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arXiv 2201.12956gr-qcmath-phmath.MP
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引力场与规范场耦合的准局域能问题

Aspects of Quasi-local energy for gravity coupled to gauge fields

Puskar Mondal, Shing-Tung Yau

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22 pages
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摘要：

我们研究了在重力场与规范场耦合的情况下，物理$3+1$维时空的类空域$Ω$与$2-$表面$Σ$的准局域能量。Wang-Yau准定域能量加上由于引力与规范场耦合而产生的附加项，构成了包含在膜$Σ=\partialΩ$中的总能量（$\mathcal{QLE}$）。我们专注于Kerr-Newman时空家族，其中包含一个U（1）规范场耦合到引力和外视界。通过显式计算，我们表明，总能量满足一个较弱的版本的Bekenstein型不等式$\mathcal{QLE}> \frac{Q^{2}}{2R}$为大的球形膜，$Q$是电荷和$R$是膜的半径。关闭角动量（Reissner Nordström），对于所有包含视界的等半径膜，得到$\mathcal{QLE}> \frac{Q^{2}}{2R}$，在这种情况下，右边出现的电荷因子正好等于贝肯斯坦不等式的电荷因子。此外，在黑洞角动量较小的假设下，我们证明了总准局域能量从外视界的$2M_{irr}+V_{Q}$（$M_{irr}$是不可约质量，$V_{Q}$是电势能）单调衰减到类空无穷远处的$M$（$M$是ADM质量）.

英文摘要：

We study the aspects of quasi-local energy associated with a $2-$surface $Σ$ bounding a space-like domain $Ω$ of a physical $3+1$ dimensional spacetime in the regime of gravity coupled to a gauge field. The Wang-Yau quasi-local energy together with an additional term arising due to the coupling of gravity to a gauge field constitutes the total energy ($\mathcal{QLE}$) contained within the membrane $Σ=\partialΩ$. We specialize in the Kerr-Newman family of spacetimes which contains a U(1) gauge field coupled to gravity and an outer horizon. Through explicit calculations, we show that the total energy satisfies a weaker version of a Bekenstein type inequality $\mathcal{QLE}> \frac{Q^{2}}{2R}$ for large spherical membranes, $Q$ is the charge and $R$ is the radius of the membrane. Turning off the angular momentum (Reissner Nordström) yields $\mathcal{QLE}> \frac{Q^{2}}{2R}$ for all constant radii membranes containing the horizon and in such case the charge factor appearing in the right-hand side exactly equals to that of Bekenstein's inequality. Moreover, we show that the total quasi-local energy monotonically decays from $2M_{irr}+V_{Q}$ ($M_{irr}$ is the irreducible mass, $V_{Q}$ is the electric potential energy) at the outer horizon to $M$ ($M$ is the ADM mass) at the space-like infinity under the assumption of a small angular momentum of the black hole.

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