Double-Kasner spacetime: Peculiar velocities and cosmic jets

In dynamic spacetimes in which asymmetric gravitational collapse/expansion is taking place, the timelike geodesic equation appears to exhibit an interesting property: Relative to the collapsing configuration, free test particles undergo gravitational “acceleration” and form a double-jet configuration parallel to the axis of collapse. We illustrate this aspect of peculiar motion in simple spatially homogeneous cosmological models such as the Kasner spacetime. To estimate the effect of spatial inhomogeneities on cosmic jets, timelike geodesics in the Ricci-flat double-Kasner spacetime are studied in detail. While spatial inhomogeneities can significantly modify the structure of cosmic jets, we find that under favorable conditions the double-jet pattern can initially persist over a finite period of time for sufficiently small inhomogeneities.

Figure 1

Conformal time diagrams of the Einstein-de Sitter model showing free particle orbits. Lightlike geodesics would be at 45 degrees. Vertical lines are worldlines of fundamental observers. The curved lines are examples of noncomoving free orbits. The behavior as $t\to 0$ (corresponding to $\eta \to 0$) shows the (negative time) acceleration towards the speed of light relative to the fundamental observers (or equivalently the homogeneous hypersurfaces). Left panel: Orbits corresponding to one value of the parameter $b$ are shown. Right panel: The family of orbits which is attracted to the fiducial observer at $r=0$ in the limit $t\to 0$ is shown. These orbits have different values of $b$ corresponding to different initial distances from the fiducial observer.

Figure 2

Plot of the Lorentz factor $\gamma$ versus proper time $\tau$ for the peculiar motion of a free test particle along the $x$ direction in Case 2. The initial conditions at $\tau =0$ are $x_0=0$ and $W_0=1$; moreover, $ϵ={10}^{-3}$ and $C_2=C_3=0$. Thus at $\tau =0$ we have $\gamma =\sqrt{2}\approx 1.4$. The figure shows that the initial peculiar acceleration reaches a peak around $\tau =50$ and then turns to deceleration. Further numerical integration shows that the particle eventually stops at around $\tau =50000$, reverses course and accelerates toward the $\xi =0$ singularity with $\gamma \to \infty$.

Figure 3

Plot of the Lorentz factor $\gamma$ versus proper time $\tau$ for the peculiar motion of a free test particle in the $(x,y)$ plane in Case 3. Note that initially $\gamma =\sqrt{2}\approx 1.4$ at $\xi =1$, but in time $\gamma \to \infty$ as the $\xi =0$ singularity is approached. The initial conditions at $\tau =0$ are $x_0=0$ and $W_0=0$; moreover, $ϵ={10}^{-3}$, $C_2=\pm 1$ and $C_3=0$.

Figure 4

Plot of the Lorentz factor $\gamma$ versus proper time $\tau$ in Case 4. The initial conditions at $\tau =0$ are $x_0=0$ and $W_0=0$; moreover, $ϵ=1$, $C_2=0$ and $C_3=\pm 1$. Note that $\gamma$ is initially $\sqrt{2}\approx 1.4$ and approaches a finite terminal value as $\tau \to \infty$.

Figure 5

Plot of the oscillatory Lorentz factor $\gamma$ versus proper time $\tau$ in Case 2. The initial conditions at $\tau =0$ are $x_0=-1$ and $W_0=0$; moreover, $ϵ=1$, $C_2=1$ and $C_3=2$. Note that the Lorentz factor is initially $\gamma \approx 2.97$. The components of the peculiar velocity are all oscillatory with increasing amplitudes for $v_x$ and $v_y$ and decreasing amplitude for $v_z$. In fact, $v_z$ tends to zero as ${\xi }_{\min }$ approaches zero.