Kerr (yes, who found the well-known Kerr black hole solutions) disagrees with Penrose's singularity theorem and its variations. Namely these theorems prove the existence of geodesics that can't be extended beyond a finite affine length, but Kerr finds numerous examples of inextensible light rays that don't contain singularities. These geodesics go all the way to the null infinity, and yet the affine parameter remains finite. And there are such light rays through every point of the Kerr spacetime. Only some geodesics hit the ring singularity, but this region can be replaced by a nonsingular one, perhaps matter can do this. Kerr thinks that his perfectly symmetric vacuum solution doesn't happen in reality (despite the "no-hair theorem", which is in fact a conjecture improperly called "theorem", stating that all black holes evolve into a Kerr solution), even though he thinks that black holes exist.
Now, how is this possible? I mean the singularity theorems, now sealed forever by a Nobel prize, prove that there are conditions that necessarily lead to singularities. That if there's a black hole, there must be a singularity beyond its horizon. Or do they?
This is a bit of a word play. There are more meanings of the word "singularity". Normally singularity means a place where the metric blows up. Or its inverse. Or the curvature, or any field that we think it's physical. But then we can think of excluding these points from spacetime. If these points are "in the way" of the physical fields, if the evolution equations can't go beyond such a place but they should, this would be a problem even if we exclude them from spacetime. But if these singularities are somewhere at the "edge" of spacetime, and the spacetime admits a nice foliation so that the evolution equations work fine across the entire spacetime, why would this be a problem? And yet, the other definition of singularity, the one that is actually the object of the singularity theorems, includes such cases as well. That is, as a diagnostic method, it gives numerous false positives.
Here's what happened. And I don't say it's a plot against General Relativity, rather an accident, perhaps welcomed by many. If your spacetime contains singularities, we can think of excluding them from spacetime. But this, as I said, doesn't solve the problem. So maybe there is a way to detect this pathology even with the singularities removed, and talk about such a spacetime as being singular anyway. And here comes into play the redefinition of singular spacetime in terms of geodesic incompleteness. And it is said in the Hawking & Ellis bible, on page 258:
I don't want to single out this great book, it explains well the adoption of this diagnosis, and others said similar things. But here I think lies the problem. Because this definition can be misunderstood (unintentionally I think) in a way that makes the singularity theorems seem about singularities even if there are no singularities in the interior of spacetime, even if the spacetime can be nicely foliated, offering a nice home to the evolution equations.
The singularity theorems prove (and they indeed prove this) that there are incomplete geodesics, where incomplete means they can't be extended beyond a finite affine length. Whether all of them deserve to be called "incomplete" is also questionable. If the affine length (which is not the same as geometric length anyway) of a timelike or null geodesic is finite, but it goes to the "real edge" of spacetime, as in Kerr's paper, why should it be called incomplete? This already seeds in our minds the idea that there's something wrong with them.
So, one on top of another, the meaning of words shifted so that now it's widely believed that General Relativity breaks down, due to the singularities. And Kerr gives nice rich counterexamples, all in the same spacetime of a Kerr black hole. I mean, his spacetime has a singularity, but the singularity theorem doesn't even predict that singularity. It predicts some singularities, but they are false positives, they don't occur on the geodesics up to the boundary of spacetime. It doesn't predict the ring singularity, because, as Kerr says, there is no trapped surface inside the inner horizon of the Kerr black hole. So, if we cut out the spacetime around that ring, and replace that region (and the "other universe" beyond the ring) with one without singularities, we get a spacetime without singularities (and from what we know matter may do this), and yet the singularity theorems as usually cited say it has singularities (outside that region)!
I'd like to add that I was convinced as well, for a long time, that the singularity
theorems imply the kind of metric singularities that are problematic. They were the reason why I worked
to save General Relativity by reformulating it in a way that doesn't
have infinities at the singularities. And I repeated numerous times the
claim that the singularity theorems prove that the metric tensor has
singularities, assuming that they are of this kind. And I might have regarded people who didn't believe in
singularities as, let's say, not very serious. Despite being aware that
there was a step in the proof of the singularity theorems that I never understood,
namely exactly the step where from inextensibility we conclude the
existence of such singularities. Despite never being able to find a place where this step is proved for a limited person like me. And that while knowing that I didn't
understand that step, and being limited, I considered that I should trust the experts about
it, or maybe just my limited understanding of what experts say. And now, after seeing Roy Kerr's counterexamples, I think I was wrong.
So yes, Kerr is right, to be able to say that General Relativity breaks down because of singularities we need a proof for exactly such singularities, and the singularity theorems alone don't do the job, and there are counterexamples showing this. But of course counterexamples are a no go mainly for the more mathematically inclined (and some of them noticed this at some times, but somehow the most spread interpretation of the singularity theorems remained unaffected). Many physicists may still use the confusion between the two notions of singular spacetimes (assuming they're aware of them) to reject classical General Relativity, and at the same time they would claim that quantum gravity doesn't have this problem, again without proof, without even a theory of quantum gravity! (The only argument is that quantum fields may violate a condition in the singularity theorems, but this doesn't prove that this avoids the alleged singularity)
But what if somebody takes notice now of Kerr's paper, and of the disambiguation of the term "singular spacetime", and finds a singularity theorem, with different conditions evidently, that is actually about such singularities? Even so, General Relativity can be formulated in terms of finite geometric objects, which can evolve beyond the singularities, as I showed some time ago https://arxiv.org/abs/1301.2231. This formulation is equivalent with the usual one outside the singularities, but it extends at the singularities too, at least in the usual cases.
So I see no reason why General Relativity is so often pronounced dead. I mean, sure, we need a quantum theory of gravity, but let's stop throwing the baby with the bathwater. There's no reason to treat like a stepchild one of the two babies, General Relativity and Quantum Theory, and favor the other one. The really naughty one ;)
Tim Maudlin has an interesting paper in which he criticizes the importance given to the black hole information paradox, and even brings arguments that it is not even a problem: (Information) Paradox Lost. I agree that the importance of the problem is perhaps exaggerated, but at the same time many consider it to be a useful benchmark to test quantum gravity solutions. This led to decades of research made by many physicists, and to many controversies. I wrote a bit about some of the proposed solutions to the problem in some older posts, for example [1,2,3]. Maudlin's paper is discussed by Sabine here.
One of the central arguments in Maudlin's paper is that the well-known spacetime illustrating the information loss can be foliated into some 3D spaces (which are Cauchy hypersurfaces that are discontinuous at the singularity). These hypersurfaces have a part outside the black hole, and another one inside it, which are not connected to one another. Cauchy hypersurfaces contain the Cauchy data necessary to solve the partial differential equations, so the information should be preserved if we consider both their part inside and their part outside the black hole.
I illustrate this with this animated gif:
I made this gif back in 2010, when I independently had the same idea and wanted to write about it, but I don't think I made it public. Probably the idea is older. The reason I didn't write about it was that I was more attracted* to another solution I found, which led to an analytic extension of the black hole spacetime, and has Cauchy hypersurfaces but no discontinuities. I reproduce a picture of the Penrose diagram from an older post in which I say more about this:
A. The standard Penrose diagram of an evaporating black hole.
B The diagram from the analytic solution I proposed.
___________________________
* The reason I preferred to work at the second solution is that it allows the information to become available after the evaporation to an external observer. The solution which relies on completing the Cauchy hypersurface with a part inside the black hole doesn't restore information and unitarity for an external observer. I don't know if this is a problem, but many physicists believe that information should be restored for an external observer, because otherwise we would observe violations of unitarity even in the most mundane cases, considering that micro black holes form and evaporate at very high energies. I don't think this argument, also given by Sabine, is very good, because there is no reason to believe that micro black holes form at high energy under normal conditions. People arrive at high energies for normal situations because they use perturbative expansions, but this is just a method of approximation. And even so, I doubt anyone who sums over Feynman diagrams includes black holes. But nevertheless, I wouldn't like information to be lost for an outside observer after evaporation, but this is just personal taste, I don't claim that there is some experiment that proved this. And the solution I preferred to research allows recovery of information and unitarity for an external observer, and other things which I explained in the mentioned posts and my PhD thesis.
I learned recently about a paper which attempts to refute one of my papers. While being sure about my proofs, I confess that I was a bit worried, you never know when you made a mistake, a silly assumption that you overlooked. But as I was reading the refutation paper, my worries dissipated, and were replaced by amusement and I actually had a lot of fun. Because that so-called refutation was something like: "I will refute Pythagoras's Theorem by showing that it doesn't apply to triangles that are not right."
My paper in cause about Big-Bang singularities is arXiv:1112.4508 (The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved). As it is known, the main mathematical tool used in General Relativity is semi-Riemannian geometry, and this works only as long as the metric is regular. The metric ceases to be regular at singularities, but I developed the extension of semi-Riemannian geometry at some degenerate metrics, so it applies to a large class of singularities, in arxiv:1105.0201. And this allowed me to find descriptions of such singularities in terms of quantities that are still invariant, but as opposed to the usual ones, they remain finite at singularities. More about this can be found in my PhD thesis arxiv:1301.2231. In the paper arXiv:1112.4508, I give a theorem that shows that, if the scaling function of the FLRW universe is smooth at the Big-Bang singularity, then I can apply the tools I developed previously, and get a finite description of both the geometry, and the physical quantities involved.
The paper attempting to refute my result is arxiv:1603.02837 (Behavior of Friedmann-Lemaitre-Robertson-Walker Singularities, by L. Fernández-Jambrina). Both my paper and this one appeared this year in International Journal of Theoretical Physics. I think F-J is a good researcher and expert in singularities. But for some reason, he didn't like my paper, and he "refuted" it. The "refutation" simply takes the case that was explicitly not covered in my theorem, namely when the scaling function of the FLRW solution is not derivable at the singularity, and checks that indeed my tools don't work in this case. Now, while my result is much more humble than Pythagoras's Theorem, I will use it for comparison, since it is well-known by everybody. You can't refute Pythagoras's Theorem by taking triangles that are not right, and proving that the sum of squares of two sides is different than the square of the third. Simply because the Theorem makes clear in its hypothesis that it refers only to right triangles. My theorem also states clearly that the result doesn't refer to FLRW models whose scaling function is not derivable at the singularity. And F-J even copies the Theorem's enounce in his paper, so how could he miss this? So what F-J said is that my theorem can't be applied to some cases, which I made clear that I leave out (I don't claim my theorem solves everything, neither that it cures cancer). Now, is the case when the scaling function is not derivable important? Yes, at least historically, because some classical solutions fit here. But the cases covered by my theorem include what we know today about inflation. So I think that my result is not only correct, but also significant. In addition to this, F-J says that I actually don't remove the Big-Bang singularity. This is also true, and stated in my paper from the beginning. I don't remove the singularities, I just try to understand them to describe them in terms of finite quantities that make sense both geometrically and physically. But he wrote it as if I claim that I try to remove them and he proves that I don't, not that I accept them and provide a finite-quantities description of them.
Vectors are present in all domains of fundamental physics, so if you want to understand physics, you will need them. You may think you know them, but the truth is that they appear in so many guises, that nobody really knows everything about them. But vectors are a gate that allows you to enter the Cathedral of physics, and once you are inside, they can guide you in all places. That is, special and general relativity, quantum mechanics, particle physics, gauge theory... all these places need vectors, and once you master the vectors, they become much simpler (if you don't know them and are interested, read this post).
The Cathedral has many gates, and vectors are just one of them. You can enter through groups, sets and relations, functions, categories, through all sorts of objects or structures from algebra, geometry, even logic. I decided to show you now the way of vectors, because I think is fast and deep in the same time, but remember, this is a matter of choice. And vectors will lead us, inevitably, to the other gates too.
I will explain some elementary and not so elementary things about vectors, but you have to read and practice, because here I just give some guidelines, a big picture. The reason I am doing this is that when you study, you may get lost in details and miss the essential.
Very basic things
A vector can be understood in many ways. One way is to see it as a
way to specify how to move from one point to another. A vector is like an
arrow, and if you place the arrow in that point, you
find the destination point. To find the new position for any point, just
place the vector in that point, and the tip of the vector will show you
the new position. You can compose more such arrows, and what you'll get
is another vector, their sum. You can also subtract them, just place
their origins in the same point, and the difference is the vector
obtained by joining their tips with another arrow.
Once you fix a reference position, an origin, you can specify any position, by the vector that tells you how to move from origin to that position. You can see that vector as being the difference between the destination, and the starting position.
You can add and subtract vectors. You can multiply them with numbers. Those numbers are from a field $\mathbb{K}$, and we can take for example $\mathbb{K}=\mathbb{R}$, or $\mathbb{K}=\mathbb{C}$, and are called scalars. A vector space is a set of vectors, so that no matter how you add them and scale them, the result is from the same set. The vector space is real (complex), if the scalars are real (complex) numbers. A sum of rescaled vectors is named linear combination. You can always pick a basis, or a frame, a set of vectors so that any vector can be written as a linear combination of the basis vectors, in a unique way.
Vectors and functions
Consider a vector $v$ in an $n$-dimensional space $V$, and suppose its components in a given basis are $(v^1,\ldots,v^n)$. You can represent any vector $v$ as a function $f:\{1,\ldots,n\}\to\mathbb{K}$ given by $f(i)=v^i$. Conversely, any such function defines a unique vector. In general, if $S$ is a set, then the set of the functions $f:S\to\mathbb{K}$ form a vector space, which we will denote by $\mathbb{K}^S$. The cardinal of $S$ gives the dimension of the vector space, so $\mathbb{K}^{\{1,\ldots,n\}}\cong\mathbb{K}^n$. So, if $S$ is an infinite set, we will have an infinite dimensional vector space. For example, the scalar fields on a three
dimensional space, that is, the functions $f:\mathbb{R}^3\to
\mathbb{R}$, form an infinite dimensional vector space. Not only the vector spaces are not limited to $2$ or $3$ dimensions, but infinite dimensional spaces are very natural too.
Dual vectors
If $V$ is a $\mathbb{K}$-vector space, a linear functions $f:V\to\mathbb{K}$ is a function satisfying $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u,v\in V,\alpha\in\mathbb{K}$. The linear functions $f:V\to\mathbb{K}$ form a vector space $V^*$ named the dual space of $V$.
Tensors
Consider now two sets, $S$ and $S'$, and a field $\mathbb{K}$. The Cartesian product $S\times S'$ is defined as the set of pairs $(s,s')$, where $s\in S$ and $s'\in S'$. The functions defined on the Cartesian product, $f:S\times S'\to\mathbb{K}$, form a vector space $\mathbb{K}^{S\times S'}$, named the tensor product of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, $\mathbb{K}^{S\times S'}=\mathbb{K}^{S}\otimes\mathbb{K}^{S'}$. If $(e_i)$ and $(e'_j)$ are bases of $\mathbb{K}^{S}$ and $\mathbb{K}^{S'}$, then $(e_ie'j)$, where $e_ie'_j(s,s')=e_i(s)e'_j(s')$, is a basis of $\mathbb{K}^{S\times S'}$. Any vector $v\in\mathbb{K}^{S_1\times S_2}$ can be uniquely written as $v=\sum_i\sum_j \alpha_{ij} e_ie'j$.
Also, the set of functions $f:S\to\mathbb{K}^{S'}$ is a vector space, which can be identified with the tensor product $\mathbb{K}^{S}\otimes(\mathbb{K}^{S'})^*$.
The vectors that belong to tensor products of vector spaces are named tensors. So, tensors are vectors with some extra structure.
The tensor product can be defined easily for any kind of vector spaces, because any vector space can be thought of as a space of functions. The tensor product is associative, so we can define it between multiple vector spaces. We denote the tensor product of $n>1$ copies of $V$ by $V^{\otimes n}$. We can check that for $m,n>1$, $V^{\otimes (m+n)}=V^{\otimes {m}}\otimes V^{\otimes {n}}$. This can work also for $m,n\geq 0$, if we define $V^1=V$, $V^0=\mathbb{K}$. So, vectors and scalars are just tensors.
Let $U$, $V$ be $\mathbb{K}$-vector spaces. A linear operator is a function $f:U\to V$ which satisfies $f(u+v)=f(u)+f(v)$, and $f(\alpha u)=\alpha f(u)$, for any $u\in U,v\in V,\alpha\in\mathbb{K}$. The operator $f:U\to V$ is in fact a tensor from $U^*\otimes V$.
Inner products
Given a basis, any vector can be expressed as a set of numbers, the components of the vector. But the vector is independent of this numerical representation. The basis can be chosen in many ways, and in fact, any non-zero vector can have any components (provided not all are zero) in a well chosen basis. This shows that any two non-zero vectors play identical roles, which may be a surprise. This is a key point, since a common misconception when talking about vectors is that they have definite intrinsic sizes and orientations, or that they can make an angle. But in fact the sizes and orientations are relative to the frame, or to the other vectors. Moreover, you can say that from two vectors, one is larger than the other, only if they are collinear. Otherwise, no matter how small is one of them, we can easily find a basis in which it becomes larger than the other. It makes no sense to speak about the size, or magnitude, or length of a vector, as an intrinsic property.
But wait, one may say, there is a way to define the size of a vector! Consider a basis in a two-dimensional vector space, and a vector $v=(v^1,v^2)$. Then, the size of the vector is given by Pythagoras's theorem, by $\sqrt{(v^1)^2+(v^2)^2}$. The problem with this definition is that, if you change the basis, you will obtain different components, and different size of the vector. To make sure that you obtain the same size, you should allow only certain bases. To speak about the size of a vector, and about the angle between two vectors, you need an additional object, which is called inner product, or scalar product. Sometimes, for example in geometry and in relativity, it is called metric.
Choosing a basis gives a default inner product. But the best way is to define the inner product, and not to pick a special basis. Once you have the inner product, you can define angles between vectors too. But size and angles are not intrinsic properties of vectors, they depend on the scalar product too.
The inner product between two vectors $u$ and $v$, defined by a basis, is $u\cdot v = u^1 v^1 + u^2 v^2 + \ldots + u^n v^n$. But in a different basis, it will have a general form $u\cdot v=\sum_i\sum_j g_{ij} u^i v^j$, where $g_{ij}=g_{ji}$ can be seen as the components of a symmetric matrix. These components change when we change the basis, they form the components of a tensor from $V^*\otimes V^*$. Einstein had the brilliant idea to omit the sum signs, so the inner product looks like $u\cdot v=g_{ij} u^i v^j$, where you know that since $i$ and $j$ appear both in upper and in lower positions, we make them run from $1$ to $n$ and sum. This is a thing that many geometers hate, but physicists find it very useful and compact in calculations, because the same summation convention appears in many different situations, which to geometers appear to be different, but in fact are very similar.
Given a basis, we can define the inner product by choosing the coefficients $g_{ij}$. And we can always find another basis, in which $g_{ij}$ is diagonal, that is, it vanishes unless $i=j$. And we can rescale the basis so that $g_{ii}$ are equal to $-1$, $1$, or $0$. Only if $g_{ii}$ are all $1$ in some basis, the size of the vector is given by the usual Pythagoras's theorem, otherwise, there will be some minus signs there, and even some terms will be omitted (corresponding to $g_{ii}=0$).
Quantum mechanics
Quantum particles are described by Schrödinger's equation. Its solutions are, for a single elementary particle, complex functions $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}$, or more general, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$, named wavefunctions. They describe completely the states of the quantum particle. They form a vector space $H$ which also has a hermitian product (a complex scalar product so that $h_{ij}=\overline{h_{ji}}$), and is named the Hilbert space (because in the infinite dimensional case also satisfies an additional property which we don't need here), or the state space. Linear transformations of $H$ which preserve the complex scalar product are named unitary transformations, and they are the complex analogous of rotations.
The wavefunctions are represented in a basis as functions of positions, $|\psi\rangle:\mathbb{R}^3\to\mathbb{C}^k$. The element of the position basis represent point particles. But we can make a unitary transformation and obtain another basis, made of functions of the form $e^{i (k_x x + k_y y + k_z z)}$, which represent pure waves. Some observations use one of the bases, some the other, and here is why there is a duality between waves and point particles.
For more elementary particles, the state space is the tensor product of the state spaces of the individual particles. A tensor product of the form $|\psi\rangle\otimes|\psi'\rangle$ represents separable states, which can be observed independently. If the system can't be written like this, but only as a sum, the particles are entangled. When we measure them, the outcomes are correlated.
The evolution of a quantum system is described by Schrödinger's equation. Basically, the state rotates, by a unitary transformation. Only such transformations conserve the probabilities associated to the wavefunction.
When you measure the quantum systems, you need an observable. One can see an observable as defining a decomposition of the state space, in perpendicular subspaces. After the observation, the state is found to be in one of the subspaces. We can only know the subspace, but not the actual state vector. This is strange, because the system can, in principle, be in any possible state, but the measurement finds it to be only in one of these subspaces (we say it collapsed). This is the measurement problem. The things become even stranger, if we realize that if we measure another property, the corresponding decomposition of the state space is different. In other words, if you look for a point particle, you find a point particle, and if you look for a wave, you find a wave. This seems as if the unitary evolution given by the Schrödinger's equation is broken during observations. Perhaps the wavefunction remains intact, but to us, only one of the components continues to exist, corresponding to the subspace we obtained after the measurement. In the many worlds interpretation the universes splits, and all outcomes continue to exist, in new created universes. So, not only the state vector contains the universe, but it actually contains many universes.
I have a proposed explanation for some strange quantum features, in [1, 2, 3], and in these videos:
Special relativity
An example when there is a minus signs in the Pythagoras's theorem is given by the theory of relativity, where the squared size of a vector is $v\cdot v=-(v^t)^2+(v^x)^2+(v^y)^2+(v^z)^2$.
This inner product is named the Lorentz metric. Special relativity takes place in the Minkowski spacetime, which has four dimensions. A vector $v$ is named timelike if $v\cdot v < 0$, spacelike if $v\cdot v > 0$, and null or lightlike if $v\cdot v = 0$. A particle moving with the speed of light is described by a lightlike vector, and one moving with an inferior speed, by a timelike vector. Spacelike vectors would describe faster than light particles, if they exist. Points in spacetime are named events. Events can be simultaneous, but this depends on the frame. Anyway, to be simultaneous in a frame, two events have to be separated by a spacelike interval. If they are separated by a lightlike or timelike interval, they can be connected causally, or joined by a particle with a speed equal to, respectively smaller than the speed of light.
In Newtonian mechanics, the laws remain unchanged to translations and rotations in space, translations in time, and inertial movements of the frame - together they form the Galilei transformations. However, electromagnetism disobeyed. In fact, this was the motivation of the research of Einstein, Poincaré, Lorentz, and FitzGerald. Their work led to the discovery of special relativity, according to which the correct transformations are not those of Galilei, but those of Poincaré, which preserve the distances given by the Lorentz metric.
Curvilinear coordinates
A basis or a frame of vectors in the Minkowski spacetime allows us to construct Cartesian coordinates. However, if the observer's motion is accelerated (hence the observer is non-inertial), her frame will rotate in time, so Cartesian coordinates will have to be replaced with curved coordinates. In curved coordinates, the coefficients $g_{ij}$ depend on the position. But in special relativity they have to satisfy a flatness condition, otherwise spacetime will be curved, and this didn't make much sense back in 1905, when special relativity was discovered.
General relativity
Einstein remarked that to a non-inertial observer, inertia looks similar to gravity. So he imagined that a proper choice of the metric $g_{ij}$ may generate gravity. This turned out indeed to be true, but the choice of $g_{ij}$ corresponds to a curved spacetime, and not a flat one.
One of the problems of general relativity is that it has singularities. Singularities are places where some of the components of $g_{ij}$ become infinite, or where $g_{ij}$ has, when diagonalized, some zero entries on the diagonal. For this reason, many physicist believe that this problem indicates that general relativity should be replaced with some other theory, to be discovered. Maybe it will be solved when we will replace it with a theory of quantum gravity, like string theory or loop quantum gravity. But until we will know what is the right theory of quantum gravity, general relativity can actually deal with its own singularities (while the ones mentioned above did not solve this problem). I will not describe this here, but you can read my articles about this, and also this essay, and these posts about the black hole information paradox [1, 2, 3]. And watch this video
Vector bundles and forces
We call fields the functions defined on the space or the spacetime. We have seen that fields valued in vector spaces are actually vector spaces. On a flat space $M$ which looks like a vector space, the fields valued in vector spaces can be thought of as being valued in the same vector space, for example $f:M\to V$. But if the space is curved, or if it has nontrivial topology, we are forced to consider that at each point there is another copy of $V$. So, such a field will be more like $f(x)\in V_x$, where $V_x$ is the copy of the vector space $V$ at the point $x$. Such fields still form a vector space. The union of all $V_x$ is called a vector bundle. The fields are also called sections, and $V_x$ is called the fiber at $x$.
Now, since $V_x$ are copies of $V$ at each point, there is no invariant way to identify each $V_x$ with $V$. In other words, $V_x$ and $V$ can be identified, for each $x$, up to a linear transformation of $V$. We need a way to move from $V_x$ to a neighboring $V_{x+d x}$. This can be done with a connection. Also, moving a vector from $V_x$ along a closed curve reveals that, when returning to $V_x$, the vector is rotated. This is explained by the presence of a curvature, which can be obtained easily from the connection.
Connections behave like potentials of force fields. And a force field corresponds to the curvature of the connection. This makes very natural to use vector bundles to describe forces, and this is what gauge theory does.
Forces in the standard model of particles are described as follows. We assume that there is a typical complex vector space $V$ of dimension $n$, endowed with a hermitian scalar product. The connection is required to preserve this hermitian product when moving among the copies $V_x$. The set of linear transformations that preserve the scalar product is named unitary group, and is denoted by $U(n)$. The subset of transformations having the determinant equal to $1$ is named the special unitary group, $SU(n)$. The electromagnetic force corresponds to $U(1)$, the weak force to $SU(2)$, and the strong force to $SU(3)$. Moreover, all particles turn out to correspond to vectors that appear in the representations of the gauge groups on vector spaces.
What's next?
Vectors are present everywhere in physics. We see that they help us understand quantum mechanics, special and general relativity, and the particles and forces. They seem to offer a unitary view of fundamental physics.
However, up to this point, we don't know how to unify
unitary evolution and the collapse of the wavefunction
the quantum level with the mundane classical level
quantum mechanics and general relativity
the electroweak and strong forces (we know though how to combine the electromagnetic and weak forces, in the unitary group $U(2)$)
At first, math seemed to show that anything that enters a black hole, is lost forever. Later, it seemed that black holes evaporate, but the secrets remain lost. But maybe it is not so.
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The absence of event horizons mean that there are no black holes - in the sense of regimes from which light can't escape to infinity.
But he continues that black holes exist, but they are not as he originally defined them:
There are however apparent horizons which persist for a period of time. This suggests that black holes should be redefined as metastable bound states of the gravitational field.
After a regular person makes a claim about something, he hardly changes his mind. Especially since that claim is part of what made him famous. We find difficult to withdraw our positions, because we are afraid to look weak. One reason I admire Hawking is that he had in several occasions the courage to change his mind, and even to admit he was wrong. He made several bets with his fellows Kip Thorne and John Preskill, concerning the existence of black holes, of naked singularities, and regarding the information loss. He eventually conceded all these bets, even though no clear cut evidence was discovered for either of the sides.
Hawking's first great discovery was the big bang singularity theorem, according to which the universe started from a singularity. It is difficult to later reject the very thing that made you famous in the first place, but Hawking, together with James Hartle, replaced the initial singularity with the famous no-boundary proposal, which doesn't have this singularity (although, technically, the positive defined metric they put at the beginning of the universe is separated by the Lorentzian one by a space slice which is in fact singular).
At various points of his career, Hawking expresses his doubts about string theory. For instance, in his debate with Penrose, he said
I think string theory has been over sold.
and
it seems we don’t need string theory even for the beginning of the universe.
and
If this is true it raises the question of whether string theory is a genuine scientific theory. Is mathematical beauty and completeness enough in the absence of distinctive observationally tested predictions. Not that string theory in its present form is either beautiful or complete.
But in few years, he became a major supporter of string theory, as follows from this paper and this book.
Arguably, most of the fame of Hawking comes from his results concerning the black holes. But I don't think it is true as it is claimed now that, after a lifetime dedicated to the study of black holes, he arrived at the conclusion that they don't exist. He only rejects the existence of black holes defined as objects surrounded by event horizons, defined in their turn in a particular way. And in fact, he rejects that notion of event horizon. The notion of event horizon exists for long time, but at some point, Hawking redefined it, as the surface separating the points in spacetime which can't be seen from the future null infinity. Before that, the event horizon was known from stationary black holes, like the Schwarzschild, the Reissner-Nordström, and the Kerr-Newmann ones, and was generalized to trapped null surfaces. Hawking opposed to this general definition, because it would depend on the observer. Such apparent horizons are therefore not invariant, and Hawking proposed a global definition. The problem with the global definition is that it depends on the entire future, to establish whether a given point will eventually be visible from the null infinity or not. But if the black holes evaporate in a way compatible with the AdS-CFT conjecture, they have to respect the CPT symmetry. Since a global notion of event horizon violates this symmetry, Hawking proposes to reject it.
Hawking did not change his mind about the existence of the black holes, but only about his own definition of black holes, as those regions in spacetime which can't be seen from the future null infinity. He proposes instead to consider again the black holes to be regions surrounded by apparent horizons.
Hawking finally uploaded the paper containing his Skype talk at the Fuzz or fire workshop, named Information Preservation and Weather Forecasting for Black Holes. The paper, whose body has two pages, is an almost verbatim transcription of the 9' talk, with a tiny paragraph inserted before the final one. The talk was very dense, with great qualitative arguments, but almost no quantitative ones, and I kind of hoped that the paper will be more detailed in this respect.
The first argument Hawking brought against firewalls is that
if the firewall were located at the event horizon, the position of the event horizon is not locally determined but is a function of the future of the spacetime.
Hawking defined long time ago the event horizon as being the surface separating the events that will eventually be seen from the future infinity, from those that will never be. Thus, we can know the event horizon only if we know the entire future history of the universe.This rules out any special structure which one may try to attach to the horizon, being it firewalls, stretched horizons, bits containing the information from the black hole etc. This argument is technically correct, but this doesn't rule out alternative local definitions of the horizon, and on which the firewall may live. I think this argument comes from the usage of different definitions.
One thing I find particularly intriguing is that Hawking doesn't discuss the singularities. Singularities are predicted by Penrose's black hole singularity theorem, which inspired Hawking in coming up with his own big bang singularity theorem. Also singularities are a necessary part of Hawking's original argument for the information loss. So, it is a bit strange that he doesn't say much about them. Well, he referred to the paper in which he proposed the resolution of the information paradox, and said that "the correlation functions from the Schwarzschild anti deSitter metric decay exponentially with real time". So, he considers that the contribution from the Schwarzschild singularities is negligible.
I find more interesting Hawking's argument that the ADS-CFT correspondence requires the black holes to be symmetric in time:
the
evaporation of a black hole is the time reverse of its formation
(modulo CP), though the conventional descriptions are very different.
Thus if one assume quantum gravity is CPT invariant, one rules out remnants, event horizons, and firewalls.
Of course, again, one can imagine a way by which the firewalls are time symmetric, and use a different definition of the event horizon. But the reason I find interesting this argument of Hawking is that it doesn't preclude singularities, only the singularities that are not time symmetric. For instance, fig. A. depicts the Penrose diagram of the evaporating black hole that is not time symmetric, while fig. B. depicts a time symmetric one, obtained by analytic extension beyond the singularity. I give more details about this in Black Hole Information Paradox 3. Look for the information where you lost it.
A. Penrose diagram for the evaporating black hole, standard scenario. B. Penrose diagram for the evaporating black hole, when the solution is analytically extended through the singularity (as in arXiv:1111.4837).
In the new solution, the geometry can be described in term of finite
quantities, without changing Einstein's equation. Fields can go through
the singularity, beyond it.
So, I think Hawking's argument based on the ADS-CFT correspondence is compatible with the approach to the black hole singularities which I proposed, and excludes the standard solution, which is not time symmetric.
Last week, on December 6, 2013, I defended my PhD Thesis.
The Thesis is named Singular General Relativity, and can be found at arXiv:1301.2231.
Thesis Abstract:
This work presents the foundations of Singular Semi-Riemannian Geometry and
Singular General Relativity, based on the author's research. An extension of
differential geometry and of Einstein's equation to singularities is reported.
Singularities of the form studied here allow a smooth extension of the Einstein
field equations, including matter. This applies to the Big-Bang singularity of
the FLRW solution. It applies to stationary black holes, in appropriate
coordinates (since the standard coordinates are singular at singularity, hiding
the smoothness of the metric). In these coordinates, charged black holes have
the electromagnetic potential regular everywhere. Implications on Penrose's
Weyl curvature hypothesis are presented. In addition, these singularities
exhibit a (geo)metric dimensional reduction, which might act as a regulator for
the quantum fields, including for quantum gravity, in the UV regime. This opens
the perspective of perturbative renormalizability of quantum gravity without
modifying General Relativity.
The Thesis is based on a series of papers, from which the following are published or accepted:
[1] C. Stoica On Singular Semi-Riemannian Manifolds, Int. J. Geom. Methods Mod. Phys., 0(0):1450041, March 2014, arXiv:1105.0201.
[2] C. Stoica Schwarzschild Singularity is Semi-Regularizable, Eur. Phys. J. Plus, 127(83):1–
8, 2012, arXiv:1111.4837.
[3] C. Stoica Analytic Reissner-Nordstrom Singularity, Phys. Scr., 85(5):055004, 2012, arXiv:1111.4332.
[4] C. Stoica Kerr-Newman Solutions with Analytic Singularity and no Closed Timelike Curves, To appear in U.P.B. Sci. Bull., Series A, arXiv:1111.7082.
[5] C. Stoica Spacetimes with Singularities, An. St. Univ. Ovidius Constanta, 20(2):213–238, July 2012, arXiv:1108.5099.
[6] C. Stoica Einstein Equation at Singularities, Cent. Eur. J. Phys., 12 (2014), 123-131, arXiv:1203.2140.
[7] C. Stoica Beyond the Friedmann-Lemaitre-Robertson-Walker Big Bang singularity, Commun. Theor. Phys., 58(4):613–616, March 2012, arXiv:1203.1819.
[8] C. Stoica On the Weyl Curvature Hypothesis, Annals of Physics, 338:186–194, November 2013, arXiv:1203.3382.
[9] C. Stoica The Geometry of Black Hole Singularities, Advances in High Energy Physics, Volume 2014 (2014), Article ID 907518 arXiv:1401.6283.
[10] C. Stoica Metric dimensional reduction at singularities with implications to Quantum Gravity, Annals of Physics 347C (2014), pp. 74-91,arXiv:1205.2586.
Others are not yet published:
[11] C. Stoica Warped Products of Singular Semi-Riemannian Manifolds, arXiv:1105.3404.
[12] C. Stoica Cartan's Structural Equations for Degenerate Metric, Balkan J. Geom. Appl., Vol. 19, No. 2, (2014), p. 118-126, arXiv:1111.0646.
[13] C. Stoica Big Bang singularity in the Friedmann-Lemaitre-Robertson-Walker spacetime, arXiv:1112.4508.
After I reviewed briefly the so-called black hole wars, and expressed my doubts about black hole complementarity, there are still many things to be said. However, I would like to skip over various solutions proposed in the last decades, and discuss the one that I consider most natural.
All the discussions taking place within the last year around black hole complementarity and firewall are concentrated near the event horizon. But why looking for the information at the event horizon, when it was supposed to be lost at the singularity?
Remember the old joke with the policeman helping a drunk man searching his lost keys under a streetlight, only to find later that the drunk man actually lost them in the park? When asked why did he search the keys under the streetlight, the drunk man replied that in the park was too dark. In science, this behavior is called the streetlight effect.
By analogy, the dark place is the singularity, because it is not well understood. The lightened place is the event horizon. This is Schwarzschild's equation describing the metric of the black hole:
where ${d}\sigma^2 = {d}\theta^2 + \sin^2\theta {d} \phi^2$ is the metric of the unit sphere $S^2$, $m$ the mass of the body, and the units were chosen so that $c=1$ and $G=1$.
Schwarzschild's metric has two singularities, one at the event horizon, and the other one at the "center".
But in coordinates like those proposed by Eddington-Finkelstein, or by Kruskal-Szekeres, the metric becomes regular at the event horizon, showing that this singularity is due to the coordinates used by Schwarzschild. Fig. 1. represents the Penrose-Carter diagram of the Schwarzschild black hole. The yellow lines represent the event horizon, and we see that the metric is regular there.
Figure 1. Penrose-Carter diagram of the Schwarzshild black hole.
While at the event horizon the darkness was dispersed by finding appropriate coordinates, it persisted at the central singularity, represented with red. This is a spacelike singularity, and it is not actually at the center of the black hole, but in the future. This kind of singularity could not be removed completely, because it was not due exclusively to the coordinates.
However, in my paper Schwarzschild Singularity is Semi-Regularizable, I showed that we can eliminate the part of the singularity due to coordinates, by the transformation $r = \tau^2$, $t = \xi\tau^4$. The Schwarzshild metric in the new coordinates becomes
The metric is still singular, because it is degenerate, but the coordinate singularity was removed. The metric extends analytically through the singularity $r=0$, and the Penrose-Carter diagram becomes as in Fig. 2.
Figure 2. Penrose-Carter diagram of the extended Schwarzshild black hole.
In the new coordinates, the singularity behaves well. Although the metric is degenerate at the singularities, in arXiv:1105.0201 I showed that this kind of metric allows the construction of invariant geometric objects in a natural way. These objects can be used to write evolution equations beyond the singularity.
The Schwarzschild metric is eternal, but in the case relevant to the problem of information loss, the black hole is created and then evaporates. The analytic extension through the singularity presented earlier also works for this case, and the Penrose-Carter diagram is shown in Fig. 3.B.
Figure 3. A. Penrose diagram for the evaporating black hole, standard scenario. B. Penrose diagram for the evaporating black hole, when the solution is analytically extended through the singularity (as in arXiv:1111.4837). In the new solution, the geometry can be described in term of finite quantities, without changing Einstein's equation. Fields can go through the singularity, beyond it.
Information is no longer blocked at the singularity. The physical fields can evolve beyond the singularity, carrying the information, which is therefore recovered if the black hole evaporates.
This is not a modification of General Relativity, it is just a change of variables. The proposed objects remain finite at singularity, and the standard equations can be rewritten in terms of these new, finite objects. These objects are natural, and don't require a modification of Einstein's General Relativity. The proposed fix is not made by changing the theory, but by changing our understanding of the mathematics expressing the theory.
One principal inspiration for me when finding this solution is the work of David Finkelstein, especially the brilliant solution to the problem of the apparent singularity on the event horizon. Imagine how happy I was when I received by email, at the end of December 2012, the following encouragements from him:
Dear Cristi Stoica, I write concerning your paper "Schwarzschild Singularity is Semi-Regularizable" (arXiv 1111.4837v2). I write first to thank you for the deep pleasure that this paper afforded me. Your regularization of the central true singularity of the Schwarzschild metric is a remarkable and beautiful example of thinking outside the box. It is a natural, generally covariant, and deep result on a problem that has drawn wide attention, that of gravitational singularities. You found your solution easily once you conceived the idea, and yet it has been overlooked for these many decades by the truly great minds in the field. [...] With good wishes for your future explorations, David Finkelstein
In this post, which continues Black Hole Paradox 1. Susskind vs. Hawking, I will explain my reasons for not accepting the black hole complementarity principle (BHC). I will argue that, in the process of inventing this principle, Susskind, Thorlacius and Uglum (STU) found an important result, but ignored it.
The principle of information conservation was in
fact what it had to be proven for the case of black hole information. To
allow conservation, STU assumed that an external observer will see
that information in Hawking evaporation. They assumed implicitly that
information remains outside the event horizon, at least for an external
observer. So, they replaced implicitly 1. with
1'. The principle of information conservation by avoiding falling in the black hole.
The equivalence principle is the fundamental principle in General Relativity. It states that inertia and gravity are two faces of the same coin. Accelerated motion behaves as gravity, and gravity is due to the fact that spacetime is curved, so that reference frames cannot be without acceleration.
The quantum xerox principle is in fact the no-cloning theorem, which states that unknown quantum states cannot be copied. If we would be able to copy an unknown quantum state, then linearity of Quantum Mechanics would be violated. It is amazing that this simple but profound result was discovered only 30 years ago, given that the proof is so simple. There is a funny story about this. Asher Peres was anonymous reviewer for Foundations of Physics, and refereed a paper in which superluminal communication was predicted in Quantum Mechanics. He explained in the report that the result must be wrong, and even the author is aware. However, realizing that this mistaken result would stimulate the research, and a more important result would follow from this, he recommended publication. His intuition was right.
Assume Alice dives into the black hole. For an external observer Bob, she never reaches the event horizon. This is how the things look, according to General Relativity, from Bob's point of view. In Bob's coordinates, Alice never reaches the horizon, because GR
predicts it gets closer slower and slower, like in a Zeno paradox. But in Alice's reference frame, she crosses the horizon in a finite time. This apparent contradiction is due to the different coordinates used by Alice and Bob. Bob's coordinates are singular at the horizon. So he is wrong, Alice crosses the horizon in finite time, but because he is accelerating continuously to avoid falling in the black hole, there is a redshift of the light coming from Alice, so that in Bob's frame, her time stops.
Moreover, because Bob sees the event horizon as being hot, he would see Alice being vaporized. This would be OK from Bob's point of view, because other wise he would experience violation of the no-cloning theorem. But this also takes place in Bob's coordinate system, which is singular at the event horizon. So, he should again be wrong. However, let's go with STU and assume that Bob is right.
But the equivalence principle implies that Alice would not experience something special when she would cross the horizon. So, in fact, the information describing her would cross the event horizon.
This amounts to an apparent contradiction between what Bob sees, and what Alice experiences. On the other hand, STU want that the information describing Alice remains outside the horizon. This can't be done, unless the information is cloned, one copy going with Alice, and the other remaining available to Bob in the Hawking radiation.
For me, this is a proof that 1' is wrong. Admitting that 1' is true, we have to choose between no-cloning and the equivalence principle. Everybody agreed that we should not contradict these two principles. This means that the hypothesis that information survives by remaining outside the horizon, was wrong. Please note that this doesn't mean that 1 is wrong, only that 1' is wrong. 1 and 1' are not equivalent, although 1' implies 1. In other words, information may be preserved, but not as STU wanted.
I think this is a great result, found by STU, but they decided to ignore it. They didn't stop here. They didn't want to give up 1', because they believed that the only way to save information is this. In other words, he believed that 1 is equivalent to 1'. Because this led them to contradiction, they decided to accept 1' together with the contradiction. The way was to admit cloning of the information so that it is shared by Alice and Bob, but to claim in the same time that this is not violation of 2.
STU saw that there is a contradiction between Alice and Bob, so they decided to
apply the solution from the Sufi joke with Mulla Nasrudin, and agree with both of them. But, unlike the dervish, they did not go beyond dualism, and proposed instead the black hole complementarity. Essentially, it said that, even though Alice has a copy, and Bob has a copy, this doesn't contradict the no-cloning theorem, because Alice can't see Bob's copy, and vice-versa.
Now, call this however you want, but to me, it's a contradiction. Susskind even claimed that in fact this is just Bohr's complementarity, applied to this new case. It is true that Bohr stretched his idea of complementarity, until he saw it everywhere, and others stretched it more. But there is no connection between Bohr's and STU's complementarity. In Bohr's complementarity, there is no contradiction. Sometimes light behaves like waves, sometimes like point particles, but this is not a contradiction. If in a particular experiment, light behaves like waves for Alice, it does the same for Bob.
STU said that Alice and Bob can never meet, to compare their notes, hence there will be no proof that the no-cloning was violated. In other words, Nature can break her own laws whenever she wants, if we can't catch her in the act.
But, a question was raised, what if Bob dives into the black hole, following Alice, to compare their observations? Susskind found relatively quickly an answer to this: before Bob meeting Alice, they will be destroyed by the singularity. Indeed, calculations for Schwarzschild black holes show that Susskind is right about this.
But what if the black hole has the tiniest electric charge or rotation? In this case, the singularity is not spacelike, as in the Schwarzshild black hole. The singularity is timelike, and Alice and Bob can, in principle, avoid for indefinite time to reach it. So, there is plenty of time to meet and compare their notes. For some reason, this situation is never mentioned, only the Schwarzschild black hole case, for which there is an answer.
There is another reason why I disagree with BHC: it violates the equivalence principle. I explained this already in 2011. Ironically, although BHC was invented to allow 2 coexist with 1' and 3, it actually contradicts 2. Here is why. According to the equivalence principle, an experiment involving gravity should give the same result as an experiment in which we replace gravity with acceleration. Consider for example that Alice is moving inertially (free-fall motion), and Bob's frame is accelerated. This can happen at the black hole, when Bob sees Alice crossing the event horizon, while his accelerated motion helps him avoid falling. But it can happen somewhere far from any black holes. In this case, due to his acceleration, Bob will see something similar to the event horizon - the Rindler horizon. If he will see Alice crossing the Rindler horizon, he will see her evaporating. This is the equivalent of what happens in the case of a black hole, according to the equivalence principle. There is one big difference from the case when Alice falls in a Schwarzschild black hole: if Bob goes after her, he will find her alive and in good health. He will realize that she was not destroyed when she crosses the Rindler horizon. So, the equivalence principle tells us that even though Bob sees Alice being destroyed near the event horizon, he is again wrong, as it was in the case of the Rindler horizon. Hence, we have to choose between BHC and the equivalence principle.
Last year (in 2012), Almheiri, Marolf, Polchinski, and Sully (AMPS) wrote the paper Black Holes: Complementarity or Firewalls?, in which they show, by a different argument, that BHC doesn't solve the problem. They propose instead that Alice is actually destroyed at the horizon, by a firewall (formerly considered by Susskind, who called it "brick wall"). The price paid is that this sacrifices the equivalence principle.
So, if AMPS are right, and the solution is to admit the firewall, then why should we keep BHC? It is sometimes answered that BHC is still needed, to explain why Bob sees Alice never crossing the horizon, while she actually crosses her, in a finite proper time. But, as I explained, this is just an effect of GR, due to the fact that Bob's coordinates are singular at the horizon.
All
the discussions taking place within the last year around black hole
complementarity and firewall are concentrated near the event horizon.
Information is supposed to be destroyed by the singularity, but it is
hoped that, somehow, the event horizon plays the major role in
recovering it. Black hole complementarity is based on the idea that
Nature makes a backup of the information on the stretched horizon. The
firewall proposal suggests that the event horizon is a shield that burns
whatever may fall in the black hole, in order to make the information
immortal.
To me, these are a Deus ex machina kind of explanations; it appears as if the supporters of
these ideas see a purpose in the universe, and that purpose is eternal
life for the information falling into the black hole, at any costs. It
looks like God found a problem after he patched together General
Relativity and Quantum Mechanics, and decided to fix it somehow. For
instance, if God was a programmer, he would make a backup of the
information on the horizon, to fix the memory leak caused by the
singularities. Or, if God was a plumber, he would connect a pipe at the
event horizon, to deviate information and prevent it leaking through the
singularity. Fixing a bug, or a leak, would reveal intention in
creating the universe, a watchmaker who made an imperfect work and then
repaired it using an improvisation.
Most part of this post I explained why I don't buy BHC. I also said that, during the process, STU found that 1' contradict 2 and 3, and that I consider this the correct result, and the attempt to remove the contradiction by embracing and giving it a name, did not actually remove it. So, my main point was to explain that in fact to save the lost information, copying it at the horizon is not the solution. I also don't think it is a solution to break the principle of equivalence, by building a firewall in a place where GR and QFT work well. In fact, as I will explain in a future post, I think that all this endeavor was misguided: why search the lost information in another place than that where was lost? Giving up this assumption will reveal that there is no contradiction between 1, 2, and 3 on the event horizon, without having to invoke mystical principles like no contradiction is a contradiction, until it is an observed contradiction. We will see this in the next post, named Look for the information where you lost it.
Stephen Hawking showed that there is a big problem with the information, if black holes are present. If singularities swallow matter without returning it, information vanishes too. If the initial state of a system is pure, after a part of the system falls in the black hole, it is entangled with the part remaining outside. After the part falling inside vanishes in the singularity, the outside part remains in a mixed state (its state depends on the state of the inside of the black hole). This was not considered to be a problem before Hawking's analysis, because it was assumed that the infalling part is there, somewhere inside the black hole (I represented this in figure 1, A., in a Penrose-Carter diagram). It became an issue after Hawking (and his precursors, Zel'dovich and Starobinski) found that the black holes evaporate. The problem was that the black hole may completely evaporate, vanishing with the information it swallowed, and the outside part remains in a mixed state. This means that the information is lost forever, and the unitary evolution is broken (fig. 1, B.).
Figure 1. A. In the case of a non-evaporating black hole, by choosing an appropriate foliation of spacetime in space+time, unitarity is not broken. B. Unitarity is broken for an evaporating black hole, because when the initial state is pure, the final state is mixed.
Hawking's result was bad news for many physicists of good-faith. Hawking obtained it by combining the most trusted and proven theories we know, General Relativity and Quantum Theory. So, his result became acknowledged as a paradox, because he seemed to force us to choose between things we considered to be true. On the one hand, Quantum Theory is based on unitary evolution, and we don't want to be broken. On the other hand, Hawking impeccably combined GR and QT to prove that the radiation obtained during evaporation doesn't carry information out of the black hole.
Two sides were formed. While many didn't accept the idea that information is lost, another side, made mostly of General Relativists (who also happens to support Quantum Field Theory), were more willing to accept that information is lost. Among those who don't think that violation of unitarity is such a catastrophe that would destroy the world, there were Hawking, Kip Thorne, Roger Penrose, who explains his position in Cycles of Time, also Robert Wald in his recent talk at the conference Fuzzorfire workshop, seems not to be bothered by this, etc. The side which rejected the possibility that information may be lost was made of John Preskill, Leonard Susskind, Gerard 't Hooft, and others. In 1997, Thorne and Hawking bet against Preskill that information is really lost in the black hole.
Damn it, I don't want to talk about CGHS or RST. It's a dead end. I want to do something that really will shake things up. Let's go way out on a limb and say something very bold that will really get their attention.
They did. They proposed a very intriguing solution to the black hole information paradox, described in the paper "The Stretched Horizon and Black Hole Complementarity". It is also explained very well in the book L. Susskind, J. Lindesay, An introduction to black holes, information and the string theory revolution, World Scientific, 2005. I will discuss in more detail this solution in the next post. Their ideas were very clever, and convinced a large part of the community of physicists that it was the correct solution.
In fact, Hawking was
probably not convinced by the black hole complementarity, but rather by Maldacena'sAdS/CFT correspondence, and wrote a paper
in which he explained his own solution saving the information. In the paper, Hawking didn't need to use black hole complementarity, but he mentioned Maldacena's results, which rely on Susskind's and 't Hooft holographic principle.
In his proposal, Hawking uses the
method of sum over histories, originated by Feynman, but developed as an
approach to quantum gravity by himself and J. B. Hartle. To solve the
information loss problem, he proposes that one should sum over
topologies of spacetime. While it is clear that the trivial topologies,
those without black holes, don't break unitarity, the non-trivial ones,
including those containing black holes, break it. So, Hawking tries to
argue that the non-trivial topologies don't contribute to the sum over
histories. In other words, if a possible history has black holes,
information is lost, but if we consider all of them, it is not lost,
because the histories losing information don't contribute to the overall
sum.
His solution was criticized, for instance for not being well supported mathematically, by John Preskill.
Indeed, Hawking's paper from 2005 looks rather like a sketch of a research program, where
the key points are merely conjectured. Not much progress was made in that direction since then.
My
main objection to his proposal is the following. In general, when we sum over
histories, we have to impose some boundary conditions. For example, in
the case of the two-slit experiment, we take into consideration only
paths that go through the slits. Similarly, if we want to see what
happens with information in the presence of black holes, using the sum
over histories, one should impose conditions compatible with the
presence of the black hole. Or, Hawking claims that the only
contributions are given precisely by the histories which are not
compatible with the presence of black holes, and ignores exactly the histories
which actually should be considered.
When making his proposal, it seems that Hawking was not aware neither that he was the "evil side" in a black hole war, nor that Susskind defeat him. Recently, at the Fuzzorfire workshop, Hawking asked for the definition of the stretch horizon, to which Susskind replied that he
already told the definition 20 years ago, then left (minute 45).
I think that, when Hawking raised the problem of information loss, he did a great job. This is a very good problem indeed, and fueled plenty of research. In following posts, I will argue that in this process, Susskind did an excellent job too, by finding something very important, in my opinion, but then he lost it, by inventing the black hole complementarity principle. Next, in Stretched Complementarity, I will explain why I don't buy Susskind's solution.
