渐近反德西特时空中的替代边界条件
Alternative Boundary Conditions in Asymptotically Anti-de Sitter Spacetimes
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中文总结 AI 辅助
本文研究渐近 AdS 时空的边界条件，推广标准 AdS 不变条件至电场磁场任意线性组合，并探讨其相空间、守恒荷及黑洞熵修正。
中文摘要 AI 辅助
反德西特（AdS）时空不是全局双曲的，因此通常需要边界条件来保证动力学的良定义性。Ishibashi 和 Wald（IW）利用非局域势和单个球谐模式，确定了对于标量场、电磁场和线性化引力场能产生具有守恒正能量且适定动力<｜begin▁of▁sentence｜># 的边界条件。在本文中，我们确定了哪些 IW 条件在主场中是局域的，哪些是 AdS 不变的。在四维时空中，标准的 AdS 不变边界条件在共形边界上对电磁学将重标度磁场设为零，对爱因斯坦引力将重标度磁 Weyl 张量设为零。我们通过将电场和磁场或相应的 Weyl 张量的任意线性组合设为零来推广这些标准边界条件，并推测这些条件穷尽了 AdS$_{1,3}$ 中局域的、AdS 不变的、保守的边界条件。我们还研究了具有非 AdS 不变性以及不同时空维数的边界条件。AdS$_{1,3}$ 中的 AdS 不变条件推广到一般 Fefferman-Graham 框架下的 Yang-Mills 理论和非线性引力。我们为这些一般边界条件构造了相空间。一般条件的协变相空间自然与包含 Pontryagin 项以及引力中的 Gauss-Bonnet 项的拉格朗日量相关联。在引力中，除了标准边界条件外，不存在非平凡的渐近规范对称性，所有守恒荷（如 ADM 质量）都恒为零，如同在闭合宇宙中一样。具有包括 AdS 在内的 Killing 场的时空随后表现出 Fischer-Marsden 和 Moncrief 意义上的线性化不稳定性。对于非标准边界条件，修正的辛结构为黑洞熵公式提供了新的表达式。
英文摘要
Anti-de Sitter (AdS) spacetimes are not globally hyperbolic, so boundary conditions are generally needed for well-defined dynamics. Ishibashi and Wald (IW) determined boundary conditions yielding well-posed dynamics with conserved positive energy for scalar, electromagnetic, and linearized gravitational fields, using nonlocal potentials and individual spherical harmonic modes. In this paper, we determine which IW conditions are local in the primary fields and which are AdS-invariant. In four spacetime dimensions, the standard AdS-invariant boundary conditions set the rescaled magnetic field to zero for electromagnetism and the rescaled magnetic Weyl tensor to zero for Einstein gravity on the conformal boundary. We generalize these standard boundary conditions by setting arbitrary linear combinations of the electric and magnetic fields, or the corresponding Weyl tensors, to zero, and we conjecture that these exhaust the local, AdS-invariant, conservative boundary conditions in AdS$_{1,3}$. Boundary conditions with non-AdS-invariance and in different spacetime dimensions are also studied. The AdS-invariant conditions in AdS$_{1,3}$ extend to Yang-Mills theory and nonlinear gravity in a general Fefferman-Graham setting. We construct phase spaces for these general boundary conditions. The covariant phase spaces for general conditions are naturally associated with Lagrangians containing Pontryagin terms and, in gravity, a Gauss-Bonnet term. In gravity, except for the standard boundary condition, no nontrivial asymptotic symmetries exist and all conserved charges such as ADM mass vanish identically, as in a closed universe. Spacetimes with Killing fields including AdS then exhibit linearization instability in the sense of Fischer and Marsden and Moncrief. For nonstandard boundary conditions, the modified symplectic structure yields new expressions for the entropy formula of black holes.
发表机构
- University of Chicago（芝加哥大学）
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