Construction Of Gravity Models And Thermodynamics Of Black Holes On Noncommutative Spacetimes

Einstein’s gravity theory, general relativity, has achieved great success in predict-ing and explaining physical phenomenons at large scale. In general relativity, the grav-ity is considered to be the geometry of spacetime, and this idea has greatly refreshedour cognition of spacetime. However, there exist irreconcilable contradictions betweengeneral relativity and quantum field theory. Any attempts on bringing the general rela-tivity into the frame of quantum field theory all fail in the renormalizability of quantumgravity. It is the dream of most physicists to look for a consistent theory of quan-tum gravity so that the four fundamental interactions can be unified, and also it is agreat challenge. At present, there exist some ansatzes of quantum gravity includingthe most promising one, string theory, and loop quantum gravity. Noncommutativityof spacetime is one of the various attempts whose essential idea is that the spacetimewill appear noncommutativity at very small scale. It is a natural generalization of theidea of noncommutativity of phase space in quantum mechanics. Because of the non-commutavity of spacetime, the concepts of point and coordinate become ambiguity,and general relativity basing on smooth manifolds is not valid any more. It is greatimportant to establish a corresponding gravity theory on noncommutative spacetimesand investigate its properties. This gravity theory could in some sense combine theeffects of gravity and quantum theory, and then reflects the behavior of gravity at verysmall scale (or at high energy), and so highlights the road to the final theory of quantumgravity.This thesis deals with the problems of gravity and thermodynamics of black holeson noncommutative spacetimes. The main contents of our work are organized as fol-lows.Einstein’s gravity theory can be reformulated as a gauge theory with gauge groupSL(2, C). With the help of the technique of covariant coordinates, we extend thisformulation to noncommutative spacetimes with symplectic structure and builda gauge invariant action in the frame of star-product. Using the Seiberg-Wittenmap, the physical degrees of freedom are expressed in terms of their commutativecounterparts up to the first order in noncommutative parameter and then the firstorder correction to the commutative action is obtained. The result shows that ingeneral the first order correction does not vanish. This is the extension of the ex-isted results in literatures. We then extend the above analysis to the construction of an U (2,2) gravity model on the same class of noncommutative spacetimes.The result also shows that the first order correction does not vanish. This part ofcontents is based on our following workY.-G. Miao and S.-J. Zhang, SL(2,C) gravity on noncommutative space with Pois-son structure, Phys. Rev. D82(2010)084017[arXiv:1004.2118[hep-th]].Y.-G. Miao, Z. Xue and S.-J. Zhang, U(2,2) gravity on noncommutative space withsymplectic structure, Phys. Rev. D83(2011)024023[arXiv:1006.4074[hep-th]].and will be given in detail in chapter4;Based on the coordinate coherent states approach, the results in the existed lit-eratures show that a”point mass” should be smeared in a finite size of rangeand should be described by a gaussian function rather than the usual Dirac deltafunction. Assuming that the Einstein equation is untack, a series of solutions ofnoncommutative inspired black holes can be obtained. We first apply the Parikh-Wilczek’s tunneling method to analyze the thermodynamics of the noncomm-tative inspired Schwarzschild black hole. Different from the existed literatures,we extend the Parikh-Wilczek method to the tunneling of massive particles. Theresults show that the tunneling rate is suppressed by the mass of the tunneling par-ticles and the effective radiation temperature is modified by noncommutativity.The well-known result that the tunneling rate equals the exponent of the differ-ence of entropy does not hold in the massive case. Moreover, we also discuss thecorrelations between tunneling particles. The result shows that there exist corre-lations which imply that the radiation may contain informations. We then analyzethe thermodynamics of the noncommutative inspired Kerr black hole using thereformulated Hamilton-Jacobi method proposed by Banerjee et al. We calculatethe WKB ansatz to the first order in order to include the effect of the angular mo-mentum of tunneling particles. The results show that the radiation temperatureis modified by noncommutativity but not by the angular momentum of tunnelingparticles. Moreover, We analyze the spectra of entropy and show that the entropyquantum is affected neither by noncommutativity or by the quantum correctionof wave function. This part of contents is based on our following workY.-G. Miao, Z. Xue and S.-J. Zhang, Massive charged particle’s tunnel-ing from spherical charged black hole, Europhys. Lett.96(2011)10008 [arXiv:1012.0390[hep-th]]; Tunneling of massive particles from noncommu-tative inspired Schwarzschild black hole, Gen. Rel. Gra.44(2012)555[arXiv:1012.2426[hep-th]]; Quantum tunneling and spectroscopy of noncom-mutative inspired Kerr black hole, Int. J. Mod. Phys. D21(2012)1250018[arXiv:1102.0074[hep-th]].and will be given in detail in chapter5;In the existing literatures, the coordinate coherent states approach is associatedwith a non-standard quantization procedure. The essential outcome of this non-standard quantization procedure is that a”point mass” now should be describedby a gaussian function rather than the usual Dirac delta function. We revisit thecoordinate coherent states approach with the standard quantization procedure onnoncommutative Minkowski plane and deduce a significantly different result: a”point mass” is still described by the Dirac delta function. This implies that theconcept of point particle is still valid if we deal the noncommutativity with thecoordinate coherent states approach associating with the standard quantizationprocedure. To investigate the physical consequences on the quantization proce-dures, we apply the two quantization procedures in discussing the Unruh effectand Hawking radiation. The results are: Under the non-standard quantizationprocedure, Unruh temperature and Unruh spectrum are not modified by noncom-mutativity, but the Hawking temperature is modified while the Hawking radiationspectrum is untack; while under the standard quantization procedure, Unruh tem-perature and Hawking temperature are untack but the both spectra are modifiedby an effective greybody factor. This grebody factor originates from noncommu-tativity. This part of contents is based on our following workY.-G. Miao and S.-J. Zhang, The coordinate coherent states approach revisited,arXiv:1105.4025[hep-th].and will be given in detail in chapter6.