Papapetrou field as the gravitoelectromagnetic field tensor in stationary spacetimes

Abstract

Introducing the well known Papapetrou field as the gravitoelectromagnetic field tensor, we express the Maxwell-type part of the 3-dimensional quasi-Maxwell form of the Einstein field equations in terms of differential forms, analogous to their electromagnetic counterparts in curved spacetimes. Using the same formalism we introduce the junction conditions on non-null hypersurfaces in terms of the introduced gravitoelectromagnetic 4-vector fields and apply them to the case of the Van Stockum interior and exterior solutions.

Appendices

Appendix 1: Calculation of the inhomogeneous quasi-Maxwell equations in terms of the GEM field tensor

In Sect. 3 we gave an expression for the quasi-Maxwell form of the Einstein field equations in terms of the GEM field tensor in the coordinate system adapted to the TKVF. Here we find the 3-dimensional inhomogeneous quasi-Maxwell equations introduced in Sect. 2, starting with Eq. (33),

$$\begin{aligned} \nabla _a {F_g}^{ab}= -16 \pi J^b. \end{aligned}$$

(69)

and for a perfect fluid whose 4-current is given by

$$\begin{aligned}&J^b= \left( T^{ab}-\frac{1}{2} T g^{ab}\right) \xi _a = \left( \frac{p + \rho }{1-v^2}(1+\sqrt{h}{A_g}_{\alpha }v^\alpha ) -\frac{\rho - p}{2}, \mathbf{J}\right) ,\nonumber \\&\mathbf{J}=\sqrt{h}\frac{p+\rho }{1-v^2}{} \mathbf{v} \end{aligned}$$

(70)

First we consider the spatial components of the 4-current so that ,

$$\begin{aligned} \nabla _a {F_g}^{a\beta }&= -16 \pi J^\beta \\&= \frac{1}{\sqrt{h} \sqrt{\gamma }}\dfrac{\partial }{\partial x^{a}} \left( \sqrt{h} \sqrt{\gamma } {F_g}^{a \beta }\right) \\&= \frac{1}{\sqrt{h} \sqrt{\gamma }}\dfrac{\partial }{\partial x^{\alpha }} \left( \sqrt{h} \sqrt{\gamma } (-h f^{\alpha \beta })\right) \end{aligned}$$

using $f_{\alpha \beta }= \sqrt{\gamma } e_{\alpha \beta \gamma } {B_g}^{\gamma }$ yields

$$\begin{aligned} \nabla _a {F_g}^{a\beta }&= - 16\pi J^\beta \\&= - \frac{1}{\sqrt{h} \sqrt{\gamma }}\dfrac{\partial }{\partial x^{\alpha }} \left( h^{3/2} e^{\alpha \beta \gamma } {B_g}_{\gamma }\right) \\&= \frac{3h}{\sqrt{\gamma }} {E_g}_{\alpha } {B_g}_{\gamma } e^{\alpha \beta \gamma }- \frac{h}{\sqrt{\gamma }} e^{\alpha \beta \gamma } \left( \dfrac{\partial }{\partial x^{\alpha }} {B_g}_{\gamma }\right) \\&= -\sqrt{h}\left( 2\mathbf {E} _g \times \sqrt{h}{} \mathbf {B} _g^{\beta }-(\nabla \times \sqrt{h}{} \mathbf {B} _g)^{\beta }\right) \end{aligned}$$

from which we obtain the following quasi-Maxwell equation in the three-space $\Sigma _3$

$$\begin{aligned} \nabla \times (\sqrt{h}{} \mathbf {B} _g)= 2 \mathbf {E} _g \times \sqrt{h}{} \mathbf {B} _g - 16\pi \frac{p+\rho }{1-v^2}\mathbf{v}. \end{aligned}$$

(71)

Now we turn our attention to the temporal component of the equation, i.e.

$$\begin{aligned} \nabla _a {F_g}^{a0}= -16 \pi J^0, \end{aligned}$$

(72)

which could be written as follows,

$$\begin{aligned} \nabla _{\alpha } {F_g}^{\alpha 0}= \frac{1}{\sqrt{h} \sqrt{\gamma }}\dfrac{\partial }{\partial x^{\alpha }} \left( \sqrt{h} \sqrt{\gamma } {F_g}^{\alpha 0}\right) = -16 \pi J^0. \end{aligned}$$

Writing the GEM field tensor in terms of the GEM vector fields, for the LHS we have,

$$\begin{aligned} \nabla _{\alpha } {F_g}^{\alpha 0}\!=\! \frac{2}{\sqrt{\gamma }}\dfrac{\partial }{\partial x^{\alpha }} \left( \sqrt{\gamma } {E_g}^{\alpha }\right) - \frac{h}{\sqrt{\gamma }}\dfrac{\partial }{\partial x^{\alpha }} [\sqrt{\gamma } (\mathbf A _g \times \mathbf B _g)^{\alpha }]\,{-}\,[2{E_g}^{\alpha }-3h(\mathbf A _g \times \mathbf {B} _g)^{\alpha }]{E_g}_{\alpha } \end{aligned}$$

(73)

in which for the second term in the right hand side we have

$$\begin{aligned} \frac{h}{\sqrt{\gamma }}\dfrac{\partial }{\partial x^{\alpha }} [\sqrt{\gamma } (\mathbf A _g \times \mathbf {B} _g)^{\alpha }]= h {B_g}^2-h \mathbf A _g \cdot (\nabla \times \mathbf {B} _g) \end{aligned}$$

so that we end up with

$$\begin{aligned} \nabla _{\alpha }{F_g}^{\alpha 0}&\!=\!2 \nabla \cdot {E_g}-2{E_g}^2-h{B_g}^2\!+\!h \mathbf A _g \cdot (\nabla \!\times \! \mathbf {B} _g)\!+\!3h \mathbf {E} _g \cdot (\mathbf A _g \!\times \! \mathbf {B} _g)\!=\!-16 \pi J^0. \end{aligned}$$

(74)

After making use of the Eq. (71) and substituting for $J^0$ from (70), we end up with the desired result

$$\begin{aligned} \nabla \cdot \mathbf {E} _g={E_g}^2+\frac{1}{2}h{B_g}^2 - 8\pi \left( \frac{p+\rho }{1-v^2} - \frac{\rho - p}{2} \right) . \end{aligned}$$

which is the other inhomogeneous quasi-Maxwell equation.

Appendix 2: Calculation of the extrinsic curvature in terms of the GEM fields

Before proceeding with the calculation of the jump in the extrinsic curvature of the boundary hypersurface, it should be noted that the following calculations are made in a gauge in which $A_{\alpha ;\beta }+A_{\beta ;\alpha }=0$. This is due to the fact that the combination $A_{\alpha ;\beta }+A_{\beta ;\alpha }$ is not an invariant under the gauge transformation ${A_g}_\alpha \rightarrow {A_g}_\alpha + \partial _{\alpha }f$ representing the freedom in choosing the time origin (3), where it is known that all the 3-dimensional objects are scalars under the spacetime transformations [1].

We begin by writing all the connection coefficients in terms of the GEM 3-vector fields and then show that the two expressions (42) and (43) are equivalent in the coordinate system adapted to the TKVF and so being 3-tensorial relation on the hypersurface they are equal in all coordinate systems.

In terms of the metric components, the connection coefficients for a stationary spacetime (in the coordinate system adapted to the TKVF) are given by [1],

$$\begin{aligned} \Gamma ^{0}_{00}\doteq & {} \frac{1}{2}{A_g}^\alpha h_{,\alpha }\nonumber \\ \Gamma ^{0}_{\alpha 0}\doteq & {} \frac{h_{,\alpha }}{2h}+\frac{h}{2}{A_g}^{\beta }f_{\alpha \beta } -\frac{1}{2}{A_g}_\alpha {A_g}_\lambda h^{,\lambda }\nonumber \\ \Gamma ^{0}_{\alpha \beta }\doteq & {} -\frac{1}{2}\left( {A_g}_{\alpha ,\beta }+{A_g}_{\beta ,\alpha }\right) -\frac{1}{2h}({A_g}_{\alpha }h_{,\beta }+{A_g}_{\beta }h_{,\alpha })\nonumber \\&+\,{A_g}_{\delta }\lambda ^{\delta }_{\alpha \beta } +\frac{1}{2}{A_g}_{\alpha }{A_g}_{\beta }{A_g}_{\delta }h^{,\delta } -\frac{h}{2}{A_g}^\lambda ({A_g}_{\alpha }f_{\beta \lambda } +{A_g}_{\beta }f_{\alpha \lambda })\nonumber \\ \Gamma ^{\alpha }_{00}\doteq & {} \frac{1}{2} h^{,\alpha }\nonumber \\ \Gamma ^{\alpha }_{0\beta }\doteq & {} \frac{h}{2}f_{\beta }^{\alpha }-\frac{1}{2}{A_g}_{\beta }h^{,\alpha }\nonumber \\ \Gamma ^{\alpha }_{\beta \gamma }\doteq & {} \lambda ^{\alpha }_{\beta \gamma }-\frac{h}{2}\left( {A_g}_{\beta }f_{\gamma }^{\alpha } +{A_g}_{\gamma }f_{\beta }^{\alpha }\right) +\frac{1}{2}{A_g}_{\beta }{A_g}_{\gamma }h^{,\alpha } \end{aligned}$$

(75)

Rewriting the above equations in terms of the GEM 3-vector fields, they are given as follows,

$$\begin{aligned} \Gamma ^{0}_{00}\doteq & {} -\,h {A_g}^\alpha {E_g}_\alpha \nonumber \\ \Gamma ^{0}_{\alpha 0}\doteq & {} -{E_g}_{\alpha }+\frac{h}{2}\sqrt{\gamma }{A_g}^{\beta } \epsilon _{\alpha \beta \gamma }{B_g}^{\gamma } +h{A_g}_{\alpha }{A_g}^{\lambda }{E_g}_{\lambda }\nonumber \\ \Gamma ^{0}_{\alpha \beta }\doteq & {} \left( {A_g}_{\alpha }{E_g}_{\beta }+{A_g}_{\beta }{E_g}_{\alpha }\right) \nonumber \\&-\,h{A_g}_{\alpha }{A_g}_{\beta }{A_g}^{\lambda }{E_g}_{\lambda } -\frac{h}{2}{A_g}^{\gamma }\left( {A_g}_{\alpha }\sqrt{\gamma } \epsilon _{\beta \gamma \lambda }{B_g}^{\lambda }+{A_g}_{\beta }\sqrt{\gamma } \epsilon _{\alpha \gamma \lambda }{B_g}^{\lambda }\right) \nonumber \\ \Gamma ^{\alpha }_{00}\doteq & {} -\,h\gamma ^{\alpha \beta }{E_g}_\beta \nonumber \\ \Gamma ^{\alpha }_{0\beta }\doteq & {} \frac{h}{2}\sqrt{\gamma }{\epsilon _{\beta }^{\alpha }}_{\gamma }{B_g}^{\gamma } +h{A_g}_{\beta }\gamma ^{\alpha \lambda }{E_g}_{\lambda }\nonumber \\ \Gamma ^{\alpha }_{\beta \gamma }\doteq & {} \lambda ^{\alpha }_{\beta \gamma }-\frac{h}{2} \left( {A_g}_{\beta }\sqrt{\gamma }{\epsilon _{\gamma }^{\alpha }}_{\lambda }{B_g}^{\lambda } +{A_g}_{\gamma }\sqrt{\gamma }{\epsilon _{\beta }^{\alpha }}_{\lambda }{B_g}^{\lambda }\right) -h{A_g}_{\beta }{A_g}_{\gamma }\gamma ^{\alpha \lambda }{E_g}_{\lambda } \end{aligned}$$

(76)

Now what we need, is to show that the following relations hold between the connection coefficients and the components of the tensor $P^a_{bc}$,

$$\begin{aligned} -\Gamma ^{0}_{00}= & {} P^0_{00}; \quad -\Gamma ^{0}_{\alpha 0}=P^0_{\alpha 0}\nonumber \\ -\Gamma ^{0}_{\alpha \beta }= & {} P^{0}_{\alpha \beta }; \quad -\Gamma ^{\alpha }_{00}=P^{\alpha }_{00}\nonumber \\ -\Gamma ^{\alpha }_{0\beta }= & {} P^{\alpha }_{0\beta }; \quad -\Gamma ^{\alpha }_{\beta \gamma }+\lambda ^{\alpha }_{\beta \gamma } =P^{\alpha }_{\beta \gamma }. \end{aligned}$$

(77)

This could be achieved by calculating the components of $P^a_{bc}$ in the coordinate system adapted to the timelike Killing vector as follows,

$$\begin{aligned} P^0_{00}\doteq & {} \frac{1}{h}{E_g}^0\xi _0\xi _0+\frac{1}{h}\left( -{E_g}_0^{0}\xi _0\right) =h {A_g}^\alpha {E_g}_\alpha \nonumber \\ P^0_{\alpha 0}\doteq & {} \frac{1}{h}{E_g}^0\xi _\alpha \xi _0+\frac{1}{2h}\left( -{F_g}_\alpha ^{0}\xi _0 -{F_g}_0^{0}\xi _\alpha \right) \nonumber \\= & {} {E_g}_{\alpha } -h{A_g}_{\alpha }{A_g}^{\beta }{E_g}_\beta -\frac{1}{2}h\sqrt{\gamma }{A_g}^{\beta }\epsilon _{\alpha \beta \lambda }{B_g}^{\lambda }\nonumber \\ P^{\alpha }_{00}\doteq & {} \frac{1}{h}{E_g}^{\alpha }\xi _0\xi _0+\frac{1}{h}\left( -{F_g}_0^{\alpha }\xi _0\right) =h\gamma ^{\alpha \beta }{E_g}_\beta \nonumber \\ P^{\alpha }_{\beta 0}\doteq & {} \frac{1}{h}{E_g}^{\alpha }\xi _\beta \xi _0+\frac{1}{2h} \left( -{F_g}_\beta ^{\alpha }\xi _0-{F_g}_0^{\alpha }\xi _\beta \right) \nonumber \\= & {} -{A_g}_{\beta }\gamma ^{\alpha \lambda }{E_g}_\lambda -\frac{1}{2}h\sqrt{\gamma }{\epsilon _{\beta }^{\alpha }}_{\gamma }{B_g}^{\gamma }\nonumber \\ P^{0}_{\alpha \beta }\doteq & {} \frac{1}{h}{E_g}^0\xi _\alpha \xi _\beta +\frac{1}{2h} \left( -{F_g}_\alpha ^{0}\xi _\beta -{F_g}_\beta ^{0}\xi _\alpha \right) \nonumber \\= & {} -\,\left( {A_g}_{\alpha }{E_g}_{\beta }+{A_g}_{\beta }{E_g}_{\alpha }\right) +h{A_g}_{\alpha }{A_g}_{\beta }{A_g}^{\lambda }{E_g}_{\lambda }\nonumber \\&+\,\frac{h}{2}{A_g}^{\gamma }\left( {A_g}_{\alpha }\sqrt{\gamma } \epsilon _{\beta \gamma \lambda }{B_g}^{\lambda }+{A_g}_{\beta }\sqrt{\gamma } \epsilon _{\alpha \gamma \lambda }{B_g}^{\lambda }\right) \nonumber \\ P^{\alpha }_{\beta \gamma }\doteq & {} \frac{1}{h}{E_g}^{\alpha }\xi _{\beta }\xi _\gamma +\frac{1}{2h} \left( -{F_g}_\beta ^{\alpha }\xi _\gamma -{F_g}_\gamma ^{\alpha }\xi _\beta \right) \nonumber \\= & {} \frac{h}{2}\left( {A_g}_{\beta }\sqrt{\gamma } {\epsilon _{\gamma }^{\alpha }}_{\lambda }{B_g}^{\lambda }+{A_g}_{\gamma } \sqrt{\gamma }{\epsilon _{\beta }^{\alpha }}_{\lambda } {B_g}^{\lambda }\right) +h{A_g}_{\beta }{A_g}_{\gamma } \gamma ^{\alpha \lambda }{E_g}_{\lambda } \end{aligned}$$

(78)

comparing the above relations with those in $(\mathrm{B}2)$ shows that the relations given in $(\mathrm{B}3)$ are satisfied in the coordinate system adapted to the TKVF.