Research by KIAS Scholar Science Café Newsletter Vol 3 Vol Newsletter The KIAS Science Café Science Entropic gravity over the whole family . Then (modulo an abuse of notation) we have Maurice Van Putten (2.6) School of Physics, KIAS where is an arbitrary point. Indeed, if , then the above claim is a consequence of the definition. If not, then we use in [3] a limit argument since the weights are upper semi-continuous. Hence the extendable sections of the bundle provide us with a bound of the singularities of over . Gravity and holography energy astrophysics, galaxy redshifts and the cosmic In everyday experience, gravity is what holds us microwave background, and long term observations on Remark 2.5. Actually, the curvature current associated to the metric can be seen as down to the floor, makes us ski down hill and take radio pulsars in neutron star-neutron binaries. simultaneous extension of the sections of , given its structure when restricted to . elevators to go back up the mountain. One of the first Unfortunately, so far we are unable to extract from it the geometric informations required by 1.1. scientific experiments on gravity was conducted by Despite its dramatic impact, general relativity has References Galileo, to observe that objects of different masses eluded a direct unification with other field theories, [1] B. Berndtsson, Curvature of Vector bundles associated to holomorphic fibrations, to appear in Ann. of take the same amount of time to fall down to Earth that have been developed independently and with Maths. (2007). from the Leaning Tower of Pisa. Gravity is also what great success to describe the forces of Nature between [2] B. Berndtsson, M. Paun,˘ Bergman kernels and the pseudo-effectivity of the relative canonical bundles, holds our solar system together, which ultimately led elementary particles. While general relativity works arXiv:math/0703344, Duke Math. Journal 2008. Newton to pose his law of gravity: the gravitational with astonishing predictive power on astronomical [3] B. Berndtsson, M. Paun,˘ Quantitative extensions of twisted pluricanonical forms and closed positive force between two massive objects is given by the scales, it somehow leaves us in the dark on small currents, arXiv 2009. product of their masses divided by the distance scales well below 0.1mm. It has been suggested raised [4] M. Paun,˘ Relative critical exponents, non-vanishing and metrics with minimal singularities arXiv 2008. between, times Newton’s constant. that general relativity, and Newton’s law to begin with, [5] Y.-T. Siu, Abundance Conjecture, arXiv 2010. is perhaps not fundamental (in apparent contrast to For some three centuries, this view prevailed, until electromagnetism) but, rather, emergent from entropic Einstein incorporated Newton’s law in a causal theory considerations [1]. of gravitation (no ‘spooky’ action at a distance), which beautifully explained the deflection of light around the The reasons for this suggestion is remarkable in its Sun and the non-Newtonian precession in the orbital own right. General relativity teaches us that given Mihai Paun˘ isaProfessorintheMathematicsDepartmentoftheUniversityHenriPoincaré.He receivedhisPh.D.fromtheUniversitéJosephFourier(Grenoble,France)in1998.Hisprimary motion of Mercury - a classic example of a minute an amount of energy, there is a limited amount of researchinterestsarecomplexandalgebraicgeometry.In2006,hereceived“SimionStoilov” deviation which points to an entirely new theory of meaningful information (or activity) that can exist awardoftheRomanianAcademyofScience.Havingpublishedover20researchpapersandseveral gravitation. Einstein’s theory of general relativity led within a bounding surface, expressed in terms of monographs,hehasbecomeamemberofInstituteUniversitairedeFrance(IUF)in2009,andhe wasaninvitedspeakeratthe2010InternationalCongressofMathematicians. to a dramatic change in our view on the Universe, by surface area rather than enclosed volume [2]. In revealing, un-anticipated, black holes, an expanding Nature, useful work space is area, not volume, Contactinformation:InstitutElieCartan(Mathématiques),UniversitéHenriPoincaré, Nancy1,B.P.70239,F-54506Vandoeuvre-les-NancyCedex,FRANCE,[email protected] Universe and gravitational radiation. Much, though not ironically in much the same way as real-estate all of these have been verified by observations in high- appraises homes and offices. 18 Korea Institute for Advanced Study | 19 Science Café Newsletter Vol 3 Vol Newsletter The KIAS Science Café Science Trying to squeeze more information within are mathematically described by different objects, by analogy to globular clusters. Radiation a given bounding surface by increasing energy line elements, which otherwise share the solves the cat-singularity paradox, since the event ultimately leads to a trapped surface of light exact same exterior black hole space-time. horizon of an evaporating black hole is no longer within, i.e., it evolves into the event horizon Black holes line-element inevitably introduce fixed but recedes at a velocity given by one-half the of a black hole [3], keeping all that is inside singularities by Liouville’s theorem [5], and black hole luminosity by conservation of energy. A cat hidden from the outside world. In this limit, we their location(s) depend on the extension of the falling onto the black hole is hereby always chasing are inevitably led to attribute (an astronomical exterior space time associated with the choice a receding event horizon, never to fall through and amount of) entropy to the event horizon of line-element to describe the exterior space- therefore it will never be in a position to “choose” Figure 2. Entropic gravity ascribes gravitational attraction between two proportional to its surface area [4], to account time. Remarkably, these singularity can be real, between being crunched at the origin or moving to bodies to an increase/decrease in entropy with a decrease/increase in for the loss of information that is within. at the origin, as for the Schwarzschild line- another universe. The finite lifetime of an evaporating separation distance. When going up by taking an elevator (left, right), there is a decrease in entropy (Sp) associated with him/her and that (SE) element, or imaginary, as in the line element black hole hereby introduces a strong form of the of the Earth. On the basis of well-defined apparent horizon surfaces of Black holes and strong cosmic of Einstein-Rosen, Brill-Lindquist or its Penrose cosmic censorship conjecture [7,9], which mass-equivalent black holes, a thermodynamic description recovers the Newtonian result of about 3000 J for bringing up a person to the KIAS tea censorship symmetrized form [6,7]. These exact solutions asserts that all singularities are hidden by an event room. For arbitrary separations including black hole mergers, entropic More generally, the event horizon of a black serve to show that the extended topology and horizon, never to be accessible by an outside observer. gravity from apparent horizon surfaces satisfiesmonotonicity in entropy hole hides whatever is ascribed to its interior location of singularities induced by a black hole as a function of separation distance, while time like holographic screens with fixed entropy surface density do not. from the outside word, its past history of are generally gauge dependent. Entropic gravity in black hole binaries formation or any connection it may have to an While general relativity remains mysteriously at odds adjacent universe. These various alternatives A cat falling onto a black hole, therefore, has with other field theories, it is remarkably consistent per unit surface area is less than ¼, so as to properly the apparent choice to get torn apart and be with thermodynamics. Recently, the question was preserve monotonicity of entropy [10]. crushed at the central singularity when choosing posed if perhaps gravitation, and Newton’s law in the Schwarzschild line-element, or to continue particular, is itself entropic in origin, and possibly so Our staff member Ms. Yunae Kim once remarked to live in an adjacent universe by changing in a holographic embedding [1]. that it may be better to take the elevator together, its mind to using any of the alternative line to save energy. I am not sure about the efficiency of elements with singularities along the imaginary Recently, we considered entropic gravity for a black the elevator system, but it does suggest the question: axis. Needless to say, this kind of choice runs hole binary. On the basis of the entropy of apparent what is what is the entropy change associated with the counter to gauge independence, which ensures horizon surfaces, two when the black hole are separate, entropic force between the Earth and a passenger in the that all observables are independent of our one following a merger, we find that Newton’s law of elevator? The surface area of the black hole equivalent Figure 1. The location of singularities associated with the continuation of the choice of coordinate systems. attraction is an asymptotic limit for large separations to the Earth is 9.9 square centimers with an entropy exterior Schwarzschild space-time depends on the choice of coordinates. A cat of S= exp(152) and temperature of 20 mK, where we falling onto a black hole may hereby be by torn apart and crushed at a central of a more general relationship that describes their singularity or continue to live in an adjacent universe by passing safely along The idea that black holes carry entropy, entropic interaction at all separation distances, put Boltzmann’s constant equal to unit. An elevator singularities in the complex plane. depending on its choice of line-element. as proposed by Bekenstein [4], naturally including zero following a bifurcation in their merger bringing a passenger of 60 kg up by 5 m to the KIAS This freedom of choice-to die or live - contradicts with gauge invariance. This paradox is solved by black hole evaporation, whereby the cat falling in traces a leads to radiation, as subsequently shown by at small separations. This rigorous analysis shows tea room reduces the entropy in the Earth-passenger continuously shrinking event horizon, never to penetrate it.