacetime. Most likely, the region near the singularity has to be analytically continued to the complex values of coordinates and connected to... the future null infinity where the Hawking radiation appears. That's a more accurate Ansatz for the "bridge" by which the early Hawking radiation may be connected to the black hole interior.
I started to believe that a physicist who wants to properly define quantum field theory or a form of quantum gravity on a spacetime background (and I mean in the Hamiltonian form, using spacelike slices) is obliged to fix the problems with the boundary conditions in some way. This way is perhaps not unique but it's a part of his definition of the "consistent histories" and quantum field theory (or its extension) on a curved spacetime gives him the tools to calculate the probability amplitudes. In some broader sense, all the frameworks are physically equivalent even if they assume a different background spacetime topology: we know that perturbations of the background spacetime may not only change the detailed metric but even the topology (at least, they may add Einstein-Rosen bridges if they're entangled).
So the future null infinity probably has to be connected to the singularity through the complexified spacetime in some way, to cancel the boundary terms in a way that is remotely analogous to the cancellation in the case of the closed string – periodic boundary conditions. When it's done, the radiation that reaches the null infinity may probably influence the behavior inside the black hole – and possibly also outside the black hole but close \(O(R)\) enough to the horizon.
Maybe we're allowed to connect a black hole interior's to any other black hole's interior to create a consistent background. We shouldn't be afraid of starting with complicated spacetimes with wormholes etc. because – in a striking contrast with another myth – wormholes may be undone by excitations added upon them, too. In particular, the ER bridge may be twisted by an angle before it's connected and the superposition over all angles produces an unentangled configuration - two isolated pure-state qubits, in our example. So the wormholes may be undone by excitations (they may be created and destroyed, like any object in a QFT-like theory) which means that we're probably not losing any generality by assuming that two black holes' interiors are connected.
This is a preliminary conclusion that may mutate later but at least once, I wanted to provide you with a snapshot of the partly crazy, partly trivial thoughts running through my biowires. And now I am going to restart the PC to install the June 2013 batch of Windows patches. ;-) This text will be proofread later, perhaps after a crime film on TV (done, but not too carefully because I don't expect the serious readership of this blog entry to be too numerous).
Some business with the M2-brane topology change and entanglement looked too obvious to me so I returned to the question how do the black hole exterior and interior really interact. The burning of the brain is composed of various steps that combine and recombine analytic continuation, diverse choices of coordinates, unusual ways to redefine the connectedness of the spacetime, and connections between previously disconnected regions through the complexified spacetime.
At each step, I try to ask not only HOW does it work but also WHY a particular trick that looks clever at a given moment should be picked and WHETHER it is inevitable or unique. Some of the partial conclusions are more convincing than others but I don't want to waste your time with an incomplete picture.
Instead, let me mention that we're probably victims of some bad habits and ultimately invalid lore related to the way how we think about certain issues in general relativity.
What do I mean?
General relativity is composed of the equations that are locally valid, Einstein's equations, and one may derive them from the equivalence principle and/or the diffemorphism symmetry that is needed for a consistent, ghost-free theory of spin-2 fields. We know that Einstein's original equations include the leading terms in a derivative expansion but there may be higher-order operators, too. All this stuff seems obvious and undeformable.
But we're also making some conceptual and "global" assumptions which may be wrong – and this wrongness may be especially harmful when one considers the black hole information puzzle and related topics.
General relativity is capable of producing spacetimes of nontrivial topology and the topology is a classical property of a spacetime. Therefore, it's been implicitly assumed that the discrete data defining the spacetime or space topology are observables that are represented by linear operators in the quantum theory.
Maldacena and Susskind have pretty much completely convinced me that this can't be the case, however. A spin-up electron here and a spin-down electron there, \(\ket{\uparrow\downarrow}\), that propagate on a flat spacetime seem to be eigenstates of the "topology" operator with the "trivial topology" eigenvalue. The same seems to hold for \(\ket{\downarrow\uparrow}\). If the "space topology" operator were linear, it would also obey\[
(\hat{\text{space topology}})\cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}) = (\text{trivial topology}) \cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}) \text{ NOT!}
\] However, the correct eigenvalue is different, Maldacena and Susskind argue:\[
(\hat{\text{space topology}})\cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}) = (\text{topology with an ER bridge}) \cdot (\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}).
\] (Please, open the mobile version of this page to see the whole equations; I don't want to change them or divide them to several lines.)
After all, this singlet state is the simplest quantum state that lives on the background with a single Einstein-Rosen bridge. So there's no linear operator that would count the Einstein-Rosen bridges. It's not a good observable. You can't use the number of ER bridges as a quantity that defines coarse-grained histories, e.g. in the Consistent Histories approach to quantum mechanics. In other words, the state \(\ket{\uparrow\downarrow}\) isn't orthogonal to the singlet state above: they're not mutually excluding although the former seems to be living on a bridge-free spacetime while the latter is living in a spacetime with a bridge.
Note that there's nothing wrong about the non-existence of an operator – observable – that would count the Einstein-Rosen bridges in a simple way. The bridges may be very thin and thin bridges are examples of those that are strongly affected by the "Planckian" details of a theory of quantum gravity. More generally, there isn't any clear operational method to determine the spacetime topology. A strategy to "see" a wormhole is to look for clumps of information and patterns that are repeated in two regions – but that's the same thing that defines correlations or entanglement. Entanglement of a state isn't associated with an observable, either.
You should notice that all the childish proposals to construct a theory of quantum gravity such as loop quantum gravity, causal dynamical triangulations, and so on are incompatible with the Maldacena+Susskind insight (not that it is the first lethal disease that has killed these stupidities, far from that). Why? Because they construct a spacetime (dreaming about its being large and nearly smooth) that clearly has some combinatorial properties and some topological invariants may be clearly extracted from their preferred basis of spin networks and other silly animals they use. So they have linear operators of "space topology".
It's easy to see why these approaches to quantum gravity have this property – which we now believe to be a pathological one. They have it because they belong among the naive models of quantum mechanics that start with a preferred basis and observables for which all the basis vectors are eigenvectors. But such theories are extremely unnatural as quantum theories because quantum theories don't come with any preferred bases or preferred operators. Viable quantum theories always offer us many different operators that don't commute with each other – which also implies that bases constructed from the eigenvectors of one set or another set are different bases.
We have to jettison this excess baggage. In quantum gravity, there aren't any linear operators that would honestly count the number of Einstein-Rosen bridges in the spacetime or other things. We must be careful about this incorrect assumption. It's a trap.
Bridges that invite you to count them with a linear operator are probably not the only trap. I became convinced that there is something wrong about the way how we think of black hole singularities.
The smart contemporary relativist – smart according to the "mainstream" – knows that black holes are defined by their event horizons. The event horizons is what separates the black hole interior from which nothing can escape – and that's why the black holes are black. And it's true. But...
The singularity plays a secondary role. That's where the infalling observer is finally killed. Things get sick around the singularity, the low-energy or low-curvature effective theory breaks down. One needs the full theory of quantum gravity but it's generally believed that it can't help us to eliminate the singularity as a violent and irregular end of the spacetime, anyway. The existence of the singularities is a robust conclusion of the low-curvature effective equations of motion that more accurate dynamical laws can't change because the singularity is too localized, after all. And it's true as well. But...
Let me say what is the problem. Singularities are hard to be avoided but we generally assume that they don't spoil the dynamics elsewhere in the spacetime. After all, all singularities are covered by event horizons so they can't spread the illness outside these black holes etc. Penrose strengthened this ideology – and yes, it's an ideology, not robust science supported by evidence – by the Cosmic Censorship Conjecture which we now kind of know to be wrong even in the most weakened versions you may think of. Classical general relativity can't save us from the singularities whose impact has the potential to be harmful. So we have to deal with them, after all.
My impression is that we're doing something wrong with the singularities and this error may be shown to be responsible – at least in one scheme of attribution – for some of the confusion we're still seeing when it comes to the black hole information puzzle.
So I was comparing how we imagine a well-defined problem in semiclassical (or more accurate) general relativity with quantized fields on a background with a computational framework that has much higher standards of physics rigor, namely with calculations on the string theory world sheet. The semiclassical gravity on the background of an evaporating black hole has a singularity and fields become crazy at that point but we don't care. Because of causality – something we believe to be "inherited" from the special relativistic core of the classical general relativity – the Armageddon near the Schwarzschild singularity won't spoil anything else.
Except that we have sort of known for quite some time that this strict version of locality doesn't exactly hold in quantum gravity. Can't the required non-local behavior be blamed upon the singularity? I guess it can.
Beginners who learn string theory are also told about the possible boundary conditions at the stringy endpoints. The requirement is that the boundary terms in \(\delta S\), the variation of the action, have to cancel. Otherwise we would just not get a consistent theory. What the boundary conditions are always matters. So we learn that there may be open strings – with Neumann and/or (as Polchinski and others taught us later) Dirichlet boundary conditions – or closed strings with some kind of periodic boundary conditions (the periodic conditions may also be twisted because closed string states may exist in the twisted sectors, too).
But we never ignore this problem in string theory. We don't sweep it under the rug.
Our habits in general relativity are different. Replace the world sheet with a spacetime – a background spacetime with an evaporating black hole. It has some boundaries, the null infinities (SCRIs), the \(r=0\) "line" (an artifact of the spherical coordinates), and the black hole singularity. If these boundaries of the Penrose diagram were boundaries of a world sheet, we would insist that consistent boundary conditions cancelling the boundary terms of \(\delta S\) are picked. In general relativity, we seem to be more sloppy. After all, we're not afraid of any singularity because of the causal protection, are we?
So I began to think that we're simply obliged to solve these problems. We're solving them sort of correctly – analogously to the open strings – in the case of the null infinities and the \(r=0\) line (well, lines – before and after the black hole evaporates). The problematic boundary of the Penrose diagram (essentially the spacetime) is the singularity.
In their 2003 paper called Black Hole Final State, Maldacena and Horowitz proposed some "final boundary conditions" at the Schwarzschild singularity. Effectively, the singularity would behave as yet another open-string-like boundary, a mirror of a kind. It led to lots of puzzling threats for causality and at least some complicated transformation had to be inserted next to this "mirror" for the behavior of the object not to be self-evidently acausal or teleological.
Today, I started to think that the right boundary conditions are different than those in the black hole final state: one must define sort of closed-string-like boundary conditions for the spacetime. Most likely, the region near the singularity has to be analytically continued to the complex values of coordinates and connected to... the future null infinity where the Hawking radiation appears. That's a more accurate Ansatz for the "bridge" by which the early Hawking radiation may be connected to the black hole interior.
I started to believe that a physicist who wants to properly define quantum field theory or a form of quantum gravity on a spacetime background (and I mean in the Hamiltonian form, using spacelike slices) is obliged to fix the problems with the boundary conditions in some way. This way is perhaps not unique but it's a part of his definition of the "consistent histories" and quantum field theory (or its extension) on a curved spacetime gives him the tools to calculate the probability amplitudes. In some broader sense, all the frameworks are physically equivalent even if they assume a different background spacetime topology: we know that perturbations of the background spacetime may not only change the detailed metric but even the topology (at least, they may add Einstein-Rosen bridges if they're entangled).
So the future null infinity probably has to be connected to the singularity through the complexified spacetime in some way, to cancel the boundary terms in a way that is remotely analogous to the cancellation in the case of the closed string – periodic boundary conditions. When it's done, the radiation that reaches the null infinity may probably influence the behavior inside the black hole – and possibly also outside the black hole but close \(O(R)\) enough to the horizon.
Maybe we're allowed to connect a black hole interior's to any other black hole's interior to create a consistent background. We shouldn't be afraid of starting with complicated spacetimes with wormholes etc. because – in a striking contrast with another myth – wormholes may be undone by excitations added upon them, too. In particular, the ER bridge may be twisted by an angle before it's connected and the superposition over all angles produces an unentangled configuration - two isolated pure-state qubits, in our example. So the wormholes may be undone by excitations (they may be created and destroyed, like any object in a QFT-like theory) which means that we're probably not losing any generality by assuming that two black holes' interiors are connected.
This is a preliminary conclusion that may mutate later but at least once, I wanted to provide you with a snapshot of the partly crazy, partly trivial thoughts running through my biowires. And now I am going to restart the PC to install the June 2013 batch of Windows patches. ;-) This text will be proofread later, perhaps after a crime film on TV (done, but not too carefully because I don't expect the serious readership of this blog entry to be too numerous).
Finding and abandoning incorrect general relativity lore
![Finding and abandoning incorrect general relativity lore]()
Reviewed by DAL
on
June 11, 2013
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