Regular Black Holes: A Short Topic Review

Abstract

The essential singularity in Einstein’s gravity can be avoidable if the preconditions of Penrose’s theorem can be bypassed, i.e., if the strong energy condition is broken in the vicinity of a black hole center. The singularity mentioned here includes two aspects: (i) the divergence of curvature invariants, and (ii) the incompleteness of geodesics. Both aspects are now taken into account in order to determine whether a black hole contains essential singularities. In this sense, black holes without essential singularities are dubbed regular (non-singular) black holes. The regular black holes have some intriguing phenomena that are different from those of singular black holes, and such phenomena have inspired numerous studies. In this review, we summarize the current topics that are associated with regular black holes.

Appendix

1.1 Segré Notation

   Segré notation (or Segré symbol) is a systematic method for studying the complete intersection of two quadrics in algebraic geometry [219]. It is extended by Petrov [128, 220, 221] to classify gravitational fields [103]. However, the original and extended notations aim at different research subjects. The former targets second-order tensors in a vector space, and acts as an algebraic classification of Ricci tensors in general relativity, called the Segré classification. The latter directs at fourth-order tensors in a bivector space in general relativity, and acts as an algebraic classification of Weyl tensors, known as the Petrov classification. The Segré notation refers to the Segré classification in our current context. It provides a guide to searching the matter source in RBHs for a given metric and gravitational theory.

For two given tensors of order two, $\mathcal {R}$ and $\mathcal {I}$, which can also be regarded as two matrixes, one considers the $\lambda$-matrix, $\mathcal {R}-\lambda \mathcal {I}$, and computes the corresponding elementary divisors. Supposing that $\det (\mathcal {R}-\lambda \mathcal {I})=0$ as an algebraic equation has n distinct roots $\lambda _1,\ldots , \lambda _n$, we have the elementary divisors for every root $\lambda _i$,

$$\begin{aligned} (\lambda -\lambda _i)^{p_{i}^{(1)}}, \ldots , (\lambda -\lambda _i)^{p_{i}^{(s_i)}},\qquad p_{i}^{(1)}\le \cdots \le p_{i}^{(s_i)}, \end{aligned}$$

(128)

where $p_{i}^{(s_j)}$ is the multiplicity of eigenvalue $\lambda _i$ in the $s_j$-th divisor. The Segré notation is the collection,

$$\begin{aligned} \left[ (p_{1}^{(1)} \ldots p_{1}^{(s_1)} ) \ldots (p_{n}^{(1)} \ldots p_{n}^{(s_n)} ) \right] . \end{aligned}$$

(129)

For a second-order EMT, one constructs an orthonormal basis $\hat{e}^\alpha _\mu$ [148] on the spacetime manifold $(\mathcal {M}, g_{\mu \nu })$, which satisfies

$$\begin{aligned} g_{\alpha \beta } \hat{e}^\alpha _\mu \hat{e}^\beta _\nu = \eta _{\mu \nu },\qquad \eta ={{\,\textrm{diag}\,}}\{-1,1,1,1\}, \end{aligned}$$

(130)

then decomposes the EMT by

$$\begin{aligned} T^{\mu \nu } = \rho \hat{e}^\mu _0 \hat{e}^\nu _0 +p_1 \hat{e}^\mu _1 \hat{e}^\nu _1 +p_2 \hat{e}^\mu _2 \hat{e}^\nu _2 +p_3 \hat{e}^\mu _3 \hat{e}^\nu _3. \end{aligned}$$

(131)

In other words, the EMT in the basis $\hat{e}^\alpha _\mu$ can be represented as a diagonal matrix. Thus, we can use the Segré notation to classify the matter contents that have the possibility to generate non-rotating or rotating RBHs.

Due to the algebraic and physical properties of (131), the multiplicity is unit, i.e. we can use 1 and the parentheses to denote every diagonal component of $T^{\mu \nu }$ and the equality among individual components. For instance, the algebraic property of EMT with $\rho =p_1$ and $p_2=p_3$ can be represented by [(11)(11)]. Here, we do not use commas to separate the time and space components.

1.2 Zakhary-Mcintosh Invariants

   We list all the 17 ZM invariants following Ref. [95].

• 1. Ricci data construed by Ricci tensors,

  $$\begin{aligned} \mathcal {I}_5=g^{\mu \nu }R_{\mu \nu }, \qquad \mathcal {I}_6=R_{\mu \nu }R^{\mu \nu }, \end{aligned}$$

  (132a)
  $$\begin{aligned} \mathcal {I}_7:= R_{\mu }^{\;\;\nu }R_{\nu }^{\;\;\alpha }R_{\alpha }^{\;\;\mu }, \qquad \mathcal {I}_8:= R_{\mu }^{\;\;\nu }R_{\nu }^{\;\;\alpha }R_{\alpha }^{\;\;\beta }R_{\beta }^{\;\;\mu }, \end{aligned}$$

  (132b)

• 2. Weyl data construed by Weyl tensors $W_{\alpha \beta _\mu \nu }$,

  $$\begin{aligned} \mathcal {I}_1:= W^{\mu \nu }_{\quad \alpha \beta }W^{\alpha \beta }_{\quad \mu \nu }, \qquad \mathcal {I}_3:= W^{\mu \nu }_{\quad \alpha \beta }W^{\alpha \beta }_{\quad \rho \sigma }W^{\rho \sigma }_{\quad \mu \nu }, \end{aligned}$$

  (133a)
  $$\begin{aligned} \mathcal {I}_2:= -W^{\mu \nu }_{\quad \alpha \beta }W^{*\alpha \beta }_{\quad \mu \nu }, \qquad \mathcal {I}_4:= -W^{\mu \nu }_{\quad \alpha \beta }W^{*\alpha \beta }_{\quad \rho \sigma }W^{\rho \sigma }_{\quad \mu \nu }, \end{aligned}$$

  (133b)

• 3. Mixed data constructed by both Ricci and Weyl tensors,

  $$\begin{aligned} \mathcal {I}_9 := W_{\mu \alpha \beta \nu }R^{\alpha \beta }R^{\nu \mu },\quad \mathcal {I}_{10} := -W^*_{\mu \alpha \beta \nu }R^{\alpha \beta }R^{\nu \mu }, \end{aligned}$$

  (134a)
  $$\begin{aligned} \mathcal {I}_{11} := R^{\mu \nu } R^{\alpha \beta } \left( W_{\rho \mu \nu }^{\quad \; \sigma } W_{\sigma \alpha \beta }^{\quad \; \rho } - W_{\rho \mu \nu }^{*\quad \sigma } W_{\sigma \alpha \beta }^{*\quad \rho } \right) , \end{aligned}$$

  (134b)
  $$\begin{aligned} \mathcal {I}_{12} := -R^{\mu \nu } R^{\alpha \beta } \left( W_{\rho \mu \nu }^{*\quad \; \sigma } W_{\sigma \alpha \beta }^{\quad \; \rho } + W_{\rho \mu \nu }^{\quad \sigma } W_{\sigma \alpha \beta }^{*\quad \rho } \right) , \end{aligned}$$

  (134c)
  $$\begin{aligned} \mathcal {I}_{13} := R^{\mu \rho } R_{\rho }^{\;\; \alpha } R^{\nu \sigma } R_{\sigma }^{\;\beta } W_{\mu \nu \alpha \beta },\quad \mathcal {I}_{14} := R^{\mu \rho } R_{\rho }^{\;\; \alpha } R^{\nu \sigma } R_{\sigma }^{\;\beta } W^*_{\mu \nu \alpha \beta } \end{aligned}$$

  (134d)
  $$\begin{aligned} \mathcal {I}_{15} := \frac{1}{16} R^{\mu \nu }R^{\alpha \beta } \left( W_{\rho \mu \nu \sigma }W_{\;\;\alpha \beta }^{\rho \quad \sigma } + W^*_{\rho \mu \nu \sigma }W_{\quad \alpha \beta }^{*\rho \quad \sigma } \right) , \end{aligned}$$

  (134e)
  $$\begin{aligned} \begin{aligned} \mathcal {I}_{16}&:= -\frac{1}{32} R^{\alpha \beta } R^{\mu \nu } \Big ( W_{\rho \sigma \gamma \zeta } W^{\rho \quad \zeta }_{\;\;\alpha \beta }W^{\sigma \quad \zeta }_{\;\;\mu \nu } +W_{\rho \sigma \gamma \zeta } W^{*\rho \quad \zeta }_{\;\;\;\;\alpha \beta }W^{\sigma \quad \zeta }_{\;\;\;\;\mu \nu }\\&\qquad \qquad \qquad -W^*_{\rho \sigma \gamma \zeta } W^{*\rho \quad \zeta }_{\;\;\;\;\alpha \beta }W^{\sigma \quad \zeta }_{\;\;\mu \nu } +W^*_{\rho \sigma \gamma \zeta } W^{\rho \quad \zeta }_{\;\;\alpha \beta }W^{*\sigma \quad \zeta }_{\;\;\;\;\mu \nu } \Big ), \end{aligned} \end{aligned}$$

  (134f)
  $$\begin{aligned} \begin{aligned} \mathcal {I}_{17}&:= \frac{1}{32} R^{\alpha \beta } R^{\mu \nu } \Big ( W^*_{\rho \sigma \gamma \zeta } W^{\rho \quad \zeta }_{\;\;\alpha \beta }W^{\sigma \quad \zeta }_{\;\;\mu \nu } +W^*_{\rho \sigma \gamma \zeta } W^{*\rho \quad \zeta }_{\;\;\;\;\alpha \beta }W^{*\sigma \quad \zeta }_{\;\;\;\;\mu \nu }\\&\qquad \qquad \qquad -W_{\rho \sigma \gamma \zeta } W^{*\rho \quad \zeta }_{\;\;\;\;\alpha \beta }W^{\sigma \quad \zeta }_{\;\;\mu \nu } +W_{\rho \sigma \gamma \zeta } W^{\rho \quad \zeta }_{\;\;\alpha \beta }W^{*\sigma \quad \zeta }_{\;\;\;\;\mu \nu } \Big ), \end{aligned} \end{aligned}$$

  (134g)

where $W_{\mu \nu \alpha \beta }$ is Weyl tensor and $W^*_{\mu \nu \alpha \beta }=\epsilon _{\mu \nu \rho \sigma }W^{\rho \sigma }_{\;\;\;\alpha \beta }/2$ denotes its dual.