广义Schwarzschild黑洞解的统一构造

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"这篇研究论文探讨了一种普遍的黑洞构造方法，适用于任何基于黎曼张量及其协变导数的标量不变量的引力度量理论。作者Sigbjørn Hervik和Marcello Ortaggio展示了如何使用广义的Schwarzschild型 ansatz 来构建d维静态真空黑洞解决方案，无论理论的具体细节如何。他们指出，这种ansatz不仅限于两个自由函数a(r)和f(r)，这两个函数分别对应于时空度规的g tt 和 g rr 分量，而且可以将任何各向同性的不可约均匀空间作为基本空间。这种方法的一个关键优点是它简化了场方程，使其通常只涉及a(r)和f(r)的两个常微分方程（ODE），从而极大地扩展了解空间和可能的层状几何形状。在论文中，作者通过构建特定的解决方案来展示他们的结果，这些解决方案涵盖了高斯-邦尼特引力、二次方引力、F(R)引力理论以及F(Lovelock)引力和某些共形引力理论。这些例子进一步证实了通用黑洞ansatz的有效性和广泛适用性。此外，论文是在2020年由Springer为SISSA出版的Open Access文章，意味着公众可以免费访问其全文。该研究对于理解黑洞理论的普遍性和引力理论的多样性具有重要意义，可能为未来的研究提供新的洞察和工具。" 这篇研究的核心知识点包括： 1. **广义Schwarzschild型ansatz**：这是一种适用于任何基于黎曼张量和其协变导数的标量不变量的引力理论的黑洞构造方法，允许构造d维静态真空黑洞解决方案。 2. **静态真空黑洞解决方案**：在这种解决方案中，时空是静态的且没有物质场，仅由引力场描述，通常由度规的对称性决定。 3. **各向同性-不可约的均匀空间**：基本空间可以是任何这样的空间，这意味着在所有方向上都具有相同的性质，这对于黑洞解的构造至关重要。 4. **场方程简化**：采用该ansatz，场方程通常可以简化为两个关于a(r)和f(r)的常微分方程，这大大简化了求解过程。 5. **应用示例**：包括高斯-邦尼特引力、二次方引力、F(R)引力理论和F(Lovelock)引力，以及共形引力，这些示例展示了ansatz在不同引力理论中的应用。 6. **Open Access**：该论文是开放获取的，意味着公众无需订阅或支付费用即可阅读全文，增加了研究的可访问性和影响力。 7. **引力理论的多样性**：研究强调了黑洞解的普适性，表明不同的引力理论可以共存并产生独特的黑洞结构。 8. **层位几何形状**：论文讨论了ansatz如何允许更广泛的层状几何结构，这可能会导致对黑洞物理性质的新理解。

JHEP02(2020)047

not aligned, unless a =const — in which case one can take a = 0, upon rescaling f and t.

This thus provides an alternative geometric interpretation of the “g

= −1” condition

discussed in [26] (also meaning that, in such a case, r is linear in v, i.e., it is an aﬃne

parameter along ` [20, 26]).

In the rest of the paper, it will be assumed that, in (1.2), the transverse Riemannian

metric h = h

)dx

is an n-dimensional universal space (thus being, in particular,

Einstein and with constant scalar invariants) — see appendix A and references therein

for a deﬁnition and more details.

This property of h will be understood from now on.

Quantities with a tilde will refer to the transverse space geometry. We normalise the

transverse metric so that

= (n − 1)Kh

, (2.1)

which obvioulsy implies

R = n(n − 1)K.

2.2 Reduced ﬁeld equations and universality

The ﬁeld equations derived from (1.3) (neglecting boundary terms) are of the form E =

0, where E is a symmetric, conserved rank-2 tensor locally constructed out of g and

its derivatives [10] (cf. also [28]). However, for the ansatz (1.2) they can be drastically

simpliﬁed thanks to the following

Proposition 2.1. Consider any symmetric 2-tensor, E, constructed from tensor prod-

ucts, sums and contractions from the metric g, the Riemann tensor R, and its covariant

derivatives. Then for any metric of the form (1.2) with h universal we have:

E = F (r)dt

+ G(r)dr

+ H(r)h

)dx

. (2.2)

Proof. First, let us utilise that the metric is invariant under time-reversal: t 7→ −t. This

implies that any curvature tensor

is purely electric (as deﬁned in [22]). In particular, E

is purely electric (and, of course, t-independent). For a symmetric 2-tensor this implies

= E

= 0. Second, if h

)dx

is universal then this is an isotropy irreducibe

(locally) homogeneous space (cf. appendix A). This means that the isotropy group acts

irreducibly on the tangent space of the transverse metric. These symmetries of h can be

lifted trivially to the total metric (1.2), and hence, the tensor E needs to be invariant

under the these symmetries as well. Using the isotropy group (which acts irreducibly on

M), we thus get E

∝ h

(cf. also [29]) and E

= 0. Finally, since h

)dx

is a locally homogeneous space, the components E

, E

can only depend on r, and

= H(r)h

)dx

More in detail, the Weyl type is generically D(bd) since here h is Einstein [21, 22] (cf. also proposi-

tion 8.16 of [23]), in which case there exist precisely two mWANDs ∂

± f ∂

(see footnote 15 of [24]) and

thus the Riemann type is G if a

6= 0 and D (aligned) if a

= 0. The spacetime is conformally ﬂat iﬀ h is

of constant curvature and the functions f and a satisfy a diﬀerential equation which can be obtained from

(40, [25]).

To make the reading of the paper more ﬂuent, let us already mention here that universal spaces turn

out to be ultimately equivalent to the well-known isotropy-irreducible homogeneous spaces, which have

been thoroughly studied (see, e.g., [27] and references therein).

I.e., any tensor constructed polynomially from the Riemann tensor and its covariant derivatives.

– 4 –

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