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Don N. Page

Bio: Don N. Page is an academic researcher from University of Alberta. The author has contributed to research in topics: Black hole & Extremal black hole. The author has an hindex of 63, co-authored 267 publications receiving 18213 citations. Previous affiliations of Don N. Page include Canadian Institute for Advanced Research & Texas A&M University.


Papers
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TL;DR: In this paper, it was shown that the canonical ensemble exists for asymptotically anti-de-Sitter space, unlike the case for the case of asymPTotically flat space.
Abstract: The Einstein equations with a negative cosmological constant admit black hole solutions which are asymptotic to anti-de Sitter space. Like black holes in asymptotically flat space, these solutions have thermodynamic properties including a characteristic temperature and an intrinsic entropy equal to one quarter of the area of the event horizon in Planck units. There are however some important differences from the asymptotically flat case. A black hole in anti-de Sitter space has a minimum temperature which occurs when its size is of the order of the characteristic radius of the anti-de Sitter space. For larger black holes the red-shifted temperature measured at infinity is greater. This means that such black holes have positive specific heat and can be in stable equilibrium with thermal radiation at a fixed temperature. It also implies that the canonical ensemble exists for asymptotically anti-de Sitter space, unlike the case for asymptotically flat space. One can also consider the microcanonical ensemble. One can avoid the problem that arises in asymptotically flat space of having to put the system in a box with unphysical perfectly reflecting walls because the gravitational potential of anti-de Sitter space acts as a box of finite volume.

2,923 citations

Journal ArticleDOI
TL;DR: There is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.
Abstract: If a quantum system of Hilbert space dimension mn is in a random pure state, the average entropy of a subsystem of dimension m\ensuremath{\le}n is conjectured to be ${\mathit{S}}_{\mathit{m},\mathit{n}}$= ${\mathit{S}}_{\mathit{k}=\mathit{n}+1}^{\mathit{mn}}$ 1/k-m-1/2n and is shown to be \ensuremath{\simeq}lnm-m/2n for 1\ensuremath{\ll}m\ensuremath{\le}n. Thus there is less than one-half unit of information, on average, in the smaller subsystem of a total system in a random pure state.

1,313 citations

Journal ArticleDOI
TL;DR: If black hole formation evaporation can be described by an S matrix, information would be expected to come out in black hole radiation, but an estimate shows that it may come out initially so slowly, or else be so spread out, that it would never show up in an analysis perturbative in M/M.
Abstract: If black hole formation evaporation can be described by an S matrix, information would be expected to come out in black hole radiation. An estimate shows that it may come out initially so slowly, or else be so spread out, that it would never show up in an analysis perturbative in ${\mathit{M}}_{\mathrm{Planck}}$/M, or in 1/N for two-dimensional dilatonic black holes with a large number N of minimally coupled scalar fields.

1,061 citations

Journal ArticleDOI
TL;DR: In this article, the authors combine Teukolsky and Press' method with the black-hole perturbation method to calculate the emission rate for the known massless particles.
Abstract: Hawking has predicted that a black hole will emit particles as if it had a temperature proportional to its surface gravity. This paper combines Hawking's quantum formalism with the black-hole perturbation methods of Teukolsky and Press to calculate the emission rate for the known massless particles. Numerical results indicate that a hole of mass $M\ensuremath{\gg}{10}^{17}$ g should emit a total power output of $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\ensuremath{\hbar}{c}^{6}{G}^{\ensuremath{-}2}{M}^{\ensuremath{-}2}$, of which 81% is in neutrinos, 17% is in photons, and 2% is in gravitons. These rates plus an estimate for the emission rates of massive particles from smaller holes allow one to infer that a primordial black hole will have decayed away within the present age of the universe if and only if its initial mass was $Ml(5\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{14}$ g.

984 citations


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TL;DR: In this article, it was shown that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of the super Yang-Mills theory in four dimensions.
Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of ${\cal N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the ${\cal N}=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

14,084 citations

Posted Content
TL;DR: In this article, a correspondence between conformal field theory observables and those of supergravity was proposed, where correlation functions in conformal fields are given by the dependence of the supergravity action on the asymptotic behavior at infinity.
Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.

8,751 citations

Journal ArticleDOI
TL;DR: In this paper, the holographic correspondence between field theories and string/M theory is discussed, focusing on the relation between compactifications of string theory on anti-de Sitter spaces and conformal field theories.

5,610 citations

01 Jan 2011
TL;DR: To understand the central claims of evolutionary psychology the authors require an understanding of some key concepts in evolutionary biology, cognitive psychology, philosophy of science and philosophy of mind.
Abstract: Evolutionary psychology is one of many biologically informed approaches to the study of human behavior. Along with cognitive psychologists, evolutionary psychologists propose that much, if not all, of our behavior can be explained by appeal to internal psychological mechanisms. What distinguishes evolutionary psychologists from many cognitive psychologists is the proposal that the relevant internal mechanisms are adaptations—products of natural selection—that helped our ancestors get around the world, survive and reproduce. To understand the central claims of evolutionary psychology we require an understanding of some key concepts in evolutionary biology, cognitive psychology, philosophy of science and philosophy of mind. Philosophers are interested in evolutionary psychology for a number of reasons. For philosophers of science —mostly philosophers of biology—evolutionary psychology provides a critical target. There is a broad consensus among philosophers of science that evolutionary psychology is a deeply flawed enterprise. For philosophers of mind and cognitive science evolutionary psychology has been a source of empirical hypotheses about cognitive architecture and specific components of that architecture. Philosophers of mind are also critical of evolutionary psychology but their criticisms are not as all-encompassing as those presented by philosophers of biology. Evolutionary psychology is also invoked by philosophers interested in moral psychology both as a source of empirical hypotheses and as a critical target.

4,670 citations

Journal ArticleDOI
TL;DR: The correspondence between supergravity and string theory on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamic properties of Schwarzschild black holes in Anti-de Sitter space as mentioned in this paper.
Abstract: The correspondence between supergravity (and string theory) on AdS space and boundary conformal eld theory relates the thermodynamics of N = 4 super Yang-Mills theory in four dimensions to the thermodynamics of Schwarzschild black holes in Anti-de Sitter space. In this description, quantum phenomena such as the spontaneous breaking of the center of the gauge group, magnetic connement, and the mass gap are coded in classical geometry. The correspondence makes it manifest that the entropy of a very large AdS Schwarzschild black hole must scale \holographically" with the volume of its horizon. By similar methods, one can also make a speculative proposal for the description of large N gauge theories in four dimensions without supersymmetry.

4,209 citations