Quevedo–Mashhoon Spacetime Overview

• Quevedo–Mashhoon spacetime is an exact, stationary, axisymmetric vacuum solution of Einstein’s equations that generalizes the Kerr metric by adding an independent quadrupole deformation.
• The metric incorporates a range of parameters—including mass, rotation, and an anomalous quadrupole parameter—that lead to degeneracies with Kerr and allow modeling both rotating deformed bodies and naked singularities.
• Astrophysical analyses using the QM spacetime explore modified ISCO properties, accretion disk emissions, and enhanced magnetic Penrose process efficiencies, providing insights into extreme gravity phenomena.

The Quevedo–Mashhoon spacetime is an exact, stationary, axisymmetric, asymptotically flat vacuum solution of Einstein’s equations that generalizes Kerr by introducing an independent deformation parameter for the mass quadrupole moment. In its broader form it is characterized by mass, rotation, a Zipoy–Voorhees parameter, and either an infinite set of multipole parameters or, in the quadrupolar subfamily used most often in recent applications, a single anomalous quadrupole parameter. It contains Kerr, Schwarzschild, Erez–Rosen, and Hartle–Thorne as special or limiting cases, and in the branches emphasized in recent studies it is used as the exterior field of a rotating deformed body and, for suitable parameter choices, as a rotating naked singularity (Kurmanov et al., 23 Mar 2025, Frutos-Alfaro et al., 2016, Boshkayev et al., 2012).

1. Solution class and exact metric structure

Stationary, axisymmetric vacuum spacetimes are commonly written in Weyl–Lewis–Papapetrou form. In cylindrical coordinates $(\rho,z,\varphi)$, or equivalently in prolate spheroidal coordinates $(x,y,\varphi)$, the line element may be written as

$$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$

or

$$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$

with

$$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$

The Papapetrou functions are $f$, $\omega$, and $\gamma$. The standard Ernst potential is $\mathcal{E}=f+i\chi$, with auxiliary potential

$$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$

and $(x,y,\varphi)$0 are reconstructed from $(x,y,\varphi)$1 and the twist potential $(x,y,\varphi)$2 by quadratures. Quevedo and Mashhoon obtained the stationary solution by applying two Hoenselaers–Kinnersley–Xanthopoulos transformations to an Erez–Rosen seed (Frutos-Alfaro et al., 2016).

In the quadrupolar subfamily used in recent work, the metric is also written in Boyer–Lindquist–like coordinates $(x,y,\varphi)$3 via

$(x,y,\varphi)$4

Then

$(x,y,\varphi)$5

with

$(x,y,\varphi)$6

and nonvanishing components

$(x,y,\varphi)$7

For the anomalous quadrupole parameter $(x,y,\varphi)$8,

$(x,y,\varphi)$9

where $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$0 and $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$1 are Legendre functions and $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$2 are given explicitly in terms of $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$3 with $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$4 (Zhang, 11 Jul 2025).

2. Parameters, multipole moments, and special limits

The QM family admits two closely related parameterizations. One description uses mass $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$5, rotation $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$6, a Zipoy–Voorhees parameter $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$7, and an infinite set $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$8 controlling deviations of the mass multipoles from Kerr. A widely used four-parameter subfamily retains only a quadrupole parameter $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$9 together with $$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$0, $$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$1, and $$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$2. In the most common asymptotically flat branch one sets $$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$3 (Frutos-Alfaro et al., 2016, Boshkayev et al., 2012).

For the four-parameter solution, equatorial reflection symmetry gives

$$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$4

and

$$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$5

The lowest nontrivial mass and current moments are

$$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$6

$$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$7

For $$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$8, $$ds^2=-f(dt-\omega\,d\varphi)^2+\frac{k^2}{f}\Bigg[e^{2\gamma}(x^2-y^2)\left(\frac{dx^2}{x^2-1}+\frac{dy^2}{1-y^2}\right)+(x^2-1)(1-y^2)\,d\varphi^2\Bigg],$$9 coincides with the total mass $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$0, $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$1 is the specific angular momentum, and $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$2 becomes an independent quadrupolar deformation parameter. With $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$3 and $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$4,

$$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$5

This makes explicit the quadrupole degeneracy: fixing $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$6 and $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$7 allows one to trade $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$8 for $$\rho=k\sqrt{(x^2-1)(1-y^2)},\qquad z=kxy,\qquad k^2=m^2-a^2.$$9, a freedom absent in Kerr (Kurmanov et al., 23 Mar 2025).

In the notation of the anomalous-quadrupole subfamily,

$f$0

so Kerr has $f$1, and the anomalous deviation is

$f$2

In that convention, $f$3 corresponds to a spacetime more prolate than Kerr and $f$4 to one more oblate than Kerr; in the extremal limit $f$5, the quadrupolar correction vanishes and the spacetime tends to extremal Kerr for any $f$6 (Zhang, 11 Jul 2025).

Limit	Parameter choice	Result
Kerr	$f$7 for all $f$8, or $f$9 and $\omega$0	Exact Kerr
Schwarzschild	$\omega$1 and $\omega$2	Static spherical limit
Erez–Rosen	$\omega$3, $\omega$4, higher $\omega$5	Static quadrupolar spacetime
Hartle–Thorne	Small $\omega$6, small $\omega$7, higher $\omega$8	Slow-rotation/quadrupolar limit

3. Horizons, naked singularities, ergoregions, and causal structure

In the branch used for accretion studies, the geometry is explicitly treated as horizonless: the QM solution is described there as an exact external metric for a rotating deformed mass “which corresponds to a naked singularity,” and the parameter sets considered do not produce an event horizon. Operationally, in stationary axisymmetry the presence or absence of a horizon is tied to the zeros of the lapse-like function $\omega$9 and the signature of $\gamma$0; in that parameter regime the spacetime is horizonless (Kurmanov et al., 23 Mar 2025).

In the Boyer–Lindquist–like representation, the would-be horizon is located by $\gamma$1,

$\gamma$2

but for $\gamma$3 the solution generically develops a naked singularity and this surface is not a regular event horizon. The ergosurface is determined by

$\gamma$4

and the ergoregion is the domain where $\gamma$5, equivalently $\gamma$6 in the sign convention used there (Zhang, 11 Jul 2025).

Large anomalous quadrupole deformations qualitatively alter the near-horizon geometry. For sufficiently large $\gamma$7, a region with $\gamma$8, permitting closed timelike curves, appears adjacent to $\gamma$9, and both the ergoregion and the CTC region develop multi-lobe structures. Representative cases reported for $\mathcal{E}=f+i\chi$0 are: for sufficiently large $\mathcal{E}=f+i\chi$1, the ergoregion splits into two disconnected lobes that do not intersect the equatorial plane while the CTC region splits into three parts; for sufficiently large $\mathcal{E}=f+i\chi$2, both the ergoregion and the CTC region split into three parts, and the ergoregion continues to intersect the equatorial plane. By contrast, the thin-disk study does not report explicit pathologies such as closed timelike curves or exotic ergoregion structure in the ranges explored; its analysis is confined to equatorial radii $\mathcal{E}=f+i\chi$3 (Zhang, 11 Jul 2025, Kurmanov et al., 23 Mar 2025).

To define a physically meaningful exterior problem in the presence of CTCs, the magnetized analysis excises the interior and places a reflecting surface at

$\mathcal{E}=f+i\chi$4

where $\mathcal{E}=f+i\chi$5 is the outer boundary of the CTC region. This implements an exterior-field interpretation analogous to a superspinar treatment (Zhang, 11 Jul 2025).

4. Slow-rotation limit, Hartle–Thorne equivalence, and interior matching

The exact QM metric admits a controlled slow-rotation, slight-deformation expansion for $\mathcal{E}=f+i\chi$6 and $\mathcal{E}=f+i\chi$7, retaining terms through $\mathcal{E}=f+i\chi$8 and $\mathcal{E}=f+i\chi$9. In that regime one may set $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$0 and choose $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$1, after which a coordinate transformation to Schwarzschild-like variables $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$2 brings the metric into the exterior Hartle–Thorne form (Boshkayev et al., 2012).

The key parameter map is

$$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$3

Thus the slow-rotation/small-deformation limit of the exact QM exterior is identical to the Hartle–Thorne vacuum exterior, with the independent quadrupole carried by $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$4. In the notation used there, the Kerr limit corresponds to $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$5, so activating $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$6 generates an independent quadrupolar deviation away from Kerr (Boshkayev et al., 2012).

The same work connects this exterior to approximate interior models constructed by Fock’s method. For the extended Fock exterior,

$$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$7

where $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$8 is the Newtonian quadrupole and $$\xi=\frac{1-\mathcal E}{1+\mathcal E},\qquad \mathcal E=\frac{1-\xi}{1+\xi},$$9 its relativistic correction. Combining this with the QM–HT map gives $(x,y,\varphi)$00 in terms of interior source parameters. For vanishing Newtonian quadrupole,

$(x,y,\varphi)$01

with $(x,y,\varphi)$02 for a liquid sphere and $(x,y,\varphi)$03 for a solid sphere. The paper’s conclusion is that standard Fock interiors cannot source the Kerr exterior already at $(x,y,\varphi)$04, whereas the extra degree of freedom in QM/HT permits consistent matching of an interior and exterior for slowly rotating, slightly deformed compact objects (Boshkayev et al., 2012).

A plausible implication is that the QM spacetime serves two distinct roles. In one regime it is used as an exact exterior metric for non-collapsed rotating bodies whose multipole structure deviates from Kerr; in another, more strongly deformed regime, it is treated as a naked-singularity exterior with the pathological interior excised. Both uses appear explicitly in the cited literature.

5. Equatorial geodesics and thin-disk emission

For neutral massive test particles on circular equatorial orbits, the accretion analysis uses the standard stationary-axisymmetric formulas

$(x,y,\varphi)$05

$(x,y,\varphi)$06

with the ISCO defined numerically by the marginal-stability condition

$(x,y,\varphi)$07

On the equatorial plane of the form used there,

$(x,y,\varphi)$08

The reported trends are monotonic: in the Kerr branch $(x,y,\varphi)$09, $(x,y,\varphi)$10 lowers $(x,y,\varphi)$11 relative to Schwarzschild and $(x,y,\varphi)$12 raises it; in the Erez–Rosen branch $(x,y,\varphi)$13, $(x,y,\varphi)$14 lowers $(x,y,\varphi)$15 and $(x,y,\varphi)$16 raises it. Likewise, $(x,y,\varphi)$17 is higher than Schwarzschild for $(x,y,\varphi)$18 and lower for $(x,y,\varphi)$19; with $(x,y,\varphi)$20, negative $(x,y,\varphi)$21 increases $(x,y,\varphi)$22 and positive $(x,y,\varphi)$23 decreases it. The dimensionless specific angular momentum $(x,y,\varphi)$24 is lower than Schwarzschild for $(x,y,\varphi)$25 or $(x,y,\varphi)$26, and higher for $(x,y,\varphi)$27 or $(x,y,\varphi)$28. The specific energy $(x,y,\varphi)$29 is lower than Schwarzschild for $(x,y,\varphi)$30, higher for $(x,y,\varphi)$31, with the corresponding $(x,y,\varphi)$32-dependence inverted as reported in that study (Kurmanov et al., 23 Mar 2025).

Disk emission is modeled with the Novikov–Thorne/Page–Thorne thin-disk formalism. The radiative flux is

$(x,y,\varphi)$33

with

$(x,y,\varphi)$34

The differential luminosity observed at infinity is

$(x,y,\varphi)$35

and, assuming local blackbody emission, the spectral luminosity is

$(x,y,\varphi)$36

where

$(x,y,\varphi)$37

and the radiative efficiency is estimated by

$(x,y,\varphi)$38

In the Erez–Rosen branch, $(x,y,\varphi)$39 decreases below the Schwarzschild value $(x,y,\varphi)$40 for $(x,y,\varphi)$41 and increases for $(x,y,\varphi)$42; in the Kerr branch, $(x,y,\varphi)$43 raises the efficiency while $(x,y,\varphi)$44 lowers it. Flux and differential luminosity follow the same ordering: in Kerr, $(x,y,\varphi)$45 produces higher values than Schwarzschild and $(x,y,\varphi)$46 lower ones; in Erez–Rosen, $(x,y,\varphi)$47 raises them and $(x,y,\varphi)$48 lowers them. Spectral luminosity displays a corresponding hardening for increasing $(x,y,\varphi)$49 in Kerr and increasing $(x,y,\varphi)$50 in Erez–Rosen, while decreasing $(x,y,\varphi)$51 softens the spectrum (Kurmanov et al., 23 Mar 2025).

A central result is the existence of strong degeneracies with Kerr. The Erez–Rosen/QM family can mimic Kerr flux profiles for suitable choices of $(x,y,\varphi)$52, and models can even be tuned to have the same $(x,y,\varphi)$53. Nonetheless, when the mass $(x,y,\varphi)$54 and quadrupole $(x,y,\varphi)$55 are matched by choosing different $(x,y,\varphi)$56 pairs consistent with

$(x,y,\varphi)$57

the resulting naked singularities can still show different flux normalizations and different radial profiles. The same study therefore proposes using combined observables—ISCO-informed efficiency, flux shape, spectral hardness, and independent quadrupole information—to distinguish Kerr black holes from rotating naked singularities, and it points to X-ray reflection spectroscopy, continuum-fitting, iron-line profiles, polarization, and timing via epicyclic frequencies as promising additional tests (Kurmanov et al., 23 Mar 2025).

6. Magnetized QM spacetime and the Penrose process

The magnetized analysis considers the QM spacetime immersed in a uniform external magnetic field $(x,y,\varphi)$58 aligned or anti-aligned with the symmetry axis, in the test-field regime $(x,y,\varphi)$59. Using Wald’s construction, the four-potential is taken as

$(x,y,\varphi)$60

with gauge $(x,y,\varphi)$61. Its nonzero components are

$(x,y,\varphi)$62

and the characteristic scale is

$(x,y,\varphi)$63

For a charged particle of mass $(x,y,\varphi)$64 and charge $(x,y,\varphi)$65,

$(x,y,\varphi)$66

Writing $(x,y,\varphi)$67, $(x,y,\varphi)$68, and $(x,y,\varphi)$69, the equatorial effective potential is

$(x,y,\varphi)$70

The negative-energy region is defined by $(x,y,\varphi)$71 (Zhang, 11 Jul 2025).

The qualitative behavior depends strongly on the product $(x,y,\varphi)$72. For $(x,y,\varphi)$73, negative-energy states require $(x,y,\varphi)$74 and occur only within the ergoregion; for large $(x,y,\varphi)$75, the negative-energy region inherits the ergoregion’s multi-lobe structure. For $(x,y,\varphi)$76, even relatively small values can push the negative-energy region outside the ergoregion; as $(x,y,\varphi)$77 increases it expands, can envelop the entire ergoregion, and for $(x,y,\varphi)$78 tends to simplify from a multi-lobe topology to a simply connected one. In that regime, negative energy may also occur for $(x,y,\varphi)$79, often far from the ergoregion and especially near the poles. For $(x,y,\varphi)$80, negative-energy states remain confined inside the ergoregion, require $(x,y,\varphi)$81, and the allowed region shrinks and may disappear as $(x,y,\varphi)$82 grows (Zhang, 11 Jul 2025).

The magnetic Penrose process is formulated by splitting a parent particle into two charged fragments at $(x,y,\varphi)$83, conserving four-momentum, charge, energy, and angular momentum. Energy extraction occurs when one fragment falls inward with $(x,y,\varphi)$84 and the escaping fragment satisfies $(x,y,\varphi)$85. The efficiency is

$(x,y,\varphi)$86

Under the maximizing approximations used there—$(x,y,\varphi)$87, $(x,y,\varphi)$88, $(x,y,\varphi)$89, $(x,y,\varphi)$90, $(x,y,\varphi)$91—this becomes

$(x,y,\varphi)$92

The first term is the purely geometric mechanical-Penrose contribution; the second is electromagnetic. Because $(x,y,\varphi)$93 and $(x,y,\varphi)$94 enter the conserved quantities, the condition $(x,y,\varphi)$95 can be satisfied even where $(x,y,\varphi)$96, allowing the magnetic Penrose process to operate far outside the classical ergoregion (Zhang, 11 Jul 2025).

The maximum efficiency exhibits a characteristic three-stage evolution as $(x,y,\varphi)$97 is varied: slow growth at moderately positive $(x,y,\varphi)$98, rapid increase across a transitional range, and slow growth again at sufficiently negative $(x,y,\varphi)$99. Without magnetic field, more negative $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$00 increases $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$01, and in the excised non-black-hole exterior the efficiency can exceed $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$02. In the electromagnetic-dominated regime the preference reverses: for fixed charge sign, $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$03 is maximized when $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$04, and larger positive $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$05 gives larger $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$06. The Kerr benchmark quoted there is $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$07 for extremal Kerr in the purely mechanical process, dropping below $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$08 for moderate spins. By contrast, in the magnetized QM spacetime the efficiency can become extremely large. For the neutron beta-decay channel $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$09, one has $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$10, so electromagnetic dominance sets in already for $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$11 in geometric units, with $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$12 and representative values of $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$13 at $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$14 or $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$15 for $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$16. The same analysis notes escaping proton energies

$$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$17

as a possible route to ultra-high-energy cosmic rays for sufficiently large $$ds^2=-f(dt-\omega\,d\varphi)^2+f^{-1}\left[e^{2\gamma}(d\rho^2+dz^2)+\rho^2 d\varphi^2\right],$$18 (Zhang, 11 Jul 2025).