[论文解读] Timelike Geodesic, Effective Komar Conserved Quantities and Entropy in Kerr-Newman black Hole

Sujoy K. Modak, Saurav Samanta|arXiv (Cornell University)|Jun 17, 2010

Astrophysical Phenomena and Observations参考文献 2被引用 1

一句话总结

本文通过计算有效Komar守恒量，推导出Kerr-Newman时空中类时测地线的有效能量（Eeff）的闭式表达式。该结果从无穷远延伸至事件视界，且在视界处通过S = E²/T可正确给出黑洞熵，从而将守恒量与弯曲时空中的热力学性质联系起来。

ABSTRACT

We find the constant of motion (effective energy) corresponding to a timelike trajectory of a physical stationary observer moving in the Kerr-Newman spacetime in a closed form. In order to find this, we explicitly calculate the effective Komar conserved quantities of this spacetime. This result for the effective energy (Eeff) holds for the entire range of timelike geodesic starting from the asymptotic infinity to the black hole event horizon. Remarkably at the event horizon, this result gives the correct value of the black hole entropy by using the identity S = E 2T . For a spacetime endowed with symmetries, one can define Killing vectors corresponding to each of these symmetry directions. One major application of these Killing vectors is to find the constants associated with the motion along some geodesic. This is often done by exploiting the Komar expressions of conserved quantities[1] which can be written in a covariant form. For example, the Kerr-Newman spacetime has two Killing vectors ∂t and ∂φ (t being the time axis and φ is the azimuthal angle). Therefore one has two conserved quantities, namely mass (M) and the angular momentum (J) corresponding to these Killing vectors. By construction, the mass and the angular momentum of a asymptotically flat spacetime are defined with respect to an observer situated far away from the horizon and not influenced by the spacetime curvature. However, as one approaches towards the event horizon the surrounding spacetime no longer remains flat and therefore the earlier results get modified. Indeed the effective mass of the Kerr-Newman black hole, as calculated by Cohen and DeFelice [2], is given by Meff = M − Q2 2r − Q 2(r2 + a2) ar2 tan (a r )

研究动机与目标

• 推导Kerr-Newman时空中类时测地线的有效能量（Eeff）的闭式表达式。
• 计算考虑视界附近曲率效应的有效Komar守恒量。
• 研究在视界处的有效能量是否能通过S = E²/T正确重现黑洞熵。
• 将标准Komar质量与角动量定义推广至弯曲时空中的局部、视界邻近观测者。
• 探讨从渐近守恒量到黑洞视界附近局部物理可观测量的过渡。

提出的方法

• 利用Kerr-Newman时空中的Killing矢量∂t和∂φ，通过Komar积分定义守恒量。
• 应用协变Komar表达式计算弯曲时空中的有效质量与角动量。
• 利用度规的对称性与Killing矢量场，沿类时测地线推导有效能量Eeff。
• 在事件视界处评估Eeff，以检验其与黑洞热力学的一致性。
• 采用恒等式S = E²/T验证Eeff在视界处是否能正确给出Bekenstein-Hawking熵。
• 考虑时空曲率对守恒量的修正，尤其在r = r+附近。

实验结果

研究问题

• RQ1 Kerr-Newman时空中类时测地线的有效能量Eeff的闭式表达式是什么？
• RQ2 当在视界附近而非无穷远处评估时，Komar守恒量如何变化？
• RQ3 在视界处，有效能量Eeff是否能通过S = E²/T正确重现Bekenstein-Hawking熵？
• RQ4 Kerr-Newman黑洞的有效质量与强引力场中渐近质量M有何不同？
• RQ5 时空对称性与曲率效应在多大程度上改变了视界附近守恒量的标准定义？

主要发现

• 有效能量Eeff以闭式推导得出，且对从无穷远到事件视界的全部类时测地线均有效。
• 在事件视界处，当使用恒等式S = E²/T时，有效能量Eeff能正确给出黑洞熵。
• Komar守恒量表达式被推广以考虑视界附近的局部曲率效应。
• Kerr-Newman黑洞的有效质量受Q²/r与Q²(r² + a²)/ar² tan(ar)等项修正，与Cohen和DeFelice先前的结果一致。
• 结果表明，测地线上的守恒量与黑洞热力学之间存在直接联系。
• 该框架为接近视界的观测者在静态、带电、旋转黑洞中提供了能量与熵的一致物理诠释。