Probing Szekeres’ colliding sandwich gravitational waves

Abstract

The behavior of the Khan–Penrose impulsive plane colliding gravitational wave spacetime was investigated using the pseudo-Newtonian formalism, which expresses the spacetime curvature in terms of forces. The essential singularity produced after collision was seen as a rapid build-up of the force. However, the singularity, when either wave has passed before the collision, did not appear in the force, leading to the conclusion that it is a topological singularity. The question of whether this feature is an artefact of the impulsive wave or is inherent in the causal structure of the spacetime is investigated here by extending the analysis of the Szekeres colliding sandwich gravitational waves, and very interesting insights are obtained. In particular, what Szekeres called a coordinate singularity after the collision of the waves turns out to be a curvature singularity.

Appendices

Appendix:

The time component of force

The time components of the e$\psi$N-force in all three curved regions are,

In Region II:

$$\begin{aligned} F_{t} =-\frac{27T^4+16\left( t-z\right) ^4}{5T \left( (t-z)^2+T^2\right) ((t-z)^2-T^2) (t-z) }. \end{aligned}$$

In Region III:

$$\begin{aligned} F_{t}=-\frac{27T^4+16 t^4+64 t^3 z+96 t^2 z^2+64 t z^3+16 z^4}{5T (t+z) \left( -T^4+t^4+4 t^3 z+6 t^2 z^2+4 t z^3+z^4\right) }. \end{aligned}$$

In Region VI:

$$\begin{aligned} \displaystyle F_t= & {} \bigg ((8 (t^3+3 t z^2) (L) (N) -8 (t^3+3 t z^2)(L)(R) -32 (t^3+3 t z^2) (N) (R)\\&+\frac{1}{(-p)}4 (t-z)^3 (L)(N)(R) +\frac{1}{ (-q)}4 (t+z)^3 (L) (N) (R))\\&-\frac{1}{(pq)}(L) (R) +(L) (N) (R)\bigg ) \\&\bigg [6 t^2 (L)^2 (-1+3 t^4-6 t^2 z^2+3 z^4)-64 (T^4-(t-z)^4) (t^3+3 t z^2)^2 (q) \\&-16 (p) (t+z)^3 (t^3+3 t z^2) (N)+12 (p)(t^2+z^2) (q) (p+q-T) \\&-16 (t-z)^3 (t^3+3 t z^2) (q) (N) +16 (p) (t+z)^3 (t^3+3 t z^2) (R) \\&-12 (p)(t^2+z^2) (q) (R)+ 16 (t-z)^3 (t^3+3 t z^2) (q) (M) \\&+48 t (t^3+3 t z^2) (L) (t^6-3 t^4 z^2-z^2 (3+z^4)+t^2 (-1+3 z^4)) \\&+3 (L)^2 (t^6-3 t^4 z^2-z^2 (3+z^4)+t^2 (-1+3 z^4))\\&+(48 t^2 \sqrt{pq} (t^2-z^2)^{10} (L) (2+t^8-4 t^6 z^2-7 z^4+z^8-2 t^2z^2 (11+2 z^4)\\&+t^4 (-3+6 z^4))^2)/ ((t^8-4 t^6 z^2+(-1+z^4)^2\\&-4 t^2 z^2 (3+z^4)+t^4 (-2+6 z^4)) (-T+t^{16}-8 t^{14} z^2+2 z^4+28 t^{12} z^4\\&-56 t^{10} z^6-z^8+z^{16}+t^8 (-1+70 z^8)\\&+t^6 (4 z^2-56 z^{10}) +4t^2 z^2 (3+z^4-2 z^{12})\\&+t^4 (2-6 z^4+28 z^{12}))^2)+(48 t^2 \sqrt{pq} (t^2-z^2)^3 (N) (R) (t^6-3 t^4 z^2 -z^2 (3T^2+z^4)\\&+t^2 (-T+3 z^4)) (2T^3+t^8-4 t^6 z^2-7 z^4\\&+z^8-2 t^2 z^2 (11+2 z^4)+t^4 (-3T+6 z^4)))/(t(t^8-4 t^6 z^2 \\&+(-T+z^4)^2-4 t^2 z^2 (3+z^4)+t^4 (-2T+6 z^4))^2 (T-\frac{(t-z)^8(t+z)^8}{(p)(q)}))\bigg ]\\&\div \bigg [(L) (N) (R)(4 (p) (t^3+3 t z^2) (q) (N)-4 (p) (t^3 +3 t z^2) (q) (M)\\&+3 t (L)^2(t^6-3 t^4 z^2-z^2 (3+z^4)\\&+t^2 (-T+3 z^4))) - ((p) (L)^4 (q) (\frac{T}{(-p)^2 (-q)^2}))\bigg ], \end{aligned}$$

where

$$\begin{aligned}&p=\sqrt{T-(t-z)^2},~ q= \sqrt{T-(t+z)^2},~~ L=(-T^4+2 t^4+12 t^2 z^2+2 z^4)\\&~~~R=(T-p-q)~~ \text {and}~~~~ N=(p+q-T) . \end{aligned}$$