Distorted Myers-Perry Black Hole

• Distorted Myers-Perry black holes are higher-dimensional rotating solutions immersed in external gravitational fields that induce horizon deformations.
• They are constructed using advanced techniques like the two-fold Bäcklund transformation applied to a five-dimensional prolate spheroidal metric.
• Despite pronounced geometric distortions, key physical properties such as the ergoregion remain unaffected under specific parameter conditions.

A distorted Myers-Perry black hole refers to a solution of the higher-dimensional vacuum Einstein equations that represents a Myers-Perry (MP) black hole—characterized by mass and angular momentum—immersed in a nontrivial external gravitational field, leading to physical and geometric distortions of the black hole’s horizon while preserving its fundamental topology. In five dimensions, such solutions are of particular interest because the interplay between rotational dynamics and external sources gives rise to distinctive phenomena: the event horizon becomes “bumpy,” with its intrinsic geometry and curvature sensitive to distortion multipoles, while key aspects such as the ergoregion can remain surprisingly unaffected under certain conditions. These solutions are constructed via advanced solution-generating methods such as the two-fold Bäcklund transformation and provide a precise, mathematically tractable model for investigating interactions between black holes and ambient gravitational environments (Abdolrahimi et al., 16 Sep 2025, Abdolrahimi et al., 2014).

1. Construction of Distorted Myers-Perry Black Holes

The five-dimensional Myers-Perry metric with a single nonvanishing rotation parameter serves as the seed solution. Distortion is introduced by superposing an external stationary, axisymmetric gravitational field, encoded by an infinite set of real parameters $\{a_n, b_n\}$ corresponding to gravitational multipole moments. The resulting exact solution is constructed using a two-fold Bäcklund transformation, applied to a prolate spheroidal coordinate chart $(x,y)$ with $x \in [1,\infty)$ and $y \in [-1,1]$:

$$ds^2 = -\frac{x-1-\hat{a}^2(1-y)}{x+1+\hat{a}^2(1+y)} \, e^{2(\widehat{U}+\widehat{W})} (dt-\omega\,d\psi)^2 + \frac{x+1+\hat{a}^2(1+y)}{x-1-\hat{a}^2(1-y)} e^{-2\widehat{W}} d\psi^2 + e^{-2\widehat{U}} d\phi^2 + C_1 [ x + 1 + \hat{a}^2(1+y) ] e^{2(\hat{\gamma} - \widehat{W})} \left[ \frac{dx^2}{x^2-1} + \frac{dy^2}{1-y^2} \right]$$

Distortion potentials $\widehat{U}$ and $\widehat{W}$ are given by infinite series in $a_n$ and $b_n$, typically as $$\widehat{U}(x,y) = \sum_{n=0}^\infty a_n R^n P_n(xy/R)$$, with $R = \sqrt{x^2 + y^2}$ and $P_n$ the Legendre polynomials. The rotation function $\hat{a}$ is modulated as $$\hat{a} = \alpha \exp\left[ -\sum_{n=0}^\infty (a_n+2b_n)(\cos^n\theta-1) \right]$$ when restricted to the horizon.

The horizon is located at $x=1$ and, in the absence of distortion ($a_n = b_n = 0$), the metric reduces to the standard Myers-Perry solution. The horizon remains topologically $S^3$, although its induced geometry and curvature are altered by the external field (Abdolrahimi et al., 2014, Abdolrahimi et al., 16 Sep 2025).

2. Horizon Geometry and Curvature Properties

Restricting the distorted metric to the horizon $(x=1)$ and expressing $y=\cos\theta$, the induced metric is

$\begin{split} ds^2_H &= \frac{4\sigma(1+\alpha^2)^2}{1+\hat{a}^2(\theta)\cos^2(\frac{\theta}{2})} \sin^2(\frac{\theta}{2})e^{-2\widehat{W}(\theta)}\,d\psi^2 \ &\quad + 4\sigma \cos^2(\frac{\theta}{2})e^{-2\widehat{U}(\theta)}\,d\phi^2 \ &\quad + \sigma \left[1+\hat{a}^2(\theta)\cos^2(\frac{\theta}{2})\right] e^{2(\hat{\gamma}(\theta)-\widehat{W}(\theta))}\,d\theta^2 \end{split}$

where

$$\widehat{U}(\theta) = \sum_{n=0}^{\infty} a_n \cos^n\theta,\qquad \widehat{W}(\theta) = \sum_{n=0}^{\infty} b_n \cos^n\theta, \qquad \hat{a}(\theta) = \alpha\exp\left[-\sum_{n=0}^{\infty}(a_n+2b_n)(\cos^n\theta-1)\right]$$

These distortion multipoles directly influence the metric components, leading to local “bumps” or deformations. The Ricci scalar $\mathcal{R}$ of the horizon can display pronounced peaks or regions with sign changes, particularly for dipole distortions that break spherical symmetry. For example, in the unperturbed case, $$\mathcal{R}= -\frac{(1+\alpha^2)(\alpha^2\cos^2(\theta/2)-3)}{2(\alpha^2\cos^2(\theta/2)+1)^3}$$ is monotonic in $\theta$, but in the presence of distortion, new local maxima or minima arise (Abdolrahimi et al., 16 Sep 2025).

In certain parameter regimes, higher-order curvature invariants such as $\mathcal{R}_{AB} \mathcal{R}^{AB}$ can develop sharp peaks, signaling intense tidal fields concentrated on parts of the horizon.

3. Isometric Embedding and Geometric Characterization

To visualize the bumpy horizon, isometric embedding of two-dimensional sections (such as $(\psi,\theta)$ or $(\phi,\theta)$ sections) into auxiliary three-dimensional (pseudo-)Euclidean spaces is performed. For a metric section

$$ds^2_{(\psi,\theta)} = A(\theta)\, d\psi^2 + B(\theta)\, d\theta^2$$

the embedding proceeds via

$$dl^2 = \left( \epsilon \left( \frac{dZ}{d\theta} \right)^2 + \left( \frac{d\rho}{d\theta} \right)^2 \right) d\theta^2 + \rho(\theta)^2\, d\psi^2$$

with $A(\theta) = \rho^2(\theta)$ and $$B(\theta) = \epsilon (dZ/d\theta)^2 + (d\rho/d\theta)^2$$. The sign $\epsilon$ ensures real embedding curves and can switch to pseudo-Euclidean if the metric section is not positive-definite. The distortion parameters can induce prolateness or oblateness—monopole or dipole distortions with $b_1>0$ may elongate the horizon along specific sections (Abdolrahimi et al., 16 Sep 2025).

4. Physical Quantities and Local Smarr Relations

On the distorted horizon, Komar integrals yield intrinsic physical quantities. The horizon mass $M_h$ and angular momentum $J_h$ are

$M_h = \frac{3\pi}{2}\sigma(1 + \alpha^2)$

$$J_h = 2\pi \sigma^{3/2}\alpha(1+\alpha^2)\exp\Big[-\sum_n(a_n+2b_n)\Big]$$

The horizon area $\mathcal{A}_h$ and surface gravity $\kappa_h$ can be derived in an analogous fashion, and these satisfy a local Smarr-like relation akin to that for the asymptotically flat Myers-Perry solution:

$$M_h = \frac{3}{16\pi}\kappa_h\, \mathcal{A}_h + \frac{3}{2} \Omega J_h$$

where $\Omega$ is the black hole's angular velocity, modified by external distortion parameters via an exponential rescaling (Abdolrahimi et al., 2014, Abdolrahimi et al., 16 Sep 2025).

A distinctive consequence is that the dimensionless ratio $(J_h^2/M_h^3)$, bounded in the undistorted case, becomes unbounded for certain choices of multipoles, reflecting the ability of the external field to “spin up” the black hole arbitrarily for fixed local horizon mass.

5. Ergosphere and Its Insensitivity to Distortion

Despite the strong deformations of the horizon, the ergoregion—a region outside the event horizon where the stationary Killing field becomes spacelike ($g_{tt} > 0$)—can remain unchanged under a particular class of distortions. When distortion multipoles satisfy $a_n + 2b_n = 0$ for all $n$, the angular velocity at the horizon and the structure of the ergosphere are unaltered from the pure Myers-Perry solution. This means that the “external gravitational hair” can deform the shape and local curvature of the event horizon while leaving the rotational (frame-dragging) properties encapsulated by the ergoregion completely intact (Abdolrahimi et al., 16 Sep 2025).

This phenomenon, where the horizon senses external sources but the ergoregion is “oblivious,” is highly nontrivial and distinguishes distorted higher-dimensional black holes from four-dimensional analogues, in which ergosurface deformations typically accompany horizon deformations.

6. Broader Implications and Connections

Distorted Myers-Perry black holes offer a controlled setting to investigate physical interactions of higher-dimensional rotating black holes with external sources, and provide potential models for black holes surrounded by matter rings or multipolar fields. The explicit construction via integrable methods connects to broader developments in solution-generating techniques and the paper of black hole uniqueness, extremality, and phase structures (Abdolrahimi et al., 2014, Abdolrahimi et al., 16 Sep 2025).

These solutions also serve as probes for the response of horizon geometry and intrinsic quantities (area, curvature, angular momentum) to ambient gravitational environments, relevant both for classical general relativity and as testing grounds for quantum or holographic corrections—especially in settings with nontrivial external gravitational multipoles.

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Key Formulas and Distortion Potentials

Quantity	Expression	Notes
Horizon metric	See $ds^2_H$ above	$\widehat{U}(\theta),\ \widehat{W}(\theta)$ control distortion
Komar $M_h$/$J_h$	See above	Exponential dependence on $a_n$, $b_n$
Special condition	$a_n + 2b_n = 0$ for all $n$	Ergosphere remains unchanged

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These results establish that in five dimensions, the event horizon of a Myers-Perry black hole is highly sensitive to external distortions, leading to pronounced and locally variable geometric features, while the ergosphere may remain robust under such deformations.