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General relativistic manifestations of orbital angular and intrinsic hyperbolic momentum in electromagnetic radiation

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Abstract

General relativistic effects in the weak field approximation are calculated for electromagnetic Laguerre–Gaussian beams. The current work is an extension of previous work on the precession of a spinning neutral particle in the weak gravitational field of an optical vortex. In the current work, the metric perturbation is extended to all coordinate configurations and includes gravitational effects from circular polarization and intrinsic hyperbolic momentum. The final metric reveals frame-dragging effects due to intrinsic spin angular momentum (SAM), orbital angular momentum (OAM), and spin–orbit coupling. When investigating the acceleration of test particles in this metric, an unreported gravitational phenomenon was found. This effect is analogous to the motion of charged particles in the magnetic field produced by a current-carrying wire. It was found that the gravitational influence of SAM and OAM affects test-rays traveling perpendicular to the intense beam and from this a gravitational Aharonov–Bohm analog is pursued.

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Appendices

Appendix 1: energy–momentum tensor

The energy–momentum for a beam of electromagnetic radiation can be calculated from

$$ T_{\mu \nu } = \left[ {\begin{array}{*{20}c} {\frac{1}{2}\left( {E^{2} + B^{2} } \right)} & { - S_{x} } & { - S_{y} } & { - S_{z} } \\ { - S_{x} } & {\sigma_{xx} } & {\sigma_{xy} } & {\sigma_{xz} } \\ { - S_{y} } & {\sigma_{yx} } & {\sigma_{yy} } & {\sigma_{yz} } \\ { - S_{z} } & {\sigma_{zx} } & {\sigma_{zy} } & {\sigma_{zz} } \\ \end{array} } \right] , $$
(A.1)

where the Poynting vector and the Maxwell stress tensor are given respectively by,

$$ \vec{S} = \frac{1}{{\mu_{0} }}\vec{E} \times \vec{B} , $$
(A.2)
$$ \sigma_{ij} = \varepsilon_{0} E_{i} E_{j} + \frac{1}{{\mu_{0} }}B_{i} B_{j} - \frac{1}{2}\left( {\varepsilon_{0} E^{2} + \frac{1}{{\mu_{0} }}B^{2} } \right)\delta_{ij} . $$
(A.3)

The electric and magnetic fields of a beam within the paraxial approximation can be found from Maxwell’s equation as outlined in Ref [13] and are given by

$$ \vec{E} = E_{0} \left[ {\alpha \hat{e}_{x} + \beta \hat{e}_{y} + \frac{i}{k}\left( {\alpha \frac{\partial }{\partial x} + \beta \frac{\partial }{\partial y}} \right)\hat{e}_{z} } \right]\left| \psi \right|^{2} , $$
(A.4)
$$ \vec{B} = B_{0} \left[ { - \beta \hat{e}_{x} + \alpha \hat{e}_{y} - \frac{i}{k}\left( {\beta \frac{\partial }{\partial x} - \alpha \frac{\partial }{\partial y}} \right)\hat{e}_{z} } \right]\left| \psi \right|^{2} . $$
(A.5)

From Eqs. A.4 and A.5, it can be seen that \( B_{y} = E_{x} \) and \( B_{x} = - E_{y} \). In calculating the energy–momentum tensor in the paraxial approximation using Eqs. A.2 and A.3, terms of second-order in wavelength are neglected (viz., \( E_{z}^{2} ,B_{z}^{2} \approx 0 \)). These relations can be substituted into Eq. A.1 which reduces to,

$$ T_{\mu \nu } = \left[ {\begin{array}{*{20}c} {S_{z} } & { - S_{x} } & { - S_{y} } & { - S_{z} } \\ { - S_{x} } & 0 & 0 & {S_{x} } \\ { - S_{y} } & 0 & 0 & {S_{y} } \\ { - S_{z} } & {S_{x} } & {S_{y} } & {S_{z} } \\ \end{array} } \right] , $$
(A.6)

wherein Cartesian coordinates the Poynting vector is given by

$$ \vec{S} = \left| {E_{0} } \right|^{2} \frac{1}{{c\mu_{0} }}\left[ \begin{aligned} \left( {\cos (\theta )\frac{r}{R(z)} - \sin (\theta )\frac{1}{k}\left( {\frac{\ell }{r} - \sigma_{z} \frac{1}{2}\frac{\partial }{\partial r}} \right)} \right)\hat{e}_{x} \hfill \\ + \left( {\sin (\theta )\frac{r}{R(z)} + \cos (\theta )\frac{1}{k}\left( {\frac{\ell }{r} - \sigma_{z} \frac{1}{2}\frac{\partial }{\partial r}} \right)} \right)\hat{e}_{y} + \hat{e}_{z} \hfill \\ \end{aligned} \right]\left| \psi \right|^{2} . $$
(A.7)

Appendix 2: derivatives of the metric perturbation

In this appendix, relevant derivatives of the metric perturbations \( h_{\mu \nu }^{P} \) and \( h_{\mu \nu }^{SO} \) are provided and plotted as a function of radial \( r/w_{0} \) and longitudinal \( z/w_{0} \) directions. To begin with, a common factor that appears frequently can be written in shorthand notation as

$$ \begin{aligned} f_{ - } (r,z) & = \sqrt {r^{2} + r_{\rho \ell }^{2} + g_{ - }^{2} } \\ f_{ + } (r,z) & = \sqrt {r^{2} + r_{\rho \ell }^{2} + g_{ + }^{2} } \\ \end{aligned} , $$
(B.1)

where \( g_{ \pm } = L \pm z \), When \( z = 0 \) these equations reduce to \( g_{ - } = g_{ + } \) and \( f_{ - } = f_{ + } \). With this handy notation, Eqs. 21 and 30 can be written as follows

$$ h_{\mu \nu }^{P} /\kappa \rho_{L} = - \tau_{\mu \nu }^{P} \ln \left| {\frac{{f_{ - } + g_{ - } }}{{f_{ + } - g_{ + } }}} \right| , $$
(B.2)
$$ h_{\mu \nu }^{SO} /\kappa \rho_{L} = - \tau_{\mu \nu }^{SO} (\theta )B_{\rho }^{\ell } (r_{\rho \ell } )\frac{1}{2k}\frac{r}{{r^{2} + r_{\rho \ell }^{2} }}\left( {\frac{{g_{ - } }}{{f_{ - } }} + \frac{{g_{ + } }}{{f_{ + } }}} \right) . $$
(B.3)

The radial and longitudinal derivatives of Eq. B.2 are (Fig. A2)

$$ \frac{1}{{\kappa \rho_{L} }}\frac{{\partial h_{\mu \nu }^{P} }}{\partial r} = - \tau_{\mu \nu }^{P} r\left( {\frac{1}{{f_{ - } \left( {f_{ - } + g_{ - } } \right)}} - \frac{1}{{f_{ + } \left( {f_{ + } - g_{ + } } \right)}}} \right) , $$
(B.4)
$$ \frac{1}{{\kappa \rho_{L} }}\frac{{\partial h_{\mu \nu }^{P} }}{\partial z} = - \tau_{\mu \nu }^{P} \left( {\frac{1}{{f_{ + } }} - \frac{1}{{f_{ - } }}} \right) , $$
(B.5)

and those for Eq. B.3 are

Fig. A2
figure 5

a and b are curves of \( h^{P} \) and their respective derivatives plotted as a function of a radial position \( r \) and b longitudinal position \( z \). c and d are the same as those for a and b except they are the curves for the metric perturbation \( h^{SO} \) and its derivatives. The dotted curves are for \( h^{P} \) and \( h^{SO} \), and the solid blue curves are for their derivatives \( dh^{P} /dr \), \( dh^{P} /dz \), \( dh^{SO} /dr \) and \( dh^{SO} /dz \). In all plots, \( w_{0} = 1 \), \( z = 5w_{0} \), \( \ell = 1 \) and \( \rho = 0 \). Curves in panels a and c have a constant longitudinal position \( z = 0 \) and those in b and d have a constant radial position \( r = w_{0} \)

$$ \frac{1}{{\kappa \rho_{L} }}\frac{{\partial h_{\mu \nu }^{SO} }}{\partial z} = - \tau_{\mu \nu }^{SO} (\theta )B_{\rho }^{\ell } (r_{\rho \ell } )\frac{1}{2k}\frac{r}{{r^{2} + r_{\rho \ell }^{2} }}\left( {\frac{1}{{f_{ + } }} - \frac{1}{{f_{ - } }} - \frac{{g_{ + }^{2} }}{{f_{ + }^{3} }} + \frac{{g_{ - }^{2} }}{{f_{ - }^{3} }}} \right) , $$
(B.6)
$$ \frac{1}{{\kappa \rho_{L} }}\frac{{\partial h_{\mu \nu }^{SO} }}{\partial r} = - \tau_{\mu \nu }^{SO} (\theta )B_{\rho }^{\ell } (r_{\rho \ell } )\frac{1}{2k}\frac{1}{{r^{2} + r_{\rho \ell }^{2} }}\left[ {\left( {1 - \frac{{2r^{2} }}{{r^{2} + r_{\rho \ell }^{2} }}} \right)\left( {\frac{{g_{ - } }}{{f_{ - } }} + \frac{{g_{ + } }}{{f_{ + } }}} \right) - r^{2} \left( {\frac{{g_{ - } }}{{f_{ - }^{3} }} + \frac{{g_{ + } }}{{f_{ + }^{3} }}} \right)} \right] . $$
(B.7)

A few special cases should be noted. In the plane \( z = 0 \) the following quantities are zero: \( \partial h^{P} /\partial r = \partial h^{P} /\partial z = 0 \) and \( \partial h^{SO} /\partial z = 0 \), and when \( r = 0 \) we have \( h^{SO} = 0 \), \( \partial h^{P} /\partial r = 0 \) and \( \partial h^{SO} /\partial z = 0 \).

Appendix 3: analysis of the geodesic equation

In this appendix, a derivation of velocity squared terms and higher of the geodesic of Eq. 31 equation are given. The geodesic equation in coordinate time is given by,

$$ \frac{{\partial^{2} x^{\mu } }}{{\partial t^{2} }} = - \varGamma_{00}^{\mu } - 2\varGamma_{0i}^{\mu } \frac{{\partial x^{i} }}{\partial t} - \varGamma_{ij}^{\mu } \frac{{\partial x^{i} }}{\partial t}\frac{{\partial x^{j} }}{\partial t} + \left( {\varGamma_{00}^{0} + 2\varGamma_{0i}^{0} \frac{{\partial x^{i} }}{\partial t} + \varGamma_{ij}^{0} \frac{{\partial x^{i} }}{\partial t}\frac{{\partial x^{j} }}{\partial t}} \right)\frac{{\partial x^{\mu } }}{\partial t} , $$
(C.1)

where the connection coefficients have been separated into time and spatial components. Using the connection coefficients \( 2\varGamma_{\sigma \rho }^{\mu } = \eta^{\mu \nu } (h_{\sigma \nu ,\rho } + h_{\rho \nu ,\sigma } - h_{\sigma \rho ,\nu } ) \), the first term on the right can be written as,

$$ \varGamma_{00}^{\mu } = \eta^{\mu \nu } \left( {\frac{{\partial h_{0\nu } }}{\partial t} - \frac{1}{2}\frac{{\partial h_{00} }}{{\partial x^{\nu } }}} \right) \to \frac{{\partial {\vec{\mathfrak{A}}}}}{\partial t} + \vec{\nabla }\varphi , $$
(C.2)

where \( \varphi = h_{00} /2 \) and \( {\vec{\mathfrak{A}}} = ( - h_{0x} , - h_{0y} , - h_{0z} ) \). The second term in Eq. C.1 is

$$ 2\varGamma_{0i}^{\mu } \frac{{\partial x^{i} }}{\partial t} = \eta^{\mu \nu } \left[ {\frac{{\partial h_{i\nu } }}{\partial t} + \left( {h_{0\nu ,i} - h_{0i,\nu } } \right)} \right]\frac{{\partial x^{i} }}{\partial t} \to - \vec{v} \cdot \frac{{\partial \vec{h}}}{\partial t} - \vec{v} \times \left( {\vec{\nabla } \times {\vec{\mathfrak{A}}}} \right) . $$
(C.3)

Here \( {\vec{\mathfrak{B}}} = \vec{\nabla } \times {\vec{\mathfrak{A}}} \) and \( \vec{h} \) a matrix equal to the spatial part of \( h_{\mu \nu } \). Exception for the term \( \vec{v} \cdot \partial \vec{h}/\partial t \), Eqs. C.2 and C.3 are mathematically equivalent to the Lorentz force. The third term in Eq. C.1 is somewhat more complex,

$$ \begin{aligned} \varGamma_{ji}^{\mu } \frac{{\partial x^{j} }}{\partial t}\frac{{\partial x^{i} }}{\partial t} & = - \frac{1}{2}\left( {h_{i\mu ,j} + h_{j\mu ,i} } \right)\frac{{\partial x^{j} }}{\partial t}\frac{{\partial x^{i} }}{\partial t} + \frac{1}{2}h_{ij,\mu } \frac{{\partial x^{j} }}{\partial t}\frac{{\partial x^{i} }}{\partial t} \\ \varGamma_{ab}^{\mu } \frac{{\partial x^{a} }}{\partial t}\frac{{\partial x^{b} }}{\partial t} & \to - \frac{1}{2}\left[ {2\left( {\vec{v} \cdot \vec{\nabla }_{h} } \right)\left( {\vec{h} \cdot \vec{v}} \right) - \vec{\nabla }_{h} \left( {\vec{v} \cdot \vec{h} \cdot \vec{v}} \right)} \right] \\ \end{aligned} . $$
(C.4)

The next two terms together are,

$$ \varGamma_{00}^{0} + 2\varGamma_{0i}^{0} \frac{{\partial x^{i} }}{\partial t} = \frac{1}{2}\frac{{\partial h_{00} }}{\partial t} + \frac{{\partial x^{i} }}{\partial t}\frac{{\partial h_{00} }}{{\partial x^{i} }} = \frac{\partial \varphi }{\partial t} + 2\left( {\vec{v} \cdot \vec{\nabla }} \right)\varphi , $$
(C.5)

and the last term is

$$ \begin{aligned} \varGamma_{ij}^{0} \frac{{\partial x^{i} }}{\partial t}\frac{{\partial x^{j} }}{\partial t} & = \frac{{\partial x^{i} }}{\partial t}\frac{{\partial x^{j} }}{\partial t}\frac{{\partial h_{i0} }}{{\partial x^{j} }} - \frac{1}{2}\frac{{\partial x^{i} }}{\partial t}\frac{{\partial x^{j} }}{\partial t}\frac{{\partial h_{ij} }}{\partial t} \\ \varGamma_{ij}^{0} \frac{{\partial x^{i} }}{\partial t}\frac{{\partial x^{j} }}{\partial t} & = - \,\left( {\vec{v} \cdot \vec{\nabla }_{{\mathfrak{A}}} } \right)\left( {\vec{v} \cdot {\vec{\mathfrak{A}}}} \right) - \frac{1}{2}\vec{v} \cdot \left( {\frac{{\partial \vec{h}}}{\partial t} \cdot \vec{v}} \right) \\ \end{aligned} $$
(C.6)

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Strohaber, J. General relativistic manifestations of orbital angular and intrinsic hyperbolic momentum in electromagnetic radiation. Gen Relativ Gravit 52, 56 (2020). https://doi.org/10.1007/s10714-020-02702-1

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