Disformal transformation of stationary and axisymmetric solutions in modified gravity

The most general stationary and axisymmetric spacetime which possesses the two commuting Killing vector fields $\xi^{\mu}=(\partial/\partial t)^{\mu}$ and $\sigma^{\mu}=(\partial/\partial\varphi)^{\mu}$, satisfying $\nabla_{(\mu}\xi_{\nu)}=\nabla_{(\mu}\sigma_{\nu)}=0$ and $\left[\xi,\sigma\right]^{\nu}=\xi^{\mu}\partial_{\mu}\sigma^{\nu}-\sigma^{\mu}\partial_{\mu}\xi^{\nu}=0$, is given by

$\displaystyle ds^{2}=g_{\mu\nu}(r,\theta)dx^{\mu}dx^{\nu},$

(4)

where $x^{\mu}=(t,r,\theta,\varphi)$ represents the time, radial, polar angular, and azimuthal angular coordinates, respectively. The metric (4) further reduces to the simpler form, if the two Killing vector fields obey the integrability conditions Kundt and Trumper (1966); Carter (1969); Wald (1984)

$\displaystyle\sigma_{[\mu}\xi_{\nu}\nabla_{\alpha}\xi_{\beta]}=0,\qquad\sigma_{[\mu}\xi_{\nu}\nabla_{\alpha}\sigma_{\beta]}=0.$

(5)

Moreover, the conditions (5) are satisfied in the case that the so-called circularity conditions

$\displaystyle\xi^{\mu}R_{\mu}{}^{[\nu}\xi^{\alpha}\sigma^{\beta]}=0,\qquad\sigma^{\mu}R_{\mu}{}^{[\nu}\xi^{\alpha}\sigma^{\beta]}=0,$

(6)

are satisfied (see Appendix C), where $R_{\mu\nu}$ is the Ricci tensor associated with the metric $g_{\mu\nu}$. In the case that the circularity conditions are satisfied, there exist the two-surfaces orthogonal to the two Killing vector fields $\xi^{\mu}$ and $\sigma^{\mu}$, and then the general stationary and axisymmetric metric (4) reduces to the block-diagonal form

$\displaystyle ds^{2}=\left(-Adt^{2}+Bd\varphi^{2}+2Cdtd\varphi\right)+\left(Ddr^{2}+Ed\theta^{2}+2Fdrd\theta\right),$

(7)

where $A,B,C,D,E$, and $F$ are the functions of $r$ and $\theta$. An appropriate coordinate transformation on the two-surfaces allows us to set $F=0$, then Eq. (7) reduces to

$\displaystyle ds^{2}=\left(-Adt^{2}+Bd\varphi^{2}+2Cdtd\varphi\right)+\left(Ddr^{2}+Ed\theta^{2}\right).$

(8)

In general relativity, the vacuum spacetime where $R_{\mu\nu}=\lambda g_{\mu\nu}$ with $\lambda$ being constant and the spacetime filled with perfect fluids with four-velocities spanned by $\xi^{\mu}$ and $\sigma^{\mu}$ satisfy the circularity conditions (6). On the other hand, in the gravitational theories with the additional scalar or vector degrees of freedom, the absence of matter or the perfect fluid form of the matter energy-momentum tensor does not ensure the circularity conditions (6) and hence the integrability (5).

We assume that our starting gravitational theory possesses a stationary and axisymmetric solution given in the form of Eq. (8). In general relativity, it is well known that the Kerr solution is the unique, asymptotically flat, stationary, and axisymmetric black hole solution in vacuum Israel (1967, 1968); Carter (1971), which can be written into the form of Eq. (8). The parametrized ‘non-Kerr’ metrics Bambi (2012a, b); Konoplya and Zhidenko (2013); Glampedakis and Babak (2006); Johannsen and Psaltis (2010a, b); Vigeland et al. (2011); Medeiros et al. (2020) employed to test the Kerr hypothesis observationally also belong to the form of Eq. (8). In the next section, we will show that the disformal transformations, such as Eqs. (22), (35), and (41) below, map a stationary and axisymmetric metric solution satisfying the circularity conditions (8) to a noncircular one given by the general metric (4).

For a given stationary and axisymmetric spacetime (8) satisfying the circularity conditions (6), the most general Ansatz of the vector field is given by

$\displaystyle A_{\mu}dx^{\mu}=A_{t}(r,\theta)dt+A_{r}(r,\theta)dr+A_{\theta}(r,\theta)d\theta+A_{\varphi}(r,\theta)d\varphi,$

(9)

for which the norm of the vector field can be written as

$\displaystyle{\cal Y}:=g^{\mu\nu}A_{\mu}A_{\nu}=\frac{A_{\varphi}^{2}A-A_{t}^{2}B+2A_{t}A_{\varphi}C}{AB+C^{2}}+\frac{A_{r}^{2}}{D}+\frac{A_{\theta}^{2}}{E}.$

(10)

In the vector-tensor theories with the $U(1)$ gauge symmetry, there is an ambiguity about the gauge transformation $A_{\mu}\to A_{\mu}+\partial_{\mu}\chi$, where $\chi$ is an arbitrary function of the spacetime coordinates $(r,\theta)$.

A solution in a class of the vector-tensor theories with $F_{\mu\nu}=0$ can be mapped to that in a class of the shift-symmetric scalar-tensor theories with the scalar field $\phi$, via the substitution of $A_{\mu}\to\partial_{\mu}\phi$ and the integration of it Minamitsuji (2016). The conditions $F_{\mu\nu}=0$ for the Ansatz (9) lead to

$\displaystyle\partial_{r}A_{t}=0,\quad\partial_{\theta}A_{t}=0,\quad\partial_{r}A_{\theta}-\partial_{\theta}A_{r}=0,\quad\partial_{r}A_{\varphi}=0,\quad\partial_{\theta}A_{\varphi}=0,$

(11)

which are integrated as

$\displaystyle A_{t}=q,\qquad A_{\varphi}=m,\qquad A_{r}=\partial_{r}\psi(r,\theta),\qquad A_{\theta}=\partial_{\theta}\psi(r,\theta),$

(12)

where $q$ and $m$ are constants and $\psi$ is the function of $(r,\theta)$. Through the replacement of $A_{\mu}\to\partial_{\mu}\phi$ and its further integration, we obtain the most general form of the scalar field in the stationary and axisymmetric spacetime (8) in terms of the shift-symmetric scalar-tensor theories,

$\displaystyle\phi=\psi(r,\theta)+qt+m\varphi.$

(13)

The norm of the vector field ${\cal Y}$ is then promoted to the kinetic term of the scalar field

$\displaystyle{\cal X}:=g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi=\frac{m^{2}A-q^{2}B+2mqC}{AB+C^{2}}+\frac{(\partial_{r}\psi)^{2}}{D}+\frac{(\partial_{\theta}\psi)^{2}}{E}.$

(14)

In the case of the scalar-tensor theories without the shift symmetry, the explicit dependence on $t$ and $\varphi$ in Eq. (13) contradicts with the stationarity and axisymmetry of the spacetime, since the equations of motion explicitly would depend on $t$ and $\varphi$. Hence the most general Ansatz of the scalar field in the stationary and axisymmetric spacetimes is given by

$\displaystyle\phi=\psi(r,\theta),$

(15)

and hence, ${\cal X}=(\partial_{r}\psi)^{2}/D+(\partial_{\theta}\psi)^{2}/E$.

In the latter part of this paper, we will also focus on the disformal transformation of the slowly rotating solutions within the first order of the Hartle-Thorne approximation Hartle (1967); Hartle and Thorne (1968). In this approach, the metric functions $A$, $B$, $C$, $D$, and $E$ in Eq. (8) are expanded with respect to the small parameter $\epsilon(\ll 1)$ which is of the order of the dimensionless angular speed of the black hole rotation $M\Omega\sim J/M^{2}(\ll 1)$ with $M$ and $J$ being the mass and the angular momentum of the black hole, respectively. At ${\cal O}(\epsilon^{0})$, we prepare for the reference static and spherically symmetric solutions. At ${\cal O}(\epsilon^{1})$, the diagonal components of the metric $A$, $B$, $D$, and $E$ in Eq. (8) remain the same as those in the reference solutions and the pure function of $r$, and the leading slow rotation corrections appear only in the function $C$. Thus, the general metric Ansatz (8) reduces to the form of

$\displaystyle ds^{2}$

$\displaystyle=$

$\displaystyle g_{\mu\nu}dx^{\mu}dx^{\nu}=-A_{(0)}(r)dt^{2}+D_{(0)}(r)dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right)+2C_{(1)}(r,\theta)dtd\varphi+{\cal O}\left(\epsilon^{2}\right),$

(16)

with

$\displaystyle C_{(1)}(r,\theta):=-r^{2}\omega_{(1)}(r)\sin^{2}\theta,$

(17)

where $A_{(0)}(r)$ and $D_{(0)}(r)$ are of ${\cal O}(\epsilon^{0})$, and $\omega_{(1)}(r)={\cal O}(\epsilon)$ represents the frame-dragging function. The slow rotation corrections to the diagonal components $A$, $B$, $D$, and $E$ appear at ${\cal O}(\epsilon^{2})$, where we denote the $n$th order quantities of the slow rotation approximation by ${\cal O}(\epsilon^{n})$. Going beyond ${\cal O}(\epsilon)$ would be much more involved, as one has to solve for all the components of the metric and matter fields. Thus, for simplicity, we will focus only on the leading order slow rotation corrections of ${\cal O}(\epsilon)$.

On the other hand, the slow rotation corrections to $A_{t}$ and $A_{r}$ in Eq. (9) appear at ${\cal O}(\epsilon^{2})$, and $A_{\theta}$ in Eq. (9) is the quantity of ${\cal O}(\epsilon^{2})$. Thus, the general Ansatz of the vector field (9) reduces to

$\displaystyle A_{\mu}dx^{\mu}=A_{t(0)}(r)dt+A_{r(0)}(r)dr+A_{\varphi(1)}(r,\theta)d\varphi+{\cal O}\left(\epsilon^{2}\right),$

(18)

with

$\displaystyle A_{\varphi(1)}(r,\theta):=r^{2}a_{3(1)}(r)\sin^{2}\theta,$

(19)

where $A_{t(0)}(r)$ and $A_{r(0)}(r)$ are of ${\cal O}(\epsilon^{0})$, and $a_{3(1)}(r)$, which is of ${\cal O}(\epsilon)$, represents the radial profile of the magnetic component of the vector field Chagoya et al. (2016); Minamitsuji (2016).

In order to obtain a slowly rotating solution in a class of the vector-tensor theories, we substitute the Ansatz of Eqs. (16) and (18) into the action and then expand it with respect to $\epsilon$. At ${\cal O}(\epsilon^{0})$, the variations of the action with respect to $A_{(0)}(r)$, $D_{(0)}(r)$, $A_{t(0)}(r)$, and $A_{r(0)}(r)$ yield the Euler-Lagrange equations which give rise to the reference static and spherically symmetric solutions. Then, at ${\cal O}(\epsilon^{2})$, the variations with respect to $C_{(1)}$ and $A_{\varphi(1)}$ provide the coupled equations, which after the substitution of Eqs. (17) and (19) reduce to the equations to determine $\omega_{(1)}(r)$ and $a_{3(1)}(r)$.

A slowly rotating solution in a class of the vector-tensor theories with $F_{\mu\nu}=0$ can be mapped to that in a class of the shift-symmetric scalar-tensor theories. Up to ${\cal O}(\epsilon)$, the conditions $F_{\mu\nu}=0$ for the Ansatz (18) lead to

$\displaystyle\partial_{r}A_{t(0)}=0,\quad a_{3(1)}=0.$

(20)

After the replacement of $A_{\mu}\to\partial_{\mu}\phi$ and the integration of it, the Ansatz for the scalar field in the slow rotation limit of the stationary and axisymmetric solutions in the shift-symmetric scalar-tensor theories is obtained as Maselli et al. (2015).

$\displaystyle\phi=qt+\psi_{(0)}(r)+{\cal O}(\epsilon^{2}).$

(21)

In the scalar-tensor theories without the shift symmetry, we have to set $q=0$, otherwise the equations of motion would explicitly depend on the time. Thus, up to ${\cal O}(\epsilon)$, the scalar field remains the same as that in the static and spherically symmetric black hole solutions Babichev and Charmousis (2014); Appleby (2015); Maselli et al. (2015).

In order to obtain a slowly rotating solution in the scalar-tensor theories, we substitute the Ansatze (16) and (21) into the action and follow the same way as in the case of the vector-tensor theories. At ${\cal O}(\epsilon^{2})$, the variation with respect to $C_{(1)}$ provides the equation for it, which after the substitution of Eq. (17) reduces to the equation to determine $\omega_{(1)}(r)$.

We consider the disformal transformation in a vector-tensor theory

$\displaystyle{\tilde{g}}_{\mu\nu}=\mathcal{P}g_{\mu\nu}+\mathcal{Q}A_{\mu}A_{\nu},$

(22)

which explicitly breaks the $U(1)$ gauge invariance, where we assume that the dependence on $x^{\mu}$ in the functions $\mathcal{P}$ and $\mathcal{Q}$ comes through the norm of the vector field ${\cal Y}$ (see Eq. (10)), $\mathcal{P}:=\mathcal{P}({\cal Y})$ and $\mathcal{Q}:=\mathcal{Q}({\cal Y})$, so that the disformally transformed theory is covariant, where the disformal transformation (22) relates a class of degenerate higher-order vector-tensor theories to another Kimura et al. (2017). ^*2*2*2 In principle, we may consider more general disformal transformations which contain both $A_{\mu}A_{\nu}$, $g^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta}$, and the dependence on ${\cal Y}$ and ${\cal F}$ into $\mathcal{P}$ and $\mathcal{Q}$ in Eq. (22). Here, for simplicity, for the vector-tensor theories without the $U(1)$ gauge symmetry, we will focus on the transformation in the form of Eq. (22) which minimally breaks the $U(1)$ gauge symmetry. The inverse disformally mapped metric of Eq. (22) is given by

$\displaystyle{\tilde{g}}^{\mu\nu}=\frac{1}{\mathcal{P}}\left(g^{\mu\nu}-\frac{\mathcal{Q}}{\mathcal{P}+{\cal Y}\mathcal{Q}}A^{\mu}A^{\nu}\right).$

(23)

We choose the functions $\mathcal{P}$ and $\mathcal{Q}$ in Eq. (22), so that the disformal transformation (22) is invertible, namely, $\mathcal{P}+{\cal Y}\mathcal{Q}\neq 0$ and $\mathcal{P}\neq 0$ Kimura et al. (2017). As long as the disformal transformation (22) is invertible, any known solution $\left({g}_{\mu\nu},A_{\mu}\right)$ in a class of the vector-tensor theories ${S}\left[{g}_{\mu\nu},A_{\mu}\right]$ is uniquely mapped to a solution $\left(g_{\mu\nu},A_{\mu}\right)$ in a new class of the vector-tensor theories given by ${\tilde{S}}\left[{\tilde{g}}_{\mu\nu},A_{\mu}\right]=S\left[g_{\mu\nu}\left({\tilde{g}}_{\rho\sigma},A_{\rho}\right),A_{\mu}\right]$. The norm of the vector field for the disformed metric ${\tilde{g}}_{\mu\nu}$ is given by

$\displaystyle{\tilde{\cal Y}}:={\tilde{g}}^{\mu\nu}A_{\mu}A_{\nu}=\frac{{\cal Y}}{\mathcal{P}+{\cal Y}\mathcal{Q}}.$

(24)

In the case that ${\cal Y}={\cal Y}_{0}={\rm const}$, $\mathcal{P}({\cal Y}_{0})$, $\mathcal{Q}({\cal Y}_{0})$, and ${\tilde{\cal Y}}_{0}:=\frac{{\cal Y}_{0}}{\mathcal{P}({\cal Y}_{0})+{\cal Y}_{0}\mathcal{Q}({\cal Y}_{0})}$ are also constants.

Following Eq. (22), we find that the stationary and axisymmetric spacetime metric (8) is disformally mapped to

	$\displaystyle d{\tilde{s}}^{2}:={\tilde{g}}_{\mu\nu}dx^{\mu}dx^{\nu}$	$\displaystyle=$	$\displaystyle-\left(A\mathcal{P}-\mathcal{Q}A_{t}^{2}\right)dt^{2}+\left(\mathcal{P}B+\mathcal{Q}A_{\varphi}^{2}\right)d\varphi^{2}+2\left(\mathcal{P}C+\mathcal{Q}A_{t}A_{\varphi}\right)dtd\varphi$		(25)
		$\displaystyle+$	$\displaystyle\left(\mathcal{P}D+\mathcal{Q}A_{r}^{2}\right)dr^{2}+\left(\mathcal{P}E+\mathcal{Q}A_{\theta}^{2}\right)d\theta^{2}+2\mathcal{Q}A_{r}A_{\theta}drd\theta$
		$\displaystyle+$	$\displaystyle 2\mathcal{Q}\left(A_{t}A_{r}dtdr+A_{t}A_{\theta}dtd\theta+A_{\varphi}A_{r}drd\varphi+A_{\varphi}A_{\theta}d\theta d\varphi\right),$

where each component in the last line violates the circularity conditions (6). Imposing them leads to

$\displaystyle A_{t}A_{r}=0,\qquad A_{t}A_{\theta}=0,\qquad A_{\varphi}A_{r}=0,\qquad A_{\varphi}A_{\theta}=0,$

(26)

which further can be solved as

$\displaystyle A_{t}=A_{\varphi}=0,\qquad{\rm or}\qquad A_{r}=A_{\theta}=0.$

(27)

In the former and latter cases, the metric of Eq. (25) reduces to

$\displaystyle d{\tilde{s}}^{2}$

$\displaystyle=$

$\displaystyle-A\mathcal{P}dt^{2}+\mathcal{P}Bd\varphi^{2}+2\mathcal{P}Cdtd\varphi+\left(\mathcal{P}D+\mathcal{Q}A_{r}^{2}\right)dr^{2}+\left(\mathcal{P}E+\mathcal{Q}A_{\theta}^{2}\right)d\theta^{2}+2\mathcal{Q}A_{r}A_{\theta}drd\theta,$

(28)

and

$\displaystyle d{\tilde{s}}^{2}$

$\displaystyle=$

$\displaystyle-\left(A\mathcal{P}-\mathcal{Q}A_{t}^{2}\right)dt^{2}+\left(\mathcal{P}B+\mathcal{Q}A_{\varphi}^{2}\right)d\varphi^{2}+2\left(\mathcal{P}C+\mathcal{Q}A_{t}A_{\varphi}\right)dtd\varphi+\mathcal{P}Ddr^{2}+\mathcal{P}Ed\theta^{2}.$

(29)

The metrics (28) and (29) belong to the class of Eqs. (7) and (8), respectively.

After the disformal transformation (22), the metric of a slowly rotating solution (16) is given by

	$\displaystyle d{\tilde{s}}^{2}={\tilde{g}}_{\mu\nu}dx^{\mu}dx^{\nu}$	$\displaystyle=$	$\displaystyle-\left(\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}\right)dt^{2}+(\mathcal{P}_{(0)}D_{(0)}+\mathcal{Q}_{(0)}A_{r(0)}^{2})dr^{2}+2\mathcal{Q}_{(0)}A_{t(0)}A_{r(0)}dtdr$		(30)
		$\displaystyle+$	$\displaystyle r^{2}\mathcal{P}_{(0)}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right)-2r^{2}\sin^{2}\theta\left[\left(\mathcal{P}_{(0)}\omega_{(1)}-\mathcal{Q}_{(0)}a_{3(1)}A_{t(0)}\right)dt-\mathcal{Q}_{(0)}a_{3(1)}A_{r(0)}dr\right]d\varphi$
		$\displaystyle+$	$\displaystyle{\cal O}(\epsilon^{2}),$

where since ${\cal Y}={\cal Y}_{(0)}(r)+{\cal O}(\epsilon^{2})$ with ${\cal Y}_{(0)}(r):=A_{r(0)}^{2}/D_{(0)}-A_{t(0)}^{2}/A_{(0)}$,

	$\displaystyle\mathcal{P}({\cal Y})$	$\displaystyle=$	$\displaystyle\mathcal{P}({\cal Y}_{(0)}(r))+{\cal O}(\epsilon^{2})=:\mathcal{P}_{(0)}(r)+{\cal O}(\epsilon^{2}),$
	$\displaystyle\mathcal{Q}({\cal Y})$	$\displaystyle=$	$\displaystyle\mathcal{Q}({\cal Y}_{(0)}(r))+{\cal O}(\epsilon^{2})=:\mathcal{Q}_{(0)}(r)+{\cal O}(\epsilon^{2}).$		(31)

Introducing the new time coordinate by

$\displaystyle dt=d{\tilde{t}}+\frac{\mathcal{Q}_{(0)}A_{t(0)}A_{r(0)}}{\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}}dr,$

(32)

the disformed metric (30) is rewritten as

	$\displaystyle d{\tilde{s}}^{2}$	$\displaystyle=$	$\displaystyle-\left(\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}\right)d{\tilde{t}}^{2}+\mathcal{P}_{(0)}\left[D_{(0)}+\frac{\mathcal{Q}_{(0)}A_{(0)}A_{r(0)}^{2}}{\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}}\right]dr^{2}+r^{2}\mathcal{P}_{(0)}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right)$		(33)
		$\displaystyle-$	$\displaystyle 2r^{2}\sin^{2}\theta\left[\left(\mathcal{P}_{(0)}\omega_{(1)}-\mathcal{Q}_{(0)}a_{3(1)}A_{t(0)}\right)d{\tilde{t}}-\mathcal{P}_{(0)}\mathcal{Q}_{(0)}A_{r(0)}\frac{A_{(0)}a_{3(1)}-\omega_{(1)}A_{t(0)}}{\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}}dr\right]d\varphi+{\cal O}(\epsilon^{2}).$

The existence of the nonzero ${\tilde{g}}_{r\varphi}\neq 0$ in Eqs. (30) and (33) indicates the violation of the circularity conditions, which explicitly appears at ${\cal O}(\epsilon^{2})$. Up to ${\cal O}(\epsilon)$, the position of the black hole event horizon in the disformed frame is determined by a positive root of $\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}=0$. The apparent divergence of $g_{r\varphi}$ in Eq. (33) at the point where $\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}=0$ is just a coordinate artifact.

In the case that the reference static and spherically symmetric solution is given by a black hole spacetime and has an event horizon at which $A_{(0)}=D_{(0)}^{-1}=0$, after the disformal transformation, the position of the event horizon is modified to the position which is fixed by the algebraic equation $\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}=0$. Since the number of the positive roots of the equation $\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}=0$ may be different from that of the original equation $A_{(0)}=D_{(0)}^{-1}=0$, the number of the horizons may be modified before and after the disformal transformation. The shift of the position of the event horizon happens in the case that the character the vector field is timelike with nonzero $A_{t(0)}\neq 0$, which will be discussed in Secs. IV and V.

The solution of the vector field rewritten in terms of the time coordinate defined in the disformed frame (32) is given by

$\displaystyle A_{\mu}dx^{\mu}=A_{t(0)}d{\tilde{t}}+\frac{\mathcal{P}_{(0)}A_{(0)}}{\mathcal{P}_{(0)}A_{(0)}-\mathcal{Q}_{(0)}A_{t(0)}^{2}}A_{r(0)}dr+{\cal O}(\epsilon),$

(34)

which will be employed to investigate the regularity of the vector field at the event and cosmological horizons of the disformed solutions.

When we consider the vector-tensor theories with the $U(1)$ gauge symmetry, instead of the vector disformal transformation (22), we consider the disformal transformation with the $U(1)$ gauge symmetry given by

$\displaystyle{\tilde{g}}_{\mu\nu}=\mathcal{R}g_{\mu\nu}+\mathcal{W}g^{\alpha\beta}F_{\mu\alpha}F_{\nu\beta},$

(35)

where $\mathcal{R}=\mathcal{R}(\mathcal{F},\mathcal{G})$ and $\mathcal{W}=\mathcal{W}(\mathcal{F},\mathcal{G})$ with

$\displaystyle\mathcal{F}:=g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma},\qquad\mathcal{G}:=g^{\mu\nu}g^{\alpha\beta}g^{\rho\sigma}g^{\gamma\delta}F_{\mu\alpha}F_{\rho\beta}F_{\sigma\gamma}F_{\nu\delta}.$

(36)

We choose the functions $\mathcal{R}$ and $\mathcal{W}$, so that the disformal transformation (35) is invertible Gumrukcuoglu and Namba (2020). Ref. Gumrukcuoglu and Namba (2020) explored the vector-tensor theories which are related to the Einstein-Maxwell theory under the disformal transformation (35). Ref. De Felice and Naruko (2020) showed that Eq. (35) corresponds to the most general disformal transformation with the $U(1)$ gauge symmetry which is built up with $F_{\mu\nu}$ and its dual tensor $F^{\ast}_{\mu\nu}:=\left(1/2\right)\epsilon_{\mu\alpha\nu\beta}g^{\alpha\rho}g^{\beta\sigma}F_{\rho\sigma}$, where $\epsilon_{\alpha\beta\gamma\delta}$ denotes the volume element. As long as the disformal transformation (35) is invertible, a solution $\left(g_{\mu\nu},F_{\mu\nu}\right)$ in a class of the vector-tensor theories with the $U(1)$ gauge symmetry ${S}\left[{g}_{\mu\nu},F_{\mu\nu}\right]$ is uniquely mapped to a solution $\left({\tilde{g}}_{\mu\nu},F_{\mu\nu}\right)$ in a new class obtained by ${\tilde{S}}\left[{\tilde{g}}_{\mu\nu},F_{\mu\nu}\right]=S\left[g_{\mu\nu}\left({\tilde{g}}_{\rho\sigma},F_{\rho\sigma}\right),F_{\mu\nu}\right]$.

Under the disformal transformation (35), the metric of the stationary and axisymmetric spacetime (8) with the vector field (9) is mapped as

	$\displaystyle d{\tilde{s}}^{2}={\tilde{g}}_{\mu\nu}dx^{\mu}dx^{\nu}$	$\displaystyle=$	$\displaystyle-\left(\mathcal{R}A-\frac{WA_{t,r}^{2}}{D}-\frac{WA_{t,\theta}^{2}}{E}\right)dt^{2}$		(37)
		$\displaystyle+$	$\displaystyle\left(\mathcal{R}B+\frac{\mathcal{W}A_{\varphi,r}^{2}}{D}+\frac{\mathcal{W}A_{\varphi,\theta}^{2}}{E}\right)d\varphi^{2}+2\left(\mathcal{R}C+\frac{\mathcal{W}}{D}A_{t,r}A_{\varphi,r}+\frac{\mathcal{W}}{E}A_{t,\theta}A_{\varphi,\theta}\right)dtd\varphi$
		$\displaystyle+$	$\displaystyle\left[\mathcal{R}D+\frac{\mathcal{W}}{E}\left(A_{r,\theta}-A_{\theta,r}\right)^{2}+\frac{\mathcal{W}}{AB+C^{2}}\left(-BA_{t,r}^{2}+A_{\varphi,r}\left(2CA_{t,r}+AA_{\varphi,r}\right)\right)\right]dr^{2}$
		$\displaystyle+$	$\displaystyle\left[\mathcal{R}E+\frac{\mathcal{W}}{D}\left(A_{r,\theta}-A_{\theta,r}\right)^{2}+\frac{\mathcal{W}}{AB+C^{2}}\left(-BA_{t,\theta}^{2}+A_{\varphi,\theta}\left(2CA_{t,\theta}+AA_{\varphi,\theta}\right)\right)\right]d\theta^{2}$
		$\displaystyle+$	$\displaystyle\frac{2\mathcal{W}}{AB+C^{2}}\left[-BA_{t,\theta}A_{t,r}+AA_{\varphi,\theta}A_{\varphi,r}+C\left(A_{\varphi,\theta}A_{t,r}+A_{\varphi,r}A_{t,\theta}\right)\right]drd\theta$
		$\displaystyle-$	$\displaystyle 2\mathcal{W}\left(A_{\theta,r}-A_{r,\theta}\right)\left(\frac{A_{t,\theta}}{E}dtdr-\frac{A_{t,r}}{D}dtd\theta+\frac{A_{\varphi,\theta}}{E}drd\varphi-\frac{A_{\varphi,r}}{D}d\theta d\varphi\right),$

where each component in the last line violates the circularity conditions (6). Imposing the circularity conditions requires the vanishing azimuthal component of the magnetic field

$\displaystyle F_{r\theta}=A_{\theta,r}-A_{r,\theta}=0.$

(38)

The metric of a slowly rotating solution (16) with the vector field (18) is disformally mapped to

	$\displaystyle d{\tilde{s}}^{2}={\tilde{g}}_{\mu\nu}dx^{\mu}dx^{\nu}$	$\displaystyle=$	$\displaystyle\left(\mathcal{R}_{(0)}-\frac{\mathcal{W}_{(0)}A_{t(0),r}{}^{2}}{A_{(0)}D_{(0)}}\right)\left(-A_{(0)}dt^{2}+D_{(0)}dr^{2}\right)+\mathcal{R}_{(0)}r^{2}\left(d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right)$		(39)
		$\displaystyle-$	$\displaystyle 2r^{2}\sin^{2}\theta\left(\mathcal{R}_{(0)}\omega_{(1)}-\frac{\mathcal{W}_{(0)}(ra_{3(1),r}+2a_{3(1)})A_{t(0),r}}{rD_{(0)}}\right)dtd\varphi+{\cal O}(\epsilon^{2}),$

where since $\mathcal{F}=\mathcal{F}_{(0)}(r)+{\cal O}(\epsilon^{2})$ and $\mathcal{G}=\mathcal{G}_{(0)}+{\cal O}(\epsilon^{2})$ with $\mathcal{F}_{(0)}(r):=-2\left(A_{t(0),r}\right)^{2}/\left(A_{(0)}D_{(0)}\right)$ and $\mathcal{G}_{(0)}(r):=2\left(A_{t(0),r}\right)^{4}/\left(A_{(0)}D_{(0)}\right)^{2}$,

	$\displaystyle\mathcal{R}(\mathcal{F})$	$\displaystyle=$	$\displaystyle\mathcal{R}\left(\mathcal{F}_{(0)}(r),\mathcal{G}_{(0)}(r)\right)+{\cal O}(\epsilon^{2})=:\mathcal{R}_{(0)}(r)+{\cal O}(\epsilon^{2}),$
	$\displaystyle\mathcal{W}(\mathcal{F})$	$\displaystyle=$	$\displaystyle\mathcal{W}\left(\mathcal{F}_{(0)}(r),\mathcal{G}_{(0)}(r)\right)+{\cal O}(\epsilon^{2})=:\mathcal{W}_{(0)}(r)+{\cal O}(\epsilon^{2}).$		(40)

Since $F_{r\theta}={\cal O}(\epsilon^{2})$, at ${\cal O}(\epsilon)$ all the components in the last line of Eq. (37) vanish. Thus, under the $U(1)$-invariant disformal transformation (35), up to ${\cal O}(\epsilon)$, the position of the black hole event horizon is not modified, and hence given by a solution of $A_{(0)}(r)=D_{(0)}(r)^{-1}=0$. This is in the contrast with the case of the disformal transformation without the $U(1)$ gauge symmetry (35), where the disformal transformation can modify the position of the black hole event horizon even at the leading order of the slow rotation approximation. We note that beyond the slow rotation approximation the position of the event horizon (and the other horizons) may be modified. In Sec. VII, we will check this point explicitly with the disformed Kerr-Newman solution.