string theory on black hole entropy

Daniel

<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>lubos motl is a string theorist who had this to say about LQG\n\n"> Black hole entropy\n>\n> Despite various claims, loop quantum gravity is not able to calculate the\n> black hole entropy, unlike string theory. The fact that the entropy is\n> proportional to the area does not follow from loop quantum gravity. It is\n> rather an assumption of the calculation. The calculation assumes that the\n> black hole interior can be neglected and the entropy comes from a new kind\n> of dynamics attached to the surface area - there is no justification of\n> this assumption. Not surprisingly, one is led to an area/entropy\n> proportionality law. The only non-trivial check could be the coefficient,\n> but it comes out incorrectly (see the Immirzi discrepancy)."\n\nonce upon a time, lubos debated steve carlip and john baez on\nsci.physics.research. i\'ve read the "flame wars"\n\nsteve carlip and john baez pointed out that thus far, string theory is\nonly able to calculate black hole entropy in the very special case of\nextremel black holes -- black holes whose electric charge is equal or\nnearly equal to its gravitational strength. such black holes almost\ncertainly do not exist, except in theory. at the time these debates\nwere made, string theory was unable to provide the black hole entropy\nfor astronomically observed black holes.\n\nso my question is\n\n1- can string theory reproduce the berkenstein-hawking entropy for all\nblack holes including electrically neutral?\n\n2- what is they physical interpretation of black hole entropy? what\ndoes string theory say is in the black hole that carries the entropy?\n(in comparison LQG interpets black hole entropy as the degrees of\nfreedom per plank area of the black hole\'s event horizon).\n\n3- does string theory itself limit how much electric charge a black\nhole can contain? (i.e is there a reason why a black hole cannot have\nan infinite amount of electric charge?)\n\n4- does string theory have something to say about LQG\'s assumption\nthat the black hole\'s volume-extensive entropy should contribute?\nLQG\'s assumption, that the interior volume extensive entropy should\nnot contribute seems reasonable, given that hawking-berkenstein is\ncaculating its entropy as observed from an outside observer, and the\ninterior is "hidden" by the event horizon. for the interior to\ncontribute, its information must be relayed, but no information can be\nrelayed as the escape velocity is greater than c.\n\n5- lqg\'s calculation of black hole entropy also predicts a fine\nstructure for hawking radiation that is potentially a testable\nimprovement over hawking semiclassical caclulations. does ST also\npredict a fine structure for hawking radiation?\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>lubos motl is a string theorist who had this to say about LQG

"> Black hole entropy
>
> Despite various claims, loop quantum gravity is not able to calculate the
> black hole entropy, unlike string theory. The fact that the entropy is
> proportional to the area does not follow from loop quantum gravity. It is
> rather an assumption of the calculation. The calculation assumes that the
> black hole interior can be neglected and the entropy comes from a new kind
> of dynamics attached to the surface area - there is no justification of
> this assumption. Not surprisingly, one is led to an area/entropy
> proportionality law. The only non-trivial check could be the coefficient,
> but it comes out incorrectly (see the Immirzi discrepancy)."

once upon a time, lubos debated steve carlip and john baez on
sci.physics.research. i've read the "flame wars"

steve carlip and john baez pointed out that thus far, string theory is
only able to calculate black hole entropy in the very special case of
extremel black holes -- black holes whose electric charge is equal or
nearly equal to its gravitational strength. such black holes almost
certainly do not exist, except in theory. at the time these debates
were made, string theory was unable to provide the black hole entropy
for astronomically observed black holes.

so my question is

1- can string theory reproduce the berkenstein-hawking entropy for all
black holes including electrically neutral?

2- what is they physical interpretation of black hole entropy? what
does string theory say is in the black hole that carries the entropy?
(in comparison LQG interpets black hole entropy as the degrees of
freedom per plank area of the black hole's event horizon).

3- does string theory itself limit how much electric charge a black
hole can contain? (i.e is there a reason why a black hole cannot have
an infinite amount of electric charge?)

4- does string theory have something to say about LQG's assumption
that the black hole's volume-extensive entropy should contribute?
LQG's assumption, that the interior volume extensive entropy should
not contribute seems reasonable, given that hawking-berkenstein is
caculating its entropy as observed from an outside observer, and the
interior is "hidden" by the event horizon. for the interior to
contribute, its information must be relayed, but no information can be
relayed as the escape velocity is greater than c.

5- lqg's calculation of black hole entropy also predicts a fine
structure for hawking radiation that is potentially a testable
improvement over hawking semiclassical caclulations. does ST also
predict a fine structure for hawking radiation?

Lubos Motl

<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>I will be happy if someone corrects my errors, if any, and adds new\nresources and information.\n\nOn Tue, 21 Sep 2004, Daniel wrote:\n\n> steve carlip and john baez pointed out that thus far, string theory is\n> only able to calculate black hole entropy in the very special case of\n> extremel black holes\n\nThat\'s not true, and you may have heard it a plenty of times. The entropy\nof non-extremal black holes of various types has been reproduced, too. For\nexample, in the near-extremal case - which means a non-extremal black hole\nwhose properties are expanded to the first order (usually) in "M-Q" -\ninvolve rather complicated functions e.g. of seven variables. This whole\nstructure matches the Bekenstein-Hawking result. See e.g.\n\nhttp://arxiv.org/abs/hep-th/9603195\n\nThe following paper by Maldacena+Horowitz+Strominger checked the entropy\nformula of a class of black holes arbitrarily far away from extremality\n\nhttp://arxiv.org/abs/hep-th/9603109\n\nThere are other, perhaps more important papers, but I am not able to\ngenerate the preprint numbers quickly enough right now.\n\n> -- black holes whose electric charge is equal or\n> nearly equal to its gravitational strength. such black holes almost\n> certainly do not exist, except in theory.\n\nAccording to the best extrapolation of the observed physical laws, such\nblack holes certainly can exist (and be created) in the real Universe, and\nmatching their entropy is definitely equally as important a consistency\ncheck as matching the entropy of other black holes.\n\nYour observational argument is clearly irrelevant from another reason:\neven though we have observed neutral black holes, we have not measured\ntheir entropy yet. The agreement with the Bekenstein-Hawking formula is a\ntheoretical consistency requirement, not a direct experimental\nrequirement, and this requirement is equally strong for neutral black\nholes as well as other black holes.\n\n> 1- can string theory reproduce the berkenstein-hawking entropy for all\n> black holes including electrically neutral?\n\nString theory has reproduced the formula for all black holes that have led\nto a simple enough calculation, and these results have assured virtually\neveryone that the agreement holds for all black holes. Other highly\nnonsupersymmetric black holes are *expected* to lead to complicated\ncomputations and the Bekenstein-Hawking semiclassical way of deriving\ntheir entropy *must* be the only simple way to do so. In this sense, I\nthink that our inability to calculate some neutral black holes\' entropy\nare another consistency check of the theory, although not a quantitative\none in this case. ;-)\n\nNevertheless, there are semi-convincing approximate arguments that\ncalculate the neutral black holes\' entropies up to the numerical\ncoefficient, based on the black hole correspondence principle, see\n\nhttp://arxiv.org/abs/hep-th/9612146\n\nThe correct dependence on various parameters is reproduced. Finally, let\nme say that the calculations very far from extremality - where\nsupersymmetry is of no help - are just too complicated, and people don\'t\nwant to go this direction especially because they know that it has to\nwork, and if their calculation does not work, everyone would think that\nthey made an error anyway. ;-)\n\n> 2- what is they physical interpretation of black hole entropy?\n\nIt is the same thing as any other entropy: the logarithm of the number of\nthe microstates that look almost indistinguishable macroscopically.\n\n> what does string theory say is in the black hole that carries the\n> entropy? (in comparison LQG interpets black hole entropy as the\n> degrees of freedom per plank area of the black hole\'s event horizon).\n\nDifferent descriptions of the same object describe the degrees of freedom\ndifferently, yet they lead to equivalent dynamics. In the weakly coupled\nstringy description, the black holes look like bound states of D-branes,\nand the entropy counts all possible excitations (families of open strings\nin various vibrational modes, moving along the D-branes) that respect the\nrequired total charges and mass. As the coupling increases, the\ngravitational description becomes more appropriate, and the entropy can be\nthought of as carried by the very UV degrees of freedom of gravity\nlocalized on the horizon - exactly this picture is heuristic, and this is\nwhy the rigorous D-brane picture is such a progress. Otherwise the people\ninside the black hole, if they survived, would almost certainly disagree\nwith your statement that the entropy lives on the horizon.\n\n> 3- does string theory itself limit how much electric charge a black\n> hole can contain? (i.e is there a reason why a black hole cannot have\n> an infinite amount of electric charge?)\n\nYes, there cannot be black holes with Q > M - those that have naked\nsingularities - because one can derive the BPS supersymmetric bound which\nsays, in this case, that |Q| <= M. This is guaranteed for any theory with\nSUSY charges Q. Otherwise if you allow the mass to go to infinity and you\nhave an infinite space for your black hole, the charge can go to infinity,\ntoo.\n\n> 4- does string theory have something to say about LQG\'s assumption\n> that the black hole\'s volume-extensive entropy should contribute?\n\nString theory does not describe the black hole as a naive geometric object\nwhich is made of naive atoms and where the naive notions of locality\nnecessarily apply. In the weakly coupled limit the entropy comes from the\nD-branes and one cannot say whether these branes are on the horizon or\ninside. In the strongly coupled limit the geometry is really GR-like, but\nstring theory does not exactly tell us where the degrees of freedom have\ndrifted - it can just be proved that they survived by supersymmetry.\n\n> LQG\'s assumption, that the interior volume extensive entropy should\n> not contribute seems reasonable,\n\nIt is not reasonable at all - it\'s like if you compute the entropy of a\nbottle of Coke, and you assume - or conclude - that the entropy only comes\nfrom the plastic cover. There are potentially people, bottles, and\neverything else inside the black hole, and they would certainly not find\nit reasonable to assume that they carry no entropy.\n\n> given that hawking-berkenstein is\n> caculating its entropy as observed from an outside observer, and the\n> interior is "hidden" by the event horizon.\n\nIf the bottle of Coke is painted dark blue, the interior is also hidden\nfrom the outside observer - nevertheless it is still equally unreasonable,\ndespite the paint, to say that it does not carry any entropy.\n\n> for the interior to contribute, its information must be relayed, but\n> no information can be relayed as the escape velocity is greater than\n> c.\n\nUsing this argument, you could also argue that the entropy can be zero or\nanything else. But it\'s not true because the black hole eventually\nevaporates. In other words, the interior is not perfectly hidden, by the\nvery laws of quantum mechanics, as understood by Hawking. It is not that\ndifferent from the bottle of Coke - the interior is only hidden\ntemporarily. Once again, your assumption that the interior can be hidden\nforever is exactly the wrong assumption that prevents you from seeing how\ndeep the black hole information problem is, and how shocking holography\nis.\n\n> 5- lqg\'s calculation of black hole entropy also predicts a fine\n> structure for hawking radiation that is potentially a testable\n\nLQG does not predict anything like Hawking\'s thermal radiation. All these\nthings are a complete mystery in LQG, even if you assume the most\noptimistic scenario about its low energy limit. You must have heard some\ninformation from an unreliable source.\n\n> improvement over hawking semiclassical caclulations. does ST also\n> predict a fine structure for hawking radiation?\n\nYes, but these improvements come in several types and we must distinguish\nthem. The radiation is not exactly thermal, but corrected by the "grey\nbody factors". This is a feature already calculable in GR, and Maldacena\nand Strominger showed that the D-brane radiation spectrum exactly matches\nthe thermal, black body GR radiation "filtered" by the "grey body filters"\npresent in GR, see\n\nhttp://arxiv.org/abs/hep-th/9609026\n\nThen you have detailed, microscopic deviations from the exact thermality.\nOf course, in the D-brane picture, it is easy to distinguish all\nmicrostates from each other and do the exact calculation. Their exact\ncontinuation to the strong coupling is not known, as far as I know.\n________________________________________________________________ ______________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 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>I will be happy if someone corrects my errors, if any, and adds new
resources and information.

On Tue, 21 Sep 2004, Daniel wrote:

> steve carlip and john baez pointed out that thus far, string theory is
> only able to calculate black hole entropy in the very special case of
> extremel black holes

That's not true, and you may have heard it a plenty of times. The entropy
of non-extremal black holes of various types has been reproduced, too. For
example, in the near-extremal case - which means a non-extremal black hole
whose properties are expanded to the first order (usually) in "[itex]M-Q" -[/itex]
involve rather complicated functions e.g. of seven variables. This whole
structure matches the Bekenstein-Hawking result. See e.g.

http://arxiv.org/abs/http://www.arxi...hep-th/9603195

The following paper by Maldacena+Horowitz+Strominger checked the entropy
formula of a class of black holes arbitrarily far away from extremality

http://arxiv.org/abs/http://www.arxi...hep-th/9603109

There are other, perhaps more important papers, but I am not able to
generate the preprint numbers quickly enough right now.

> -- black holes whose electric charge is equal or
> nearly equal to its gravitational strength. such black holes almost
> certainly do not exist, except in theory.

According to the best extrapolation of the observed physical laws, such
black holes certainly can exist (and be created) in the real Universe, and
matching their entropy is definitely equally as important a consistency
check as matching the entropy of other black holes.

Your observational argument is clearly irrelevant from another reason:
even though we have observed neutral black holes, we have not measured
their entropy yet. The agreement with the Bekenstein-Hawking formula is a
theoretical consistency requirement, not a direct experimental
requirement, and this requirement is equally strong for neutral black
holes as well as other black holes.

> 1- can string theory reproduce the berkenstein-hawking entropy for all
> black holes including electrically neutral?

String theory has reproduced the formula for all black holes that have led
to a simple enough calculation, and these results have assured virtually
everyone that the agreement holds for all black holes. Other highly
nonsupersymmetric black holes are *expected* to lead to complicated
computations and the Bekenstein-Hawking semiclassical way of deriving
their entropy *must* be the only simple way to do so. In this sense, I
think that our inability to calculate some neutral black holes' entropy
are another consistency check of the theory, although not a quantitative
one in this case. ;-)

Nevertheless, there are semi-convincing approximate arguments that
calculate the neutral black holes' entropies up to the numerical
coefficient, based on the black hole correspondence principle, see

http://arxiv.org/abs/http://www.arxi...hep-th/9612146

The correct dependence on various parameters is reproduced. Finally, let
me say that the calculations very far from extremality - where
supersymmetry is of no help - are just too complicated, and people don't
want to go this direction especially because they know that it has to
work, and if their calculation does not work, everyone would think that
they made an error anyway. ;-)

> 2- what is they physical interpretation of black hole entropy?

It is the same thing as any other entropy: the logarithm of the number of
the microstates that look almost indistinguishable macroscopically.

> what does string theory say is in the black hole that carries the
> entropy? (in comparison LQG interpets black hole entropy as the
> degrees of freedom per plank area of the black hole's event horizon).

Different descriptions of the same object describe the degrees of freedom
differently, yet they lead to equivalent dynamics. In the weakly coupled
stringy description, the black holes look like bound states of D-branes,
and the entropy counts all possible excitations (families of open strings
in various vibrational modes, moving along the D-branes) that respect the
required total charges and mass. As the coupling increases, the
gravitational description becomes more appropriate, and the entropy can be
thought of as carried by the very UV degrees of freedom of gravity
localized on the horizon - exactly this picture is heuristic, and this is
why the rigorous D-brane picture is such a progress. Otherwise the people
inside the black hole, if they survived, would almost certainly disagree
with your statement that the entropy lives on the horizon.

> 3- does string theory itself limit how much electric charge a black
> hole can contain? (i.e is there a reason why a black hole cannot have
> an infinite amount of electric charge?)

Yes, there cannot be black holes with Q > M - those that have naked
singularities - because one can derive the BPS supersymmetric bound which
says, in this case, that [itex]|Q| <= M[/itex]. This is guaranteed for any theory with
SUSY charges Q. Otherwise if you allow the mass to go to infinity and you
have an infinite space for your black hole, the charge can go to infinity,
too.

> 4- does string theory have something to say about LQG's assumption
> that the black hole's volume-extensive entropy should contribute?

String theory does not describe the black hole as a naive geometric object
which is made of naive atoms and where the naive notions of locality
necessarily apply. In the weakly coupled limit the entropy comes from the
D-branes and one cannot say whether these branes are on the horizon or
inside. In the strongly coupled limit the geometry is really GR-like, but
string theory does not exactly tell us where the degrees of freedom have
drifted [itex]- it[/itex] can just be proved that they survived by supersymmetry.

> LQG's assumption, that the interior volume extensive entropy should
> not contribute seems reasonable,

It is not reasonable at all - it's like if you compute the entropy of a
bottle of Coke, and you assume - or conclude - that the entropy only comes
from the plastic cover. There are potentially people, bottles, and
everything else inside the black hole, and they would certainly not find
it reasonable to assume that they carry no entropy.

> given that hawking-berkenstein is
> caculating its entropy as observed from an outside observer, and the
> interior is "hidden" by the event horizon.

If the bottle of Coke is painted dark blue, the interior is also hidden
from the outside observer - nevertheless it is still equally unreasonable,
despite the paint, to say that it does not carry any entropy.

> for the interior to contribute, its information must be relayed, but
> no information can be relayed as the escape velocity is greater than
> c.

Using this argument, you could also argue that the entropy can be zero or
anything else. But it's not true because the black hole eventually
evaporates. In other words, the interior is not perfectly hidden, by the
very laws of quantum mechanics, as understood by Hawking. It is not that
different from the bottle of Coke - the interior is only hidden
temporarily. Once again, your assumption that the interior can be hidden
forever is exactly the wrong assumption that prevents you from seeing how
deep the black hole information problem is, and how shocking holography
is.

> 5- lqg's calculation of black hole entropy also predicts a fine
> structure for hawking radiation that is potentially a testable

LQG does not predict anything like Hawking's thermal radiation. All these
things are a complete mystery in LQG, even if you assume the most
optimistic scenario about its low energy limit. You must have heard some
information from an unreliable source.

> improvement over hawking semiclassical caclulations. does ST also
> predict a fine structure for hawking radiation?

Yes, but these improvements come in several types and we must distinguish
them. The radiation is not exactly thermal, but corrected by the "grey
body factors". This is a feature already calculable in GR, and Maldacena
and Strominger showed that the D-brane radiation spectrum exactly matches
the thermal, black body GR radiation "filtered" by the "grey body filters"
present in GR, see

http://arxiv.org/abs/http://www.arxi...hep-th/9609026

Then you have detailed, microscopic deviations from the exact thermality.
Of course, in the D-brane picture, it is easy to distinguish all
microstates from each other and do the exact calculation. Their exact
continuation to the strong coupling is not known, as far as I know.
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/384-9488[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^

Rufus Anton

<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>> Yes, there cannot be black holes with Q > M - those that have naked\n> singularities - because one can derive the BPS supersymmetric bound which\n> says, in this case, that |Q| <= M. This is guaranteed for any theory with\n> SUSY charges Q. Otherwise if you allow the mass to go to infinity and you\n> have an infinite space for your black hole, the charge can go to infinity,\n> too.\n\nI have a question about this. The BPS bound stems from the central\ncharges of various BPS objects in the SUSY algebra. But some branes\n(mysteriously?) do not seem to have corresponding central charges. I\nbelieve type IIA\'s D6-brane is an example. Am I right? How does this\nhappen and what does it imply for black holes involving such branes?\nMany thanks. Rufus\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>> Yes, there cannot be black holes with Q > M - those that have naked
> singularities - because one can derive the BPS supersymmetric bound which
> says, in this case, that [itex]|Q| <= M[/itex]. This is guaranteed for any theory with
> SUSY charges Q. Otherwise if you allow the mass to go to infinity and you
> have an infinite space for your black hole, the charge can go to infinity,
> too.

I have a question about this. The BPS bound stems from the central
charges of various BPS objects in the SUSY algebra. But some branes
(mysteriously?) do not seem to have corresponding central charges. I
believe type IIA's D6-brane is an example. Am I right? How does this
happen and what does it imply for black holes involving such branes?
Many thanks. Rufus

Lubos Motl

<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>On Wed, 22 Sep 2004, Rufus Anton wrote:\n\n> I have a question about this. The BPS bound stems from the central\n> charges of various BPS objects in the SUSY algebra. But some branes\n> (mysteriously?) do not seem to have corresponding central charges. I\n> believe type IIA\'s D6-brane is an example. Am I right? How does this\n> happen and what does it imply for black holes involving such branes?\n> Many thanks. Rufus\n\nIt\'s a good question, even though I hope that you will sleep better when\nyou learn that IIA D6-brane *has* a central charge. Type IIA/IIB string\ntheories have all possible branes of even/odd dimensions. Each of them is\nassociated with a Ramond-Ramond gauge field that can have a field strength\nand a flux - and in fact, all these D-branes are T-dual to each other.\n(T-duality changes the dimension of the D-branes.) If one of them\npreserves some supersymmetry, and it does, it is guaranteed that all\neven/odd-dimensional branes in IIA/IIB preserve the same amount of\nsupersymmetry (one half).\n\nIf something preserves a supersymmetry and it is massive, it *must* carry\nnonzero charges - it\'s because the square/anticommutators of these SUSY\ngenerators look like energy plus something else (the central charges), and\nbecause the generators (and their anticommutators), by assumption of\nsupersymmetry, annihilate the state, but energy does not, the central\ncharges must be those who cancel the energy on the right hand side.\n\n{Q1,Q2} = Energy + Charge annihilates the BPS states\n\nI think that the similar fact that you might have in mind is that there\nis no BPS bound state of D0-branes and D6-branes - in fact, they repel.\n\nQuite generally, a brane or another object would be unstable if it carried\nno charge like that; it is the case of the non-BPS D-branes, for example\nthose odd/even-dimensional D-branes in IIA/IIB - note that I switched odd\nand even. An object can only be stable if it is the lightest object with\nthe same charges - if someone knows an exception, tell me about it -, and\ntherefore there is nothing else that it can decay to. Non-BPS\n(non-supersymmetric) stable objects are very rare.\n\nAnother special feature of D6-branes that may have led you to the\nconclusion is that the M-theory limit of D2-branes and D4-branes are\nbranes - namely M2 and M5-branes - but the limit of D6-branes is a\ngeometric configuration, the Kaluza-Klein monopole - some sort of magnetic\nmonopole, magnetically charged under the U(1) Kaluza-Klein symmetry\nrotating the 11th circular dimension, stretched in extra 6+1 dimensions.\nIn M-theory, D6-branes are associated with the metric tensor itself, and\ntheir charge looks "topological". It is an example how M-theory\ngeometerizes some other fields - well, the more dimensions you can work\nwith, the more geometrical the concepts may become and the less\n"non-geometric" extra fields are included in your toolkit. Nevertheless,\nthe relevant components of the metric - like g_{i,11} - are also fields,\nthey exist, and they are dual to regular Ramond-Ramond fields in type IIA\nstring theory.\n______________________________________________________________ ________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 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>On Wed, 22 Sep 2004, Rufus Anton wrote:

> I have a question about this. The BPS bound stems from the central
> charges of various BPS objects in the SUSY algebra. But some branes
> (mysteriously?) do not seem to have corresponding central charges. I
> believe type IIA's D6-brane is an example. Am I right? How does this
> happen and what does it imply for black holes involving such branes?
> Many thanks. Rufus

It's a good question, even though I hope that you will sleep better when
you learn that IIA D6-brane [itex]*has* a[/itex] central charge. Type [itex]IIA/IIB[/itex] string
theories have all possible branes of [itex]even/odd[/itex] dimensions. Each of them is
associated with a Ramond-Ramond gauge field that can have a field strength
and a flux - and in fact, all these D-branes are T-dual to each other.
(T-duality changes the dimension of the D-branes.) If one of them
preserves some supersymmetry, and it does, it is guaranteed that all
even/odd-dimensional branes in [itex]IIA/IIB[/itex] preserve the same amount of
supersymmetry (one half).

If something preserves a supersymmetry and it is massive, [itex]it *must*[/itex] carry
nonzero charges - it's because the square/anticommutators of these SUSY
generators look like energy plus something else (the central charges), and
because the generators (and their anticommutators), by assumption of
supersymmetry, annihilate the state, but energy does not, the central
charges must be those who cancel the energy on the right hand side.

[itex]{Q1,Q2} =[/itex] Energy + Charge annihilates the BPS states

I think that the similar fact that you might have in mind is that there
is no BPS bound state of D0-branes and D6-branes - in fact, they repel.

Quite generally, a brane or another object would be unstable if it carried
no charge like that; it is the case of the non-BPS D-branes, for example
those odd/even-dimensional D-branes in [itex]IIA/IIB -[/itex] note that I switched odd
and even. An object can only be stable if it is the lightest object with
the same charges - if someone knows an exception, tell me about [itex]it -,[/itex] and
therefore there is nothing else that it can decay to. Non-BPS
(non-supersymmetric) stable objects are very rare.

Another special feature of D6-branes that may have led you to the
conclusion is that the M-theory limit of D2-branes and D4-branes are
branes - namely M2 and M5-branes - but the limit of D6-branes is a
geometric configuration, the Kaluza-Klein monopole - some sort of magnetic
monopole, magnetically charged under the U(1) Kaluza-Klein symmetry
rotating the 11th circular dimension, stretched in extra 6+1 dimensions.
In M-theory, D6-branes are associated with the metric tensor itself, and
their charge looks "topological". It is an example how M-theory
geometerizes some other fields - well, the more dimensions you can work
with, the more geometrical the concepts may become and the less
"non-geometric" extra fields are included in your toolkit. Nevertheless,
the relevant components of the metric - like [itex]g_{i,11} -[/itex] are also fields,
they exist, and they are dual to regular Ramond-Ramond fields in type IIA
string theory.
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/384-9488[/itex] home: [itex]+1-617/868-4487[/itex] (call)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^

Steve Carlip

<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>> These lines were written by Lubos Motl <motl@feynman.harvard.edu>.\n> > These lines were written by Daniel on 21 Sep 2004.\n\nSteve Carlip is me.\n\n> > steve carlip and john baez pointed out that thus far, string theory is\n> > only able to calculate black hole entropy in the very special case of\n> > extremal black holes\n\n> That\'s not true, and you may have heard it a plenty of times. The entropy\n> of non-extremal black holes of various types has been reproduced, too.\n\nString theory gives the correct entropy for extremal black holes, for\nreasons that are well understood. The computation is done in a weak-\ncoupling regime, in which it is straightforward to count brane states,\nand for BPS black holes, nonrenormalization theorems guarantee that\nthe entropy is unchanged when one then goes to strong coupling.\n\nString theory also gives the correct entropy for near-extremal black\nholes, for reasons that are much less well understood. Again, one\ncan do the counting at weak coupling, but here there are no obvious\nnonrenormalization theorems, so it is rather mysterious why one can\nextrapolate to strong coupling. Note that there are certainly some\ncases in which renormalization does matter -- the nonextremal black\n3-brane, for example, in which the entropy comes out wrong by a factor\nof (4/3)^{1/4} (see Gubser et al., hep-th/9602135).\n\nWhat\'s going on may be this. The near-extremal black holes where the\ncomputation works almost all have near-horizon regions that look like\n(BTZ black hole)x(something trivial). One can use AdS/CFT to compute\nthe BTZ black hole entropy via the Cardy formula, and then simply\ntranslate the results into an expression involving the full black hole.\n\nNote, though, that this way of doing things doesn\'t tell you much about\nwhat states are being counted. It is mainly a symmetry argument -- the\nCardy formula gives an identical answer for any two conformal field\ntheories with the same central charge, even if the actual states are\nvery different. A large majority of the successful string theory\ncomputations of exact black hole entropy fall into this category.\n\n> For example, in the near-extremal case - which means a non-extremal\n> black hole whose properties are expanded to the first order (usually)\n> in "M-Q" - involve rather complicated functions e.g. of seven\n> variables. This whole structure matches the Bekenstein-Hawking result.\n> See e.g. http://arxiv.org/abs/hep-th/9603195\n>\n> The following paper by Maldacena+Horowitz+Strominger checked the entropy\n> formula of a class of black holes arbitrarily far away from extremality\n> http://arxiv.org/abs/hep-th/9603109\n\nThese are bad examples. The entropy is not computed from string theory,\nbut from ``a simple but mysterious duality-invariant extension of\npreviously derived formulae for the number of D-brane states in string\ntheory.\'\' In the second paper, the authors state explicitly, very early\non, ``We have not been able to obtain a stringy derivation of the full\nexpression. This would require more than a weak-coupling analysis and\nseems to be quite challenging.\'\' It would be wonderful if one could derive\nthe formulae of these papers from first principles, but for now they are\n``string-inspired,\'\' not string theoretical.\n\n[...]\n\n> > -- black holes whose electric charge is equal or\n> > nearly equal to its gravitational strength. such black holes almost\n> > certainly do not exist, except in theory.\n>\n> According to the best extrapolation of the observed physical laws, such\n> black holes certainly can exist (and be created) in the real Universe,\n\nThis is very unlikely. Highly charged black holes discharge extremely\nrapidly by Schwinger pair production. Moreover, if you compute the\nenergy it takes to overcome electrical repulsion in order to add an\nadditional charge to a near-extremal black hole, you will find that it\nincreases the black hole mass enough to keep it from reaching extremality.\n\nThere may be some way around this last argument, but I have never seen\na convincing one, and the existence of extremal black holes in our real\nUniverse seems extremely unlikely.\n\n> and matching their entropy is definitely equally as important a consistency\n> check as matching the entropy of other black holes.\n\nThat\'s certainly true. Just don\'t exaggerate what string theory currently\ndoes -- it meets an important consistency check, but so far only in a very\nlimited and physically unrealistic class of examples.\n\nThere is another deep problem, the ``problem of universality,\'\' which\nneither string theory nor any other microscopic approach has yet solved.\nIf you look at a detailed string theory computation of black hole entropy,\nit typically goes like this. First, you find a black hole solution in\nsome suitable supergravity theory, and identify the charges of various\ngauge fields. Next, you build a system of strings and D-branes with the\nsame charges. In the brane system, in the weak coupling limit, you count\nthe number of states, and express the result in terms of the charges. In\nthe supergravity theory, you compute the horizon area, and write the\nresult in terms of the charges. Then you compare, and find that the two\nexpressions match correctly.\n\nThis is, as I said, a very good consistency check. But the trouble is\nthat for each new black hole, the density of states and the horizon area\ndepend on a new set of charges, and have to be recalculated from scratch.\n\nIf you give me a brand new near-extremal black hole, I would certainly\nexpect to get the right agreement again. But this expectation is based on\npast success, and not on any general result that\'s yet been discovered.\n\nIn other words, no one has yet really derived the Bekenstein-Hawking\nrelation between entropy and area for black holes in string theory.\nPeople have derived expressions for entropy of particular black holes,\nand expressions for horizon area of the same black holes, and so far they\nhave all matched the Bekenstein-Hawking relation. But no one knows, in\nany deep way, *why* they match.\n\nI don\'t want to belittle string theory\'s successes in understanding black\nhole thermodynamics. Some of the results -- in particular, the computations\nnot just of entropy, but of gray body factors -- are very impressive. But\nI think it\'s important not to exaggerate the successes, or to ignore the\nvery deep problems that remain unanswered.\n\nSteve Carlip\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>> These lines were written by Lubos Motl <motl@feynman.harvard.edu>.
> > These lines were written by Daniel on 21 Sep 2004.

Steve Carlip is me.

> > steve carlip and john baez pointed out that thus far, string theory is
> > only able to calculate black hole entropy in the very special case of
> > extremal black holes

> That's not true, and you may have heard it a plenty of times. The entropy
> of non-extremal black holes of various types has been reproduced, too.

String theory gives the correct entropy for extremal black holes, for
reasons that are well understood. The computation is done in a weak-
coupling regime, in which it is straightforward to count brane states,
and for BPS black holes, nonrenormalization theorems guarantee that
the entropy is unchanged when one then goes to strong coupling.

String theory also gives the correct entropy for near-extremal black
holes, for reasons that are much less well understood. Again, one
can do the counting at weak coupling, but here there are no obvious
nonrenormalization theorems, so it is rather mysterious why one can
extrapolate to strong coupling. Note that there are certainly some
cases in which renormalization does matter -- the nonextremal black
3-brane, for example, in which the entropy comes out wrong by a factor
of [itex](4/3)^{1/4}[/itex] (see Gubser et al., http://www.arxiv.org/abs/hep-th/9602135).

What's going on may be this. The near-extremal black holes where the
computation works almost all have near-horizon regions that look like
(BTZ black hole)x(something trivial). One can use [itex]AdS/CFT[/itex] to compute
the BTZ black hole entropy via the Cardy formula, and then simply
translate the results into an expression involving the full black hole.

Note, though, that this way of doing things doesn't tell you much about
what states are being counted. It is mainly a symmetry argument -- the
Cardy formula gives an identical answer for any two conformal field
theories with the same central charge, even if the actual states are
very different. A large majority of the successful string theory
computations of exact black hole entropy fall into this category.

> For example, in the near-extremal case - which means a non-extremal
> black hole whose properties are expanded to the first order (usually)
> in "[itex]M-Q" -[/itex] involve rather complicated functions e.g. of seven
> variables. This whole structure matches the Bekenstein-Hawking result.
> See e.g. http://arxiv.org/abs/http://www.arxi...hep-th/9603195
>
> The following paper by Maldacena+Horowitz+Strominger checked the entropy
> formula of a class of black holes arbitrarily far away from extremality
> http://arxiv.org/abs/http://www.arxi...hep-th/9603109

These are bad examples. The entropy is not computed from string theory,
but from ``a simple but mysterious duality-invariant extension of
previously derived formulae for the number of D-brane states in string
theory.'' In the second paper, the authors state explicitly, very early
on, ``We have not been able to obtain a stringy derivation of the full
expression. This would require more than a weak-coupling analysis and
seems to be quite challenging.'' It would be wonderful if one could derive
the formulae of these papers from first principles, but for now they are
``string-inspired,'' not string theoretical.

[...]

> > -- black holes whose electric charge is equal or
> > nearly equal to its gravitational strength. such black holes almost
> > certainly do not exist, except in theory.
>
> According to the best extrapolation of the observed physical laws, such
> black holes certainly can exist (and be created) in the real Universe,

This is very unlikely. Highly charged black holes discharge extremely
rapidly by Schwinger pair production. Moreover, if you compute the
energy it takes to overcome electrical repulsion in order to add an
additional charge to a near-extremal black hole, you will find that it
increases the black hole mass enough to keep it from reaching extremality.

There may be some way around this last argument, but I have never seen
a convincing one, and the existence of extremal black holes in our real
Universe seems extremely unlikely.

> and matching their entropy is definitely equally as important a consistency
> check as matching the entropy of other black holes.

That's certainly true. Just don't exaggerate what string theory currently
does -- it meets an important consistency check, but so far only in a very
limited and physically unrealistic class of examples.

There is another deep problem, the ``problem of universality,'' which
neither string theory nor any other microscopic approach has yet solved.
If you look at a detailed string theory computation of black hole entropy,
it typically goes like this. First, you find a black hole solution in
some suitable supergravity theory, and identify the charges of various
gauge fields. Next, you build a system of strings and D-branes with the
same charges. In the brane system, in the weak coupling limit, you count
the number of states, and express the result in terms of the charges. In
the supergravity theory, you compute the horizon area, and write the
result in terms of the charges. Then you compare, and find that the two
expressions match correctly.

This is, as I said, a very good consistency check. But the trouble is
that for each new black hole, the density of states and the horizon area
depend on a new set of charges, and have to be recalculated from scratch.

If you give me a brand new near-extremal black hole, I would certainly
expect to get the right agreement again. But this expectation is based on
past success, and not on any general result that's yet been discovered.

In other words, no one has yet really derived the Bekenstein-Hawking
relation between entropy and area for black holes in string theory.
People have derived expressions for entropy of particular black holes,
and expressions for horizon area of the same black holes, and so far they
have all matched the Bekenstein-Hawking relation. But no one knows, in
any deep way, *why* they match.

I don't want to belittle string theory's successes in understanding black
hole thermodynamics. Some of the results -- in particular, the computations
not just of entropy, but of gray body factors -- are very impressive. But
I think it's important not to exaggerate the successes, or to ignore the
very deep problems that remain unanswered.

Steve Carlip

Lubos Motl

<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>On Sun, 26 Sep 2004, Steve Carlip wrote:\n\n> String theory also gives the correct entropy for near-extremal black\n> holes, for reasons that are much less well understood. Again, one\n> can do the counting at weak coupling, but here there are no obvious\n> nonrenormalization theorems, so it is rather mysterious why one can\n> extrapolate to strong coupling. Note that there are certainly some\n> cases in which renormalization does matter -- the nonextremal black\n> 3-brane, for example, in which the entropy comes out wrong by a factor\n> of (4/3)^{1/4} (see Gubser et al., hep-th/9602135).\n\nI certainly agree that some results confirming the Bekenstein-Hawking\nentropy by methods of string theory seem to be surprising and they seem to\ngo beyond the known nonrenormalization theorems. Sometimes it is\nconceivable that some new, unknown non-renornalization theorems exist,\nsometimes the agreement can be just a coincidence - a consequence of a\nsimple enough form of the result. Of course, there can exist another (and\nconceivable deep) explanation of the match - one that does not deserve the\nname "nonrenormalization theorem".\n\n> What\'s going on may be this. The near-extremal black holes where the\n> computation works almost all have near-horizon regions that look like\n> (BTZ black hole)x(something trivial). One can use AdS/CFT to compute\n> the BTZ black hole entropy via the Cardy formula, and then simply\n> translate the results into an expression involving the full black hole.\n\nIncidentally, when you talk about Cardy\'s formula, let me mention that I\nenjoyed your recent paper, much like most of the previous ones.\n\n> Note, though, that this way of doing things doesn\'t tell you much about\n> what states are being counted.\n\nI don\'t quite understand what you\'re exactly missing here. They are states\nin a specific CFT at a given energy level - something that you are using\nin your own papers all the time. ;-) You can call the states "Peter",\n"Paul", and so on, but I am not sure how useful it is given the\nexponentially huge number of these states.\n\nYes, if you mean that the CFT states don\'t tell us what is the right\ngeometric interpretation of these states at stronger coupling (are they\nlocalized at the horizon, inside, or near the singularity?), I agree. The\nanswer to this question might be observer-dependent.\n\n> It is mainly a symmetry argument -- the Cardy formula gives an\n> identical answer for any two conformal field theories with the same\n> central charge, even if the actual states are very different.\n\nIsn\'t this the whole point of Cardy\'s formula? ;-) If you think that not\nonly the BH formula is universal, but the precise structure of the CFT\nstates is universal as well - then I think that your conjecture has\nalready been ruled out. This is one of the things that make string theory\nnice - the same states look very differently in different extreme points\nof the moduli space. They have different interpretations which are "dual"\nto each other.\n\nMoreover, even in the purely perturbative case of CFTs, we just know many\nequivalences between seemingly different CFTs - bosonization,\nfermionization are basic examples. The central charge is one of the few\n"universal" properties of a CFT that are independent of the choice of\n(equivalent) variables.\n\nI often have the feeling that various people often criticize string theory\nfor not confirming their prejudices - the universal interpretation of\nthese states is certainly such a prejudice. And it seems clear to me that\nthe truth, as string theory teaches us, is even "nicer" than these\nprejudices.\n\n> A large majority of the successful string theory\n> computations of exact black hole entropy fall into this category.\n\nThat\'s right. Is there a problem with it?\n\n> These are bad examples. The entropy is not computed from string theory,\n> but from ``a simple but mysterious duality-invariant extension of\n> previously derived formulae for the number of D-brane states in string\n> theory.\'\'\n\nRight. Well, the duality group is viewed by the majority as a nice,\nnonperturbative feature of string theory which is supported by its beauty\nand a huge amount of circumstantial evidence. Yes, the counting of states\nhas been extrapolated from some rigorous examples, without making a\nnew, complete calculation in each case.\n\n> This would require more than a weak-coupling analysis and\n> seems to be quite challenging.\'\' It would be wonderful if one could derive\n> the formulae of these papers from first principles, but for now they are\n> ``string-inspired,\'\' not string theoretical.\n\nWell, it may be wonderful, but I am afraid that it would not be wonderful\nenough to revolutionize the field. ;-)\n\n> This is very unlikely. Highly charged black holes discharge extremely\n> rapidly by Schwinger pair production. Moreover, if you compute the\n> energy it takes to overcome electrical repulsion in order to add an\n> additional charge to a near-extremal black hole, you will find that it\n> increases the black hole mass enough to keep it from reaching extremality.\n\nWell, the force may be increasing, but it does not increase enough to stop\nyou from going arbitrarily close to extremality. If you think that there\nexists another critical value of Q/M, smaller than 1 (which stands for\n"extremal") that cannot be achieved, could you please specify what the\nvalue is and why is it exactly this value? I agree that very close to the\nextremality, we will need accelerators to approach even closer :-), but\nthe existence of accelerators is not ruled out by any general physical\nprinciples (a budget constraint is not a physical principle).\n\n> > and matching their entropy is definitely equally as important a consistency\n> > check as matching the entropy of other black holes.\n>\n> That\'s certainly true. Just don\'t exaggerate what string theory currently\n> does -- it meets an important consistency check, but so far only in a very\n> limited and physically unrealistic class of examples.\n\nI am not exaggerating anything here. On the other hand, you are\nexaggerating the importance of checking some other black holes which lead\nto much harder calculations - and perhaps, in some cases, calculations\nthat we are not yet able to do even in principle. If someone proposed a\nstringy calculation of the entropy of a Schwarzschild black hole in a\ncertain number of dimensions, she would definitely show that she\nunderstands the stringy machinery in detail, but I don\'t think that she\nwould be viewed as an extremely important contributor to string theory\nbecause of this paper - unless her methods could also be useful for some\ncompletely different questions.\n\nThe agreement between string theory and the BH formula is just obviously\ntrue and no one is too interested in verification of yet another class of\nblack holes; this might have been a popular topic eight years ago, but\nnot now. On the other hand, the *universal* explanation of the BH formula\nin any context can probably depend on the low energy physics of GR only -\nthis is what makes it universal. However, the same feature also makes such\nan argument non-stringy and indirect. A universal proof of the BH formula\nin string theory would have to start with explaining why the object looks\nlike a black hole at low energies, and then it would use an\neffective-field-theoretical based universal justification. However, the\nexplicit stringy checks based on D-branes are much more non-trivial.\nHowever, enough of them have been verified and today, the topic is simply\nnot as hot as you seem to think.\n\nA verification of yet another black hole may be nice, but it is basically\njust a pretty small addition to Strominger\'s and Vafa\'s pioneering\ncontribution.\n\n> This is, as I said, a very good consistency check. But the trouble is\n> that for each new black hole, the density of states and the horizon area\n> depend on a new set of charges, and have to be recalculated from scratch.\n\nThat\'s a necessary requirement for each such new calculation to be\nnon-trivial and stringy.\n\n> If you give me a brand new near-extremal black hole, I would certainly\n> expect to get the right agreement again. But this expectation is based on\n> past success, and not on any general result that\'s yet been discovered.\n\nIt\'s because the universal law is not really a consequence of very special\nstringy effects, but a result of a consistent treatment of the low-energy\nlimit. Because Bekenstein and Hawking could have derived these quantities\n*without* knowing anything specific about string theory, by thermodynamic\nconsiderations, it is reasonable to expect that the universal proof of the\nBH formula does not have to be too stringy either.\n\nTwo of us are almost certainly thinking about very similar forms of the\nuniversal proof - I\'ve been trying to convince myself to write TeX notes\nabout "my" proof involving the delta-function-like contribution to the\nEinstein-Hilbert action, coming from "integrating out" the Planckian\nthroat inside the black hole.\n\nLet me summarize the main objection of mine against your comments: you\nseem to require (or assume, or whatever it exactly is) that some more\ndetailed qualitative features of the black hole states, not just the\ncounting, are universal in string theory (or in any theory of quantum\ngravity in any description). I think that this is simply not true, and\nquantum gravity would be boring if it were true. On the other hand, you\nseem to criticize string theory for not giving a universal proof of the BH\nformula - but a universal proof would have to start with deciding whether\na system of states can be thought of as a black hole with certain area -\nand this can only be decided in the low-energy approximation. Therefore\nthe universal proof is unlikely to be too similar to these "very stringy"\ncalculations of D-brane states - it will be more similar to Bekenstein\'s\nand Hawking\'s thermodynamic arguments rooted in semiclassical GR.\n\nAll the best,\nLubos\n_________________________________________________________ _____________________\nE-mail: lumo@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/\neFax: +1-801/454-1858 work: +1-617/384-9488 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>On Sun, 26 Sep 2004, Steve Carlip wrote:

> String theory also gives the correct entropy for near-extremal black
> holes, for reasons that are much less well understood. Again, one
> can do the counting at weak coupling, but here there are no obvious
> nonrenormalization theorems, so it is rather mysterious why one can
> extrapolate to strong coupling. Note that there are certainly some
> cases in which renormalization does matter -- the nonextremal black
> 3-brane, for example, in which the entropy comes out wrong by a factor
> of [itex](4/3)^{1/4}[/itex] (see Gubser et al., http://www.arxiv.org/abs/hep-th/9602135).

I certainly agree that some results confirming the Bekenstein-Hawking
entropy by methods of string theory seem to be surprising and they seem to
go beyond the known nonrenormalization theorems. Sometimes it is
conceivable that some new, unknown non-renornalization theorems exist,
sometimes the agreement can be just a coincidence - a consequence of a
simple enough form of the result. Of course, there can exist another (and
conceivable deep) explanation of the match - one that does not deserve the
name "nonrenormalization theorem".

> What's going on may be this. The near-extremal black holes where the
> computation works almost all have near-horizon regions that look like
> (BTZ black hole)x(something trivial). One can use [itex]AdS/CFT[/itex] to compute
> the BTZ black hole entropy via the Cardy formula, and then simply
> translate the results into an expression involving the full black hole.

Incidentally, when you talk about Cardy's formula, let me mention that I
enjoyed your recent paper, much like most of the previous ones.

> Note, though, that this way of doing things doesn't tell you much about
> what states are being counted.

I don't quite understand what you're exactly missing here. They are states
in a specific CFT at a given energy level - something that you are using
in your own papers all the time. ;-) You can call the states "Peter",
"Paul", and so on, but I am not sure how useful it is given the
exponentially huge number of these states.

Yes, if you mean that the CFT states don't tell us what is the right
geometric interpretation of these states at stronger coupling (are they
localized at the horizon, inside, or near the singularity?), I agree. The
answer to this question might be observer-dependent.

> It is mainly a symmetry argument -- the Cardy formula gives an
> identical answer for any two conformal field theories with the same
> central charge, even if the actual states are very different.

Isn't this the whole point of Cardy's formula? ;-) If you think that not
only the BH formula is universal, but the precise structure of the CFT
states is universal as well - then I think that your conjecture has
already been ruled out. This is one of the things that make string theory
nice - the same states look very differently in different extreme points
of the moduli space. They have different interpretations which are "dual"
to each other.

Moreover, even in the purely perturbative case of CFTs, we just know many
equivalences between seemingly different CFTs - bosonization,
fermionization are basic examples. The central charge is one of the few
"universal" properties of a CFT that are independent of the choice of
(equivalent) variables.

I often have the feeling that various people often criticize string theory
for not confirming their prejudices - the universal interpretation of
these states is certainly such a prejudice. And it seems clear to me that
the truth, as string theory teaches us, is even "nicer" than these
prejudices.

> A large majority of the successful string theory
> computations of exact black hole entropy fall into this category.

That's right. Is there a problem with it?

> These are bad examples. The entropy is not computed from string theory,
> but from ``a simple but mysterious duality-invariant extension of
> previously derived formulae for the number of D-brane states in string
> theory.''

Right. Well, the duality group is viewed by the majority as a nice,
nonperturbative feature of string theory which is supported by its beauty
and a huge amount of circumstantial evidence. Yes, the counting of states
has been extrapolated from some rigorous examples, without making a
new, complete calculation in each case.

> This would require more than a weak-coupling analysis and
> seems to be quite challenging.'' It would be wonderful if one could derive
> the formulae of these papers from first principles, but for now they are
> ``string-inspired,'' not string theoretical.

Well, it may be wonderful, but I am afraid that it would not be wonderful
enough to revolutionize the field. ;-)

> This is very unlikely. Highly charged black holes discharge extremely
> rapidly by Schwinger pair production. Moreover, if you compute the
> energy it takes to overcome electrical repulsion in order to add an
> additional charge to a near-extremal black hole, you will find that it
> increases the black hole mass enough to keep it from reaching extremality.

Well, the force may be increasing, but it does not increase enough to stop
you from going arbitrarily close to extremality. If you think that there
exists another critical value of [itex]Q/M,[/itex] smaller than 1 (which stands for
"extremal") that cannot be achieved, could you please specify what the
value is and why is it exactly this value? I agree that very close to the
extremality, we will need accelerators to approach even closer :-), but
the existence of accelerators is not ruled out by any general physical
principles (a budget constraint is not a physical principle).

> > and matching their entropy is definitely equally as important a consistency
> > check as matching the entropy of other black holes.
>
> That's certainly true. Just don't exaggerate what string theory currently
> does -- it meets an important consistency check, but so far only in a very
> limited and physically unrealistic class of examples.

I am not exaggerating anything here. On the other hand, you are
exaggerating the importance of checking some other black holes which lead
to much harder calculations - and perhaps, in some cases, calculations
that we are not yet able to do even in principle. If someone proposed a
stringy calculation of the entropy of a Schwarzschild black hole in a
certain number of dimensions, she would definitely show that she
understands the stringy machinery in detail, but I don't think that she
would be viewed as an extremely important contributor to string theory
because of this paper - unless her methods could also be useful for some
completely different questions.

The agreement between string theory and the BH formula is just obviously
true and no one is too interested in verification of yet another class of
black holes; this might have been a popular topic eight years ago, but
not now. On the other hand, the *universal* explanation of the BH formula
in any context can probably depend on the low energy physics of GR only -
this is what makes it universal. However, the same feature also makes such
an argument non-stringy and indirect. A universal proof of the BH formula
in string theory would have to start with explaining why the object looks
like a black hole at low energies, and then it would use an
effective-field-theoretical based universal justification. However, the
explicit stringy checks based on D-branes are much more non-trivial.
However, enough of them have been verified and today, the topic is simply
not as hot as you seem to think.

A verification of yet another black hole may be nice, but it is basically
just a pretty small addition to Strominger's and Vafa's pioneering
contribution.

> This is, as I said, a very good consistency check. But the trouble is
> that for each new black hole, the density of states and the horizon area
> depend on a new set of charges, and have to be recalculated from scratch.

That's a necessary requirement for each such new calculation to be
non-trivial and stringy.

> If you give me a brand new near-extremal black hole, I would certainly
> expect to get the right agreement again. But this expectation is based on
> past success, and not on any general result that's yet been discovered.

It's because the universal law is not really a consequence of very special
stringy effects, but a result of a consistent treatment of the low-energy
limit. Because Bekenstein and Hawking could have derived these quantities
*without* knowing anything specific about string theory, by thermodynamic
considerations, it is reasonable to expect that the universal proof of the
BH formula does not have to be too stringy either.

Two of us are almost certainly thinking about very similar forms of the
universal proof - I've been trying to convince myself to write TeX notes
about "my" proof involving the [itex]\delta-function-like[/itex] contribution to the
Einstein-Hilbert action, coming from "integrating out" the Planckian
throat inside the black hole.

Let me summarize the main objection of mine against your comments: you
seem to require (or assume, or whatever it exactly is) that some more
detailed qualitative features of the black hole states, not just the
counting, are universal in string theory (or in any theory of quantum
gravity in any description). I think that this is simply not true, and
quantum gravity would be boring if it were true. On the other hand, you
seem to criticize string theory for not giving a universal proof of the BH
formula - but a universal proof would have to start with deciding whether
a system of states can be thought of as a black hole with certain area -
and this can only be decided in the low-energy approximation. Therefore
the universal proof is unlikely to be too similar to these "very stringy"
calculations of D-brane states [itex]- it[/itex] will be more similar to Bekenstein's
and Hawking's thermodynamic arguments rooted in semiclassical GR.

All the best,
Lubos
__{____________________________________________________________________ ________}
E-mail: lumo@matfyz.cz fax: [itex]+1-617/496-0110[/itex] Web: http://lumo.matfyz.cz/
eFax: [itex]+1-801/454-1858[/itex] work: [itex]+1-617/384-9488[/itex] home: [itex]+1-617/868-4487[/itex] (call)
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