# Nonsingular, Lump-like, Scalar Compact Objects in (2 + 1)-Dimensional Einstein Gravity

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## Abstract

**:**

## 1. Introduction

## 2. $(\mathbf{2}+\mathbf{1})$-Einstein Theory with Nonlinear Scalar Field

#### Static and Circularly Symmetric Solutions

## 3. Families of Solutions

- $\frac{1}{2}<\alpha <1$ corresponds to the interval $\gamma >0$, with $\gamma =0$ identified with $\alpha =1/2$ and $\gamma \to +\infty $ with $\alpha \to {1}^{-}$.
- $\alpha >1$ leads to $-\infty <\gamma <-2$, with $\alpha \to {1}^{+}$ corresponding to $\gamma \to -\infty $ and $\alpha \to \infty $ leading to $\gamma \to -2$.
- $\alpha <0$ is mapped into $-2<\gamma <-1$, with $\alpha =0$ leading to $\gamma =-1$ and $\alpha \to -\infty $ to $\gamma =-2$.
- $0<\alpha <\frac{1}{2}$ is mapped into $-1<\gamma <0$.

#### Cases $\alpha =0,1/2$, and 1

## 4. Asymptotically Flat Solutions

#### Geodesics

- $\gamma =1\Rightarrow \hspace{1em}\frac{\widehat{r}}{2-2{\widehat{r}}^{2}}+\widehat{r}+\frac{3}{4}log\frac{(\widehat{r}-1)}{(\widehat{r}+1)}$
- $\gamma =2\Rightarrow \hspace{1em}\widehat{r}+\frac{5-6\widehat{r}}{2{(\widehat{r}-1)}^{2}}+3log(\widehat{r}-1)$
- $\gamma =3\Rightarrow \hspace{1em}\frac{1}{16}\left(\frac{-693{\widehat{r}}^{5/3}+144{\widehat{r}}^{7/3}+16{\widehat{r}}^{3}-315\sqrt[3]{\widehat{r}}+840\widehat{r}}{{\left({\widehat{r}}^{2/3}-1\right)}^{3}}\right.$$\left.-315\left[{tanh}^{-1}\left(\sqrt[3]{\widehat{r}}\right)+\frac{i\pi}{2}\right]\right)$

## 5. Energy Density Distribution

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Note

1 | For the static scenarios we are considering, the kinetic term Y is always positive. In more general settings, and in order to prevent problems if Y becomes negative, one could consider a redefinition of $\alpha $ as $\alpha \to 2\tilde{\alpha}$ to force that the Lagrangian is indeed a real quantity. |

## References

- Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; Herlt, E. Exact Solutions of Einstein’s Field Equations; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Penrose, R. Gravitational collapse: The role of general relativity. Riv. Nuovo Cim.
**1969**, 1, 252. [Google Scholar] - Carter, B. Axisymmetric Black Hole Has Only Two Degrees of Freedom. Phys. Rev. Lett.
**1971**, 26, 331. [Google Scholar] [CrossRef] - Kerr, R.P. Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett.
**1963**, 11, 237. [Google Scholar] [CrossRef] - Newman, E.T.; Couch, E.; Chinnapared, K.; Exton, A.; Prakash, A.; Torrence, R. Metric of a Rotating, Charged Mass. J. Math. Phys.
**1965**, 6, 918. [Google Scholar] [CrossRef] - Event Horizon Telescope; Akiyama, K.; Alberdi, A.; Alef, W.; Asada, K.; Azuly, R. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett.
**2019**, 875, L1. [Google Scholar] - Akiyama, K.; Alberdi, A.; Alef, W.; Algaba, J.C.; Anantua, R.; Asada, K.; Azulay, R.; Bach, U.; Baczko, A.K.; Ball, D.; et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. Astrophys. J. Lett.
**2022**, 930, L12. [Google Scholar] - Maggiore, M. Theory and Experiments. In Gravitational Waves; Oxford University Press: Oxford, UK, 2007; Volume 1. [Google Scholar]
- Senovilla, J.M.; Garfinkle, D. The 1965 Penrose singularity theorem. Class. Quant. Grav.
**2015**, 32, 124008. [Google Scholar] [CrossRef] - Cardoso, V.; Pani, P. Testing the nature of dark compact objects: A status report. Living Rev. Rel.
**2019**, 22, 4. [Google Scholar] [CrossRef] - Carlip, S. The (2+1)-Dimensional black hole. Class. Quant. Grav.
**1995**, 12, 2853. [Google Scholar] [CrossRef] - Banados, M.; Henneaux, M.; Teitelboim, C.; Zanelli, J. Geometry of the (2+1) black hole. Phys. Rev. D
**1993**, 48, 1506, Erratum in Phys. Rev. D**2013**, 88, 069902.. [Google Scholar] [CrossRef] - Banados, M.; Teitelboim, C.; Zanelli, J. The Black hole in three-dimensional space-time. Phys. Rev. Lett.
**1992**, 69, 1849. [Google Scholar] [CrossRef] - Martinez, C.; Teitelboim, C.; Zanelli, J. Charged rotating black hole in three space-time dimensions. Phys. Rev. D
**2000**, 61, 104013. [Google Scholar] [CrossRef] - Carlip, S. Conformal field theory, (2+1)-dimensional gravity, and the BTZ black hole. Class. Quant. Grav.
**2005**, 22, R85. [Google Scholar] [CrossRef] - Sahoo, B.; Sen, A. BTZ black hole with Chern-Simons and higher derivative terms. J. High Energy Phys.
**2006**, 7, 008. [Google Scholar] [CrossRef] - Li, R.; Ren, J.R. Dirac particles tunneling from BTZ black hole. Phys. Lett. B
**2008**, 661, 370. [Google Scholar] [CrossRef] - He, Y.; Ma, M.S. (2+1)-dimensional regular black holes with nonlinear electrodynamics sources. Phys. Lett. B
**2017**, 774, 229. [Google Scholar] [CrossRef] - Bueno, P.; Cano, P.A.; Moreno, J.; van der Velde, G. Regular black holes in three dimensions. Phys. Rev. D
**2021**, 104, L021501. [Google Scholar] [CrossRef] - Estrada, M.; Tello-Ortiz, F. A new model of regular black hole in (2+1) dimensions. EPL
**2021**, 135, 20001. [Google Scholar] [CrossRef] - Maluf, R.V.; Muniz, C.R.; Santos, A.C.L.; Estrada, M. A new class of regular black hole solutions with quasi-localized sources of matter in (2 + 1) dimensions. Phys. Lett. B
**2022**, 835, 137581. [Google Scholar] [CrossRef] - Alencar, G.; Bezerra, V.B.; Muniz, C.R. Casimir wormholes in 2+1 dimensions with applications to the graphene. Eur. Phys. J. C
**2021**, 81, 924. [Google Scholar] [CrossRef] - Santos, A.C.; Muniz, C.R.; Maluf, R.V. Yang-Mills Casimir wormholes in D = 2 + 1. J. Cosmol. Astropart. Phys.
**2023**, 9, 22. [Google Scholar] [CrossRef] - Wheeler, J.A. Geons. Phys. Rev.
**1955**, 97, 511. [Google Scholar] [CrossRef] - Misner, C.W.; Wheeler, J.A. Classical physics as geometry: Gravitation, electromagnetism, unquantized charge, and mass as properties of curved empty space. Ann. Phys.
**1957**, 2, 525. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Damour, T.; Mukhanov, V. k-inflation. Phys. Lett. B
**1999**, 458, 209. [Google Scholar] [CrossRef] - Armendariz-Picon, C.; Mukhanov, V.; Steinhardt, P.J. Essentials of k essence. Phys. Rev. D
**2001**, 63, 103510. [Google Scholar] [CrossRef] - Liebling, S.L.; Palenzuela, C. Dynamical boson stars. Living Rev. Rel.
**2023**, 26, 1. [Google Scholar] [CrossRef] - Vincent, F.H.; Meliani, Z.; Grandclément, P.; Gourgoulhon, E.; Straub, O. Imaging a boson star at the Galactic center. Class. Quant. Grav.
**2016**, 33, 105015. [Google Scholar] [CrossRef] - Palenzuela, C.; Pani, P.; Bezares, M.; Cardoso, V.; Lehner, L.; Liebling, S. Gravitational Wave Signatures of Highly Compact Boson Star Binaries. Phys. Rev. D
**2017**, 96, 104058. [Google Scholar] [CrossRef] - Olivares, H.; Younsi, Z.; Fromm, C.M.; De Laurentis, M.; Porth, O.; Mizuno, Y.; Falcke, H.; Kramer, M.; Rezzolla, L. How to tell an accreting boson star from a black hole. Mon. Not. Roy. Astron. Soc.
**2020**, 497, 521. [Google Scholar] [CrossRef] - Wyman, M. Static Spherically Symmetric Scalar Fields in General Relativity. Phys. Rev. D
**1981**, 24, 839. [Google Scholar] [CrossRef] - Magalhães, R.B.; Crispino, L.C.; Olmo, G.J. Compact objects in quadratic Palatini gravity generated by a free scalar field. Phys. Rev. D
**2022**, 105, 064007. [Google Scholar] [CrossRef] - Bambi, C.; Cardenas-Avendano, A.; Olmo, G.J.; Rubiera-Garcia, D. Wormholes and nonsingular spacetimes in Palatini f(R) gravity. Phys. Rev. D
**2016**, 93, 064016. [Google Scholar] [CrossRef] - Bazeia, D.; Losano, L.; Marques, M.A.; Menezes, R. From Kinks to Compactons. Phys. Lett. B
**2014**, 736, 515–521. [Google Scholar] [CrossRef] - Brax, P.; Mourad, J.; Steer, D.A. Tachyon kinks on nonBPS D-branes. Phys. Lett. B
**2003**, 575, 115–125. [Google Scholar] [CrossRef]

**Figure 1.**Representation of the (normalized) affine parameter $\frac{E}{{R}_{0}{\sigma}_{0}^{1/2}}\tau $ as a function of the radial coordinate $\widehat{r}$ for ingoing (blue) and outgoing (red) geodesics. The dashed lines depict the trajectories $\pm \widehat{r}$ that represent the Minkowskian geodesics (for illustration). Note that for $0<\gamma <1$, the convergence to the Minkowskian value is very fast, being fastest in the limit $\gamma \to 0$. The divergence of the affine parameter as $\widehat{r}\to 1$ shows that this region is a boundary of the manifold that cannot be reached by any observer or light signal.

**Figure 2.**Representation of the kinetic term $Y={g}^{\mu \nu}{\partial}_{\mu}\varphi {\partial}_{\nu}\varphi $ for the cases $\gamma =1/2,1,$ and 2 (recall that $\alpha =(1+\gamma )/(2+\gamma )$) when $\lambda <0$. Vertical dashed lines indicate the location of the maximum. The localized nature of these solutions is evident. Note that the smaller the value of $\gamma $, the higher the peak and the more compact the structure.

**Figure 3.**Left: three-dimensional representation of the kinetic term $Y={g}^{\mu \nu}{\partial}_{\mu}\varphi {\partial}_{\nu}\varphi $ when $\lambda <0$ for the case $\gamma =1$. Right: same representation for $\gamma =1/2$ (blue) and $\gamma =2$ (orange). Note how the more compact solution $\gamma =1/2$ is always hidden by the $\gamma =2$ one except at the innermost region. The different amplitudes of the maxima are also evident in this plot.

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**MDPI and ACS Style**

Maluf, R.V.; Mora-Pérez, G.; Olmo, G.J.; Rubiera-Garcia, D.
Nonsingular, Lump-like, Scalar Compact Objects in (2 + 1)-Dimensional Einstein Gravity. *Universe* **2024**, *10*, 258.
https://doi.org/10.3390/universe10060258

**AMA Style**

Maluf RV, Mora-Pérez G, Olmo GJ, Rubiera-Garcia D.
Nonsingular, Lump-like, Scalar Compact Objects in (2 + 1)-Dimensional Einstein Gravity. *Universe*. 2024; 10(6):258.
https://doi.org/10.3390/universe10060258

**Chicago/Turabian Style**

Maluf, Roberto V., Gerardo Mora-Pérez, Gonzalo J. Olmo, and Diego Rubiera-Garcia.
2024. "Nonsingular, Lump-like, Scalar Compact Objects in (2 + 1)-Dimensional Einstein Gravity" *Universe* 10, no. 6: 258.
https://doi.org/10.3390/universe10060258