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Wave optics and weak gravitational lensing of lights from spherically symmetric static scalar vector tensor Brans Dicke wormholes


This paper has three parts. In first step we use modified scalar tensor vector Brans Dicke gravity (Ghaffarnejad in Gen Relativ Gravit 40:2229, 2008) to obtain metric solution of a spherically symmetric static curved space time in two kind isotropic and anisotropic coordinates system in which stress tensor behaves as anisotropic spherically symmetric perfect fluid. Metric solution is similar to a wormhole line element with a power law mass function in the anisotropic coordinates system. Exponent in the mass function is ratio of directional barotropic indexes. Second part of the paper is dedicated to review Fresnel–Kirchhoff diffraction theory for massless scalar field instead of the vector electromagnetic waves. In third part of the work we consider the obtained wormhole metric solution to be gravitational lens and we study weak gravitational lensing of moving light which is originated from a point star. Production of stationary images is investigated via both interference of waves and geometric optics approaches. At last we check correspondence of produced images between two approaches and give some outlooks.

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This work was supported in part by the Semnan University Grant No.1398-07-061108 for Scientific Research.

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Correspondence to Hossein Ghaffarnejad.

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Appendix A

To reproduce the equations given by [1] which we applied in this work for Lorentzian signature (-+++) we should consider important published paper [22] which is in fact generalization of paper [23] where its author Fernando Barbero generalized the Einstein Hilbert action via a unit dynamical timelike vector field \(\eta _\alpha \) with two real parameter \(\alpha , \beta \) as \(S(g_{ab};\alpha ,\beta )=\int _M \mathrm{d}x^4\sqrt{g}[\alpha G^{ab}+\beta R^{ab}]\eta _a\eta _b.\) Then he generalized this model by his collaborator at [22] by considering a suitable particular relation between an Euclidean signature metric field with corresponding Lorentzian signature \((+---)\) as follows.

$$\begin{aligned} g^E_{ab}=-\frac{1}{2}\sqrt{|\alpha (\alpha +2\beta )|}\bigg [g_{ab}^L-2 \bigg (\frac{\alpha +\beta }{\alpha +2\beta }\bigg )u_au_b\bigg ] \end{aligned}$$

where \(g^{E(L)}_{ab}\) is Euclidean (Lorentzian) signature metric field components and \(u_a\) is a unit timelike dynamical vector field satisfied by \(g_{ab}^Lu^au^b=1.\) If we set

$$\begin{aligned} \xi =\frac{\alpha +\beta }{\alpha +2\beta },~~~\alpha =\pm 2\sqrt{|2\xi -1|},~~~u_a=N_a \end{aligned}$$

then the above transformation reduces to the equation (2.3) in ref. [1]. Applying the definition (123) the determinant of the metric field given by Eq. 3 in [22] reduces to

$$\begin{aligned} g^E=det g^E_{ab}=-|2\xi -1|det g^L_{ab}=-|2\xi -1|g^L. \end{aligned}$$

Substituting (123) the Euclidean Christoffel symbols given by Eq. (5) in ref. [22] reads

$$\begin{aligned} \Gamma _{bc}^{a (E)}= & {} \Gamma _{bc}^{a(L)}- \xi [\nabla _{b}(u^au_c)+\nabla _c(u^au_b)-\nabla ^a(u_bu_c)] \nonumber \\&+\frac{2\xi ^2}{2\xi -1}[u^a\nabla _bu_c+u^a\nabla _cu_b-u^au^d\nabla _b(u_bu_c)]. \end{aligned}$$

If we substitute definitions (4) given by the present paper for which we can write

$$\begin{aligned} \nabla _au_b=\frac{F_{ab}+\Omega _{ab}}{4} \end{aligned}$$

and by replacing \(N_{\mu }\rightarrow u_a,\) the relation (125) reduces to the following form.

$$\begin{aligned} \Gamma _{bc}^{a(E)}= & {} \gamma _{bc}^{a(L)}-\frac{\xi }{2}[u_cF_b^a+u_bF_c^a+u^a\Omega _{bc}] \nonumber \\&-\frac{\xi ^2}{(2\xi -1)}u^a\bigg \{\frac{u_cu^d}{2}( F_{db}+\Omega _{db})+\frac{u_bu^d}{2}(F_{dc}+\Omega _{dc})-\Omega _{ab}\bigg \}. \end{aligned}$$

I should notice that this equation is different with F terms at last part with respect to Eq. (2.6) given by ref. [1] regretfully and should be considered as corrections on the gravity model given by [1]. Now we seek that how can be transformed an Euclidean Ricci scalar \(R^E\) to its Lorentzian version \(R^L\) under the transformation (122). To do so we consider action functional (6) given by [22] in which Lagrangian density is

$$\begin{aligned} \sqrt{|g^E|}R^E=\mathrm{sgn}(\alpha )\sqrt{|g^L|}\bigg [- \frac{\alpha }{2}R^L+(\alpha +\beta )u^au^bR^L_{ab}-\frac{(\alpha +\beta )^2}{\alpha +2\beta }g_{L}^{ab}t_at_b \bigg ] \end{aligned}$$

with \(t_a\) to be twist of \(u^a\) given by Eq. (7) in ref. [22]. This satisfies the identity

$$\begin{aligned} g^{ab}_Lt_at_b=\frac{1}{2}F_{ab}F^{ab}-(u^c\nabla _cu_a)(u^d\nabla _du^a) \end{aligned}$$

given by Eq. (8) in ref. [22]. Also sgn\((\alpha )=+1(-1)\) for \(\alpha >0(<0)\) respectively. Now we substitute (123) and (124) into the above Lagrangian density to obtain

$$\begin{aligned} R^E=\frac{\mathrm{sgn}(\alpha )}{2(2\xi -1)}\{2(1-2\xi )R^L-4\xi u^au^bR^L_{ab}-2\xi ^2F_{ab}F^{ab} +4\xi ^2u^cu^d\nabla _cu_a\nabla _du^a\}. \end{aligned}$$

If we substitute the identity (126) into the last term of the above relation we will have

$$\begin{aligned} R^E= & {} \frac{\mathrm{sgn}(\alpha )}{2(2\xi -1)}\{2(1-2\xi )R^L-4\xi u^au^bR^L_{ab}-2\xi ^2F_{ab}F^{ab} \nonumber \\&+\frac{\xi ^2u^cu_d}{4}[F_{ca}F^{\mathrm{d}a}+2\Omega _{ca}F^{\mathrm{d}a}+\Omega _{ca}\Omega ^{\mathrm{d}a}]\} \end{aligned}$$

which its coefficients for \(R_{ab}\) and last term is different with respect to Eq. (2.9) given by ref. [1] regretfully and they should be considered as corrections when we want to study metric signature transition property of a curved spacetime in quantum level.

We should remember that the above calculations were produced for a first time by Barbero and Villasenor for Lorentzian signature \((+---)\) while we like to reproduce them for Lorentzian signature \((-+++)\) in this work. To do so we should replace \(g_{ab}^L\rightarrow -g_{ab}^L\) to infer that \(\Gamma _{bc}^{a(L)}(+---)=\Gamma _{bc}^{a(L)}(-+++),\) \(R_{ab}^L(+---)=R_{ab}^L(-+++),\) \(R_L(+---)=-R_L(-+++)\) \(g_{ab}^E=-g_{ab}^L+2\xi u_au_b\rightarrow g_{ab}^E=g_{ab}^L+2\xi u_au_b\), \(g_L^{ab}u_au_b=1\rightarrow g_L^{ab}u_au_b=-1\) and so (131) reads

$$\begin{aligned} R^E= & {} \frac{\mathrm{sgn}(\alpha )}{2(2\xi -1)}\{2(2\xi -1)R^L-4\xi u^au^bR^L_{ab}-2\xi ^2F_{ab}F^{ab} \nonumber \\&+\frac{\xi ^2u^cu_d}{4}[F_{ca}F^{\mathrm{d}a}+2\Omega _{ca}F^{\mathrm{d}a}+\Omega _{ca}\Omega ^{\mathrm{d}a}]\}. \end{aligned}$$

Now we are in position to generate Lorentzian version of Brans Dicke action functional from its Euclidean version under the metric transformation \(g_{ab}^E=g_{ab}^L+2\xi u_au_b\). Vacuum sector of Brans Dicke scalar tensor gravity in a Lorentzian signature \((-+++)\) background metric is given by [19] which we assume it is true for an Euclidean signature (++++) Riemannian manifold (torsion free \(2\tau _{bc}^a=\Gamma ^a_{bc}-\Gamma ^a_{cb}=0\)) as follows.

$$\begin{aligned} I^E_{\mathrm{BD}}=\frac{1}{16\pi }\int \mathrm{d}x^4\sqrt{|g^E|}\{\phi ^E R^E-\frac{\omega ^E}{\phi ^E}g_E^{ab}\partial _a\phi ^E\partial _b\phi ^E\}. \end{aligned}$$

Applying \(g_{ab}^E=g_{ab}^L+2\xi u_au_b\) and definition Kerenoker delta function \(\delta _c^{a(E)}=g^{ab}_{E}g_{bc}^{E}=(g^{ab}_L+su^au^b)(g^L_{ab}+2\xi u_bu_c)\) and \(\delta _c^{a(L)}=g^{ab}_{L}g_{bc}^L\) we obtain

$$\begin{aligned} g^{ab}_E= g^{ab}_L+\bigg (\frac{2\xi }{2\xi -1}\bigg )u^au^b \end{aligned}$$

which satisfies equation (2.5) in ref. [1] with signature \((-+++)\). By substituting (124), (131) and (134) into the above action functional we obtain

$$\begin{aligned} I_{\mathrm{BD}}^E=I_{\mathrm{BD}}^L+I_{\mathrm{signature}} \end{aligned}$$

in which we defined

$$\begin{aligned} I_{\mathrm{signature}}= & {} \frac{1}{16\pi }\bigg (\frac{2\xi }{2\xi -1}\bigg )\int \mathrm{d}x^4\sqrt{|g^L|}\phi ^L\bigg \{\frac{\xi }{16}u^cu_d(F_{ca}F^{\mathrm{d}a}+2F^{\mathrm{d}a}\Omega _{ca}+\Omega _{ca}\Omega ^{\mathrm{d}a}) \nonumber \\&-\frac{\xi }{2}F_{ab}F^{ab}-u^au^bR_{ab}^L-\frac{\omega ^L}{\phi _L^2}u^au^b\partial _a\phi ^L\partial _b\phi ^L\bigg \} \end{aligned}$$


$$\begin{aligned} \frac{\phi ^L}{\phi ^E}=\mathrm{sgn}(\alpha )\sqrt{|2\xi -1|},~~~\frac{\omega ^E}{\omega ^L}=\mathrm{sgn}(\alpha ). \end{aligned}$$

By setting \(\xi =1\) and \(u_a\rightarrow N_{\mu }\) and by dropping the index L,  the action functional (135) reads to the equation (1) in this paper in which \(\zeta (x^\mu )\) and \(U(\phi ,N^\mu )\) are additional terms.

Appendix B

We defined

$$\begin{aligned} P_{\mu \nu }= & {} -\zeta (x^\mu )N_\mu N_\nu +\phi (F_{\mu \beta }F_\nu ^{~\beta }+F_{\alpha \mu }F_{~\nu }^\beta )+g_{\rho \mu }g_{\sigma \nu }\nabla _\alpha \nabla _\beta (\phi g^{\rho \sigma }g^{\alpha \beta }) \nonumber \\&+g_{\rho \mu }g_{\alpha \nu }\nabla _\sigma \nabla _\beta (\phi g^{\rho \sigma }g^{\alpha \beta })+\frac{\omega }{\phi }\nabla _\mu \phi \nabla _\nu \phi \nonumber \\&-\frac{\phi }{4}[N^\xi N_\nu \Omega _\mu ^{~\eta }F_{\xi \eta }+N_\mu N_{\theta }\Omega ^{\theta \eta }F_{\nu \eta }+N^\xi N_{\theta }\Omega _{~\mu }^{\theta }F_{\xi \nu }] \nonumber \\&-\frac{\phi }{8}[N^\xi N_\nu F_\mu ^{~\eta }F_{\xi \eta }+N_\mu N_{\theta }F^{\theta \eta }F_{\nu \eta }+N^\xi N_{\theta }F_{~\mu }^{\theta }F_{\xi \nu }] \nonumber \\&-\frac{\phi }{4}[N^\xi N_\nu \Omega _\mu ^{~\eta }\Omega _{\xi \eta }+N_\mu N_{\theta }\Omega ^{\theta \eta }\Omega _{\nu \eta }+N^\xi N_{\theta }\Omega _{~\mu }^{\theta }\Omega _{\xi \nu }]+4\phi N^\theta N_\nu R_{\mu \theta } \nonumber \\&-\frac{g_{\mu \nu }}{\sqrt{g}}\nabla _\alpha \nabla _\beta (\phi \sqrt{g}N^\alpha N^\beta )+\frac{g_{\beta \mu }g_{\eta \nu }}{\sqrt{g}}\nabla _\alpha \nabla ^\eta (\phi \sqrt{g}N^\alpha N^\beta ) \nonumber \\&+\frac{g_{\alpha \mu }g_{\eta \nu }}{\sqrt{g}}\nabla _\beta \nabla ^\eta (\phi \sqrt{g}N^\alpha N^\beta )-\frac{g_{\alpha \mu }g_{\beta \nu }}{\sqrt{g}}\nabla _\eta \nabla ^\eta (\phi \sqrt{g}N^\alpha N^\beta ) \end{aligned}$$


$$\begin{aligned} Q= & {} U(\phi ,N^\mu )-\phi F_{\alpha \beta }F^{\alpha \beta }+\frac{\phi }{8} N_\alpha N^\beta \Omega ^{\alpha \lambda }\Omega _{\beta \lambda } \nonumber \\&-\frac{\omega }{\phi }g^{\alpha \beta }\nabla _\alpha \phi \nabla _\beta \phi -2\phi N_\alpha N^\beta R^\alpha _\beta -\frac{2\omega }{\phi }N_\alpha N^\beta \nabla ^\alpha \phi \nabla _\beta \phi \end{aligned}$$

which they obtained by varying the total action (1) with respect to \(g^{\mu \nu }\) and by dropping covariant divergence-less terms which are generated after integration by parts and use of the following identities.

$$\begin{aligned} \delta \sqrt{g}= & {} -\frac{1}{2}\sqrt{g}g_{\mu \nu }\delta g^{\mu \nu } \end{aligned}$$
$$\begin{aligned} F_{\alpha \beta }\delta (F^{\alpha \beta })= & {} (F_{\mu \beta }F_{\nu }^\beta +F_{\alpha \mu }F^{\alpha }_{\nu })\delta g^{\mu \nu } \end{aligned}$$
$$\begin{aligned} \delta (N_{\alpha }N^{\beta }F^{\alpha \lambda }\Omega _{\beta \lambda })= & {} \{N^\xi N_{\nu }\Omega _\mu ^\eta F_{\xi \eta }+N_\mu N_\theta \Omega ^{ \theta \eta }F_{\nu \eta } +N^\xi N_\theta \Omega ^\theta _\mu F_{\xi \nu }\}\delta g^{\mu \nu }\nonumber \\ \end{aligned}$$
$$\begin{aligned} \delta (N_{\alpha }N^{\beta }F^{\alpha \lambda }F_{\beta \lambda })= & {} \{N^\xi N_{\nu }F_\mu ^\eta F_{\xi \eta }+N_\mu N_\theta F^{\theta \eta }F_{\nu \eta } +N^\xi N_\theta F^\theta _\mu F_{\xi \nu }\}\delta g^{\mu \nu }\nonumber \\ \end{aligned}$$
$$\begin{aligned} \delta (N_{\alpha }N^{\beta }\Omega ^{\alpha \lambda }\Omega _{\beta \lambda })= & {} \{N^\xi N_{\nu }\Omega _\mu ^\eta \Omega _{\xi \eta }+N_\mu N_\theta \Omega ^{ \theta \eta }\Omega _{\nu \eta } +N^\xi N_\theta \Omega ^\theta _\mu \Omega _{\xi \nu }\}\delta g^{\mu \nu }\nonumber \\ \end{aligned}$$
$$\begin{aligned} \delta (N^\alpha N^\beta \partial _\alpha \phi \partial _\beta \phi )= & {} (2N_\nu N^\alpha \partial _\mu \phi \partial _\alpha \phi )\delta g^{\mu \nu } \end{aligned}$$
$$\begin{aligned} \delta (\phi N^\alpha N^\beta R_{\alpha \beta })= & {} \phi R_{\alpha \beta }\delta (N^\alpha N^\beta )+\phi N^\alpha N^\beta \delta (R_{\alpha \beta }). \end{aligned}$$

In the above relations we have

$$\begin{aligned} R_{\alpha \beta }\delta (N^\alpha N^\beta )=2R_{\mu \theta }N^\theta N_\nu \delta g^{\mu \nu } \end{aligned}$$

and from [24]

$$\begin{aligned} \delta R_{\alpha \beta }=\nabla _\beta (\delta \Gamma _{\alpha \lambda }^\lambda )-\nabla _\lambda (\delta \Gamma _{\alpha \beta }^\lambda ) \end{aligned}$$

for which

$$\begin{aligned} \delta \Gamma _{\alpha \beta }^\lambda= & {} \frac{1}{2}\delta g^{\lambda \eta } (\nabla _\alpha g_{\beta \eta }+\nabla _\beta g_{\alpha \eta }-\nabla _\eta g_{\alpha \beta }) \nonumber \\&+\frac{1}{2}g^{\lambda \eta }(\nabla _\alpha \delta g_{\beta \eta }+\nabla _\beta \delta g_{\alpha \eta }-\nabla _\eta \delta g_{\alpha \beta }) \end{aligned}$$

and from [25]

$$\begin{aligned} \delta g_{\beta \eta }=-g_{\beta \mu }g_{\eta \nu }\delta g^{\mu \nu } \end{aligned}$$

we obtain after integration by parts

$$\begin{aligned} -2\sqrt{g}\phi N^\alpha N^\beta \delta R_{\alpha \beta }= & {} \frac{\delta g^{\mu \nu }}{\sqrt{g}}\{g_{\mu \nu }\nabla _\alpha \nabla _\beta (\phi \sqrt{g}N^\alpha N^\beta ) \nonumber \\&-g_{\beta \mu }g_{\eta \nu }\nabla _\alpha \nabla ^\eta (\phi \sqrt{g}N^\alpha N^\beta )-g_{\alpha \mu }g_{\eta \nu }\nabla _\beta \nabla ^\eta (\phi \sqrt{g}N^\alpha N^\beta ) \nonumber \\&+g_{\alpha \mu }g_{\beta \nu }\nabla _\eta \nabla ^\eta (\phi \sqrt{g}N^\alpha N^\beta )\} \end{aligned}$$

and from [25] we use

$$\begin{aligned} \delta R=-R^{\alpha \beta }\delta g_{\alpha \beta }+g^{\rho \sigma }g^{\alpha \beta }(\nabla _\nu \nabla _\mu \delta g_{\rho \sigma }+\nabla _\nu \nabla _\sigma \delta g_{\rho \mu }) \end{aligned}$$

and calculate

$$\begin{aligned} \sqrt{g}\phi \delta R= & {} \frac{\delta g^{\mu \nu }}{\sqrt{g}}\{ \phi \sqrt{g}R_{\mu \nu }-g_{\rho \mu }g_{\sigma \nu }\nabla _\alpha \nabla _\beta (\phi \sqrt{g} g^{\rho \sigma }g^{\alpha \beta })- \nonumber \\&g_{\rho \mu }g_{\alpha \nu }\nabla _\beta \nabla _\sigma ( \phi \sqrt{g}g^{\rho \sigma }g^{\alpha \beta })\} \end{aligned}$$

after integration by parts.

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Ghaffarnejad, H. Wave optics and weak gravitational lensing of lights from spherically symmetric static scalar vector tensor Brans Dicke wormholes. Gen Relativ Gravit 54, 13 (2022).

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  • Fresnel-Kirchhoff diffraction theory
  • Waves
  • Ghravitational lensing
  • Althernative gravities
  • Vector field
  • Brans Dicke scalar field
  • acceleration