Geodesic congruences in 5D warped Ellis–Bronnikov spacetimes

Abstract

We study the timelike geodesic congruences in the generalised Ellis–Bronnikov spacetime (4D-GEB) and in recently proposed 5D model where a 4D-GEB is embedded in a warped geometry (5D-WGEB) and conduct a comparative study. Analytical expressions of ESR variables (for 4D geometries) are found which reveal the role of the wormhole parameter. In more general 4D and 5D scenarios geodesic equation, geodesic deviation equation and Raychaudhuri equations are solved numerically. The evolution of cross-sectional area of the congruences of timelike geodesics (orthogonal to the geodesic flow lines) projected on 2D-surfaces yield an interesting perspective and shows the effects of the wormhole parameter and growing/decaying warp factors. The presence of war** factor triggers rotation or accretion even in the absence of initial congruence rotation. The presence of rotation in the congruence is also found to be playing a crucial role which we discuss in detail.

Appendix

1.1 Boundary conditions to determine ESR profiles in 4D-GEB spacetimes

$k = \sqrt{3}$, $h = 0$; $t(0) = 0$, $l(0) = 0$, $\theta (0) = \pi /2$, $\phi (0) = 0$ ;

$\dot{t}(0) = k$, $\dot{l}(0) = 1.41421$, $\dot{\theta }(0) = 0$, $\dot{\phi }(0) = \frac{h}{(1 + l(0)^{m})^{2/m}} = 0$ ;

$B_{AB}(0) = 0$ for without rotation and (initially $\theta$, $\Sigma _{AB}$ and $\Omega _{AB}$ is zero) ;

For with rotation case: $\theta$ and $\Sigma _{AB}$ are zero at $\lambda =0$ but nonzero $\Omega _{AB}$ are chosen such that $\Omega _{AB}u^{B} = 0 = u^{A}\Omega _{AB}$ at $\lambda = 0$

1.2 Boundary conditions to determine ESR profiles in 5D-WGEB spacetimes with growing warp factor

$T = \sqrt{3}$, $H = 0$; $t(0) = 0$, $l(0) = 0$, $\theta (0) = \pi /2$, $\phi (0) = 0$, $y(0) = 0.1$ ;

$\dot{t}(0) = \frac{T}{e^{2f(y)}} = 1.71485$, $\dot{l}(0) = 1.39665$, $\dot{\theta }(0) = 0$, $\dot{\phi }(0) = \frac{H}{(1 + l(0)^{m})^{2/m}} = 0$,$\dot{y}(0) = 0$ ;

$B_{AB}(0) = 0$ for without rotation that implies $\theta (0)$, $\Sigma _{AB}(0)$ and $\Omega _{AB}(0)$ is zero ;

For with rotation case: $\theta$ and $\Sigma _{AB}$ are zero at $\lambda =0$ but nonzero $\Omega _{AB}$ are chosen such that $\Omega _{AB}u^{B} = 0 = u^{A}\Omega _{AB}$ at $\lambda = 0$

1.3 Boundary conditions to determine ESR profiles in 5D-WGEB spacetimes with decaying warp factor

$T = \sqrt{3}$, $H = 0$ ; $t(0) = 0$, $l(0) = 0$, $\theta (0) = \pi /2$, $\phi (0) = 0$, $y(0) = 0.1$ ;

$\dot{t}(0) = \frac{T}{e^{2f(y)}} = 1.74943$, $\dot{l}(0) = 1.43195$, $\dot{\theta }(0) = 0$, $\dot{\phi }(0) = \frac{H}{(1 + l(0)^{m})^{2/m}} = 0$,$\dot{y}(0) = 0$ ;

$B_{AB}(0) = 0$ for without rotation that implies $\theta (0)$, $\Sigma _{AB}(0)$ and $\Omega _{AB}(0)$ is zero ;

For with rotation case: initially $\theta$ and $\Sigma _{AB}$ are still zero but nonzero $\Omega _{AB}$ are chosen such that $\Omega _{AB}u^{B} = 0 = u^{A}\Omega _{AB}$ at $\lambda = 0$

1.4 Boundary conditions for evolution of cross-sectional area in 4D-GEB spacetimes

$k = \sqrt{3}$, $h = \sqrt{\frac{k^{2} - 1 }{2}} = 1$ ; $t(0) = 0$, $l(0) = 0$, $\theta (0) = \pi /2$, $\phi (0) = 0$ ;

$\dot{t}(0) = k$, $\dot{l}(0) = 1$, $\dot{\theta }(0) = 0$, $\dot{\phi }(0) = \frac{h}{(1 + l(0)^{m})^{2/m}} = h = 1$ ;

$\dot{l}(0)$ is calculated from the timelike geodesic constraint ;

$B_{AB}(0) = 0$ ; $\xi ^{1}(0) = \pm 1$, $\xi ^{2}(0) = 0$, $\xi ^{3}(0) = \pm 1$ ;

$\xi ^{0}(0) = \pm \frac{2}{\sqrt{3}}, 0$ is calculated from $u_{A}\xi ^{A} = 0$.

1.5 Boundary conditions to determine evolution of cross-sectional area in 5D-WGEB spacetimes with growing warp factor

$t(0) = 0$, $l(0) = 0$, $\theta (0) = \pi /2$, $\phi (0) = 0$, $y(0) = a = 0.1$ ;

$\dot{t}(0) = T$, $\dot{l}(0) = 0.98758$ , $\dot{\theta }(0) = 0$, $\dot{\phi }(0) = \frac{H}{\cosh (a)^{2}(1 + l(0)^{m})^{2/m}} = 0.98758$, $\dot{y}(0) = b = 0$ ;

$T = \sqrt{3}$, $H = \sqrt{\frac{k^{2} - \cosh (a)^{2}(b^{2} + 1)}{2}} = 0.997489$ ;

$B_{\mu \nu }(0) = 0$ , where $\mu$ and $\nu$ runs from 0 to 4 ;

$\xi ^{1}(0) = \pm \frac{1}{\cosh (a)} = \pm 0.995021$, $\xi ^{2}(0) = 0$, $\xi ^{3}(0) = \pm \frac{1}{\cosh (a)} = \pm 0.995021$, $\xi ^{4}(0) = \pm \frac{1}{\cosh (a)} = \pm 0.995021$ ;

$\xi ^{0}(0) = \pm 1.19213, \pm 0.0574475$ is calculated from $u_{A}\xi ^{A} = 0$.

1.6 Boundary conditions to determine evolution of cross-sectional area in 5D-WGEB spacetimes with decaying warp factor

$t(0) = 0$, $l(0) = 0$, $\theta (0) = \pi /2$, $\phi (0) = 0$, $y(0) = a = 0.1$ ;

$\dot{t}(0) = T$, $\dot{l}(0) = 1.01254$ , $\dot{\theta }(0) = 0$, $\dot{\phi }(0) = \frac{H}{\mathrm{sech}(a)^{2} (1 + l(0)^{m})^{2/m}} = 1.01254$, $\dot{y}(0) = b = 0$ ;

$T = \sqrt{3}$, $H = \sqrt{\frac{k^{2} - \mathrm{sech}(a)^{2}(b^{2} + 1)}{2}} = 1.00248$ ;

$\xi ^{1}(0) = \pm \frac{1}{\mathrm{sech}(a)} = \pm 1.005$, $\xi ^{2}(0) = 0$, $\xi ^{3}(0) = \pm \frac{1}{\mathrm{sech}(a)} = \pm 1.005 = \xi ^{4}(0)$ ;

$\xi ^{0}(0) = \pm 1.23305, \pm 0.0580239$ is calculated from $u_{A}\xi ^{A} = 0$.