Hopf不变量的拓扑结构和黑洞量子辐射的研究
摘要
本论文在拓扑场论和广义相对论两个领域做了深入的研究工作.拓扑场论方面,运用段氏拓扑流理论研究了微分几何和拓扑学中的一个重要的拓扑不变量—Hopf不变量的内部拓扑结构,得出了表征Hopf不变量与扭结缠绕数之间关系的准确表达式。广义相对论方面,主要研究了黑洞的量子辐射,分别利用量子遂穿的机制和量子反常的方法研究了一系列黑洞时空和FRW宇宙表观视界的Hawking辐射问题。
首先,本论文研究了Hopf不变量的拓扑结构,以及它在规范场理论中的应用.对于最简单的情况,我们从Hopf映射的旋量表示入手,得到了两个重要结论,一是Hopf不变量实际上是Hopf映射的映射度,二是表征Hopf不变量与扭结不变量之间关系的准确表达式。对于高阶Hopf不变量的情况,利用段氏拓扑流理论,我们推导出了关于Hopf不变量的一个比较繁杂的表达式,根据数学方面关于高阶扭结的文献,从高维扭结缠绕数的定义出发推导出了其表达式,最终发现了Hopf不变量与高维扭结缠绕数之间的关系。基于这些工作,我们还研究了Skyrme-Faddeev模型中扭结状拓扑缺陷的拓扑根源,解释了Hopf不变量是描述其扭结状拓扑缺陷的合适的拓扑不变量.
其次,本论文利用量子遂穿的机制研究了标量粒子和费米子在黑洞时空中的Hawk-ing辐射问题。对于标量粒子或者光子的Hawking辐射,分别利用类光测地线方法和Hamilton-Jacobi方法,我们计算了Dilaton引力理论中的球对称黑洞,转动的Kaluza-Klein黑洞和Kerr-Sen黑洞的Hawking温度,由于未考虑辐射标量粒子之间的自引力相互作用,得到了纯热谱形式的辐射谱.我们还运用量子遂穿的模型研究了稳态转动的BTZ黑洞和Kerr黑洞时空中费米子的Hawking辐射.通过对相应背景时空中的广义协变Dirac方程作WKB近似,我们推导出了Hawking热辐射的费米谱.
第三,我们利用量子反常取消的机制分别研究了转动带电荷的Godel黑洞背景中标量场和一般稳态(2+1)维黑洞背景中费米场的Hawking辐射问题.通过对Godel黑洞背景中标量场的作用量和一般稳态(2+1)维黑洞背景中费米场的作用量进行维数约化,我们发现它们均可以由两维等效理论所描述。通过对等效的两维时空背景中反常方程的求解,我们得到了遵守Hawking分布的角动量通量和能量通量。
最后,我们研究了FRW宇宙表观视界的Hawking辐射问题,分别利用费米子遂穿的机制和表观视界引力反常的方法推导出了表观视界的Hawking温度,解决了在研究FRW宇宙的热力学时对表观视界Hawking温度的假设问题.
In this dissertation, we have made in-depth researches on two topics, i.e. topological field theory and general relativity. In the field of topological field theory, in terms of Duan's topological current theory, we study the inner topological structure of Hopf invariant, which is an important topological invariant in differential geometry and topology, and obtain the exact expression for revealing the relationship between Hopf invariant and linking number of knots. In the field of general relativity and black hole physics, we study the Hawking radiation for a series of balck hole background spacetimes and FRW universe by using the methods of quantum tunneling and quantum anomaly, respectively.
Firstly, based on Duan's topological current theory, the inner topological structure of Hopf invariant is studied, and we also discuss its applications in gauge field theory. For the simplest case, starting with the spinor representation of Hopf mapping, we derive two important results, which shows that Hopf invariant is actually the degree of Hopf mapping and Hopf invariant is also the linking number of knots. In the case of higher dimensional Hopf invariant, we firstly deduce a complicated expression fot it. Then, according to the definition of linking number for higher dimensional knots in the mathematical literatures, an explicit expression of linking number for higher dimensional knots is derived. Finally, after comparing the two expressions, the relationship between Hopf invariant and the linking number of higher dimensional knots is constructed. Based on these research work, we further study the topological origin of knot-like topological defects in Skyrme-Faddeev model, and explain that Hopf invariant is the proper topological invariant to describe the topology of these knot-like defects.
Secondly, we study the scalar and fermonic particle's Hawking radiations in black hole spacetime backgrounds by using the mechanism of quantum tunneling. In terms of the null geodesic method and Hamilton-Jacobi method, we calculate the Hawking temperatures of the spherical symmetric GHS black hole, stationary rotating Kaluza-Klein black hole and Kerr-Sen black hole, respectively. Because the self-gravitational interaction of radiated scalar particles are not taken into account, the radiation spectrums are all pure thermal. We also investigate the fermionic particle's Hawking radiations from the BTZ black hole and Kerr black hole by using the tunneling formalism. Applying WKB approximation to the covariant Dirac equations, we obtain the radiation spectrums for fermions in these spacetime backgrounds.
Thirdly, using the the mechanism of quantum anomaly cancellation, we study the Hawking radiation for scalar field in the rotating charged Godel black hole and the Hawk-ing radiation for fermionic field in the general stationary (2+1) dimensional black hole. By performing the dimensional reduction procedure for the action of scalar field in Godel black hole background and the action of fermionic field in (2+1) dimensional black hole back-ground, we find that the original field theories can be approximated by two dimensional effective theories, respectively. By solving the anomaly equation in the two dimensional spacetime background, we obtain the angular flux and energy-momentum flux satisfying the Hawking distribution.
Finally, by using the fermion tunneling formalism and the method of gravitational anomaly, we study the Hawking radiation of apparent horizon in FRW universe, and derive the Hawking temperature associated with the apparent horizon of FRW universe, which is extensively applied in investigating the relationship between the first law of thermodynamics and Friedmann equations.
首先,本论文研究了Hopf不变量的拓扑结构,以及它在规范场理论中的应用.对于最简单的情况,我们从Hopf映射的旋量表示入手,得到了两个重要结论,一是Hopf不变量实际上是Hopf映射的映射度,二是表征Hopf不变量与扭结不变量之间关系的准确表达式。对于高阶Hopf不变量的情况,利用段氏拓扑流理论,我们推导出了关于Hopf不变量的一个比较繁杂的表达式,根据数学方面关于高阶扭结的文献,从高维扭结缠绕数的定义出发推导出了其表达式,最终发现了Hopf不变量与高维扭结缠绕数之间的关系。基于这些工作,我们还研究了Skyrme-Faddeev模型中扭结状拓扑缺陷的拓扑根源,解释了Hopf不变量是描述其扭结状拓扑缺陷的合适的拓扑不变量.
其次,本论文利用量子遂穿的机制研究了标量粒子和费米子在黑洞时空中的Hawk-ing辐射问题。对于标量粒子或者光子的Hawking辐射,分别利用类光测地线方法和Hamilton-Jacobi方法,我们计算了Dilaton引力理论中的球对称黑洞,转动的Kaluza-Klein黑洞和Kerr-Sen黑洞的Hawking温度,由于未考虑辐射标量粒子之间的自引力相互作用,得到了纯热谱形式的辐射谱.我们还运用量子遂穿的模型研究了稳态转动的BTZ黑洞和Kerr黑洞时空中费米子的Hawking辐射.通过对相应背景时空中的广义协变Dirac方程作WKB近似,我们推导出了Hawking热辐射的费米谱.
第三,我们利用量子反常取消的机制分别研究了转动带电荷的Godel黑洞背景中标量场和一般稳态(2+1)维黑洞背景中费米场的Hawking辐射问题.通过对Godel黑洞背景中标量场的作用量和一般稳态(2+1)维黑洞背景中费米场的作用量进行维数约化,我们发现它们均可以由两维等效理论所描述。通过对等效的两维时空背景中反常方程的求解,我们得到了遵守Hawking分布的角动量通量和能量通量。
最后,我们研究了FRW宇宙表观视界的Hawking辐射问题,分别利用费米子遂穿的机制和表观视界引力反常的方法推导出了表观视界的Hawking温度,解决了在研究FRW宇宙的热力学时对表观视界Hawking温度的假设问题.
In this dissertation, we have made in-depth researches on two topics, i.e. topological field theory and general relativity. In the field of topological field theory, in terms of Duan's topological current theory, we study the inner topological structure of Hopf invariant, which is an important topological invariant in differential geometry and topology, and obtain the exact expression for revealing the relationship between Hopf invariant and linking number of knots. In the field of general relativity and black hole physics, we study the Hawking radiation for a series of balck hole background spacetimes and FRW universe by using the methods of quantum tunneling and quantum anomaly, respectively.
Firstly, based on Duan's topological current theory, the inner topological structure of Hopf invariant is studied, and we also discuss its applications in gauge field theory. For the simplest case, starting with the spinor representation of Hopf mapping, we derive two important results, which shows that Hopf invariant is actually the degree of Hopf mapping and Hopf invariant is also the linking number of knots. In the case of higher dimensional Hopf invariant, we firstly deduce a complicated expression fot it. Then, according to the definition of linking number for higher dimensional knots in the mathematical literatures, an explicit expression of linking number for higher dimensional knots is derived. Finally, after comparing the two expressions, the relationship between Hopf invariant and the linking number of higher dimensional knots is constructed. Based on these research work, we further study the topological origin of knot-like topological defects in Skyrme-Faddeev model, and explain that Hopf invariant is the proper topological invariant to describe the topology of these knot-like defects.
Secondly, we study the scalar and fermonic particle's Hawking radiations in black hole spacetime backgrounds by using the mechanism of quantum tunneling. In terms of the null geodesic method and Hamilton-Jacobi method, we calculate the Hawking temperatures of the spherical symmetric GHS black hole, stationary rotating Kaluza-Klein black hole and Kerr-Sen black hole, respectively. Because the self-gravitational interaction of radiated scalar particles are not taken into account, the radiation spectrums are all pure thermal. We also investigate the fermionic particle's Hawking radiations from the BTZ black hole and Kerr black hole by using the tunneling formalism. Applying WKB approximation to the covariant Dirac equations, we obtain the radiation spectrums for fermions in these spacetime backgrounds.
Thirdly, using the the mechanism of quantum anomaly cancellation, we study the Hawking radiation for scalar field in the rotating charged Godel black hole and the Hawk-ing radiation for fermionic field in the general stationary (2+1) dimensional black hole. By performing the dimensional reduction procedure for the action of scalar field in Godel black hole background and the action of fermionic field in (2+1) dimensional black hole back-ground, we find that the original field theories can be approximated by two dimensional effective theories, respectively. By solving the anomaly equation in the two dimensional spacetime background, we obtain the angular flux and energy-momentum flux satisfying the Hawking distribution.
Finally, by using the fermion tunneling formalism and the method of gravitational anomaly, we study the Hawking radiation of apparent horizon in FRW universe, and derive the Hawking temperature associated with the apparent horizon of FRW universe, which is extensively applied in investigating the relationship between the first law of thermodynamics and Friedmann equations.
引文
[1]Y. S. Duan and M. L. Ge, KEXUE TONGBAO,26(1976) 282.
[2]Y. S. Duan and M. L. Ge, Scientia Sinica,11(1979) 1072.
[3]Y.S. Duan and S.C. Zhao, Commun. Theor. Phys.2(1983)1553.
[4]Y. S. Duan, SLAC-PUB-3301/84.
[5]Duan Y S, In Proceedings Symposium on Yang-Mills gauge theories (Beijing), Commun. Theor. Phys. 4(1984)661.
[6]Y. S. Duan and Z. P. Duan, Int. J.Eng. Sci.24 (1986) 513.
[7]Y. S. Duan and J. C. Liu, In Proceedings of the 11th Johns Hopkins Workshop on Current Problem in Particle theory,11(1987)183, Lanzhou, China, edited by Y. S. Duan, G. Domokos, and S. Kovesi-Domokos, World Scientific, Singapore, (1988).
[8]Y. S. Duan and S. L. Zhang, Int. J. Eng. Sci.28(1990)689; 29(1991)153; 29(1991)1593; 30(1992)153.
[9]Y. S. Duan and X. H. Meng, Int. J. Engng. Sci.31(1993)1173.
[10]Y.S. Duan, S.L. Zhang, and S.S. Feng, J. Math. Phys.35(1994)4463.
[11]Y. S. Duan, G. H. Yang and Y. Jiang, Gen. Rel. Grav.29(1997)715.
[12]Ying Jiang, Yishi Duan, J. Math. Phys.24(2000)6463.
[13]Duan Y. S., Meng X. H., J. Math. Phys.34(1993)1149.
[14]Duan Y. S., Li S., Yang G. H.:Nucl. Phys. B514(1998)705.
[15]Y.S.Duan, S.Li, Phys.Lett.A246(1998)172.
[16]Y.S.Duan, P.M.Zhang, H.Zhang, Phys. Lett. A291(2001)296.
[17]S.Li, Y.S.Duan, Astro. and Space Sci.268(1999)455.
[18]Duan Y. S., Zhang H., Li S.:Phys. Rev. B 58(1998)125.
[19]Y.S.Duan, H,Zhang, G.Jia, Phys. Lett. A 253(1999)57; Y.S.Duan, G.Jia, H.Zhang, J.Phys.A 32(1999)4943.
[20]Duan, YS; Fu, LB; Jia, G, J. Math. Phys.,41(2000)4379.
[21]Li-Bin Fu, Yi-Shi Duan, Hong Zhang,Phys. Rev. D 61(2000)045004.
[22]J.R.Ren. R.Li and Y.S.Duan, J. Math. Phys.48,073502(2007).
[23]Ji-rong Ren, Ran Li, Yi-shi Duan, Commun.Theor.Phys.50(2008)53.
[24]Ren Ji-Rong, Li Ran, Duan Yi-Shi, Int.J.Mod.Phys.A,23 (2008)1447.
[25]Yi-Shi Duan, Ji-Rong Ren, Ran Li, Commun.Theor.Phys.47(2007)875.
[26]H. Hopf, Math. Ann.104,639(1931); H. Hopf, Fund. Math.25,427(1935).
[27]H. K. Urbantke, J. Geom. Phys.46,125(2003).
[28]N. D. Mermin and T. L. Ho, Phys. Rev. Lett.9236,594(1976).
[29]Yi-shi Duan, Peng-ming Zhang, Mod.Phys.Lett. A16 (2001) 2483.
[30]James H. White, American Journal of Mathematics,91,693(1969).
[31]Alberto S. Cattaneo and Carlo A. Rossi, Commun. Math. Phys.256, (2005)513. Alberto S. Cattaneo, J.Math.Phys.37 (1996) 3684. Alberto S:Cattaneo, Commun.Math.Phys.221 (2001) 591.
[32]E.Witten, Commun.Math.Phys.121,351(1989).
[33]Duan Yi-Shi, Liu Xin and Fu Li-Bin, Commun. Theor. Phys. (Beijing, China) 40,447(2003).
[34]L. Faddeev and A.J. Niemi, Phys. Rev. Lett.82,1624(1999).
[35]L. Faddeev and A.J. Niemi, Phys. Lett. B 449,214(1999).
[36]L. Faddeev and A.J. Niemi, Phys. Lett. B 464,90(1999).
[37]Y. M. Cho, Phys. Rev. Lett.46,302(1981).
[38]Y. M. Cho, Phys. Rev. D 21,1080(1980).
[39]Y. M. Cho, Phys. Rev. Lett.87,252001(2001).
[40]L. Faddeev and A. J. Niemi, Nature,387,58(1997).
[41]T.T. Wu, C.N. Yang, Phys. Rev. D 12,3845(1975); T.T. Wu, C.N. Yang, Phys. Rev. D 14,437(1975).
[42]Bardeen, J. M., Carter, B., and Hawking, S. W., Commun. math. Phys.,31,161(1973).
[43]Beckenstein, J. D., Phys. Rev. D 7,2333(1973).
[44]S.W.Hawking, Nature 248(1974)30; Commun.Math.Phys.43(1975)199.
[45]Per Kraus, Frank Wilczek, Nucl.Phys.B 433(1995)403.
[46]Per Kraus, Frank Wilczek, Nucl.Phys. B 437 (1995) 231.
[47]Maulik K. Parikh, Frank Wilczek, Phys.Rev.Lett.85 (2000) 5042.
[48]J.Zhang and Z.Zhao, JHEP 10,055(2005).
[49]J.Zhang and Z.Zhao, Phys.Lett.B 618,14(2005); J.Zhang and Z.Zhao, Phys.Lett.B 638,110(2006).
[50]Qing-Quan Jiang, Shuang-Qing Wu, Xu Cai, Phys.Rev. D73 (2006) 064003.
[51]M.Angheben, M.Nadalini, L.Vanzo and S.zerbini, JHEP 05,014(2005).
[52]K.Srinivasan and T.Padmanabhan, Phys.Rev.D 60,24007(1999); S.Shankaranarayanan, T.Padmanabhan and K.Srinivasan, Class.Quant.Grav.19,2671(2002).
[53]R.Kerner and R.B.Mann, Phys.Rev.D 73,104010(2006).
[54]R.Kerner and R.B.Mann, Class.Quant.Grav.25(2008)095014.
[55]Ji-Rong Ren, Ran Li, Fei-Hu Liu, Mod.Phys.Lett.A23(2009)3419.
[56]Ran Li, Ji-Rong Ren, Phys.Lett.B 661(2008)370.
[57]Ran Li, Ji-Rong Ren, Shao-Wen Wei, Class.Quant.Grav.25(2008)125016.
[58]D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D43(1991)3140.
[59]J. B. Hartle and S. W. Hawking Phys. Rev. D 13(1976)2188.
[60]G. W. Gibbons and D. L. Wiltshire, Ann. Phys. (N.Y.) 167(1986)201.
[61]J. I. Koga and K. I. Maeda, Phys. Rev. D 52(1995)7066.
[62]A. Sen, Phys. Rev. Lett.69(1992)1006.
[63]M.banados, C.Teitelboim and J.Zanelli, Phys.Rev.Lett.69,1849(1992).
[64]X.He and W.Liu, Phys.Lett.B 653,330(2007).
[65]Roberto Di Criscienzo, Luciano Vanzo, Europhys.Lett.82(2008)60001.
[66]Ryan Kerner, R.B. Mann, Phys.Lett.B 665(2008)277.
[67]De-You Chen, Qing-Quan Jiang, Xiao-Tao Zu, Phys.Lett.B665(2008)106.
[68]Qing-Quan Jiang, Phys.Rev.D78(2008)044009.
[69]Kai Lin, Shu-Zheng Yang, Phys. Rev. D 79 (2009)064035.
[70]Alexandre Yale, Robert B. Mann, Phys. Lett. B 673(2009)168.
[71]Rabin Banerjee, Phys.Rev.D79(2009)044005.
[72]Rabin Banerjee, Sujoy Kumar Modak, JHEP 0905(2009)063.
[73]S. Christensen and S. Fulling, Phys.Rev.D 15,2088(1977).
[74]S. P. Robinson and F.Wilczek, Phys.Rev.Lett.95,011303(2005).
[75]S. Iso, H. Umetsu and F. Wilczek, Phys.Rev.Lett.96,151302(2006).
[76]Satoshi Iso, Hiroshi Umetsu, Frank Wilczek, Phys.Rev.D 74,044017(2006).
[77]Keiju Murata, Jiro Soda, Phys.Rev. D 74,044018(2006);
[78]M. R. Setare, Eur.Phys.J.C,49,865(2007);
[79]Satoshi Iso, Takeshi Morita, Hiroshi Umetsu, JHEP,04,068(2007);
[80]Qing-Quan Jiang, Shuang-Qing Wu, Phys.Lett.B 647,200(2007);
[81]Qing-Quan Jiang, Shuang-Qing Wu, Xu Cai, Phys.Rev.D 75,064029(2007);
[82]Kui Xiao, Wenbiao Liu, Hongbao Zhang, Phys.Lett.B 647,482(2007);
[83]Hyeonjoon Shin, Wontae Kim, JHEP,06,012(2007);
[84]Qing-Quan Jiang, Shuang-Qing Wu, Xu Cai, Phys.Lett.B 651,65(2007);
[85]Shuang-Qing Wu, Jun-Jin Peng, Class.Quant.Grav.24,5123(2007);
[86]Wontae Kim, Hyeonjoon Shin, JHEP,07,070(2007);
[87]Rabin Banerjee, Shailesh Kulkarni, Phys.Rev.D 77,024018(2008);
[88]Rabin Banerjee, Shailesh Kulkarni, Phys.Lett.B 659,827(2008);
[89]Sunandan Gangopadhyay, Shailesh Kulkarni, Phys.Rev.D 77,024038(2008);
[90]Jun-Jin Peng, Shuang-Qing Wu, Phys.Lett.B 661,300(2008);
[91]Xiaoning Wu, Chao-Guang Huang, Jia-Rui Sun, Phys.Rev.D 77,124023(2008);
[92]Shuang-Qing Wu, Jun-Jin Peng, Zhan-Yue Zhao, Class.Quant.Grav.25,135001(2008);
[93]Sunandan Gangopadhyay, Phys.Rev.D 78,044026(2008);
[94]Sunandan Gangopadhyay, Europhys.Lett.85,10004(2009);
[95]Shao-Wen Wei, Ran Li, Yu-Xiao Liu, Ji-Rong Ren, arXiv:0901.2614;
[96]S. Nam and J.-D. Park, Class.Quant.Grav.26,145015(2009).
[97]A. P. Porfyriadis, Phys.Rev. D 79,084039(2009);
[98]A. P. Porfyriadis, Phys.Lett.B 675,235(2009);
[99]Shao-Wen Wei, Ran Li, Yu-Xiao Liu, Ji-Rong Ren, arXiv:0904.2915.
[100]Takeshi Morita, Phys.Lett.B 677,88(2009).
[101]Ran Li, Ji-Rong Ren, Hawking radiation via quantum anomaly from the rotating charged Godel black hole in five dimensions.
[102]S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 75,124004(2007);
[103]S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 76,064015(2007);
[104]S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 77,045007(2008);
[105]S. Iso, T. Morita and H. Umetsu, Nucl. Phys. B 799,60(2008);
[106]S. Iso, Int. J. Mod. Phys. A 23,2082(2008);
[107]L. Bonora, M. Cvitan, JHEP G805,071(2008);
[108]L.Bonora, M.Cvitan, S.Pallua, I.Smolic, JHEP 0812,021(2008);
[109]Takeshi Morita, JHEP 0901,037(2009).
[110]Keiju Murata, Umpei Miyamoto, Phys.Rev.D 76,084038(2007);
[111]L. Bonora, M. Cvitan, S. Pallua, I. Smolic, Phys. Rev. D 80,084034(2009).
[112]K. Godel, Rev. Mod. Phys.21,447 (1949).
[113]Eric G. Gimon, Akikazu Hashimoto, Phys.Rev.Lett.91 (2003) 021601.
[114]C.A.R.Herdeiro, Class.Quant.Grav.20 (2003) 4891.
[115]D. Brecher, U..H. Danielsson, J. P. Gregory, M. E. Olsson, JHEP 0311:033,2003.
[116]Glenn Barnich, Geoffrey Compere Phys.Rev.Lett.95 (2005) 031302.
[117]Shuang-Qing Wu, Phys. Rev. Lett.100:121301,2008.
[118]E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett.102,201301(2009).
[119]J. Oliva, D. Tempo and R. Troncoso, JHEP 0907,011(2009); G. Giribetl, J. Oliva, D. Tempo and R. Troncoso, arXiv:0909.2564.
[120]T. Jacobson, Phys. Rev. Lett.75,1260 (1995).
[121]C. Eling, R. Guedens and T. Jacobson, Phys. Rev. Lett.96,121301 (2006).
[122]R. G. Cai and L. M. Cao, Phys. Rev. D 75,064008 (2007).
[123]Shao-Feng Wu, Guo-Hong Yang, Peng-Ming Zhang, Prog.Theor.Phys.120,615(2008).
[124]T.Padmanabhan, Phys.Rept.406,49(2005).
[125]Aseem Paranjape, Sudipta Sarkar, T. Padmanabhan, Phys.Rev.D74,104015(2006).
[126]Dawood Kothawala, Sudipta Sarkar, T. Padmanabhan, Phys.Lett.B652,338(2007).
[127]T. Padmanabhan, Entropy density of spacetime and thermodynamic interpretation of field equations of gravity in any diffeomorphism invariant theory, arXiv:0903.1254.
[128]Dawood Kothawala, T. Padmanabhan, Phys.Rev.D79,104020(2009).
[129]T.Padmanabhan, Rep. Prog. Phys.73 (2010) 046901.
[130]T.Padmanabhan,Equipartition of energy in the horizon degrees of freedom and the emergence of gravity, arXiv:0912.3165.
[131]T.Padmanabhan, Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity, arXiv:1003.5665.
[132]Erik P. Verlinde, On the Origin of Gravity and the Laws of Newton, arXiv:1001.0785.
[133]Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta, Phys. Rev. D 81,061501(R) (2010).
[134]Lee Smolin, Newtonian gravity in loop quantum gravity, arXiv:1001.3668.
[135]Miao Li, Yi Wang, Quantum UV/IR Relations and Holographic Dark Energy from Entropic Force, arXiv:1001.4466.
[136]Yu Tian, Xiaoning Wu, Thermodynamics of Black Holes from Equipartition of Energy and Holography, arXiv:1002.1275.
[137]Jae-Weon Lee, On the Origin of Entropic Gravity and Inertia, arXiv:1003.4464.
[138]Rong-Gen Cai, Sang Pyo Kim, JHEP 0502 (2005) 050.
[139]M. Akbar, Rong-Gen Cai, Phys.Lett.B635,7(2006).
[140]M. Akbar, Rong-Gen Cai, Phys.Rev.D75,084003(2007).
[141]Shao-Feng Wu, Bin Wang, Guo-Hong Yang, Nucl.Phys.B799,330(2008).
[142]Rong-Gen Cai, Li-Ming Cao, Ya-Peng Hu, JHEP0808,090(2008).
[143]Rong-Gen Cai, Li-Ming Cao, Nucl.Phys.B785,135(2007).
[144]Ahmad Sheykhi, Bin Wang, Rong-Gen Cai, Nucl.Phys.B779,1(2007).
[145]Tao Zhu, Ji-Rong Ren, Shu-Fan Mo, Int.J.Mod.Phys.A24,5877(2009).
[146]Rong-Gen Cai, Li-Ming Cao, Ya-Peng Hu, Class.Quant.Grav.26,155018(2009).
[147]Ran Li, Ji-Rong Ren, Dun-Fu Shi, Phys.Lett.B 670,446(2009).
[148]Ran Li, Ji-Rong Ren, Shao-Wen Wei, Eur. Phys. J. C 62,455(2009).
[149]Sean A. Hayward, Phys. Rev. D 49,6467 (1994)
[150]Sean A. Hayward, Class.Quant.Grav.15,3147(1998).
[151]Tao Zhu, Ji-Rong Ren, Euro. Phys. J. C 62 (2009) 413.
[152]Tao Zhu, Ji-Rong Ren, Ming-Fan Li, JCAP 0908,010(2009).
[153]Ke-Xia Jiang, Tsun Feng, Dan-Tao Peng, Int J Theor Phys 48,2112(2009).
[154]Fu-Wen Shu, Yungui Gong, Equipartition of energy and the first law of thermodynamics at the apparent horizon, arXiv:1001.3237.
[2]Y. S. Duan and M. L. Ge, Scientia Sinica,11(1979) 1072.
[3]Y.S. Duan and S.C. Zhao, Commun. Theor. Phys.2(1983)1553.
[4]Y. S. Duan, SLAC-PUB-3301/84.
[5]Duan Y S, In Proceedings Symposium on Yang-Mills gauge theories (Beijing), Commun. Theor. Phys. 4(1984)661.
[6]Y. S. Duan and Z. P. Duan, Int. J.Eng. Sci.24 (1986) 513.
[7]Y. S. Duan and J. C. Liu, In Proceedings of the 11th Johns Hopkins Workshop on Current Problem in Particle theory,11(1987)183, Lanzhou, China, edited by Y. S. Duan, G. Domokos, and S. Kovesi-Domokos, World Scientific, Singapore, (1988).
[8]Y. S. Duan and S. L. Zhang, Int. J. Eng. Sci.28(1990)689; 29(1991)153; 29(1991)1593; 30(1992)153.
[9]Y. S. Duan and X. H. Meng, Int. J. Engng. Sci.31(1993)1173.
[10]Y.S. Duan, S.L. Zhang, and S.S. Feng, J. Math. Phys.35(1994)4463.
[11]Y. S. Duan, G. H. Yang and Y. Jiang, Gen. Rel. Grav.29(1997)715.
[12]Ying Jiang, Yishi Duan, J. Math. Phys.24(2000)6463.
[13]Duan Y. S., Meng X. H., J. Math. Phys.34(1993)1149.
[14]Duan Y. S., Li S., Yang G. H.:Nucl. Phys. B514(1998)705.
[15]Y.S.Duan, S.Li, Phys.Lett.A246(1998)172.
[16]Y.S.Duan, P.M.Zhang, H.Zhang, Phys. Lett. A291(2001)296.
[17]S.Li, Y.S.Duan, Astro. and Space Sci.268(1999)455.
[18]Duan Y. S., Zhang H., Li S.:Phys. Rev. B 58(1998)125.
[19]Y.S.Duan, H,Zhang, G.Jia, Phys. Lett. A 253(1999)57; Y.S.Duan, G.Jia, H.Zhang, J.Phys.A 32(1999)4943.
[20]Duan, YS; Fu, LB; Jia, G, J. Math. Phys.,41(2000)4379.
[21]Li-Bin Fu, Yi-Shi Duan, Hong Zhang,Phys. Rev. D 61(2000)045004.
[22]J.R.Ren. R.Li and Y.S.Duan, J. Math. Phys.48,073502(2007).
[23]Ji-rong Ren, Ran Li, Yi-shi Duan, Commun.Theor.Phys.50(2008)53.
[24]Ren Ji-Rong, Li Ran, Duan Yi-Shi, Int.J.Mod.Phys.A,23 (2008)1447.
[25]Yi-Shi Duan, Ji-Rong Ren, Ran Li, Commun.Theor.Phys.47(2007)875.
[26]H. Hopf, Math. Ann.104,639(1931); H. Hopf, Fund. Math.25,427(1935).
[27]H. K. Urbantke, J. Geom. Phys.46,125(2003).
[28]N. D. Mermin and T. L. Ho, Phys. Rev. Lett.9236,594(1976).
[29]Yi-shi Duan, Peng-ming Zhang, Mod.Phys.Lett. A16 (2001) 2483.
[30]James H. White, American Journal of Mathematics,91,693(1969).
[31]Alberto S. Cattaneo and Carlo A. Rossi, Commun. Math. Phys.256, (2005)513. Alberto S. Cattaneo, J.Math.Phys.37 (1996) 3684. Alberto S:Cattaneo, Commun.Math.Phys.221 (2001) 591.
[32]E.Witten, Commun.Math.Phys.121,351(1989).
[33]Duan Yi-Shi, Liu Xin and Fu Li-Bin, Commun. Theor. Phys. (Beijing, China) 40,447(2003).
[34]L. Faddeev and A.J. Niemi, Phys. Rev. Lett.82,1624(1999).
[35]L. Faddeev and A.J. Niemi, Phys. Lett. B 449,214(1999).
[36]L. Faddeev and A.J. Niemi, Phys. Lett. B 464,90(1999).
[37]Y. M. Cho, Phys. Rev. Lett.46,302(1981).
[38]Y. M. Cho, Phys. Rev. D 21,1080(1980).
[39]Y. M. Cho, Phys. Rev. Lett.87,252001(2001).
[40]L. Faddeev and A. J. Niemi, Nature,387,58(1997).
[41]T.T. Wu, C.N. Yang, Phys. Rev. D 12,3845(1975); T.T. Wu, C.N. Yang, Phys. Rev. D 14,437(1975).
[42]Bardeen, J. M., Carter, B., and Hawking, S. W., Commun. math. Phys.,31,161(1973).
[43]Beckenstein, J. D., Phys. Rev. D 7,2333(1973).
[44]S.W.Hawking, Nature 248(1974)30; Commun.Math.Phys.43(1975)199.
[45]Per Kraus, Frank Wilczek, Nucl.Phys.B 433(1995)403.
[46]Per Kraus, Frank Wilczek, Nucl.Phys. B 437 (1995) 231.
[47]Maulik K. Parikh, Frank Wilczek, Phys.Rev.Lett.85 (2000) 5042.
[48]J.Zhang and Z.Zhao, JHEP 10,055(2005).
[49]J.Zhang and Z.Zhao, Phys.Lett.B 618,14(2005); J.Zhang and Z.Zhao, Phys.Lett.B 638,110(2006).
[50]Qing-Quan Jiang, Shuang-Qing Wu, Xu Cai, Phys.Rev. D73 (2006) 064003.
[51]M.Angheben, M.Nadalini, L.Vanzo and S.zerbini, JHEP 05,014(2005).
[52]K.Srinivasan and T.Padmanabhan, Phys.Rev.D 60,24007(1999); S.Shankaranarayanan, T.Padmanabhan and K.Srinivasan, Class.Quant.Grav.19,2671(2002).
[53]R.Kerner and R.B.Mann, Phys.Rev.D 73,104010(2006).
[54]R.Kerner and R.B.Mann, Class.Quant.Grav.25(2008)095014.
[55]Ji-Rong Ren, Ran Li, Fei-Hu Liu, Mod.Phys.Lett.A23(2009)3419.
[56]Ran Li, Ji-Rong Ren, Phys.Lett.B 661(2008)370.
[57]Ran Li, Ji-Rong Ren, Shao-Wen Wei, Class.Quant.Grav.25(2008)125016.
[58]D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D43(1991)3140.
[59]J. B. Hartle and S. W. Hawking Phys. Rev. D 13(1976)2188.
[60]G. W. Gibbons and D. L. Wiltshire, Ann. Phys. (N.Y.) 167(1986)201.
[61]J. I. Koga and K. I. Maeda, Phys. Rev. D 52(1995)7066.
[62]A. Sen, Phys. Rev. Lett.69(1992)1006.
[63]M.banados, C.Teitelboim and J.Zanelli, Phys.Rev.Lett.69,1849(1992).
[64]X.He and W.Liu, Phys.Lett.B 653,330(2007).
[65]Roberto Di Criscienzo, Luciano Vanzo, Europhys.Lett.82(2008)60001.
[66]Ryan Kerner, R.B. Mann, Phys.Lett.B 665(2008)277.
[67]De-You Chen, Qing-Quan Jiang, Xiao-Tao Zu, Phys.Lett.B665(2008)106.
[68]Qing-Quan Jiang, Phys.Rev.D78(2008)044009.
[69]Kai Lin, Shu-Zheng Yang, Phys. Rev. D 79 (2009)064035.
[70]Alexandre Yale, Robert B. Mann, Phys. Lett. B 673(2009)168.
[71]Rabin Banerjee, Phys.Rev.D79(2009)044005.
[72]Rabin Banerjee, Sujoy Kumar Modak, JHEP 0905(2009)063.
[73]S. Christensen and S. Fulling, Phys.Rev.D 15,2088(1977).
[74]S. P. Robinson and F.Wilczek, Phys.Rev.Lett.95,011303(2005).
[75]S. Iso, H. Umetsu and F. Wilczek, Phys.Rev.Lett.96,151302(2006).
[76]Satoshi Iso, Hiroshi Umetsu, Frank Wilczek, Phys.Rev.D 74,044017(2006).
[77]Keiju Murata, Jiro Soda, Phys.Rev. D 74,044018(2006);
[78]M. R. Setare, Eur.Phys.J.C,49,865(2007);
[79]Satoshi Iso, Takeshi Morita, Hiroshi Umetsu, JHEP,04,068(2007);
[80]Qing-Quan Jiang, Shuang-Qing Wu, Phys.Lett.B 647,200(2007);
[81]Qing-Quan Jiang, Shuang-Qing Wu, Xu Cai, Phys.Rev.D 75,064029(2007);
[82]Kui Xiao, Wenbiao Liu, Hongbao Zhang, Phys.Lett.B 647,482(2007);
[83]Hyeonjoon Shin, Wontae Kim, JHEP,06,012(2007);
[84]Qing-Quan Jiang, Shuang-Qing Wu, Xu Cai, Phys.Lett.B 651,65(2007);
[85]Shuang-Qing Wu, Jun-Jin Peng, Class.Quant.Grav.24,5123(2007);
[86]Wontae Kim, Hyeonjoon Shin, JHEP,07,070(2007);
[87]Rabin Banerjee, Shailesh Kulkarni, Phys.Rev.D 77,024018(2008);
[88]Rabin Banerjee, Shailesh Kulkarni, Phys.Lett.B 659,827(2008);
[89]Sunandan Gangopadhyay, Shailesh Kulkarni, Phys.Rev.D 77,024038(2008);
[90]Jun-Jin Peng, Shuang-Qing Wu, Phys.Lett.B 661,300(2008);
[91]Xiaoning Wu, Chao-Guang Huang, Jia-Rui Sun, Phys.Rev.D 77,124023(2008);
[92]Shuang-Qing Wu, Jun-Jin Peng, Zhan-Yue Zhao, Class.Quant.Grav.25,135001(2008);
[93]Sunandan Gangopadhyay, Phys.Rev.D 78,044026(2008);
[94]Sunandan Gangopadhyay, Europhys.Lett.85,10004(2009);
[95]Shao-Wen Wei, Ran Li, Yu-Xiao Liu, Ji-Rong Ren, arXiv:0901.2614;
[96]S. Nam and J.-D. Park, Class.Quant.Grav.26,145015(2009).
[97]A. P. Porfyriadis, Phys.Rev. D 79,084039(2009);
[98]A. P. Porfyriadis, Phys.Lett.B 675,235(2009);
[99]Shao-Wen Wei, Ran Li, Yu-Xiao Liu, Ji-Rong Ren, arXiv:0904.2915.
[100]Takeshi Morita, Phys.Lett.B 677,88(2009).
[101]Ran Li, Ji-Rong Ren, Hawking radiation via quantum anomaly from the rotating charged Godel black hole in five dimensions.
[102]S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 75,124004(2007);
[103]S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 76,064015(2007);
[104]S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 77,045007(2008);
[105]S. Iso, T. Morita and H. Umetsu, Nucl. Phys. B 799,60(2008);
[106]S. Iso, Int. J. Mod. Phys. A 23,2082(2008);
[107]L. Bonora, M. Cvitan, JHEP G805,071(2008);
[108]L.Bonora, M.Cvitan, S.Pallua, I.Smolic, JHEP 0812,021(2008);
[109]Takeshi Morita, JHEP 0901,037(2009).
[110]Keiju Murata, Umpei Miyamoto, Phys.Rev.D 76,084038(2007);
[111]L. Bonora, M. Cvitan, S. Pallua, I. Smolic, Phys. Rev. D 80,084034(2009).
[112]K. Godel, Rev. Mod. Phys.21,447 (1949).
[113]Eric G. Gimon, Akikazu Hashimoto, Phys.Rev.Lett.91 (2003) 021601.
[114]C.A.R.Herdeiro, Class.Quant.Grav.20 (2003) 4891.
[115]D. Brecher, U..H. Danielsson, J. P. Gregory, M. E. Olsson, JHEP 0311:033,2003.
[116]Glenn Barnich, Geoffrey Compere Phys.Rev.Lett.95 (2005) 031302.
[117]Shuang-Qing Wu, Phys. Rev. Lett.100:121301,2008.
[118]E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett.102,201301(2009).
[119]J. Oliva, D. Tempo and R. Troncoso, JHEP 0907,011(2009); G. Giribetl, J. Oliva, D. Tempo and R. Troncoso, arXiv:0909.2564.
[120]T. Jacobson, Phys. Rev. Lett.75,1260 (1995).
[121]C. Eling, R. Guedens and T. Jacobson, Phys. Rev. Lett.96,121301 (2006).
[122]R. G. Cai and L. M. Cao, Phys. Rev. D 75,064008 (2007).
[123]Shao-Feng Wu, Guo-Hong Yang, Peng-Ming Zhang, Prog.Theor.Phys.120,615(2008).
[124]T.Padmanabhan, Phys.Rept.406,49(2005).
[125]Aseem Paranjape, Sudipta Sarkar, T. Padmanabhan, Phys.Rev.D74,104015(2006).
[126]Dawood Kothawala, Sudipta Sarkar, T. Padmanabhan, Phys.Lett.B652,338(2007).
[127]T. Padmanabhan, Entropy density of spacetime and thermodynamic interpretation of field equations of gravity in any diffeomorphism invariant theory, arXiv:0903.1254.
[128]Dawood Kothawala, T. Padmanabhan, Phys.Rev.D79,104020(2009).
[129]T.Padmanabhan, Rep. Prog. Phys.73 (2010) 046901.
[130]T.Padmanabhan,Equipartition of energy in the horizon degrees of freedom and the emergence of gravity, arXiv:0912.3165.
[131]T.Padmanabhan, Surface Density of Spacetime Degrees of Freedom from Equipartition Law in theories of Gravity, arXiv:1003.5665.
[132]Erik P. Verlinde, On the Origin of Gravity and the Laws of Newton, arXiv:1001.0785.
[133]Rong-Gen Cai, Li-Ming Cao, Nobuyoshi Ohta, Phys. Rev. D 81,061501(R) (2010).
[134]Lee Smolin, Newtonian gravity in loop quantum gravity, arXiv:1001.3668.
[135]Miao Li, Yi Wang, Quantum UV/IR Relations and Holographic Dark Energy from Entropic Force, arXiv:1001.4466.
[136]Yu Tian, Xiaoning Wu, Thermodynamics of Black Holes from Equipartition of Energy and Holography, arXiv:1002.1275.
[137]Jae-Weon Lee, On the Origin of Entropic Gravity and Inertia, arXiv:1003.4464.
[138]Rong-Gen Cai, Sang Pyo Kim, JHEP 0502 (2005) 050.
[139]M. Akbar, Rong-Gen Cai, Phys.Lett.B635,7(2006).
[140]M. Akbar, Rong-Gen Cai, Phys.Rev.D75,084003(2007).
[141]Shao-Feng Wu, Bin Wang, Guo-Hong Yang, Nucl.Phys.B799,330(2008).
[142]Rong-Gen Cai, Li-Ming Cao, Ya-Peng Hu, JHEP0808,090(2008).
[143]Rong-Gen Cai, Li-Ming Cao, Nucl.Phys.B785,135(2007).
[144]Ahmad Sheykhi, Bin Wang, Rong-Gen Cai, Nucl.Phys.B779,1(2007).
[145]Tao Zhu, Ji-Rong Ren, Shu-Fan Mo, Int.J.Mod.Phys.A24,5877(2009).
[146]Rong-Gen Cai, Li-Ming Cao, Ya-Peng Hu, Class.Quant.Grav.26,155018(2009).
[147]Ran Li, Ji-Rong Ren, Dun-Fu Shi, Phys.Lett.B 670,446(2009).
[148]Ran Li, Ji-Rong Ren, Shao-Wen Wei, Eur. Phys. J. C 62,455(2009).
[149]Sean A. Hayward, Phys. Rev. D 49,6467 (1994)
[150]Sean A. Hayward, Class.Quant.Grav.15,3147(1998).
[151]Tao Zhu, Ji-Rong Ren, Euro. Phys. J. C 62 (2009) 413.
[152]Tao Zhu, Ji-Rong Ren, Ming-Fan Li, JCAP 0908,010(2009).
[153]Ke-Xia Jiang, Tsun Feng, Dan-Tao Peng, Int J Theor Phys 48,2112(2009).
[154]Fu-Wen Shu, Yungui Gong, Equipartition of energy and the first law of thermodynamics at the apparent horizon, arXiv:1001.3237.