We review the (2 + 1)-dimensional Baňados-Teitelboim-Zanelli black hole solution in conformally invariant gravity, uplifted to (3 + 1)-dimensional spacetime. For the matter content we use a scalar-gauge field. The me...We review the (2 + 1)-dimensional Baňados-Teitelboim-Zanelli black hole solution in conformally invariant gravity, uplifted to (3 + 1)-dimensional spacetime. For the matter content we use a scalar-gauge field. The metric is written as <img src="Edit_be2cdfd9-fda6-4846-b64d-4d1062f9964e.bmp" alt="" /> where the <em>dilaton</em> field <span style="white-space:nowrap;"><span style="white-space:nowrap;">ω</span></span> contains all the scale dependencies and where <img src="Edit_ffd065ec-fc7e-41cd-b2c6-05b86c3b566a.bmp" alt="" /> represents the “un-physical” spacetime. A numerical solution is presented and shows how the dilaton can be treated on equal footing with the scalar field. The location of the apparent horizon and ergo-surface depends critically on the parameters and initial values of the model. It is not a hard task to find suitable initial parameters in order to obtain a regular and singular free <img src="Edit_5d830100-019b-4a6a-82e7-deefdf327ecc.bmp" alt="" /> out of a BTZ-type solution for <img src="Edit_ffd065ec-fc7e-41cd-b2c6-05b86c3b566a.bmp" alt="" style="white-space:normal;" />. In the vacuum situation, an exact time-dependent solution in the Eddington-Finkelstein coordinates is found, which is valid for the (2 + 1)-dimensional BTZ spacetime as well as for the uplifted (3 + 1)-dimensional BTZ spacetime. While <img src="Edit_ffd065ec-fc7e-41cd-b2c6-05b86c3b566a.bmp" alt="" style="white-space:normal;" /> resembles the standard BTZ solution with its horizons, <img src="Edit_5d830100-019b-4a6a-82e7-deefdf327ecc.bmp" alt="" style="white-space:normal;" /> is flat. The dilaton field becomes an infinitesimal renormalizable quantum field, which switches on and off Hawking radiation. This solution can be used to investigate the small distance scale of the model and the black hole complementarity issues. It can also be used to describe the problem of how to map the quantum states of the outgoing radiation as seen by a distant observer and the ingoing by a local observer in a one-to-one way. The two observers will use a different conformal gauge. A possible connection is made with the antipodal identification and unitarity issues. This research shows the power of conformally invariant gravity and can be applied to bridge the gap between general relativity and quantum field theory in the vicinity of the horizons of black holes.展开更多
文摘We review the (2 + 1)-dimensional Baňados-Teitelboim-Zanelli black hole solution in conformally invariant gravity, uplifted to (3 + 1)-dimensional spacetime. For the matter content we use a scalar-gauge field. The metric is written as <img src="Edit_be2cdfd9-fda6-4846-b64d-4d1062f9964e.bmp" alt="" /> where the <em>dilaton</em> field <span style="white-space:nowrap;"><span style="white-space:nowrap;">ω</span></span> contains all the scale dependencies and where <img src="Edit_ffd065ec-fc7e-41cd-b2c6-05b86c3b566a.bmp" alt="" /> represents the “un-physical” spacetime. A numerical solution is presented and shows how the dilaton can be treated on equal footing with the scalar field. The location of the apparent horizon and ergo-surface depends critically on the parameters and initial values of the model. It is not a hard task to find suitable initial parameters in order to obtain a regular and singular free <img src="Edit_5d830100-019b-4a6a-82e7-deefdf327ecc.bmp" alt="" /> out of a BTZ-type solution for <img src="Edit_ffd065ec-fc7e-41cd-b2c6-05b86c3b566a.bmp" alt="" style="white-space:normal;" />. In the vacuum situation, an exact time-dependent solution in the Eddington-Finkelstein coordinates is found, which is valid for the (2 + 1)-dimensional BTZ spacetime as well as for the uplifted (3 + 1)-dimensional BTZ spacetime. While <img src="Edit_ffd065ec-fc7e-41cd-b2c6-05b86c3b566a.bmp" alt="" style="white-space:normal;" /> resembles the standard BTZ solution with its horizons, <img src="Edit_5d830100-019b-4a6a-82e7-deefdf327ecc.bmp" alt="" style="white-space:normal;" /> is flat. The dilaton field becomes an infinitesimal renormalizable quantum field, which switches on and off Hawking radiation. This solution can be used to investigate the small distance scale of the model and the black hole complementarity issues. It can also be used to describe the problem of how to map the quantum states of the outgoing radiation as seen by a distant observer and the ingoing by a local observer in a one-to-one way. The two observers will use a different conformal gauge. A possible connection is made with the antipodal identification and unitarity issues. This research shows the power of conformally invariant gravity and can be applied to bridge the gap between general relativity and quantum field theory in the vicinity of the horizons of black holes.
