Construction and application of variations on the cylindrical gravitational waves of Weber, Wheeler, and Bonnor

To clarify certain nonlinear properties of strong gravitational field, we investigate cylindrically symmetric gravitational waves that are localized as regular wave packets in the space of radial and time coordinates. The waves are constructed by applying a certain kind of harmonic mapping method to the seed solutions with linear polarization, which are generalizations of the solution representing a cylindrical gravitational pulse wave discussed by Weber, Wheeler, and Bonnor. The solutions obtained here, though their form is rather simple, show occurrence of strong mutual conversion between a linear mode and a cross mode apparently. The single localized wave shows the conversion in the vicinity of the symmetric axis where the self-interaction is strengthened, and the collision between multiple waves also causes the conversion. These phenomena can be thought to be the emergence of genuine nonlinearity that the Einstein gravity holds. Finally we discuss a simple, but interesting application of the solutions to the case of the Einstein-Maxwell system.

Figure 1

Time dependence of the energy densities $\mathcal{E}$, ${\mathcal{E}}_{+}$, and ${\mathcal{E}}_{\times }$ for $(a,c,A,\delta )=(1,2,0.05,0)$. Each figure displays the snapshots of $\mathcal{E}$, ${\mathcal{E}}_{+}$, and ${\mathcal{E}}_{\times }$ in the order from left to right, where the gray, red, and blue graphs show the pulses corresponding to $t=-15$, 0, 14, respectively.

Figure 2

Time dependence of the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ for $(a,c,A,\delta )=(1,2,0.05,0)$. In the figures from left to right, the red graph display the ratio at $t=-10$, 0, 10

Figure 3

Time dependence of the energy densities $\mathcal{E}$, ${\mathcal{E}}_{+}$, and ${\mathcal{E}}_{\times }$ for $(a,c,A,\delta )=(1,2,1,0)$. Each figure displays the snapshots of $\mathcal{E}$, ${\mathcal{E}}_{+}$, and ${\mathcal{E}}_{\times }$ in the order from left to right, where the gray, red, and blue graphs show the pulses corresponding to $t=-15$, 0, 14, respectively.

Figure 4

Time dependence of the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ for $(a,c,A,\delta )=(1,2,1,0)$. In the figures from left to right, the graphs display the ratio at $t=-10$, 0, 10.

Figure 5

The left and right figures display the time dependence of the ratio $E_{+}(t,1000)/E(t,1000)$ for $(a,c,A,\delta )=(1,2,0.05,0)$ and $(a,c,A,\delta )=(1,2,1,0)$, respectively.

Figure 6

The upper graphs show the snapshots of pulses at the collision for $(a_1,a_2,c_1,c_2,t_1,t_2,A,\delta )=(1,1,2,3,-45,55,0.05,0)$, where from left to right, the three graphs correspond to $t=0$, 4.6, 10. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.

Figure 7

The time dependence of the ratio $E_{+}(t,1000)/E(t,1000)$ when an outgoing pulse with $(a_1,c_1,t_1)=(1,2,-45)$ and an ingoing pulse with $(a_2,c_2,t_2)=(1,3,55)$ collide at $t\simeq 5$, where the left and right graphs display the ratio for $(A,\delta )=(0.05,0)$ and $(A,\delta )=(1,0)$, respectively.

Figure 8

The snapshots at the collision of a single large gravitational pulse with $(t_0,a_0,c_0)=(-40,1/7,2)$ and six small gravitational pulses with $(t_i,a_i,c_i)=(56+4i,1,3)(i=1,\dots ,6)$ for $A=0.05$ and $\delta =0$, where from left to right, the three graphs correspond to $t=0$, 13, 20. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.

Figure 9

The upper graphs show the snapshots at the collision of a single large gravitational pulse with $(t_0,a_0,c_0)=(-40,1/7,2)$ and six small gravitational pulses with $(t_i,a_i,c_i)=(58+2i,1,3)(i=1,\dots ,6)$ for $A=0.05$ and $\delta =0$, where from left to right, the three graphs correspond to $t=0$, 13, 20. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.

Figure 10

The time dependence of the ratio $E_{+}(t,1000)/E(t,1000)$ when an outgoing pulse with $(a_0,c_0,t_0)=(1/7,2,-40)$ and six ingoing pulses with $(a_i,c_i)=(1,3)$ collide for ($A=0.05$, 0), where the left and right graphs display the ratio for $t_i=56+4i$ and $t_i=58+2i$ ($i=1,\dots ,6$), respectively.

Figure 11

The upper graphs show the snapshots of pulses at the collision for $(a_1,a_2,c_1,c_2,t_1,t_2,A,\delta )=(1,1,2,3,-45,55,1,0)$, where from left to right, the three graphs correspond to $t=0$, 4.6, 10. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.

Figure 12

The upper graphs show the snapshots at the collision of a single large gravitational pulse with $(t_0,a_0,c_0)=(-40,1/7,2)$ and six small gravitational pulses with $(t_i,a_i,c_i)=(56+4i,1,3)(i=1,\dots ,6)$ for $A=1$ and $\delta =0$, where from left to right, the three graphs correspond to $t=0$, 15, 30. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.

Figure 13

The upper graphs show the snapshots at the collision of a single large gravitational pulse with $(t_0,a_0,c_0)=(-40,1/7,2)$ and six small gravitational pulses with $(t_i,a_i,c_i)=(58+2i,1,3)(i=1,\dots ,6)$ for $A=1$ and $\delta =0$, where from left to right, the three graphs correspond to $t=0$, 13, 30. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.

Figure 14

The time dependence of the ratio $E_{+}(t,1000)/E(t,1000)$ when an outgoing pulse with $(a_0,c_0,t_0)=(1/7,2,-40)$ and six ingoing pulses with $(a_i,c_i)=(1,3)$ collide for ($A=1$, 0), where the left and right graphs display the ratio for $t_i=56+4i$ and $t_i=58+2i$ ($i=1,\dots ,6$), respectively.

Figure 15

The upper graphs show the plots of $\mathcal{E}$ at the collision of an outgoing pulse with $(a_1,c_1,{\alpha }_1)=(1,\pi /2,1)$ and an ingoing pulse with $(a_2,c_2,{\alpha }_2)=(1,-\pi /2,1)$ for $A={10}^{-7}$, where from left to right, the three graphs correspond to $t=0$, 6.5, 15. The lower graphs show the ratio ${\mathcal{E}}_{+}/\mathcal{E}$ at each time.