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String theory
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== Black holes == In general relativity, a black hole is defined as a region of spacetime in which the gravitational field is so strong that no particle or radiation can escape. In the currently accepted models of stellar evolution, black holes are thought to arise when massive stars undergo [[gravitational collapse]], and many [[galaxies]] are thought to contain [[supermassive black hole]]s at their centers. Black holes are also important for theoretical reasons, as they present profound challenges for theorists attempting to understand the quantum aspects of gravity. String theory has proved to be an important tool for investigating the theoretical properties of black holes because it provides a framework in which theorists can study their [[black hole thermodynamics|thermodynamics]].<ref name="de Haro et al. 2013, p.2"/> === Bekenstein–Hawking formula === In the branch of physics called [[statistical mechanics]], [[entropy]] is a measure of the randomness or disorder of a physical system. This concept was studied in the 1870s by the Austrian physicist [[Ludwig Boltzmann]], who showed that the [[thermodynamics|thermodynamic]] properties of a [[gas]] could be derived from the combined properties of its many constituent [[molecule]]s. Boltzmann argued that by averaging the behaviors of all the different molecules in a gas, one can understand macroscopic properties such as volume, temperature, and pressure. In addition, this perspective led him to give a precise definition of entropy as the [[natural logarithm]] of the number of different states of the molecules (also called ''microstates'') that give rise to the same macroscopic features.<ref>[[#Yau|Yau and Nadis]], pp. 187–188</ref> In the twentieth century, physicists began to apply the same concepts to black holes. In most systems such as gases, the entropy scales with the volume. In the 1970s, the physicist [[Jacob Bekenstein]] suggested that the entropy of a black hole is instead proportional to the ''surface area'' of its [[event horizon]], the boundary beyond which matter and radiation may escape its gravitational attraction.<ref name=Bekenstein/> When combined with ideas of the physicist [[Stephen Hawking]],<ref name=Hawking1975/> Bekenstein's work yielded a precise formula for the entropy of a black hole. The [[Bekenstein–Hawking formula]] expresses the entropy {{math|''S''}} as : <math>S= \frac{c^3kA}{4\hbar G}</math> where {{math|''c''}} is the [[speed of light]], {{math|''k''}} is the [[Boltzmann constant]], {{math|''ħ''}} is the [[reduced Planck constant]], {{math|''G''}} is [[Newton's constant]], and {{math|''A''}} is the surface area of the event horizon.<ref>[[#Wald|Wald]], p. 417</ref> Like any physical system, a black hole has an entropy defined in terms of the number of different microstates that lead to the same macroscopic features. The Bekenstein–Hawking entropy formula gives the expected value of the entropy of a black hole, but by the 1990s, physicists still lacked a derivation of this formula by counting microstates in a theory of quantum gravity. Finding such a derivation of this formula was considered an important test of the viability of any theory of quantum gravity such as string theory.<ref>[[#Yau|Yau and Nadis]], p. 189</ref> === Derivation within string theory === In a paper from 1996, [[Andrew Strominger]] and [[Cumrun Vafa]] showed how to derive the Bekenstein–Hawking formula for certain black holes in string theory.<ref name="Strominger and Vafa 1996"/> Their calculation was based on the observation that D-branes—which look like fluctuating membranes when they are weakly interacting—become dense, massive objects with event horizons when the interactions are strong. In other words, a system of strongly interacting D-branes in string theory is indistinguishable from a black hole. Strominger and Vafa analyzed such D-brane systems and calculated the number of different ways of placing D-branes in spacetime so that their combined mass and charge is equal to a given mass and charge for the resulting black hole. Their calculation reproduced the Bekenstein–Hawking formula exactly, including the factor of {{math|1/4}}.<ref>[[#Yau|Yau and Nadis]], pp. 190–192</ref> Subsequent work by Strominger, Vafa, and others refined the original calculations and gave the precise values of the "quantum corrections" needed to describe very small black holes.<ref name=MSW/><ref name=OST/> The black holes that Strominger and Vafa considered in their original work were quite different from real astrophysical black holes. One difference was that Strominger and Vafa considered only [[extremal black hole]]s in order to make the calculation tractable. These are defined as black holes with the lowest possible mass compatible with a given charge.<ref>[[#Yau|Yau and Nadis]], pp. 192–193</ref> Strominger and Vafa also restricted attention to black holes in five-dimensional spacetime with unphysical supersymmetry.<ref>[[#Yau|Yau and Nadis]], pp. 194–195</ref> Although it was originally developed in this very particular and physically unrealistic context in string theory, the entropy calculation of Strominger and Vafa has led to a qualitative understanding of how black hole entropy can be accounted for in any theory of quantum gravity. Indeed, in 1998, Strominger argued that the original result could be generalized to an arbitrary consistent theory of quantum gravity without relying on strings or supersymmetry.<ref name=Strominger1998/> In collaboration with several other authors in 2010, he showed that some results on black hole entropy could be extended to non-extremal astrophysical black holes.<ref name=Guica/><ref name=CMS/>
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