Ultrarelativistic spinning objects in non-local ghost-free gravity

Our goal is to obtain the metric for an ultrarelativistic spinning object (gyraton) in infinite-derivative gravity. The method of solving the problem is the following. First one finds a solution of the gravitational field equations in the object’s rest frame. Then one transforms this solution to a new reference frame moving with a constant velocity $v$ with respect to the original one. Finally, one takes the limit $v\to 1$ while keeping the energy $E=m\gamma=m/\sqrt{1-v^{2}}$ fixed. In this Penrose limit the original mass $m$ of the object effectively approaches zero, which implies that in order to obtain the corresponding gyraton metric one may start with a solution with very small mass $m$. One can expect that higher-order curvature corrections—which are proportional to second and higher orders in the mass $m$—are therefore small. In this section we will discuss the field equations of linearized infinite-derivative gravity and present their solutions for an extended, slowly spinning object of small mass.

We denote by $X^{\mu}=(t,x^{\alpha})$ Cartesian coordinates in $(d+1)$-dimensional Minkowski spacetime and use indices $\alpha,\beta,\ldots=1,2,\ldots,d$ from the beginning of the Greek alphabet to label spatial coordinates. The Minkowski metric in the $D=d+1$ dimensional spacetime is

$\displaystyle\mbox{d}s_{0}^{2}=\eta_{\mu\nu}\mbox{d}X^{\mu}\mbox{d}X^{\nu}=-\mbox{d}t^{2}+\delta_{\alpha\beta}\mbox{d}x^{\alpha}\mbox{d}x^{\beta}\,,$

(1)

where $\delta{}_{\alpha\beta}$ denotes the flat $d$-dimensional metric. We denote by $h{}_{\mu\nu}$ a small deviation of the metric from the flat background,

$\displaystyle g{}_{\mu\nu}=\eta{}_{\mu\nu}+h{}_{\mu\nu}\,.$

(2)

One can show that the most general linearized action in a Lorentz invariant theory with an arbitrary number of derivatives and quadratic in the perturbation $h_{\mu\nu}$ can be written in the form Biswas et al. (2012)

	$\displaystyle S$	$\displaystyle=\frac{1}{2\kappa}\int\mbox{d}^{D}x\Big{(}\frac{1}{2}h^{\mu\nu}\,a(\Box)\Box\,h_{\mu\nu}-h^{\mu\nu}\,a(\Box)\partial_{\mu}\partial_{\alpha}\,h^{\alpha}{}_{\nu}$
		$\displaystyle\hskip 75.0pt+h^{\mu\nu}\,c(\Box)\partial_{\mu}\partial_{\nu}h-\frac{1}{2}h\,c(\Box)\Box h$		(3)
		$\displaystyle\hskip 75.0pt+\frac{1}{2}h^{\mu\nu}\,\frac{a(\Box)-c(\Box)}{\Box}\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}\,h^{\alpha\beta}\Big{)}\,,$

where $\Box$ is the d’Alembert operator of Minkowski space, $\Box=\eta{}^{\mu\nu}\partial_{\mu}\partial_{\nu}$. The functions $a(\Box)$ and $c(\Box)$ can be chosen freely to parametrize different Lorentz-invariant modifications of gravity, subject only to the constraint

$\displaystyle a(0)=c(0)=1$

(4)

which guarantees the proper Newtonian limit; see also the related discussions in Refs. Biswas et al. (2012); Buoninfante et al. (2019b). In the case of $a(\Box)=c(\Box)=1$ one recovers the Fierz–Pauli action and linearized General Relativity.

The field equations corresponding to the action (3) are

$\displaystyle\begin{split}&\hskip 11.0pta(\Box)\big{[}\Box\,h_{\mu\nu}-\partial_{\sigma}\big{(}\partial_{\nu}\,h_{\mu}{}^{\sigma}+\partial_{\mu}h_{\nu}{}^{\sigma}\big{)}\big{]}\\ &+c(\Box)\big{[}\eta_{\mu\nu}\big{(}\partial_{\rho}\partial_{\sigma}h^{\rho\sigma}-\Box h\big{)}+\partial_{\mu}\partial_{\nu}h\big{]}\\ &+{a(\Box)-c(\Box)\over\Box}\partial_{\mu}\partial_{\nu}\partial_{\rho}\partial_{\sigma}h^{\rho\sigma}=-2\kappa T_{\mu\nu}\,,\end{split}$

(5)

where $T{}_{\mu\nu}$ is the energy-momentum tensor of matter, and $h=\eta{}^{\alpha\beta}h{}_{\alpha\beta}$ denotes the trace of $h{}_{\mu\nu}$. From now on we shall restrict ourselves to the case of

$\displaystyle c(\Box)=a(\Box)\,.$

(6)

This condition guarantees that no extra scalar modes are present in the theory Biswas et al. (2012). We denote

$\displaystyle\hat{h}_{\mu\nu}=h{}_{\mu\nu}-\frac{1}{2}h\eta{}_{\mu\nu}\,.$

(7)

The inverse transformation is

$\displaystyle h{}_{\mu\nu}=\hat{h}{}_{\mu\nu}-\frac{1}{d-1}\hat{h}\eta{}_{\mu\nu}\,.$

(8)

We also impose the gauge conditions $\partial{}_{\mu}\hat{h}{}^{\mu\nu}=0$. Then, Eq. (LABEL:EQN) simplifies greatly and takes the form

$\displaystyle a(\Box)\Box\hat{h}{}_{\mu\nu}$

$\displaystyle=-2\kappa T_{\mu\nu}\,.$

(9)

The conservation law $\partial{}_{\mu}T^{\mu\nu}=0$ implies that the imposed gauge conditions are consistent.

We assume that $T_{\mu\nu}$ does not depend on time. For a stationary metric generated by such a stress-energy tensor the $\Box$-operator reduces to the $d$-dimensional Laplace operator $\triangle=\delta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$. We denote

$\displaystyle\mathcal{D}=a(\triangle)\triangle\,.$

(10)

Then, we can solve the field equations (9) by using the static Green function

$\displaystyle\mathcal{D}\mathcal{G}_{d}({\boldsymbol{x}},{\boldsymbol{x^{\prime}}})=-\delta{}^{(d)}({\boldsymbol{x}}-{\boldsymbol{x^{\prime}}})\,.$

(11)

The solution then takes the form

$\displaystyle\hat{h}{}_{\mu\nu}({\boldsymbol{x}})=2\kappa\int\mbox{d}^{d}y\,\mathcal{G}_{d}({\boldsymbol{x}}-{\boldsymbol{y}})T{}_{\mu\nu}({\boldsymbol{y}})\,.$

(12)

The expression for the perturbation of the metric $h_{\mu\nu}$ can be found from (12) by using relation (8). For the stress-energy tensor (87) given in the Appendix one has

$\displaystyle T_{\mu\nu}=\rho({\boldsymbol{x}})\delta^{t}_{\mu}\delta^{t}_{\nu}+\delta^{t}_{(\mu}\delta^{\alpha}_{\nu)}{\partial\over\partial x^{\beta}}j_{\alpha}{}^{\beta}({\boldsymbol{x}})\,.$

(13)

A solution $h_{\mu\nu}$ of the field equations (9) for this source can be written as follows:

	$\displaystyle{\boldsymbol{h}}$	$\displaystyle=h_{\mu\nu}\mbox{d}X^{\mu}\mbox{d}X^{\nu}\,,$		(14)
	$\displaystyle{\boldsymbol{h}}$	$\displaystyle=\phi\left(\mbox{d}t^{2}+{1\over d-2}\delta_{\alpha\beta}\mbox{d}x^{\alpha}\mbox{d}x^{\beta}\right)+2A_{\alpha}\mbox{d}x^{\alpha}\mbox{d}t\,,$
	$\displaystyle{\phi}({\boldsymbol{x}})$	$\displaystyle=2\kappa\frac{d-2}{d-1}\int\mbox{d}^{d}y\,{\rho}({\boldsymbol{y}})\mathcal{G}_{d}({\boldsymbol{x}}-{\boldsymbol{y}})\,,$		(15)
	$\displaystyle{A}_{\alpha}({\boldsymbol{x}})$	$\displaystyle=\kappa\int\mbox{d}^{d}y\,{j}{}_{\alpha}{{}^{\beta}({\boldsymbol{y}})\frac{\partial\mathcal{G}_{d}({\boldsymbol{x}}-{\boldsymbol{y}})}{\partial x{}^{\beta}}}\,.$		(16)

Due to the translational symmetry of Eq. (11), the Green function $\mathcal{G}_{d}({\boldsymbol{x}},{\boldsymbol{x^{\prime}}})$ is a function of ${\boldsymbol{x}}-{\boldsymbol{x^{\prime}}}$, while due to the spherical symmetry it depends on the radius variable $r=|{\boldsymbol{x}}-{\boldsymbol{x^{\prime}}}|$ alone. Thus one has²²2In order to keep the notation somewhat manageable we shall use the same symbol for the Green function with vectorial arguments $\mathcal{G}_{d}({\boldsymbol{x}})$ and with the scalar radius argument $\mathcal{G}_{d}(r)$.

$\displaystyle\mathcal{G}_{d}({\boldsymbol{x}}-{\boldsymbol{x^{\prime}}})=\mathcal{G}_{d}(r)\,.$

(17)

As has been shown previously Boos et al. (2018a), the Green function in $d+2$ spatial dimensions is related to the Green function in $d$ spatial dimensions via the recursion formulas

	$\displaystyle\mathcal{G}_{d}(r)$	$\displaystyle=-2\pi\int\mbox{d}\tilde{r}\,\tilde{r}\,\mathcal{G}_{d+2}(\tilde{r})\,,$		(18)
	$\displaystyle\mathcal{G}_{d+2}(r)$	$\displaystyle=-\frac{1}{2\pi r}\frac{\partial\mathcal{G}_{d}(r)}{\partial r}\,.$		(19)

Using this relation one can rewrite (16) in the form

$\displaystyle A_{\alpha}({\boldsymbol{x}})=-2\pi\kappa\int\mbox{d}^{d}y\,{j}{}_{\alpha\beta}({\boldsymbol{y}})(x^{\beta}-y^{\beta})\mathcal{G}_{d+2}({\boldsymbol{x}}-{\boldsymbol{y}})\,.$

(20)

For calculations it is convenient to use the following representation of the static Green function given in Frolov and Zelnikov (2016a),

$\displaystyle\begin{split}\mathcal{G}_{d}(r)&=\frac{1}{(2\pi)^{\tfrac{d}{2}}r^{d-2}}\int\limits_{0}^{\infty}\mbox{d}\zeta\frac{\zeta^{\tfrac{d-4}{2}}}{a(-\zeta^{2}/r^{2})}J_{\tfrac{d}{2}-1}(\zeta)\,,\\ r^{2}&=|{\boldsymbol{x}}-{\boldsymbol{x^{\prime}}}|^{2}\,,\quad d\geq 3\,.\end{split}$

(21)

Last, let us mention that in the limit $r\rightarrow\infty$ the above representation (21) gives

$\displaystyle\begin{split}\mathcal{G}_{d}(r)&\sim G_{d}(r)\quad\text{as }~{}r\rightarrow\infty\,,\\ G_{d}(r)&=\frac{1}{(2\pi)^{\tfrac{d}{2}}r^{d-2}}\,\lim\limits_{\epsilon\rightarrow 0}\int\limits_{0}^{\infty}\mbox{d}\zeta\zeta^{\tfrac{d-4}{2}}e^{-\epsilon\zeta}J_{\tfrac{d}{2}-1}(\zeta)\\ &=\frac{\Gamma\left(\tfrac{d}{2}-1\right)}{4\pi^{\tfrac{d}{2}}}\frac{1}{r^{d-2}}\,.\end{split}$

(22)

In the above we have made use of the constraint (4).³³3Note that the $\epsilon$-regularization is only required for $d\geq 5$. If one instead calculates the full expression (21) for a given choice of the function $a(\Box)$, see a detailed description in Appendix C, this regularization is not required, and one may simply take the limit $\ell\rightarrow 0$ to recover (22). $G_{d}(r)$ is the static Green function of linearized General Relativity Myers and Perry (1986) in $d$ spatial dimensions, which guarantees that for isolated sources in the far field regime one reproduces the standard asymptotics of General Relativity.

The stress-energy of a point-like spinning particle can be written in the form

$\displaystyle{T}{}_{\mu\nu}=\delta{}^{t}_{\mu}\delta{}^{t}_{\nu}\,{m}\delta{}^{(d)}({\boldsymbol{x}})+\delta{}^{t}_{(\mu}\delta{}^{\alpha}_{\nu)}\,{j}_{\alpha}{}^{\beta}\frac{\partial}{\partial x{}^{\beta}}\delta{}^{(d)}({\boldsymbol{x}})\,,$

(23)

where $m$ is the mass of the particle and ${j}_{\alpha\beta}$ is a constant antisymmetric matrix parametrizing its angular momentum. A solution for the perturbed metric (14)–(16) for such a source takes the form⁴⁴4In four-dimensional spacetime, this solution can be used to obtain a metric for a spinning ring discussed in Buoninfante et al. (2018c).

$\displaystyle\begin{split}{\phi}(r)&=2\kappa\frac{d-2}{d-1}{m}\,\mathcal{G}_{d}(r)\,,\\ {A}_{\alpha}({\boldsymbol{x}})&=-2\pi\kappa{j}{}_{\alpha\beta}x^{\beta}\,\mathcal{G}_{d+2}(r)\,.\end{split}$

(24)

At large distances one recovers the standard expressions known from linearized General Relativity Myers and Perry (1986):

	$\displaystyle{\phi}(r)$	$\displaystyle\sim\frac{\Gamma\left(\tfrac{d}{2}\right)}{(d-1)\pi^{\tfrac{d}{2}}}\frac{\kappa m}{r^{d-2}}\,,$		(25)
	$\displaystyle{A}_{\alpha}({\boldsymbol{x}})$	$\displaystyle\sim-\frac{\Gamma\left(\tfrac{d}{2}\right)}{2\pi^{\tfrac{d}{2}}}\frac{\kappa{j}_{\alpha\beta}x{}^{\beta}}{r^{d}}\,.$		(26)

We choose the sign of ${A}_{\alpha}$ such that in the three-dimensional case $d=3$ one obtains the standard Lense–Thirring expression (${j}_{xy}=j$ and $\kappa=8\pi G$)

$\displaystyle{A}_{\alpha}({\boldsymbol{x}})\mbox{d}x{}^{\alpha}$

$\displaystyle\sim\frac{2Gj}{r^{3}}(x\mbox{d}y-y\mbox{d}x)=\frac{2Gj}{r}\sin^{2}\theta\,\mbox{d}\varphi\,.$

(27)

In order to simplify our presentation further, let us consider a special type of spinning objects. That is, we assume that it has finite extension in one spatial directions, while its transverse size is zero. We call such an object a thin spinning pencil or simply “pencil.”

Before performing the boost, and, subsequently, the Penrose limit, let us briefly mention a useful representation of the static Green function $\mathcal{G}_{d}(r)$ given by

	$\displaystyle\mathcal{G}_{d}(r)$	$\displaystyle=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}\frac{\mbox{d}\eta}{a(-\eta\ell^{2})\eta}\int\limits_{-\infty}^{\infty}\mbox{d}\tau\,K_{d}(r|\tau)\,e^{i\eta\tau}\,,$		(41)
	$\displaystyle K_{d}(r|\tau)$	$\displaystyle=\frac{1}{(4\pi i\tau)^{\tfrac{d}{2}}}e^{i\tfrac{r^{2}}{4\tau}}\,.$		(42)

The function $K_{d}(r|\tau)$ is the $d$-dimensional heat kernel in imaginary time $\tau=-it$ and therefore satisfies

	$\displaystyle\triangle K_{d}(r|\tau)$	$\displaystyle=-i\partial_{\tau}K_{d}(r|\tau)\,,$		(43)
	$\displaystyle\lim\limits_{\tau\rightarrow 0}K_{d}(r|\tau)$	$\displaystyle=\delta{}^{(d)}({\boldsymbol{r}})\,.$		(44)

The derivation of the representation (41) for the static Green function is given in Appendix B. The following property makes this representation very useful for the study of the Penrose limit of solutions: Relation (41) expresses the Green function $\mathcal{G}_{d}(\bar{r})$ as a double Fourier transform, wherein the radius $\bar{r}$ enters only quadratically via the exponential function $\sim\exp[i\bar{r}^{2}/(4\tau)]$. Since $\bar{r}^{2}=\bar{\xi}^{2}+{\boldsymbol{x}}_{\perp}^{2}$, this exponent can be factorized, which in turn allows one to separate the dependence of the integrand on $\bar{\xi}$ as $\sim\exp[i\bar{\xi}^{2}/(4\tau)]$. Hence, when applying the boost, only this factor is affected. As we shall demonstrate now, this observation allows us to perform the Penrose limit procedure in a very general and convenient form.

We now apply the Penrose limit to our previously described linearized potentials of a spinning “pencil.” We parametrize the boost in the $\overline{\xi}$-direction via

$\displaystyle\overline{t}=\gamma\left(t-\beta\xi\right)\,,\quad\overline{\xi}=\gamma\left(\xi-\beta t\right)\,.$

(45)

Let us first make a simple remark concerning the scaling properties of the pencil characteristics under a boost transformation (45). We assume that both mass and angular momentum are uniformly distributed along the pencil and their densities in the rest $\bar{S}$ frame, $\bar{\lambda}=\bar{m}/\bar{L}$ and $\bar{j}=\bar{J}/\bar{L}$ are constant. Because of the Lorentz contraction, the length of the same pencil, as measured in the moving frame $S$ is $L=\bar{L}/\gamma$, while its energy is $m=\gamma\bar{m}$. As a result, the linear energy density of the pencil in $S$ frame is $\lambda=\gamma^{2}\bar{\lambda}$. In the Penrose limit the energy $m$ is taken to be fixed. Thus the energy density $\lambda$ grows to infinity as $\gamma\to\infty$. To keep it finite, one needs to rescale $\bar{L}\to\gamma\bar{L}$ in the boost process, such that the length $L$ remains unchanged. It is easy to check that under such rescalings the components of the angular momentum remain the same and finite. Note that this is a result of our assumption that the angular momentum density is orthogonal to the direction of motion, because in that case its components are not affected by the boost.

For fixed $\xi$, that is, for a fixed point in frame $S$ one has $\bar{\xi}=-\gamma\beta t+\text{const}$. This means that the frame $S$ moves in the negative direction of $\bar{\xi}$ (“left”) with respect to the rest frame $\bar{S}$. In other words, a pencil which is at rest with respect to $\bar{S}$ moves with a positive velocity in $S$ frame.

We introduce the retarded and advanced null coordinates in the $S$ frame defined as follows:

$\displaystyle u=\frac{t-\xi}{\sqrt{2}}\,,\quad v=\frac{t+\xi}{\sqrt{2}}\,.$

(46)

Then (45) implies

	$\displaystyle\overline{t}$	$\displaystyle={\gamma\over\sqrt{2}}[(1+\beta)u+(1-\beta)v]\,,$		(47)
	$\displaystyle\overline{\xi}$	$\displaystyle={\gamma\over\sqrt{2}}[-(1+\beta)u+(1-\beta)v]\,.$		(48)

In the ultrarelativistic limit, $\beta\rightarrow 1$, one has

$\displaystyle\overline{t}\to\sqrt{2}\gamma u\,,\quad\overline{\xi}\to-\sqrt{2}\gamma u\,,$

(49)

This implies that the matter distribution of such an ultrarelativistec pencil is located in the strip between $u=-L/\sqrt{2}$ and $u=0$ of spacetime; see Fig. 1.

Because we keep the ratio $\bar{L}/\gamma$ constant during the Penrose limit, the linear density scales as follows:

$\displaystyle\lambda(u)=\lim\limits_{\gamma\rightarrow\infty}\sqrt{2}\gamma^{2}\,\overline{\lambda}(-\sqrt{2}\gamma u)\,.$

(50)

This guarantees that in the Penrose limit the product $\overline{m}\gamma$ and the ratio $\bar{L}/\gamma$ remain constant,

$\displaystyle\gamma\,\overline{m}=\gamma\int\limits_{-\infty}^{\infty}\mbox{d}\overline{\xi}\,\overline{\lambda}(\overline{\xi})=\int\limits_{-\infty}^{\infty}\mbox{d}u\,\lambda(u)=\text{const}\,.$

(51)

The angular momentum line density $\overline{j}_{ij}(\bar{\xi})$ “lives” in transverse space and its tensorial structure is unaffected from the boost. Using this property we define the boosted linear density of the angular momentum in the $S$ frame as follows:

	$\displaystyle j_{ij}(u)$	$\displaystyle=\lim\limits_{\gamma\rightarrow\infty}\sqrt{2}\gamma\,\overline{j}_{ij}(-\sqrt{2}\gamma u)\,,$		(52)
	$\displaystyle j_{a}(u)$	$\displaystyle=\lim\limits_{\gamma\rightarrow\infty}\sqrt{2}\gamma\,\bar{j}_{a}(-\sqrt{2}\gamma u)\,.$		(53)

The total angular momentum of the boosted pencil therefore remains finite and has the form

$\displaystyle\begin{split}J_{ij}&=\int\limits_{-\infty}^{\infty}\mbox{d}\overline{\xi}\,\overline{j}_{ij}(\overline{\xi})=\int\limits_{-\infty}^{\infty}\mbox{d}uj{}_{ij}(u)\,.\end{split}$

(54)

Under this boost and the Penrose limit, as defined above, the resulting metric takes the form

$\displaystyle\begin{split}{\boldsymbol{g}}&=\left(\eta{}_{\mu\nu}+h{}_{\mu\nu}\right)\mbox{d}X{}^{\mu}\mbox{d}X{}^{\nu}\\ &=-2\mbox{d}u\mbox{d}v+\phi\mbox{d}u^{2}+2A{}_{i}\mbox{d}x{}^{i}_{\perp}\mbox{d}u+\mbox{d}{\boldsymbol{x}}_{\perp}^{2}\,,\end{split}$

(55)

where we defined

$\displaystyle\phi=\lim\limits_{\gamma\rightarrow\infty}2\gamma^{2}\frac{d-1}{d-2}\bar{\phi}\,,\quad A_{i}=\lim\limits_{\gamma\rightarrow\infty}\sqrt{2}\gamma\bar{A}_{i}\,.$

(56)

Here, $\bar{\phi}$ and $\bar{A}_{i}$ are given by (37) and (38), respectively. The integrands in their representations contain the Green function $\mathcal{G}_{d}(\bar{r})$. In order to understand their behavior under the Penrose limit we make use of relation (41). In this representation the only quantity which is “sensitive” to the boost is the heat kernel $K_{d}$. It factorizes such that the boost-sensitive factor is the exponent of the form $\sim\exp[i(\bar{\xi}-\bar{\xi}^{\prime})^{2}/4\tau]$, which for large $\gamma$ factors takes the form $\sim\exp[i\gamma(u-u^{\prime})^{2}/2\tau]$. To take the Penrose limit we use the following relation (see also Shankar (1994)):

$\displaystyle\delta(u)=\lim\limits_{\epsilon\rightarrow 0}\frac{1}{\sqrt{2\pi i\epsilon}}e^{i\tfrac{u^{2}}{2\epsilon}}\,.$

(57)

Denote $\epsilon=\tau/\gamma^{2}$ and apply this relation to (41) to obtain

$\displaystyle\lim\limits_{\gamma\rightarrow\infty}\gamma\,\mathcal{G}_{d}(\bar{r})$

$\displaystyle=\frac{1}{\sqrt{2}}\mathcal{G}_{d-1}(r_{\perp})\delta(u-u^{\prime})\,,$

(58)

where $r_{\perp}^{2}=\delta{}_{ij}x^{i}_{\perp}x^{j}_{\perp}$. Performing the limit $\gamma\to\infty$ in the relations (56) for the potential $\phi$ and the gravitomagnetic potential $A_{i}$ finally yields

	$\displaystyle\phi$	$\displaystyle=2\sqrt{2}\kappa\lambda(u)\mathcal{G}_{d-1}(r_{\perp})\,,$		(59)
	$\displaystyle A_{i}$	$\displaystyle=-2\pi\kappa j{}_{ij}(u)x^{j}_{\perp}\mathcal{G}_{d+1}(r_{\perp})\,.$		(60)

Introducing polar coordinates $\{\rho_{a},\varphi_{a}\}$ in each Darboux plane $\Pi_{a}$ such that

$\displaystyle y^{a}=\rho_{a}\cos\varphi_{a}\,,\quad\hat{y}^{a}=\rho_{a}\sin\varphi_{a}\,,$

(61)

one may use the relation

$\displaystyle j_{ij}\,x_{\perp}^{i}\mbox{d}x_{\perp}^{j}=\sum\limits_{a=1}^{n}j_{a}\rho_{a}^{2}\mbox{d}\varphi_{a}$

(62)

to rewrite the gravitomagnetic potential 1-form as

$\displaystyle A_{i}({\boldsymbol{x}}_{\perp})\mbox{d}x{}^{i}_{\perp}=2\pi\kappa\mathcal{G}_{d+1}(r_{\perp})\sum\limits_{a=1}^{n}j_{a}(u)\rho_{a}^{2}\mbox{d}\varphi_{a}\,,$

(63)

which makes the rotational symmetry in each Darboux plane manifest.