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In accordance with the holographic principle， by calculating the entropy of the quantum field just on the event horizon of the Gibbons-Maeda dilaton black hole， the holographic entropy and the Bekenstein-Hawking entropy of the black hole are obtained. By using the non-commutative quantum field theory， the divergence of the state density near the event horizon in usual quantum field theory is removed and the ultraviolet cutoff in the heat gas method of black hole entropy is avoided.Using the residue theorem, the integral difficulty in the calculation is overcome and the results here are obtained quantitatively. The results show that black hole entropy is identical with the statistical entropy of the quantum states at the horizon. Black hole entropy may be obtained by calculating the quantum states only at the event horizon, and in the calculation the influences of quantum states outside the horizon should be avoided.
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Keywords:
- black hole entropy /
- holographic principle /
- event horizon /
- non-commutative quantum field theory
[1] ［1］Bekenstein J D 1973 Phys. Rev. D 7 2333
[2] ［2］Hawking S W 1975 Commun. Math. Phys. 43 199
[3] ［3］Susskind L 1995 J. Math. Phys. 36 6377
[4] ［4］Hooft Gt 1996 Int. J. Mod. Phys. A 11 4623
[5] ［5］Khriplovich I B 2005 Int. J. Mod. Phys. D 14 181
[6] ［6］Hooft Gt 1985 Nucl. Phys. B 256 727
[7] ［7］Jing J L 1998 Int. J. Theor. Phys. 37 1441
[8] ［8］Ghosh A, Mitra P 1994 Phys. Rev. Lett. 73 2521
[9] ［9］Mukohyama S W, Israel W 1998 Phys. Rev. D 58 104005
[10] ］Liu W B, Zhu J Y, Zhao Z 2000 Acta Phys.Sin. 49 581 (in Chinese) ［刘文彪、朱建阳、赵峥 2000 49 581］
[11] ］Li X, Zhao Z 2001 Chin.Phys.Lett. 18 463
[12] ］Liu W B, Zhao Z 2001 Chin. Phys. Lett. 18 310
[13] ］Shen Y G 2002 Phys. Lett. B 537 187
[14] ］Zhu B,Yao G Z,Zhao Z 2002 Acta Phys.Sin. 51 2656 (in Chinese) ［朱斌、姚国政、赵峥 2002 51 2656］
[15] ］Song T P, Yao G Z 2002 Acta Phys. Sin. 51 1144 (in Chinese) ［宋太平、姚国政 2002 51 1144］
[16] ］Sun M C 2003 Acta Phys. Sin. 52 1350 (in Chinese) ［孙鸣超 2003 52 1350］
[17] ］Song T P, Hou C X 2002 Acta Phys. Sin. 51 1398 (in Chinese) ［宋太平、侯晨霞 2002 51 1398］
[18] ］Carlip S 2001 Rep. Prog. Phys. 64 885
[19] ］Plchinski J 1996 Rev. Mod. Phys. 68 1245
[20] ］Plchinski J 1996 Prog. Theor. Phys. 123(Suppl.) 9
[21] ］Koga J I, Maeda K I 1995 Phys. Rev. D 52 7066
[22] ］Garfinkle D, Horowitz G T, Strominger A 1991 Phys. Rev. D 43 3140
[23] ］Witten E 1998 Adv. Theor. Math. Phys. 2 253
[24] ］Maldacena J M 1998 Adv. Theor. Math. Phys. 2 231
[25] ］Snyder H S 1947 Phys. Rev. 71 38
[26] ］Witten E 1996 Nucl. Phys. B 460 335
[27] ］Kempt A, Mangano G, Mann R B 1995 Phys. Rev. D 52 1108
[28] ］Garay L J 1995 Int. J. Mod. Phys. A 10 145
[29] ］Cheng L N, Minic D, Okamura N, Takeuchi T 2002 Phys. Rev. D 65 125028
[30] ］Li X 2002 Phys. Lett. B 537 306
[31] ］Li X 2002 Phys. Lett. B 540 9
[32] ］Liu C Z 2004 Gen. Rel. Grav. 36 1135
[33] ］Sun X F, Liu W B 2004 Mod. Phys. Lett. A 19 677
[34] ］Bombelli L, Koul R K, Lee J, Sorkin R D 1986 Phys. Rev. D 34 373
[35] ］Page D N 2005 New J. Phys. 7 203
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[1] ［1］Bekenstein J D 1973 Phys. Rev. D 7 2333
[2] ［2］Hawking S W 1975 Commun. Math. Phys. 43 199
[3] ［3］Susskind L 1995 J. Math. Phys. 36 6377
[4] ［4］Hooft Gt 1996 Int. J. Mod. Phys. A 11 4623
[5] ［5］Khriplovich I B 2005 Int. J. Mod. Phys. D 14 181
[6] ［6］Hooft Gt 1985 Nucl. Phys. B 256 727
[7] ［7］Jing J L 1998 Int. J. Theor. Phys. 37 1441
[8] ［8］Ghosh A, Mitra P 1994 Phys. Rev. Lett. 73 2521
[9] ［9］Mukohyama S W, Israel W 1998 Phys. Rev. D 58 104005
[10] ］Liu W B, Zhu J Y, Zhao Z 2000 Acta Phys.Sin. 49 581 (in Chinese) ［刘文彪、朱建阳、赵峥 2000 49 581］
[11] ］Li X, Zhao Z 2001 Chin.Phys.Lett. 18 463
[12] ］Liu W B, Zhao Z 2001 Chin. Phys. Lett. 18 310
[13] ］Shen Y G 2002 Phys. Lett. B 537 187
[14] ］Zhu B,Yao G Z,Zhao Z 2002 Acta Phys.Sin. 51 2656 (in Chinese) ［朱斌、姚国政、赵峥 2002 51 2656］
[15] ］Song T P, Yao G Z 2002 Acta Phys. Sin. 51 1144 (in Chinese) ［宋太平、姚国政 2002 51 1144］
[16] ］Sun M C 2003 Acta Phys. Sin. 52 1350 (in Chinese) ［孙鸣超 2003 52 1350］
[17] ］Song T P, Hou C X 2002 Acta Phys. Sin. 51 1398 (in Chinese) ［宋太平、侯晨霞 2002 51 1398］
[18] ］Carlip S 2001 Rep. Prog. Phys. 64 885
[19] ］Plchinski J 1996 Rev. Mod. Phys. 68 1245
[20] ］Plchinski J 1996 Prog. Theor. Phys. 123(Suppl.) 9
[21] ］Koga J I, Maeda K I 1995 Phys. Rev. D 52 7066
[22] ］Garfinkle D, Horowitz G T, Strominger A 1991 Phys. Rev. D 43 3140
[23] ］Witten E 1998 Adv. Theor. Math. Phys. 2 253
[24] ］Maldacena J M 1998 Adv. Theor. Math. Phys. 2 231
[25] ］Snyder H S 1947 Phys. Rev. 71 38
[26] ］Witten E 1996 Nucl. Phys. B 460 335
[27] ］Kempt A, Mangano G, Mann R B 1995 Phys. Rev. D 52 1108
[28] ］Garay L J 1995 Int. J. Mod. Phys. A 10 145
[29] ］Cheng L N, Minic D, Okamura N, Takeuchi T 2002 Phys. Rev. D 65 125028
[30] ］Li X 2002 Phys. Lett. B 537 306
[31] ］Li X 2002 Phys. Lett. B 540 9
[32] ］Liu C Z 2004 Gen. Rel. Grav. 36 1135
[33] ］Sun X F, Liu W B 2004 Mod. Phys. Lett. A 19 677
[34] ］Bombelli L, Koul R K, Lee J, Sorkin R D 1986 Phys. Rev. D 34 373
[35] ］Page D N 2005 New J. Phys. 7 203
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