Lubos Motl
Jul27-04, 03:07 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>Hawking\'s 2004 solution of the information loss puzzle\n========================================== ============\n\nThis text is based on John Baez\'s (week 207) description of Hawking\'s\nlecture. John Baez\'s essays are available on sci.physics.research and\nelsewhere on the web. Everyone is welcome to reply this message.\n\nI will try to extract the important points, and comment on them. Because\nHawking.s answers seem to be much more sensible than what I\'ve expected\n(although they may be disappointing for many because they sound rather\nordinary), I will also try to describe them as if I agree with them.\n\nHawking used his lecture to surrender his bet against John Preskill - a\nbet described in The Elegant Universe as well as in other places - Hawking\nand Kip Thorne thought that information was lost, while Preskill said it\nwas not lost (much like other physicists who respect the laws of quantum\nmechanics as something that is extremely hard to modify or generalize).\nPreskill has now received a Baseball Encyclopedia from Hawking, in front\nof hundreds of witnesses, but both Preskill as well as Thorne need more\ntime to study Hawking\'s 2004 answer. Thorne has not given up yet.\n\n\nThe rough answer\n================\n\nHawking was obviously affected by Maldacena\'s AdS/CFT correspondence which\nis (together with Matrix theory) a very explicit framework in which it\nseems obvious that the information must be preserved after black holes\nform and evaporate: according to AdS/CFT, a gravitational theory (string\ntheory in anti de Sitter space times something) is equivalent to a\nconformal field theory (usually gauge theory) defined on the boundary of\nthe anti de Sitter space. Because the conformal field theory is manifestly\nunitary (information preserving), so must be the gravitational theory.\nTherefore, Hawking changed his mind. Today he agrees that the information\nis preserved - much like most of us who respect quantum mechanics as a\nprinciple that will stay with us. It is not hard to argue that a gauge\ntheory is unitary, but the real issue is why the information is conserved\neven from the viewpoint of the gravitational theory.\n\n\nThe problem - basics\n====================\n\nThe simplest context is the calculation of the scattering amplitudes\nassociated with a process in which incoming large energy particles produce\na large black hole (in Minkowski space, to be specific). The black hole\nlives for a long time, but eventually it evaporates completely. The final\nproduct is a large number of Hawking (nearly thermal) particles.\n\nAccording to Hawking\'s opinions in the 1970s, the black hole, after it is\ncreated, is determined by its mass, charge, and angular momentum (well,\nthis no-hair-theorem is due to many others). It carries no information and\nit decays into a precise thermal (mixed) state, as Hawking thought. An\nevolution of the initial pure state into the final mixed state is\nnon-unitary and it deletes most of the information - which is what Hawking\nused to believe to happen.\n\nString theory, on the other hand, was picturing the processes involving\nthe black holes simply as new examples of quantum mechanical processes in\nwhich all principles of quantum mechanics may be preserved and in which\ngravity can be treated, once again, much like other forces. The correct\nentropy can be computed as the logarithm of the number of microstates, for\nexample (as Strominger and Vafa showed), and string theory did not seem to\nrequire any modifications of quantum mechanics. (Andy Strominger would\ndisagree with this reasoning, but most others agree that Andy did not\nappreciate the importance of his finding, much like Planck, Einstein, and\nothers, and these findings can be viewed as strong evidence that quantum\nmechanics should be preserved as it is.) The formation and evaporation of\na black hole is just another scattering process, described by the S-matrix\n- the (hopefully unitary) table containing the scattering amplitudes\ndefined between the initial states at t=-infinity and t=+infinity.\n\nSuch an S-matrix can in principle be calculated in string theory (perhaps\nusing AdS/CFT or Matrix theory, if we need a nonperturbative answer).\nString theory does not really know whether there was a black hole inside.\nFor string theory, a black hole is a metastable, long-lived, degenerate\nintermediate resonance whose contributions to the S-matrix are\nautomatically included.\n\nThe real problem occurs from a geometric viewpoint. If we imagine that the\nscattering process takes place on the background of an evaporating black\nhole, we may visualize the causal relations in such a spacetime by a\nspecific (classical) Penrose diagram. Let me draw what a usual physicist\nwould draw at this point.\n\n|\\\n+------+ \\ Penrose diagram for an evaporating Schw. black hole:\n| / \\ All diagonal lines should be tilted by 45 degrees.\n| /\n| / / The diagonal line in the middle is the horizon.\n| / The hor. line is the final singularity inside the hole.\n| / /\n| / Once you cross the horizon, it seems that you\n| / and the information about you can never get outside\n| / (to the diagonal line in the upper-future right corner).\n| /\n| / Bottom is the past; top is the future.\n|/ The triangle in the left upper corner is the BH interior\n\nIn the case of this process, we used to believe that the amplitude was\ndominated by the configurations close to the Penrose diagram of an\nevaporating black hole. Such a diagram makes it clear that the information\nthat has crossed the horizon has no chance to escape to infinity unless\ncausality is violated. Therefore Hawking and others used to say that the\ninformation must be lost; it is eaten by the final singularity inside the\nblack hole. The final radiation can carry no information about the initial\nstate, he used to say.\n\nWhat about his new statement?\n\n\nFirst Hawking.s critical comment\n================================\n\nLet me first say that I fully agree with the sentences below.\n\nQuantum mechanics (Feynman\'s edition) dictates us to sum over all\nhistories that connect the initial state with the final state. Even if we\nbelieve, based on our classical intuition, that we are describing a\nprocess dominated by the black hole topology (see the Penrose diagram\nabove), in the quantized-metric approach to gravity we must still\ncarefully sum over all acceptable topologies.\n\nIf the initial and the final state is constructed from a finite number of\nparticles in empty space, the simplest spacetime topology is the flat\nMinkowski space. We are not allowed to forget the trivial topology,\ncorresponding to processes where no real black hole seems to be created in\nthe middle.\n\n\nSecond Hawking\'s step\n=====================\n\nWell, the histories where no black hole is created, are those that behave\nmuch like in ordinary quantum field theory in the Minkowski space. These\nhistories themselves don\'t erase any information because the spacetime can\nbe sliced beautifully, and the evolution between the slices is described\nby a hermitean Hamiltonian. (Hawking made a comment that Maldacena did not\nunderstand that the possibility to slice spacetime to spacelike slices in\na topologically trivial fashion implies unitarity via the Hamiltonian, but\nI am not sure whether I would believe that Hawking understood Maldacena\nwell.)\n\nThe real problem are the non-trivial spacetime topologies - with the black\nholes - that we used to consider as the dominating ones. Hawking now wants\nto argue that the nontrivial topologies actually contribute zero to the\nS-matrix between all acceptable initial and final states. If this is the\ncase, tbe information is obviously conserved because it is only the\ntopologically trivial histories (spacetimes) that contribute to the\nS-matrix, and these histories are slicable and unitary much like in the\nmost ordinary quantum field theories. (Hawking also discusses the case of\nthe eternal black holes where the non-trivial topology DO contribute, and\ntherefore he claims that the information IS lost in these cases.)\n\n\nA technicality: Euclidean gravity\n=================================\n\nAlth ough I was talking about "topologically trivial" spacetimes as if they\nwere spacetimes with signature -+++, it is not how Hawking wants to think\nabout it. The topology of an object with Lorentzian signature is a subtle\nthing. It is much easier to define a topology of a purely Euclidean\nmanifold i.e. a manifold with signature ++++. We are using technology of\nthe Wick rotation everywhere in quantum field theories. Although it is\nmuch more subtle in gravitational theories, we are using the summation\nover Euclidean spacetime topologies even in string theory. More\nconcretely, the stringy worldsheet is described by a two-dimensional\ngravitational theory, and the spacetime S-matrix is obtained as a\nsummation over Riemann surfaces of all genera. Note that it would be much\nmore controversial to define the Lorentzian worldsheets and their\ntopologies.\n\nTherefore, the word "nontrivial" in the previous sections should always be\nunderstood as "nontrivial topologies of a Euclidean ++++ spacetime". Note\nthat we focused on a process that asymptotically looks like a flat\nMinkowski spacetime both in the past and the future. This is a situation\nanalogous to quantum field theory where the Wick rotation is a legitimate\nstep that actually makes the objects better-defined mathematically.\nNevertheless it opens a plethora of physical questions: can we still\nidentify a classical Euclidean spacetime configuration with a classical\nMinkowski spacetime configuration, simply by a sort of analytical\ncontinuation? Is not there a lot of ambiguities which contours determine\nsuch analytical continuations?\n\n\nDo the nontrivial topologies contribute zero?\n=========================================== ==\n\nI am open-minded about this important step, but I find it very acceptable.\nIn order to show that the information is preserved, Hawking needs to show\nthat the contribution of the ugly, black-hole-like, non-trivial\n(Euclidean) spacetime topologies to reasonable scattering amplitudes\nvanishes.\n\nRecall that a Schwarzschild black hole, for example, quickly becomes\nspherical after it is created. The deviations from sphericity decrease\nexponentially with time. In fact, this process may be, roughly speaking,\ndescribed by the quasinormal (ringing) modes; the mode with the smallest\nimaginary part is the most important (slowest-decaying) one. The deviation\nfrom the sphericity can be thought of as the amount of information that\nhas not disappeared, and we see it decreases exponentially. This is a\nvisualization of the process how the information was supposed to\ndisappear.\n\nBut Hawking now argues that the same exponential decrease of the\ndeviations from sphericity - and the related exponential decrease of many\ncorrelation functions in time - can be used to argue that the non-trivial\ntopologies simply do not contribute - or perhaps contribute a tiny\ncontribution comparable to the exponential of minus the black hole\nlifetime, so to say. There are too many exponentially small and\nexponentially large factors in the game, and I would prefer to see a more\nrigorous argument, too. But Hawking seems to say that because the\ndeviations (the initial field configuration) decay exponentially with\ntime, and the time is large enough (comparable to the hole\'s lifetime),\nthey must have a (nearly) zero overlap with a collection of the final\nparticles that are normalized properly. Well, maybe.\n\nHawking has another pretty original interpretation of AdS/CFT and some of\nits concepts. He wants to talk about gravity coupled to a large number N\nof matter fields - the AdS/CFT correspondence is a well-known example but\nnot necessarily the example he wants to think about primarily - and\nanalyze the simplifications in the large N limit. The first\nsimplification, he claims, is that the gravitational fluctuations go to\nzero as N goes to infinity because a different scaling of the loops with\nN. In this limit, he argues (I think), it is therefore possible to\nintegrate out gravity first because it is treated as a heavy, mildly\noscillating field (do I understand him?), giving the effective action of\nthe CFT. As a byproduct, Hawking seems to claim that he has a new argument\nwhy the computation of the CFT effective action is equivalent to solving\nthe classical SUGRA equations - both of these things can be obtained from\na system of gravity coupled to a large number of fields where the gravity\ncan be integrated out (because the number of matter fields is large). I\nstill have not absorbed how it works but this insight itself might be a\npretty novel way how to justify that AdS/CFT (especially the counting of\nthe correlators) works, even though at the end it may turn out to contain\nonly as much as we know.\n\n\nSo which configurations dominate the path integral?\n======================================= ============\n\nI am not sure whether Hawking has an answer to this question. Once we\nimagine that the S-matrix is determined by the histories with trivial\ntopology only, how do we satisfy our classical intuition that simply tells\nus that the spacetime "mostly" looks like the non-trivial Penrose diagram\nof an evaporating black hole? Does it mean that the topologically trivial\nspacetime that are "close" to the topologically non-trivial ones\ncontribute most of the amplitude? Well, it is possible. In fact, one can\nchange the spacetime metric a little bit, and the Penrose diagram changes\ndrastically. Because it is the spacetime metric that is more physical, one\nshould be careful in deriving too far-reaching implications from the\nPenrose diagram. (I would tend to agree with this statement, too, if\nsomeone said it explicitly.) A spacetime that is pretty close to the\nevaporating black hole - at least outside the hole - might have a\ntopologically trivial Penrose diagram, much like the flat spacetime. If\nHawking says this, I am the first one who will try to believe it.\n\n\nWhat about the observers in the BH interior?\n======================================= =====\n\nI think that it is pretty obvious that Hawking\'s current approach does not\nallow one to say anything about the observations of the unlucky observers\nwho fell inside the black hole. Will they survive the fall behind the\nhorizon? Will they be able to detect it? If they survive it, what about\nthe rest of their finite-time physics because they\'re killed near the\nsingularity? These are questions that probably can\'t be addressed in this\nHawking\'s picture, and these special observers would have to adopt a new\npicture of physics - perhaps one that is not exact and that could involve\ninformation loss.\n\n\nIncompatibility with competing explanations?\n=================================== =========\n\nLet me mention two alternatives that have recently appeared in literature.\nMaldacena+Horowitz argued that the black hole final state is unique, and\nthere is some anti-causal behavior inside the black hole that allows the\ninformation to go back in time, so to say, recoil from the singularity,\nand return to the horizon and outside. Preskill and his collaborator (I\nreally apologize for forgetting the name) argued that once the\ninteractions between the particles inside the black hole that go forward\nAND backward in time - once these interactions are taken into account, the\nunitarity would be violated anyway, and Maldacena+Horowitz are therefore\nin trouble.\n\nSamir Mathur, Oleg Lunin (?) and perhaps some collaborators argued - by\nfinding impressive nontrivial solutions - for the existence of totally new\ndegenerate stringy classical configurations describing black holes that\nwould imply - if generalized to more usual black holes - that even the\nexistence of the horizon is problematic and the black hole interior looks\ntotally differently than we thought (something that needs nonlocal physics\nin the case of more conventional black holes). It is easy to see that\nMathur is incompatible with Maldacena+Horowitz because Maldacena+Horowitz\nrequire a pretty old-fashioned, "empty" black hole interior, while Mathur\net al. change the interior completely.\n\nIs Hawking\'s current picture compatible with these pictures? I can imagine\nthat Hawking 2004 is compatible with Maldacena+Horowitz if the "black hole\nfinal state. is something that undoes all knots on spacetime and returns\nthe trivial topology - although I see no consistent way how to draw such a\npossibility in the Penrose diagram. Hawking\'s proposal, as it gives no\ntool to probe the interior, also can be compatible with Mathur et al. who\npropose a very dense and modified black hole interior.\n\nSome answers might have been given, but they have opened many other\nquestions, I think. Your reactions will be appreciated, Lubos Motl\n____________________________________________ __________________________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)\n^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>Hawking's 2004 solution of the information loss puzzle
================================================== ====
This text is based on John Baez's (week 207) description of Hawking's
lecture. John Baez's essays are available on sci.physics.research and
elsewhere on the web. Everyone is welcome to reply this message.
I will try to extract the important points, and comment on them. Because
Hawking.s answers seem to be much more sensible than what I've expected
(although they may be disappointing for many because they sound rather
ordinary), I will also try to describe them as if I agree with them.
Hawking used his lecture to surrender his bet against John Preskill - a
bet described in The Elegant Universe as well as in other places - Hawking
and Kip Thorne thought that information was lost, while Preskill said it
was not lost (much like other physicists who respect the laws of quantum
mechanics as something that is extremely hard to modify or generalize).
Preskill has now received a Baseball Encyclopedia from Hawking, in front
of hundreds of witnesses, but both Preskill as well as Thorne need more
time to study Hawking's 2004 answer. Thorne has not given up yet.
The rough answer
================
Hawking was obviously affected by Maldacena's AdS/CFT correspondence which
is (together with Matrix theory) a very explicit framework in which it
seems obvious that the information must be preserved after black holes
form and evaporate: according to AdS/CFT, a gravitational theory (string
theory in anti de Sitter space times something) is equivalent to a
conformal field theory (usually gauge theory) defined on the boundary of
the anti de Sitter space. Because the conformal field theory is manifestly
unitary (information preserving), so must be the gravitational theory.
Therefore, Hawking changed his mind. Today he agrees that the information
is preserved - much like most of us who respect quantum mechanics as a
principle that will stay with us. It is not hard to argue that a gauge
theory is unitary, but the real issue is why the information is conserved
even from the viewpoint of the gravitational theory.
The problem - basics
====================
The simplest context is the calculation of the scattering amplitudes
associated with a process in which incoming large energy particles produce
a large black hole (in Minkowski space, to be specific). The black hole
lives for a long time, but eventually it evaporates completely. The final
product is a large number of Hawking (nearly thermal) particles.
According to Hawking's opinions in the 1970s, the black hole, after it is
created, is determined by its mass, charge, and angular momentum (well,
this no-hair-theorem is due to many others). It carries no information and
it decays into a precise thermal (mixed) state, as Hawking thought. An
evolution of the initial pure state into the final mixed state is
non-unitary and it deletes most of the information - which is what Hawking
used to believe to happen.
String theory, on the other hand, was picturing the processes involving
the black holes simply as new examples of quantum mechanical processes in
which all principles of quantum mechanics may be preserved and in which
gravity can be treated, once again, much like other forces. The correct
entropy can be computed as the logarithm of the number of microstates, for
example (as Strominger and Vafa showed), and string theory did not seem to
require any modifications of quantum mechanics. (Andy Strominger would
disagree with this reasoning, but most others agree that Andy did not
appreciate the importance of his finding, much like Planck, Einstein, and
others, and these findings can be viewed as strong evidence that quantum
mechanics should be preserved as it is.) The formation and evaporation of
a black hole is just another scattering process, described by the S-matrix
- the (hopefully unitary) table containing the scattering amplitudes
defined between the initial states at t=-infinity and t=+infinity.
Such an S-matrix can in principle be calculated in string theory (perhaps
using AdS/CFT or Matrix theory, if we need a nonperturbative answer).
String theory does not really know whether there was a black hole inside.
For string theory, a black hole is a metastable, long-lived, degenerate
intermediate resonance whose contributions to the S-matrix are
automatically included.
The real problem occurs from a geometric viewpoint. If we imagine that the
scattering process takes place on the background of an evaporating black
hole, we may visualize the causal relations in such a spacetime by a
specific (classical) Penrose diagram. Let me draw what a usual physicist
would draw at this point.
|\
+------+ \ Penrose diagram for an evaporating Schw. black hole:
| / \ All diagonal lines should be tilted by 45 degrees.
| /| / / The diagonal line in the middle is the horizon.
| / The hor. line is the final singularity inside the hole.
| / /| / Once you cross the horizon, it seems that you
| / and the information about you can never get outside
| / (to the diagonal line in the upper-future right corner).
| /
| / Bottom is the past; top is the future.
|/ The triangle in the left upper corner is the BH interior
In the case of this process, we used to believe that the amplitude was
dominated by the configurations close to the Penrose diagram of an
evaporating black hole. Such a diagram makes it clear that the information
that has crossed the horizon has no chance to escape to infinity unless
causality is violated. Therefore Hawking and others used to say that the
information must be lost; it is eaten by the final singularity inside the
black hole. The final radiation can carry no information about the initial
state, he used to say.
What about his new statement?
First Hawking.s critical comment
================================
Let me first say that I fully agree with the sentences below.
Quantum mechanics (Feynman's edition) dictates us to sum over all
histories that connect the initial state with the final state. Even if we
believe, based on our classical intuition, that we are describing a
process dominated by the black hole topology (see the Penrose diagram
above), in the quantized-metric approach to gravity we must still
carefully sum over all acceptable topologies.
If the initial and the final state is constructed from a finite number of
particles in empty space, the simplest spacetime topology is the flat
Minkowski space. We are not allowed to forget the trivial topology,
corresponding to processes where no real black hole seems to be created in
the middle.
Second Hawking's step
=====================
Well, the histories where no black hole is created, are those that behave
much like in ordinary quantum field theory in the Minkowski space. These
histories themselves don't erase any information because the spacetime can
be sliced beautifully, and the evolution between the slices is described
by a hermitean Hamiltonian. (Hawking made a comment that Maldacena did not
understand that the possibility to slice spacetime to spacelike slices in
a topologically trivial fashion implies unitarity via the Hamiltonian, but
I am not sure whether I would believe that Hawking understood Maldacena
well.)
The real problem are the non-trivial spacetime topologies - with the black
holes - that we used to consider as the dominating ones. Hawking now wants
to argue that the nontrivial topologies actually contribute zero to the
S-matrix between all acceptable initial and final states. If this is the
case, tbe information is obviously conserved because it is only the
topologically trivial histories (spacetimes) that contribute to the
S-matrix, and these histories are slicable and unitary much like in the
most ordinary quantum field theories. (Hawking also discusses the case of
the eternal black holes where the non-trivial topology DO contribute, and
therefore he claims that the information IS lost in these cases.)
A technicality: Euclidean gravity
=================================
Although I was talking about "topologically trivial" spacetimes as if they
were spacetimes with signature -+++, it is not how Hawking wants to think
about it. The topology of an object with Lorentzian signature is a subtle
thing. It is much easier to define a topology of a purely Euclidean
manifold i.e. a manifold with signature ++++. We are using technology of
the Wick rotation everywhere in quantum field theories. Although it is
much more subtle in gravitational theories, we are using the summation
over Euclidean spacetime topologies even in string theory. More
concretely, the stringy worldsheet is described by a two-dimensional
gravitational theory, and the spacetime S-matrix is obtained as a
summation over Riemann surfaces of all genera. Note that it would be much
more controversial to define the Lorentzian worldsheets and their
topologies.
Therefore, the word "nontrivial" in the previous sections should always be
understood as "nontrivial topologies of a Euclidean ++++ spacetime". Note
that we focused on a process that asymptotically looks like a flat
Minkowski spacetime both in the past and the future. This is a situation
analogous to quantum field theory where the Wick rotation is a legitimate
step that actually makes the objects better-defined mathematically.
Nevertheless it opens a plethora of physical questions: can we still
identify a classical Euclidean spacetime configuration with a classical
Minkowski spacetime configuration, simply by a sort of analytical
continuation? Is not there a lot of ambiguities which contours determine
such analytical continuations?
Do the nontrivial topologies contribute zero?
=============================================
I am open-minded about this important step, but I find it very acceptable.
In order to show that the information is preserved, Hawking needs to show
that the contribution of the ugly, black-hole-like, non-trivial
(Euclidean) spacetime topologies to reasonable scattering amplitudes
vanishes.
Recall that a Schwarzschild black hole, for example, quickly becomes
spherical after it is created. The deviations from sphericity decrease
exponentially with time. In fact, this process may be, roughly speaking,
described by the quasinormal (ringing) modes; the mode with the smallest
imaginary part is the most important (slowest-decaying) one. The deviation
from the sphericity can be thought of as the amount of information that
has not disappeared, and we see it decreases exponentially. This is a
visualization of the process how the information was supposed to
disappear.
But Hawking now argues that the same exponential decrease of the
deviations from sphericity - and the related exponential decrease of many
correlation functions in time - can be used to argue that the non-trivial
topologies simply do not contribute - or perhaps contribute a tiny
contribution comparable to the exponential of minus the black hole
lifetime, so to say. There are too many exponentially small and
exponentially large factors in the game, and I would prefer to see a more
rigorous argument, too. But Hawking seems to say that because the
deviations (the initial field configuration) decay exponentially with
time, and the time is large enough (comparable to the hole's lifetime),
they must have a (nearly) zero overlap with a collection of the final
particles that are normalized properly. Well, maybe.
Hawking has another pretty original interpretation of AdS/CFT and some of
its concepts. He wants to talk about gravity coupled to a large number N
of matter fields - the AdS/CFT correspondence is a well-known example but
not necessarily the example he wants to think about primarily - and
analyze the simplifications in the large N limit. The first
simplification, he claims, is that the gravitational fluctuations go to
zero as N goes to infinity because a different scaling of the loops with
N. In this limit, he argues (I think), it is therefore possible to
integrate out gravity first because it is treated as a heavy, mildly
oscillating field (do I understand him?), giving the effective action of
the CFT. As a byproduct, Hawking seems to claim that he has a new argument
why the computation of the CFT effective action is equivalent to solving
the classical SUGRA equations - both of these things can be obtained from
a system of gravity coupled to a large number of fields where the gravity
can be integrated out (because the number of matter fields is large). I
still have not absorbed how it works but this insight itself might be a
pretty novel way how to justify that AdS/CFT (especially the counting of
the correlators) works, even though at the end it may turn out to contain
only as much as we know.
So which configurations dominate the path integral?
================================================== =
I am not sure whether Hawking has an answer to this question. Once we
imagine that the S-matrix is determined by the histories with trivial
topology only, how do we satisfy our classical intuition that simply tells
us that the spacetime "mostly" looks like the non-trivial Penrose diagram
of an evaporating black hole? Does it mean that the topologically trivial
spacetime that are "close" to the topologically non-trivial ones
contribute most of the amplitude? Well, it is possible. In fact, one can
change the spacetime metric a little bit, and the Penrose diagram changes
drastically. Because it is the spacetime metric that is more physical, one
should be careful in deriving too far-reaching implications from the
Penrose diagram. (I would tend to agree with this statement, too, if
someone said it explicitly.) A spacetime that is pretty close to the
evaporating black hole - at least outside the hole - might have a
topologically trivial Penrose diagram, much like the flat spacetime. If
Hawking says this, I am the first one who will try to believe it.
What about the observers in the BH interior?
============================================
I think that it is pretty obvious that Hawking's current approach does not
allow one to say anything about the observations of the unlucky observers
who fell inside the black hole. Will they survive the fall behind the
horizon? Will they be able to detect it? If they survive it, what about
the rest of their finite-time physics because they're killed near the
singularity? These are questions that probably can't be addressed in this
Hawking's picture, and these special observers would have to adopt a new
picture of physics - perhaps one that is not exact and that could involve
information loss.
Incompatibility with competing explanations?
============================================
Let me mention two alternatives that have recently appeared in literature.
Maldacena+Horowitz argued that the black hole final state is unique, and
there is some anti-causal behavior inside the black hole that allows the
information to go back in time, so to say, recoil from the singularity,
and return to the horizon and outside. Preskill and his collaborator (I
really apologize for forgetting the name) argued that once the
interactions between the particles inside the black hole that go forward
AND backward in time - once these interactions are taken into account, the
unitarity would be violated anyway, and Maldacena+Horowitz are therefore
in trouble.
Samir Mathur, Oleg Lunin (?) and perhaps some collaborators argued - by
finding impressive nontrivial solutions - for the existence of totally new
degenerate stringy classical configurations describing black holes that
would imply - if generalized to more usual black holes - that even the
existence of the horizon is problematic and the black hole interior looks
totally differently than we thought (something that needs nonlocal physics
in the case of more conventional black holes). It is easy to see that
Mathur is incompatible with Maldacena+Horowitz because Maldacena+Horowitz
require a pretty old-fashioned, "empty" black hole interior, while Mathur
et al. change the interior completely.
Is Hawking's current picture compatible with these pictures? I can imagine
that Hawking 2004 is compatible with Maldacena+Horowitz if the "black hole
final state. is something that undoes all knots on spacetime and returns
the trivial topology - although I see no consistent way how to draw such a
possibility in the Penrose diagram. Hawking's proposal, as it gives no
tool to probe the interior, also can be compatible with Mathur et al. who
propose a very dense and modified black hole interior.
Some answers might have been given, but they have opened many other
questions, I think. Your reactions will be appreciated, Lubos Motl
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
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================================================== ====
This text is based on John Baez's (week 207) description of Hawking's
lecture. John Baez's essays are available on sci.physics.research and
elsewhere on the web. Everyone is welcome to reply this message.
I will try to extract the important points, and comment on them. Because
Hawking.s answers seem to be much more sensible than what I've expected
(although they may be disappointing for many because they sound rather
ordinary), I will also try to describe them as if I agree with them.
Hawking used his lecture to surrender his bet against John Preskill - a
bet described in The Elegant Universe as well as in other places - Hawking
and Kip Thorne thought that information was lost, while Preskill said it
was not lost (much like other physicists who respect the laws of quantum
mechanics as something that is extremely hard to modify or generalize).
Preskill has now received a Baseball Encyclopedia from Hawking, in front
of hundreds of witnesses, but both Preskill as well as Thorne need more
time to study Hawking's 2004 answer. Thorne has not given up yet.
The rough answer
================
Hawking was obviously affected by Maldacena's AdS/CFT correspondence which
is (together with Matrix theory) a very explicit framework in which it
seems obvious that the information must be preserved after black holes
form and evaporate: according to AdS/CFT, a gravitational theory (string
theory in anti de Sitter space times something) is equivalent to a
conformal field theory (usually gauge theory) defined on the boundary of
the anti de Sitter space. Because the conformal field theory is manifestly
unitary (information preserving), so must be the gravitational theory.
Therefore, Hawking changed his mind. Today he agrees that the information
is preserved - much like most of us who respect quantum mechanics as a
principle that will stay with us. It is not hard to argue that a gauge
theory is unitary, but the real issue is why the information is conserved
even from the viewpoint of the gravitational theory.
The problem - basics
====================
The simplest context is the calculation of the scattering amplitudes
associated with a process in which incoming large energy particles produce
a large black hole (in Minkowski space, to be specific). The black hole
lives for a long time, but eventually it evaporates completely. The final
product is a large number of Hawking (nearly thermal) particles.
According to Hawking's opinions in the 1970s, the black hole, after it is
created, is determined by its mass, charge, and angular momentum (well,
this no-hair-theorem is due to many others). It carries no information and
it decays into a precise thermal (mixed) state, as Hawking thought. An
evolution of the initial pure state into the final mixed state is
non-unitary and it deletes most of the information - which is what Hawking
used to believe to happen.
String theory, on the other hand, was picturing the processes involving
the black holes simply as new examples of quantum mechanical processes in
which all principles of quantum mechanics may be preserved and in which
gravity can be treated, once again, much like other forces. The correct
entropy can be computed as the logarithm of the number of microstates, for
example (as Strominger and Vafa showed), and string theory did not seem to
require any modifications of quantum mechanics. (Andy Strominger would
disagree with this reasoning, but most others agree that Andy did not
appreciate the importance of his finding, much like Planck, Einstein, and
others, and these findings can be viewed as strong evidence that quantum
mechanics should be preserved as it is.) The formation and evaporation of
a black hole is just another scattering process, described by the S-matrix
- the (hopefully unitary) table containing the scattering amplitudes
defined between the initial states at t=-infinity and t=+infinity.
Such an S-matrix can in principle be calculated in string theory (perhaps
using AdS/CFT or Matrix theory, if we need a nonperturbative answer).
String theory does not really know whether there was a black hole inside.
For string theory, a black hole is a metastable, long-lived, degenerate
intermediate resonance whose contributions to the S-matrix are
automatically included.
The real problem occurs from a geometric viewpoint. If we imagine that the
scattering process takes place on the background of an evaporating black
hole, we may visualize the causal relations in such a spacetime by a
specific (classical) Penrose diagram. Let me draw what a usual physicist
would draw at this point.
|\
+------+ \ Penrose diagram for an evaporating Schw. black hole:
| / \ All diagonal lines should be tilted by 45 degrees.
| /| / / The diagonal line in the middle is the horizon.
| / The hor. line is the final singularity inside the hole.
| / /| / Once you cross the horizon, it seems that you
| / and the information about you can never get outside
| / (to the diagonal line in the upper-future right corner).
| /
| / Bottom is the past; top is the future.
|/ The triangle in the left upper corner is the BH interior
In the case of this process, we used to believe that the amplitude was
dominated by the configurations close to the Penrose diagram of an
evaporating black hole. Such a diagram makes it clear that the information
that has crossed the horizon has no chance to escape to infinity unless
causality is violated. Therefore Hawking and others used to say that the
information must be lost; it is eaten by the final singularity inside the
black hole. The final radiation can carry no information about the initial
state, he used to say.
What about his new statement?
First Hawking.s critical comment
================================
Let me first say that I fully agree with the sentences below.
Quantum mechanics (Feynman's edition) dictates us to sum over all
histories that connect the initial state with the final state. Even if we
believe, based on our classical intuition, that we are describing a
process dominated by the black hole topology (see the Penrose diagram
above), in the quantized-metric approach to gravity we must still
carefully sum over all acceptable topologies.
If the initial and the final state is constructed from a finite number of
particles in empty space, the simplest spacetime topology is the flat
Minkowski space. We are not allowed to forget the trivial topology,
corresponding to processes where no real black hole seems to be created in
the middle.
Second Hawking's step
=====================
Well, the histories where no black hole is created, are those that behave
much like in ordinary quantum field theory in the Minkowski space. These
histories themselves don't erase any information because the spacetime can
be sliced beautifully, and the evolution between the slices is described
by a hermitean Hamiltonian. (Hawking made a comment that Maldacena did not
understand that the possibility to slice spacetime to spacelike slices in
a topologically trivial fashion implies unitarity via the Hamiltonian, but
I am not sure whether I would believe that Hawking understood Maldacena
well.)
The real problem are the non-trivial spacetime topologies - with the black
holes - that we used to consider as the dominating ones. Hawking now wants
to argue that the nontrivial topologies actually contribute zero to the
S-matrix between all acceptable initial and final states. If this is the
case, tbe information is obviously conserved because it is only the
topologically trivial histories (spacetimes) that contribute to the
S-matrix, and these histories are slicable and unitary much like in the
most ordinary quantum field theories. (Hawking also discusses the case of
the eternal black holes where the non-trivial topology DO contribute, and
therefore he claims that the information IS lost in these cases.)
A technicality: Euclidean gravity
=================================
Although I was talking about "topologically trivial" spacetimes as if they
were spacetimes with signature -+++, it is not how Hawking wants to think
about it. The topology of an object with Lorentzian signature is a subtle
thing. It is much easier to define a topology of a purely Euclidean
manifold i.e. a manifold with signature ++++. We are using technology of
the Wick rotation everywhere in quantum field theories. Although it is
much more subtle in gravitational theories, we are using the summation
over Euclidean spacetime topologies even in string theory. More
concretely, the stringy worldsheet is described by a two-dimensional
gravitational theory, and the spacetime S-matrix is obtained as a
summation over Riemann surfaces of all genera. Note that it would be much
more controversial to define the Lorentzian worldsheets and their
topologies.
Therefore, the word "nontrivial" in the previous sections should always be
understood as "nontrivial topologies of a Euclidean ++++ spacetime". Note
that we focused on a process that asymptotically looks like a flat
Minkowski spacetime both in the past and the future. This is a situation
analogous to quantum field theory where the Wick rotation is a legitimate
step that actually makes the objects better-defined mathematically.
Nevertheless it opens a plethora of physical questions: can we still
identify a classical Euclidean spacetime configuration with a classical
Minkowski spacetime configuration, simply by a sort of analytical
continuation? Is not there a lot of ambiguities which contours determine
such analytical continuations?
Do the nontrivial topologies contribute zero?
=============================================
I am open-minded about this important step, but I find it very acceptable.
In order to show that the information is preserved, Hawking needs to show
that the contribution of the ugly, black-hole-like, non-trivial
(Euclidean) spacetime topologies to reasonable scattering amplitudes
vanishes.
Recall that a Schwarzschild black hole, for example, quickly becomes
spherical after it is created. The deviations from sphericity decrease
exponentially with time. In fact, this process may be, roughly speaking,
described by the quasinormal (ringing) modes; the mode with the smallest
imaginary part is the most important (slowest-decaying) one. The deviation
from the sphericity can be thought of as the amount of information that
has not disappeared, and we see it decreases exponentially. This is a
visualization of the process how the information was supposed to
disappear.
But Hawking now argues that the same exponential decrease of the
deviations from sphericity - and the related exponential decrease of many
correlation functions in time - can be used to argue that the non-trivial
topologies simply do not contribute - or perhaps contribute a tiny
contribution comparable to the exponential of minus the black hole
lifetime, so to say. There are too many exponentially small and
exponentially large factors in the game, and I would prefer to see a more
rigorous argument, too. But Hawking seems to say that because the
deviations (the initial field configuration) decay exponentially with
time, and the time is large enough (comparable to the hole's lifetime),
they must have a (nearly) zero overlap with a collection of the final
particles that are normalized properly. Well, maybe.
Hawking has another pretty original interpretation of AdS/CFT and some of
its concepts. He wants to talk about gravity coupled to a large number N
of matter fields - the AdS/CFT correspondence is a well-known example but
not necessarily the example he wants to think about primarily - and
analyze the simplifications in the large N limit. The first
simplification, he claims, is that the gravitational fluctuations go to
zero as N goes to infinity because a different scaling of the loops with
N. In this limit, he argues (I think), it is therefore possible to
integrate out gravity first because it is treated as a heavy, mildly
oscillating field (do I understand him?), giving the effective action of
the CFT. As a byproduct, Hawking seems to claim that he has a new argument
why the computation of the CFT effective action is equivalent to solving
the classical SUGRA equations - both of these things can be obtained from
a system of gravity coupled to a large number of fields where the gravity
can be integrated out (because the number of matter fields is large). I
still have not absorbed how it works but this insight itself might be a
pretty novel way how to justify that AdS/CFT (especially the counting of
the correlators) works, even though at the end it may turn out to contain
only as much as we know.
So which configurations dominate the path integral?
================================================== =
I am not sure whether Hawking has an answer to this question. Once we
imagine that the S-matrix is determined by the histories with trivial
topology only, how do we satisfy our classical intuition that simply tells
us that the spacetime "mostly" looks like the non-trivial Penrose diagram
of an evaporating black hole? Does it mean that the topologically trivial
spacetime that are "close" to the topologically non-trivial ones
contribute most of the amplitude? Well, it is possible. In fact, one can
change the spacetime metric a little bit, and the Penrose diagram changes
drastically. Because it is the spacetime metric that is more physical, one
should be careful in deriving too far-reaching implications from the
Penrose diagram. (I would tend to agree with this statement, too, if
someone said it explicitly.) A spacetime that is pretty close to the
evaporating black hole - at least outside the hole - might have a
topologically trivial Penrose diagram, much like the flat spacetime. If
Hawking says this, I am the first one who will try to believe it.
What about the observers in the BH interior?
============================================
I think that it is pretty obvious that Hawking's current approach does not
allow one to say anything about the observations of the unlucky observers
who fell inside the black hole. Will they survive the fall behind the
horizon? Will they be able to detect it? If they survive it, what about
the rest of their finite-time physics because they're killed near the
singularity? These are questions that probably can't be addressed in this
Hawking's picture, and these special observers would have to adopt a new
picture of physics - perhaps one that is not exact and that could involve
information loss.
Incompatibility with competing explanations?
============================================
Let me mention two alternatives that have recently appeared in literature.
Maldacena+Horowitz argued that the black hole final state is unique, and
there is some anti-causal behavior inside the black hole that allows the
information to go back in time, so to say, recoil from the singularity,
and return to the horizon and outside. Preskill and his collaborator (I
really apologize for forgetting the name) argued that once the
interactions between the particles inside the black hole that go forward
AND backward in time - once these interactions are taken into account, the
unitarity would be violated anyway, and Maldacena+Horowitz are therefore
in trouble.
Samir Mathur, Oleg Lunin (?) and perhaps some collaborators argued - by
finding impressive nontrivial solutions - for the existence of totally new
degenerate stringy classical configurations describing black holes that
would imply - if generalized to more usual black holes - that even the
existence of the horizon is problematic and the black hole interior looks
totally differently than we thought (something that needs nonlocal physics
in the case of more conventional black holes). It is easy to see that
Mathur is incompatible with Maldacena+Horowitz because Maldacena+Horowitz
require a pretty old-fashioned, "empty" black hole interior, while Mathur
et al. change the interior completely.
Is Hawking's current picture compatible with these pictures? I can imagine
that Hawking 2004 is compatible with Maldacena+Horowitz if the "black hole
final state. is something that undoes all knots on spacetime and returns
the trivial topology - although I see no consistent way how to draw such a
possibility in the Penrose diagram. Hawking's proposal, as it gives no
tool to probe the interior, also can be compatible with Mathur et al. who
propose a very dense and modified black hole interior.
Some answers might have been given, but they have opened many other
questions, I think. Your reactions will be appreciated, Lubos Motl
__{_______________________________________________ _____________________________}
E-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
eFax: +1-801/454-1858 work: +1-617/496-8199 home: +1-617/868-4487 (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^