BPS多中心黑洞的Smarr质量公式
Smarr Mass formulas for BPS multicenter Black Holes
摘要：
给出了多中心BPS四维黑洞的质量公式。例如，两个中心BPS解的ADM质量可以通过$M_{ADM}^2 =A\left (1+ αJ^2\left(1+\frac{2M_{ADM}}{r}+\frac{A}{r^2}\right)\right)$与中心间距离$r$、角动量$J^2$、双子电荷矢量$q_i$和无穷远处的标量模量（$z_\infty$）的值相关，其中$A(Q),α(q_i)$是辛不变量（$Q$，总电荷矢量），这取决于定义理论的特殊几何预势。该公式预言了对于固定的电荷值，存在一个连续类的间距为$r\in (0,\infty)$和$M_{ADM}\in (\infty,M_\infty)$的BH。Smarr类似的表达式，包括中心间的距离，从它获得：$$ dM\equivΩd J+Φ_i d q_i+ F dr,$$除了一个有效的角速度$Ω$和电磁势$Φ_i$，该方程允许定义一个有效的“力”，$F$，作用在中心之间。这个有效力总是负的：在无穷远处，我们恢复熟悉的牛顿定律$F\sim 1/r^2$，而在短距离$F\sim f_0+f_1/r^2$。对于更一般的模型和中心数量，可以很容易地获得类似的结果。
英文摘要：
Mass formulas for multicenter BPS 4D black holes are presented. For example, ADM mass for a two center BPS solution can be related to the intercencenter distance $r$, the angular momentum $J^2$, the dyonic charge vectors $q_i$ and the value of the scalar moduli at infinity ($z_\infty$)by $M_{ADM}^2 =A\left (1+ αJ^2\left(1+\frac{2M_{ADM}}{r}+\frac{A}{r^2}\right)\right)$ where $A(Q),α(q_i)$ are symplectic invariant quantities ($Q$, the total charge vector) depending on the special geometry prepotential defining the theory. The formula predicts the existence of a continuos class, for fixed value of the charges, of BH's with interdistances $r\in (0,\infty)$ and $M_{ADM}\in (\infty,M_\infty)$. Smarr-like expressions incorporating the intercenter distance are obtained from it: $$ dM\equivΩd J+Φ_i d q_i+ F dr,$$ in addition to an effective angular velocity $Ω$ and electromagnetic potentials $Φ_i$, the equation allows to define an effective "force", $F$, acting between the centers. This effective force is always negative: at infinity we recover the familiar Newton law $F\sim 1/r^2$ while at short distances $F\sim f_0+f_1/r^2$. Similar results can be easily obtained for more general models and number of centers.
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