--- title: Accelerating Kerr–Newman–NUT–de Sitter Spacetime url: https://www.emergentmind.com/topics/accelerating-kerr-newman-nut-de-sitter-spacetime type: topic --- # Accelerating Kerr–Newman–NUT–de Sitter Spacetime Accelerating Kerr–Newman–NUT–de Sitter spacetime denotes exact Einstein–Maxwell solutions with cosmological constant that combine black-hole mass, electric charge, rotation, uniform acceleration, and gravitomagnetic mass. In recent formulations, these geometries appear as Plebański–Demiański-class or closely related spacetimes: a type D accelerating Kerr–Newman–NUT–(A)dS branch with aligned Maxwell field [2404.06551], a generalized accelerating Kerr–Newman–NUT–de Sitter solution with parameters $(M,Q,a,\alpha,\lambda;S_1,S_2,S_3)$ [2509.16647], and an Ehlers-generated accelerating double-NUT family for which the full $\Lambda\neq0$ type I extension is not constructed, although its $c=0$ type D Plebański–Demiański subclass with $\Lambda>0$ is explicit [2305.03744]. ## 1. Metric structure and parameterizations A generalized accelerating Kerr–Newman–NUT–de Sitter solution is written in coordinates $(t,r,\theta,\phi)$ as $$ \begin{aligned} ds^2 &= \frac{1}{\Omega^2}\Bigg\{ -\frac{\Sigma - \delta\,a^2\sin^2\theta}{\Delta}\,dt^2 +\frac{\Delta}{\Sigma}\,dr^2 +\frac{\Delta}{\delta}\,d\theta^2 -\frac{2a\sin^2\theta\left[\delta\left(r^2+a^2\right)-\Sigma\right]}{\Delta}\,dt\,d\phi \\ &\quad +\frac{\left[\delta\left(r^2+a^2\right)^2-\Sigma\,a^2\sin^2\theta\right]\sin^2\theta}{\Delta}\,d\phi^2 \Bigg\}, \end{aligned} $$ with $$ \Delta = r^2 + a^2\cos^2\theta,\qquad \Omega = 1 - \alpha\,r\cos\theta, $$ $$ \delta(\theta) = 1 + \frac{\lambda}{3}\,a^2\cos^2\theta +\frac{S_1 + S_3\cos\theta - S_2\cos^2\theta}{\sin^2\theta} +\frac{2\alpha M\cos^3\theta}{\sin^2\theta} -\frac{\alpha^2\,(a^2 S_1 + a^2 + Q^2)\cos^4\theta}{\sin^2\theta}, $$ and $$ \Sigma(r,\theta) = (r^2+a^2)\Big(1 - \tfrac{\lambda}{3} r^2\Big) - 2Mr + Q^2 + S_1 a^2 + S_2 r^2 - S_3\,\alpha r^3 - \alpha^2(1+S_1)\,r^4. $$ Here $M$ is the physical mass, $Q$ the electric charge, $a$ the specific angular momentum, $\alpha$ the acceleration parameter, and $\lambda$ the cosmological constant parameter, with $\Lambda=3\lambda$. The parameter $S_3$ maps to the NUT charge through $$ n=-\frac{a}{2}S_3, $$ $S_1$ plays the role of a rescaled conicity parameter $C$ for Kerr–NUT, and $S_2$ is a new dimensionless parameter which has the same physical meaning of $C$ in the weak field limit but differs from $C$ in general [2509.16647]. A complementary formulation uses coordinates $(t,r,x,\phi)$, with $x$ a latitude-like coordinate, and presents the spacetime in Lewis–Weyl–Papapetrou form with commuting Killing fields $\partial_t$ and $\partial_\phi$. In the $\Lambda=0$ baseline, the radial and angular structure functions are $$ \Delta_r(r)=(1-\alpha^2 r^2)\big[(r-m)^2-\sigma^2\big],\qquad \Delta_x(x)=(1-x^2)\big[(1+m\alpha x)^2-\alpha^2x^2\sigma^2\big], $$ with $$ \sigma=\sqrt{m^2+\ell^2-a^2-e^2-p^2}. $$ The cosmological constant is included by polynomial shifts of $\Delta_r$ and $\Delta_x$; for $\Lambda>0$ these shifts introduce a cosmological horizon and modify the positions of the inner and outer horizons compared to the $\Lambda=0$ case [2404.06551]. ## 2. Einstein–Maxwell sector In the generalized de Sitter solution, the Maxwell potential is purely electric, $$ A_\mu dx^\mu = A_t\,dt + A_\phi\,d\phi,\qquad A_t(r,\theta)=\frac{Qr}{\Delta},\qquad A_\phi(r,\theta)=-a\sin^2\theta\,A_t, $$ so that $$ F=dA. $$ The nonvanishing coordinate-basis components listed explicitly are $$ F_{rt}=Q\,\frac{\Delta-2r^2}{\Delta^2},\qquad F_{\theta t}=2Qr\,\frac{a^2\cos\theta\sin\theta}{\Delta^2}, $$ $$ F_{r\phi}=-a\sin^2\theta\,\partial_r A_t,\qquad F_{\theta\phi}=-2a\sin\theta\cos\theta\,A_t-a\sin^2\theta\,\partial_\theta A_t. $$ The field equations are $$ G_{\mu\nu} = -2F_{\mu\alpha}F_{\nu}{}^{\alpha} + \frac{1}{2}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta} + \lambda\,g_{\mu\nu},\qquad \Lambda=3\lambda, $$ together with $\nabla_\mu F^{\mu\nu}=0$ [2509.16647]. A different accelerating Kerr–Newman–NUT–(A)dS branch carries both electric and magnetic monopole charges. There the vector potential has the form $$ A_\mu dx^\mu=A_t(r,x)\,dt+A_\phi(r,x)\,d\phi, $$ with $F=dA$, and the Maxwell field is aligned with the two expanding repeated principal null directions of the Weyl tensor. The parameters $e$ and $p$ are the electric and magnetic monopole charges, respectively [2404.06551]. The Ehlers-based construction supplies a further distinction between the intrinsic NUT parameter $\ell$ of the seed and a second, independent NUT parameter encoded by the Ehlers parameter $c$. In the enhanced Ehlers map, $$ (\bar{M},\bar{B})=\left(\frac{(1-c^2)M-2 B c}{1+c^2},\,\frac{(1-c^2)B + 2 M c}{1+c^2}\right), $$ while $(Q_e,Q_m)$ are unchanged asymptotically. The full double-NUT, type I accelerated family is not given with $\Lambda$, because the symmetry of the Ernst equations is broken when $\Lambda\neq0$ [2305.03744]. ## 3. Horizons, thermodynamic quantities, and curvature singularities In the generalized de Sitter solution, horizons are Killing horizons of $$ K^\mu=\xi^\mu+\Omega_H\phi^\mu,\qquad \xi\equiv\partial_t,\quad \phi\equiv\partial_\phi, $$ and are determined by $$ \Sigma(r,\theta)=0 $$ viewed as a quartic equation in $r$ at fixed $\theta$. Generically there can be four positive roots, $$ r_10$, the additional real root of the modified $\Delta_r$ gives a cosmological horizon, and the acceleration horizon is no longer exactly $r_\alpha=1/\alpha$ once the $\Lambda$-shifts are included [2404.06551]. ## 4. Axial structure, Misner strings, and conical defects The NUT parameter introduces string-like axial defects. In the generalized accelerating Kerr–Newman–NUT–de Sitter solution, the axis behavior is governed by the $\sin^{-2}\theta$ terms in $\delta(\theta)$. Near the north and south poles, the numerators are $$ \mathcal{N}_N=S_1+S_3-S_2+2\alpha M-\alpha^2(a^2S_1+a^2+Q^2), $$ $$ \mathcal{N}_S=S_1-S_3-S_2-2\alpha M-\alpha^2(a^2S_1+a^2+Q^2). $$ If these coefficients do not vanish, $\delta$ diverges like $1/\sin^2\theta$ and the axis is singular. One can tune parameters to cancel the divergence on one axis and obtain a purely conical behavior there, but in an accelerating spacetime it is generically impossible to cancel both simultaneously. The parameter $S_1$ acts like the rescaled conicity parameter $C$ and controls whether the north or south axis carries a Misner string, while $S_2$ measures a conical defect in the static limit [2509.16647]. In that static limit, the metric reduces to $$ ds^2 = -\left(1-\frac{2M}{(1+S_2)\,r}\right)dt^2 + \left(1-\frac{2M}{(1+S_2)\,r}\right)^{-1}dr^2 + r^2\left(d\chi^2 + (1+S_2)\sin^2\chi\,d\phi^2\right), $$ which is a Schwarzschild black hole threaded by a cosmic string. The conical deficit and string tension are $$ \delta_0=2\pi\left(1-\sqrt{1+S_2}\right),\qquad \mu=\frac{\delta_0}{8\pi}=\frac{1}{4}\left(1-\sqrt{1+S_2}\right), $$ in units with $8\pi G=1$ [2509.16647]. The Ehlers-generated double-NUT accelerating family exhibits a different mechanism for axial regularity. There the jump of the rotation function is $$ \Delta\omega=\lim_{x\to1}\omega-\lim_{x\to-1}\omega=-4\ell+4c(2m+c\ell), $$ and the regularity condition $$ \Delta\omega=0 $$ is equivalent, after reparameterization, to $$ n=\ell. $$ The paper states that tuning $n=\ell$ removes the axial discontinuity in $\omega$ and $g_{t\varphi}$ without imposing periodic time, provided $a\neq0$ and $\alpha\neq0$. However, the double-NUT type I extension with $\Lambda$ is not constructed there [2305.03744]. In the type D accelerating branch with $\Lambda=0$, the exact difference of $\omega$ across the axes is $$ \Delta\omega=-\frac{4\ell}{1-a^2\alpha^2+\ell^2\alpha^2}, $$ so the only clean way to remove the Misner string globally is $\ell=0$. The $\Lambda\neq0$ case modifies that condition and constrains the allowed integration constants [2404.06551]. ## 5. Limiting procedures and relation to known families Several standard limits organize the accelerating Kerr–Newman–NUT–de Sitter family. In the generalized solution, $\alpha\to0$ removes acceleration and yields a generalized Kerr–Newman–(NUT)–(A)dS geometry with $$ \Sigma(\alpha=0)=(r^2+a^2)\left(1-\frac{\lambda}{3}r^2\right)-2Mr+Q^2+S_1a^2+S_2r^2. $$ The limit $S_3\to0$ removes the NUT charge and Misner strings; $Q\to0$ yields an accelerating Kerr–NUT–de Sitter solution; $a\to0$ yields an accelerating Reissner–Nordström–NUT–de Sitter solution; and $\lambda\to0$ gives a spacetime that is asymptotically flat only when $\alpha=0$. In the weak-field regime, with $S_2=-1+b^2$, comparison with Kerr–NUT gives $$ b^4=1-2a\,\tilde C, $$ so $S_2$ plays the role of the conicity parameter in the weak-field limit, but beyond weak fields it modifies both radial and angular functions and mass normalization [2509.16647]. The type D accelerating Kerr–Newman–NUT–(A)dS branch has straightforward specializations. The limit $a\to0$ gives the accelerating Reissner–Nordström–NUT–(A)dS metric, and the paper states that this branch was unknown in the literature before that work. The further limit $\ell\to0$ yields the standard accelerating Reissner–Nordström–(A)dS C-metric, while $\alpha\to0$ in the $a=0$ sector gives Reissner–Nordström–NUT–(A)dS [2404.06551]. The Ehlers-generated family clarifies the relation between the algebraically general double-NUT branch and the type D Plebański–Demiański sector. Switching off the Ehlers NUT by setting $c=0$, equivalently $n=2\ell$ in the $n$-parametrization, recovers a convenient type D Plebański–Demiański black hole. With $\Lambda>0$, the paper gives the full type D Plebański–Demiański black hole in the same parametrization, where $\Lambda$ enters only in $\Delta_r$ and $\Delta_x$; the gauge field remains as in the $c=0$ case [2305.03744]. ## 6. Algebraic classification, interpretation, and open issues The generalized accelerating Kerr–Newman–NUT–de Sitter spacetime is described as a member of the Plebański–Demiański class and is algebraically special of Petrov type D. Standard separability structures familiar from Kerr–(A)dS are said to persist in this broader class, but they are not explicitly analyzed. The NUT parameter $n=-aS_3/2$ introduces Misner strings, and in Kerr–NUT spacetimes the usual removal by periodic identification of $t$ with period $8\pi n$ introduces closed timelike curves [2509.16647]. The type D accelerating Kerr–Newman–NUT–(A)dS branch is explicitly checked to satisfy the invariant relation $$ I^3=27J^2, $$ with two expanding repeated principal null directions and an aligned Maxwell field. The paper emphasizes that standard Plebański–Demiański parameterizations split into disjoint accelerating and NUTty branches, so that turning off angular momentum tends to kill acceleration in the NUT sector; the construction under discussion supplies the missing branch with $a=0$, $\ell\neq0$, and $\alpha\neq0$ [2404.06551]. The Ehlers-generated double-NUT family has a different algebraic status. The full nine-parameter accelerating Kerr–Newman family with two NUT charges is Petrov type I rather than type D, and the two NUT parameters are interpreted as an intrinsic black-hole NUT $\ell$ and an Ehlers/background NUT $n$ associated with the accelerating Rindler background. The same work states that accelerating single-black-hole metrics arise as a limit of binary systems in which the second horizon is pushed to infinity, leaving a Rindler or Rindler–NUT background [2305.03744]. Thermodynamics remains only partially settled in the most general accelerating, NUT-charged, de Sitter setting. A consistent Smarr relation and first law are established in the special case $$ \alpha=0,\qquad \lambda=0,\qquad S_2=S_1, $$ for which $$ M=2TS+2\Omega_HJ+Q\Phi_H,\qquad dM=T\,dS+\Omega_H\,dJ+\Phi_H\,dQ. $$ In the generic accelerating case, the angle-dependent $\Omega_\infty$ and axial defects are identified as the main obstacle to a consistent first law including rotation at infinity and string or strut tensions [2509.16647]. Source: https://www.emergentmind.com/topics/accelerating-kerr-newman-nut-de-sitter-spacetime