# Curvature Invariants for Charged and Rotating Black Holes

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## Abstract

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## 1. Introduction

## 2. Preliminaries

## 3. Results

- Weyl invariants:$$\begin{array}{cc}\hfill {I}_{1}& ={C}_{ij}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}{C}_{kl}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}ij}={C}^{ijkl}{C}_{ijkl}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{2}& =-{C}_{ij}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}{C}_{kl}^{*\phantom{\rule{0.277778em}{0ex}}ij}=-{\textstyle \frac{1}{2}}{E}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{ij}{C}^{klop}{C}_{ijop}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{3}& ={C}_{ij}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}{C}_{kl}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}op}{C}_{op}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}ij}={C}_{ijkl}{C}^{klop}{C}_{op}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}ij}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{4}& =-{C}_{ij}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}{C}_{kl}^{*\phantom{\rule{0.277778em}{0ex}}op}{C}_{op}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}ij}=-{\textstyle \frac{1}{2}}{C}_{ijkl}{C}^{klop}{E}_{op}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}uv}{C}_{uv}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}ij}\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$
- Ricci invariants:$$\begin{array}{cc}\hfill {I}_{5}& =R={g}_{ij}{R}^{ij}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{6}& ={R}_{ij}{R}^{ij}={R}_{ij}{g}^{ki}{g}^{lj}{R}_{kl}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{7}& ={R}_{i}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j}{R}_{j}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k}{R}_{k}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{8}& ={R}_{i}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j}{R}_{j}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k}{R}_{k}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}l}{R}_{l}^{\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}i}\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$
- Mixed invariants:$$\begin{array}{cc}\hfill {I}_{9}& ={C}_{ikl}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j}{R}^{kl}{R}_{j}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}i}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{10}& =-{C}_{\phantom{\rule{0.277778em}{0ex}}ikl}^{*\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j}{R}^{kl}{R}_{j}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}i}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{11}& ={R}^{ij}{R}^{kl}({C}_{oij}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{pkl}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}o}-{C}_{oij}^{*\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}pkl}^{*\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}o})\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{12}& =-{R}^{ij}{R}^{kl}({C}_{oij}^{*\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p}{C}_{pkl}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}o}+{C}_{olj}^{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}pkl}^{*\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}o})\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{15}& ={\textstyle \frac{1}{16}}{R}^{ij}{R}^{kl}({C}_{oijp}{C}_{\phantom{\rule{0.277778em}{0ex}}kl}^{o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p}+{C}_{oijp}^{*}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{*o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p})\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{16}& =-{\textstyle \frac{1}{32}}{R}^{ij}{R}^{kl}({C}_{ouvp}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v}+{C}_{ouvp}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{*o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{11.38109pt}{0ex}}-{C}_{ouvp}^{*}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{*o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v}+{C}_{ouvp}^{*}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{*u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v})\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {I}_{17}& ={\textstyle \frac{1}{32}}{R}^{ij}{R}^{kl}({C}_{ouvp}^{*}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v}+{C}_{ouvp}^{*}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{*o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{*u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{11.38109pt}{0ex}}-{C}_{ouvp}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{*o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v}+{C}_{ouvp}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.166667em}{0ex}}ij}^{o\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}p}{C}_{\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}kl}^{*u\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}v})\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References and Note

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**Figure 1.**(

**a**) The Weyl invariants ${I}_{1}$ (yellow) and ${I}_{2}$ (blue); and (

**b**) the Ricci invariant ${I}_{6}$ (yellow) and mixed invariants ${I}_{9}$ (blue) and ${I}_{10}$ (green), all plotted as a function the Boyer–Lindquist coordinates r and $\theta $ for a black hole of mass $m=1$, angular momentum per unit mass $a=0.6$ and charge $q=0.8$.

**Figure 2.**(

**a**) A typical textbook representation of the interior structure of a rotating black hole. We argue that such coordinate-dependent depictions can be misleading, and should be supplemented by illustrations involving invariants. (

**b**) The magnitude of the Weyl invariant ${I}_{1}$ in ordinary spherical coordinates ($\varphi ,\theta $), plotted for several values of the Boyer–Lindquist radial coordinate r.

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**MDPI and ACS Style**

Overduin, J.; Coplan, M.; Wilcomb, K.; Henry, R.C.
Curvature Invariants for Charged and Rotating Black Holes. *Universe* **2020**, *6*, 22.
https://doi.org/10.3390/universe6020022

**AMA Style**

Overduin J, Coplan M, Wilcomb K, Henry RC.
Curvature Invariants for Charged and Rotating Black Holes. *Universe*. 2020; 6(2):22.
https://doi.org/10.3390/universe6020022

**Chicago/Turabian Style**

Overduin, James, Max Coplan, Kielan Wilcomb, and Richard Conn Henry.
2020. "Curvature Invariants for Charged and Rotating Black Holes" *Universe* 6, no. 2: 22.
https://doi.org/10.3390/universe6020022