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  \begin{document}
  	
  \title{Colliding Plane Gravitational Waves of Unequal Strength}% Force line breaks with \\
  \author{Kamran Qadir Abbasi}
  
  	

  \affiliation{School of Natural Sciences $($SNS$)$,\\
  	National University of Sciences and Technology $($NUST$)$,\\
  	Sector H-12, Islamabad 44000, Pakistan\\
  	email: kamran.qadir@sns.nust.edu.pk\\
  	Telephone number: +923465300068}



  \author{Asghar Qadir}

  \affiliation{Abdus Salam School of Mathematical Sciences $ (ASSMS) $\\
  	68B New Muslim Town\\
  	Government College University, Lahore, Pakistan\\
  email: asgharqadir46@gmail.com\\
  Telephone number: +923318554537}



  \date{\today}
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\begin{abstract}
	Khan and Penrose, and separately Szekeres, presented solutions of colliding
	impulsive and sandwich plane gravitational waves, respectively. They only
	considered equal strength waves, which would impart no momentum to test
	particles in their path on the plane of collision at the time. We had earlier
	probed both spacetimes using the pseudo-Newtonian ($ \psi $N) formalism. In
	neither case was it really clear what parameter represented the ``strength''
	of the wave. In this paper we identify the required parameter and extend the
	colliding sandwich plane gravitational wave solution to unequal strengths.
	Since the impulsive plane waves of Khan and Penrose correspond to a limit of
	the sandwich waves with the pulse duration tending to zero, we have used the
	procedure to obtain the generalization of the Khan-Penrose solution to
	unequal strengths. The resulting spacetimes are exact solutions of vacuum
	field equations.
\end{abstract}
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 \section{Introduction}\setlength{\parskip}{0.5cm}
 
 General Relativity (GR) predicts    the existence of gravitational waves
 (GWs) traveling at the speed of light \cite{ae}.  However, a question was
 raised about their reality by Scheidegger \cite{aes} in 1953, as they are
 solutions of vacuum filed equations, and hence have zero stress-energy
 tensor. This was satisfactorily answered by Weber and Wheeler \cite{ww}, who
 argued that since they have no time translation symmetry, energy is not
 conserved. The matter was fully laid to rest when GWs from the merger of
 binary stellar-mass black holes (BHs) were detected in 2016 by    the
 Advanced LIGO (Laser Interferometer Gravitational-Wave Observatory)
 \cite{pba}. By definition, GWs are non-static vacuum solutions of Einstein's
 field equations (EFEs). The first exact solution of the EFEs was constructed
 by Einstein and Rosen in 1937 for cylindrical GWs \cite{en} followed by Bondi
 and Robinson for plane GWs \cite{bpr} in 1957.  In the early  1970s new exact
 GW solutions describing the scattering of plane sandwich and impulsive GWs
 were found by P. Szekeres \cite{ps1, ps2}, and by Khan and Penrose \cite{kp}
 respectively.
 
 Various approaches for obtaining the energy of GWs were proposed in the
 literature \cite{lbs,ll,ap}. One was to consider linearized gravity, so that
 we get the normal wave equation for the GWs. For this purpose, the metric
 tensor is expanded about the Minkowski metric with a perturbation,
 $h_{\mu\nu}$. Now the nonlinear terms for the Ricci tensor will provide a
 source for the wave equation, which is called a ``stress-energy pseudo
 tensor'', $\tau_{\mu\nu}$, \cite{aqb}.  Weber and Wheeler (WW) \cite{ww}
 addressed this problem by considering a sphere of test particles in the path
 of cylindrical GWs and obtained first and second order approximate formulae
 for the momentum imparted to test particles by cylindrical gravitational
 waves. Ehlers and Kundt \cite{ek} found that plane waves impart a constant
 momentum. Qadir and Sharif \cite{qs1} gave a general closed form expression
 for the momentum imparted, which gives the WW results when used for
 cylindrical waves to the second order.  For this purpose they used an
 extension of a formalism, called the pseudo-Newtonian ($\psi$N) formalism,
 which reintroduced forces into relativity \cite{mqv, jq, qq}.
 
 In this paper we try to identify the ``strength" of Szekeres' sandwich waves
 and apply the WW method for sandwich waves, so that those with greater
 ``strength'' would impart greater momentum. We are now in a position to
 discuss colliding sandwich waves of different arbitrary strengths. For equal
 strengths there will be a place and time where {\it no} momentum is imparted.
 Gravitational waves are generally expressed in dimensionless coordinates. To
 be able to identify the strength, we need to reintroduce dimensions into the
 coordinates. For this purpose we insert a constant, which we take to be of
 dimension of time, say $T$, as we will use units with the speed of light,
 $c=1$. For a pulse, the strength would be inversely related to its width.
 The pulse profile for sandwich GWs is depicted in Fig. 1. The consequences of
 different strengths are best seen by using the  extended pseudo-Newtonian
 (e$\psi$N) formalism of Qadir and Sharif \cite{qs}. This had been applied to
 probe the Khan-Penrose and sandwich wave solutions of equal strength
 \cite{kq,kq1}.
 
 The plan of the paper is as follows. In the next section we will revisit the
 WW method for cylindrical GWs. In section III, we discuss the
 $e\psi$N-formalism briefly. In the subsequent section the generalization of
 Szekeres' sandwich GWs  will be given. Next, we give the application of the
 $e\psi$N-formalism to a  colliding sandwich GW spacetime of unequal
 strengths, and end with a discussion and conclusion.
\begin{figure}[H]
	\centering
	\includegraphics[width=8cm]{F1}
	\caption{{\small{The pulse profile for sandwich GWs. The sandwich wave is
				bounded by two plane null hypersurfaces, the time duration for the pulse is
				2$T$, and the width of the wave is $\sqrt{2}T$. For $(t-z)<-T$ the pulse has
				not yet arrived; $-T<(t-z)<T$ is the interior of the pulse; and $(t-z)>T$
				after the pulse has passed.}}}
\end{figure}

\section{A Review of the WW Method}

Einstein and Rosen \cite{en} obtained the exact cylindrical GW solution given
by
\begin{equation}%1
ds^2= e^{2(\gamma-\psi)}(dt^2-d\rho^2)-\rho^2 e^{-2\psi} d\phi^2-
e^{2\psi} dz^2,
\end{equation}%1
where $\psi$ and $\gamma$ are functions of $\rho$ and $t$, satisfying the
vacuum field equations
\begin{equation}%2
\psi^{''}+\frac{1}{\rho}\psi^{'}-\ddot{\psi}=0~~\gamma^{'}=
\rho(\psi^{'2}+\dot{\psi^{2}})~,~\dot{\gamma}=2\rho\psi^{'} \dot{\psi},
\end{equation}%2
and prime and dot denote differentiation with respect to $\rho$ and $t$
respectively. Notice that the first of eqs. (2) is the Bessel equation for
$\psi$, which has the solution,
\begin{equation}%3
\begin{split}
\psi&= -2C \int_{0}^{\infty} e^{-a\omega}J_{0}(\omega\rho)\cos\omega Td\omega,\\
&=C[\rho^{2}+(a-iT)^{2}]^{-1/2}+C[\rho^{2}+(a+iT)^{2}]^{-1/2},
\end{split}
\end{equation}%3
\begin{equation}%4
\begin{split}
\gamma&=\frac{1}{2}C^{2}\{a^{-2}-\rho^{2}[(a-iT)^{2}+
\rho^{2}]^{-2}-\rho^{2}[(a-iT)^{2}+\rho^{2}]^{-2}\\
&-a^{-2}(-\rho^{2}+a^{2}+T^{2})[T^{4}+2T^{2}(a^{2}-
\rho^{2})+(a^{2}+\rho^{2})]^{-1/2}\},
\end{split}
\end{equation}%4
They now used the linearized geodesic equation for $\rho$ and then used the
perturbative method to obtain the acceleration of a test particle
\begin{eqnarray}%5
\begin{split}
\frac{d^2\rho}{dT^{2}}&=-\frac{\partial\gamma }{\partial\rho}+
\frac{\partial \psi }{\partial\rho}\\
&=C^{2}\rho\bigg\{\frac{\rho^{2}-(a-iT)^{2}}{[\rho^{2}+
	(a-iT)^{2}]^{3}}+\frac{\rho^{2}-(a+iT)^{2}}{[\rho^{2}+(a-iT)^{2}]^{3}}\\
&+2\frac{\rho^{2}+a^{2}+T^{2}}{[\rho^{2}+(a-iT)^{2}]^{3/2}
	[\rho^{2}+(a+iT)^{2}]^{3/2}}\bigg\}\\
&+C\rho\{[\rho^{2}+(a-iT)^{2}]^{-3/2}+[\rho^{2}+(a+iT)^{2}]^{-3/2}\},
\end{split}
\end{eqnarray}%5
Multiplying this by the mass of the test particle gives the force it
experiences. Integration of the force gives a non-zero momentum imparted to
the test particle. For our purposes, the Ehlers and Kundt analysis \cite{ek}
for plane waves \cite{bpr} will be more relevant.

\section{Review of the Extended Pseudo-Newtonian Formalism}

General Relativity replaces the bending of paths due to forces by the
curvature of spacetime. While retaining the path predicted by GR, we can
flatten the background spacetime and ask what force gives the required
curvature. This relativistic analogue of the gravitational force has been
called \cite{mqv,qq,jq} the pseudo-Newtonian ($\psi$N) force. This formalism
required that the tidal force be the gradient of the ($\psi$N) force, and
also that the spacetime be static. This formalism was extended to include
non-static spacetimes, \cite{qs,ms}. This formalism uses a continuous
re-setting of the zero of the force with the time variation. It was used to
provide a general formula for the momentum imparted to test particles by GWs
\cite{qs1,ms}, which gives the WW formula as a second order approximation.
The momentum 4-vector is just the time integral of the force 4-vector,
\begin{eqnarray}%6
F_0=m [\{\ln(Af)\}_{,0} +\frac{ g^{ij}_{~,0}g_{ij,0}}{4A}]~, \nonumber \\
F_i=m(\ln\sqrt{g_{00}})_{,i}~, ~~~~~(i=1,2,3)
\end{eqnarray}%6
where $A=(\ln\sqrt{-g})_{,0}$, $g=\det(g_{ij})$ and $f=1/\sqrt{g_{00}}$. The
quantity whose proper time derivative is $F_{\mu}$, is called the momentum
four-vector for the test particle. Thus the momentum four-vector, $P_{\mu}$,
is
\begin{equation}%7
P_{\mu}~=~\int F_{\mu}dt~ ,~~~~~~~(\mu=0,1,2,3).
\end{equation}%7
Since the energy is anyhow re-scaled, the physical significance of $P_{0}$ is
not clear, and only the 3-momentum as seen is the $\psi$N-frame (in which
$g_{0i}= 0$), is relevant. This is the momentum imparted to test particles.

\section{Extension of  Colliding Plane Sandwich Gravitational Waves to
	Unequal Strengths and their Analysis}

The plane sandwich GW is a curved region bounded by two plane null
hypersurfaces outside of which the spacetime is flat. The sandwich GW
spacetime \cite{ps1} is given by the line element,
\begin{equation}%8
ds^2=e^{-M}T^2du dv-e^{-U}\left(e^Vdx^2+ e^{-V}dy^2\right),
\end{equation}%8
where $ M $, $ U $ and $ V $ are functions of $ u $ alone,
$u=\frac{t-z}{T}$ and $v=\frac{t+z}{T}$ are retarded and advanced time, and
$T$ is a constant with dimensions of time. The solution for the region
bounded by $0<u<1$ (or $0<t-z<T$) is \cite{ps1},
\begin{equation}%9
\begin{aligned}
U=-\ln(1-u^4)~,~V=\sqrt{6}~\tanh^{-1}u^2~,~M=-\frac{1}{4}\ln(1-u^4)~. \\
\end{aligned}
\end{equation}%9
The geodesic equation in the $z$-direction is
\begin{equation}%10
\begin{split}
2\frac{d^2z}{ds^2}&-(\frac{dx}{ds}+\frac{dy}{ds})\frac{2e^{\sqrt{6}
		\arctan[\frac{(t-z)^2}{T^2}]}(t-z)(t^2+z^2-2tz-(1+\sqrt{6})T^2))}
{T(T^4-(t-z)^4)(T^4+(t-z)^4)}\\
&+\frac{(t-z)^3}{(T^4-(t-z)^4)}(\frac{dt}{ds}\frac{dz}{ds})+
\frac{(t-z)^3}{T^4-(t-z)^4}(\frac{dt}{ds})^2+\frac{(t-z)^3}
{(T^4-(t-z)^4)}(\frac{dz}{ds})^2=0.
\end{split}
\end{equation}%10
In  the above equation $\frac{dt}{ds}$ is unknown but can be obtained from
the metric (8), to get
\begin{equation}%11
\frac{dt}{ds}=[1- \frac{1}{8}(\ln(1-\frac{(t-z)^4}{T^4})-
\frac{1}{64}\ln(1-\frac{(t-z)^4}{T^4})^2..........(\frac{dz}{ds})^2 ].
\end{equation}%11
Taking $M(t-z)=\ln(1-\frac{(t-z)^4}{T^4})$ as a first order quantity,
$V(t-z)=\sqrt{6}\arctan[\frac{(t-z)^2}{T^2}]$ will be second order. Writing
\begin{equation}
z=z_{(0)}+z_{(1)}+z_{(2)}+...~,
\end{equation}%12
eqs. (10) and  (11) give the following approximate equations,
\begin{equation}%13
\frac{d^2z_{(0)}}{ds^2}-\frac{(t-z)^3}
{T^4-(t-z)^4}\frac{dt}{ds}\frac{dz_{(0)}}{ds}=0,
\end{equation}%13
\begin{equation}%14
\frac{d^2z_{(1)}}{ds^2}-\frac{(t-z)^3}{T^4-(t-z)^4}
\frac{dt}{ds}\frac{dz_{(1)}}{ds}+\frac{(t-z)^3}
{T^4-(t-z)^4}\frac{dz_{(0)}}{ds}\frac{dz_{(1)}}{ds}=0,
\end{equation}%14
\begin{equation}%15
\frac{d^2z_{(2)}}{ds^2}+\frac{(t-z)^3}{T^4-(t-z)^4}(\frac{dt}{ds}-
\frac{dz_{(0)}}{ds})\frac{dz_{(2)}}{ds}-\frac{(t-z)^3}
{T^4-(t-z)^4}(\frac{dz_{(1)}}{ds})^2=0.
\end{equation}%15
Eqs. (13), (14) and (15) are the zeroth, first and second order approximation
equations respectively. Integrating in successive order yields,
\begin{equation}%16
\begin{aligned}
\frac{dz_{(0)}}{ds}&=\frac{-2(t-z)^3}{T^4-(t-z)^4}
(1-\ln(1-\frac{(t-z)^4}{T^4})z_{(0)}+c_{1},\\
\frac{dz_{(1)}}{ds}&=\frac{-2c_{2}(t-z)^3}{T^4-(t-z)^4}z_{(1)}+c_{3},\\
\frac{dz_{(2)}}{ds}&=\frac{-2c_{4}(t-z)^3}{T^4-(t-z)^4}z_{(2)}+
\frac{2c_{5}(t-z)^3}{T^4-(t-z)^4} s+c_{6}.\\
\end{aligned}
\end{equation}%16
Notice that the momentum imparted to test particles at each order $\sim
T^{-4}$, so that the larger the width of the sandwich GW, the smaller the
momentum imparted to test particles by them will be.

The colliding sandwich wave spacetime for different arbitrary strengths
contains six regions. Region I is flat,  regions II and III contain sandwich
waves (obviously non-flat regions, the curvature tensor is nonzero in both)
prior to the collision \cite{jbg}. Regions IV and V, adjacent to regions II
and III respectively, are flat. Region VI is the post collision region which
develops the future curvature singularity \cite{kq1}. For arbitrary strengths
we take the constants $T_{2}$ and $T_{3}$ in regions II and III to be
unequal.  With this modification region VI is no longer symmetric. This is
depicted in Fig. 2. It is obvious that when $ T_{2}= T_{3}=T$, the width of
regions IV and V will be equal, and it will reduce to the case of  collision
of equal strength sandwich GWs \cite{kq1}.  We will apply the e$\psi$N
formalism in regions II, III and VI and investigate the behavior of the
e$\psi$N force in these modified regions. The modified metric for the Szekeres Sandwich waves in the three regions where
the waves are present is
\begin{equation}%1
ds^2=e^{-M}T_2T_3dudv-e^{-U}\left(e^Vdx^2+ e^{-V}dy^2\right).
\end{equation}%1
For regions II and III, it will be $ du_{2,3}$ and $dv_{2,3} $. For region VI, the metric (17) will be $ du_{2} $ and $ dv_{3} $.
Since the ``thickness of the sandwich wave determines the life of that
portion of the ``colliding wave universe'', $v_0/u_0=T_3/T_2$.
Obviously, the narrower region (III here), will have greater curvature and end
earlier than the other (II here). 

{\bf A. Analysis of Regions II and III}

In order to use the $e\psi N$-formalism the metric has been put in block
diagonal form by using timelike and spacelike coordinates,
$t=(uT_{2}+vT_{3})/2$, $z=(vT_{3}-uT_{2})/2$:
\begin{equation}%18
U_{2,3}=-\ln(1-(\frac{t\mp z}{T_{2,3}})^4),
~V_{2,3}=\sqrt{6}\tanh^{-1}(\frac{t\mp z}{T_{2,3}})^2,~
M_{2,3}=-\frac{1}{4}\ln(1-(\frac{t\mp z}{T_{2,3}})^4);
\end{equation}%18
This is depicted in Fig. 2, where we have taken $T_{3}/T_{2}=4/5$ for
illustrative purposes.
\begin{figure}[H]
	\centering
	\includegraphics[width=20cm]{F2}
	\caption{{\small{ The modified colliding sandwich  gravitational wave
				spacetime. Region I, IV and V are flat, while the other three regions are
				curved. The strength of regions II and III are determined by the constants $
				T_{2}$ and $T_{3}$ respectively, where we have chosen the duration ratio
				$T_{3}/T_{2}=4/5$, and so region VI is asymmetric.}}}\end{figure}

The spatial components of the e$\psi$N force give the spatial rate of change
of its potential energy. The momentum imparted to test particles is only
given by the time integral of the spatial part of the $e\psi N$-force. The
e$\psi$N-force for a test particle of unit mass is
\begin{equation}%19
F_{\mp z}=(\frac{t\mp z}{T_{2,3}})^3/[2\left(1-(\frac{t\mp z}{T_{2,3}})^4\right)],
\end{equation}%19	
and $F_{x}=F_{y}=0$. The relevant quantity is the magnitude of the spatial
part of the force 4-vector, i.e. the above expression multiplied by
$\sqrt|{g^{zz}}|$,
\begin{equation}%20
\mid F_{\mp}\mid =(\frac{t\mp z}{T_{2,3}})^3
/[2\sqrt2\left(1-(\frac{t\pm z}{T_{2,3}})^4\right)^{9/8}]~.
\end{equation}%20
The magnitude of the $e\psi$N-force increases in both regions II and III. This is shown
graphically in Fig. 3.

\begin{figure}[H]
	\centering
	\begin{minipage}[t]{0.5\textwidth}
		\centering
		\includegraphics[width=1\linewidth]{F3}
		\caption*{Fig. 3}
	\end{minipage}%%
	\caption{The magnitude of the e$\psi$N force increases in regions II and III.}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\hat{}
{ \bf B. Analysis of Region VI}

Using $u_2=(t-z)/T_2$ and $v_3=(t+z)/T_3$ for ease of writing, the line
element for region VI in terms of time and space coordinates is,
\begin{equation}%21
\begin{aligned}
U&= -\ln(1-u_2^4-v_3^4),\\~~
V&=-\sqrt6 [\tanh^{-1}[u_2^2(1-v_3^4)^{-1/2}]+
\tanh^{-1}[v_3^2(1-u_2^4)^{-1/2}]],\\
M&=(3/2)[\ln(1-u_2^4)(1-v_3^4)]-(5/2)[\ln(1-u_2^4-v_3^4)]\\
&+3\ln[(u_2v_3)^2+(1-u_2^4)^{1/2}(1-v_3^4)^{1/2}]~.
\end{aligned}
\end{equation}%21

It is easily checked, using CAS Maple, that this is a vacuum solution of the
Einstein field equations, as might have been expected from the fact that the
divergence of the stress-energy tensor is zero. However, that would not rule
out the possibility of a constant value for it after the collision. The
curvature singularity in Region VI is at $u_2^4+v_3^4=1$. The apex of this
boundary is where the rate of change of $t$ with $z$ is zero, subject to the
condition that the point lie on the boundary. Writing $T_3=\alpha^3T_2$, this
will be at
\begin{equation}%5
t=\frac{1}{2}\frac{\alpha^{-2}+\alpha^2}{\alpha^{-1}+\alpha}T~,
z=\frac{1}{2}\frac{\alpha^{-2}-\alpha^2}{\alpha^{-1}+\alpha}T~.
\end{equation}%5
where $T$ is the geometric mean of the two ``doomsdays''.
Clearly, if $\alpha<1$ the apex will be at a positive $z$, if $\alpha>1$
it will be at negative $z$ and if $\alpha=1$, it will be at $z=0$. Also, the
greater the asymmetry the greater the value of $t$ will be, with the minimum
value $T/2$. The post collision region is depicted in Fig. 4.
\begin{figure*}[h!]
	\centering
	\centering
	\includegraphics[width=0.8\linewidth]{F4}
	\caption{\small{The post collision region by colliding plane sandwich GWs of unequal strengths.The vector gives the direction of the maximum e$ \psi $N force in this spacetime, at an angle, $ \theta^{*} \approx  1.40 $.}}
\end{figure*}

The magnitude of the force for this metric (for a unit mass) is:
\begin{equation}%21
|F|=\frac{180c^{13/2}(v_3^3T_2-u_2^3T_3)
	(a^2v_3^3T_2+2abu_2v_3zT_2^2T_3^2-bu_2^3T_3)
	(v_3^3T_2-u_2^3T_3+2u_2^3v_3^3zT_2^3T_3^3)}
{a^2b^{13/2}T_2^{12}T_3^{12}(u_2^2v_3^2+ab)^2}
\end{equation}%21
where
\begin{equation}%22
a=(1-u_2^4),~~b=(1-v_3^4),~~c=(1-u_2^4-v_3^4)~.
\end{equation}%22
To probe the spacetime, in Table 1(a), we vary the angle
from $\pi/4$ to $-\pi/4$ for the distance  $d\approx0.99T$.  The force increases slowly till $\theta=\theta^{*} \approx1.2$,
after which it rapidly increases and reaches a maximum at $\theta=\theta^{*}  \approx1.40$
and then decreases again. Note that the
force never reaches zero. This is displayed graphically in Fig. 5(a).
Next, in Table 1(b), we vary the distance from the origin for the angle $\theta^{*}$, where the force is at its maximum. Obviously, the force initially increases slowly and then rapidly near the singularity. This is depicted  in Fig. 5(b).

\begin{minipage}{0.5\textwidth}
	\begin{table}[H]
		\hspace{-1cm}%\begin{minipage}{.1\textwidth}
		\centering
		%\noindent\pmb{
		\small{
			$\begin{array}{|c|c||c|c|}
			\hline
			\theta & |F| & \theta & |F| \\
			\hline
			0.7805 & 0.5986 & 1.3722 & 3.8928 \\
			\hline
			0.8717 & 0.5957 & 1.40 & 6.0885 \\
			\hline
			0.9479 & 0.5794 & 1.4692 & 4.8970 \\
			\hline
			1.1192 & 0.546383 & 1.8443 & 2.5672 \\
			\hline
			1.2029 & 0.809021 & 2.0984 & 0.4956 \\
			\hline
			1.3522 & 3.17844 & 2.3355 & 0.0020 \\
			\hline
			\end{array}$}
		\caption*{{Table 1(a):\small{$|F(\theta)|$ for $d\approx0.99\sqrt{T_2T_3.}$}}}
	\end{table}	
	\end{minipage}
	\begin{minipage}{0.6\textwidth}
		\begin{table}[H]
			\hspace{-1cm}%\begin{minipage}{.1\textwidth}
			\centering
			%\noindent\pmb{
			\small{
				$\begin{array}{|c|c||c|c|}
				\hline
				d & |F| & d & |F| \\
				\hline
				0.0228 & 0.0320 & 0.5721 & 3.0164 \\
				\hline
				0.1193 & 0.0662 & 0.7110 & 4.0509 \\
				\hline
				0.159101 & 0.1384 & 0.8551 & 5.0271 \\
				\hline
				0.28313 & 0.2154 & 0.9887 & 5.4431 \\
				\hline
				0.37191 & 1.3636 & 0.9906 & 5.31628 \\
				\hline
				0.55198 & 2.7456 & 0.9997 & 5.9506 \\
				\hline
				\end{array}$}
			\caption*{{Table 1(b):\small{$|F(d)|$ for $\theta^{*}\approx1.40$}}}
		\end{table}	
	\end{minipage}
	\begin{figure}[H]
		\centering
		\begin{minipage}[t]{0.5\textwidth}
			\centering
			\includegraphics[width=1\linewidth]{F5}
			\caption*{Fig. 5(a)}
		\end{minipage}%
		\begin{minipage}[t]{0.5\textwidth}
			\centering
			\includegraphics[width=1\linewidth]{F6}
			\caption*{Fig. 5(b)}
		\end{minipage}%
		\caption{{\small The variation of $|F|$ with the ``angle'' parameter,
				$\theta$, and ``distance'' parameter, $d$. In Fig. 5(a), the distance is
				fixed at $d=0.99T$, and the maximum magnitude of the force occurs at
				$\theta=\theta^{*} \approx 1.40$. In Fig. 5(b), for $\theta=\theta^{*}$, the
				magnitude of the force increases slowly at first, but rapidly near the
				singularity.}}
	\end{figure}	
\section{Colliding Impulsive Plane Gravitational Waves of Unequal Strength}

The impulsive force can be regarded as the limit of the usual force as its
duration tends to zero, while increasing the intensity of the force applied.
As such, the Khan-Penrose colliding impulsive plane gravitational waves
should be the same limit of the colliding impulsive sandwich gravitational
waves and the Khan-Penrose solution should be obtainable from the Szekeres
solution in the appropriate limit. The problem here is that the duration
enters into the metric, and at first sight it seems that in the limit one
would get Minkowski space. However, the Szekeres solution is normally written
in dimensionless coordinates, which can be made dimensional by multiplication
by a constant with units of time. This constant can be used as a proxy for
the ``duration parameter'', with the ``strength'' built into it. 
\begin{figure}[H]
	\centering
	\includegraphics[width=16cm]{F7}
	\caption{\small{The modified colliding Khan-Penrose impulsive plane GW
			spacetime of unequal strength. Regions I, II and III are flat, while region
			IV is curved. The strength of the impulsive GW in regions II and III,
			obtained by taking the limit of sandwich GWs with unequal widths, results in
			unequal impulses for the colliding impulsive wave. Thus the strength of the
			unequal impulses will vary as $T_2^{-4}$ and $T_3^{-4}$, where $T_2\neq T_3$.
			Obviously region IV is asymmetric, just as region VI is for the Szekeres
			solution. In Region IV ``doomsday'' occurs at $T $.}}
\end{figure}
In dimensionless coordinates the singularity occurs when the sum of the squares
of the coordinates becomes unity. When we put in the time parameter the
singularity at the plane of collision develops when the coordinate time
equals that time parameter. (In the papers discussing this matter
\cite{kq,kq1} it was called ``doomsday'', as this is when this colliding plane
wave Universe ends.) This same ``doomsday parameter'' enters into the
Khan-Penrose solution. Here we need two different ``doomsday parameters'',
$T_2$ and $T_3$, and the strengths will vary as $T_2^{-4}$ and $T_3^{-4}$,
respectively, and ``doomsday'' will occur at $T=[(u_2^2+v_3^2)T_2T_3]^{1/2}$
which depends on the spatial position and time. The putative metric for the
extended Khan-Penrose colliding impulsive plane wave solution, depicted in
Fig. 8, is:
\begin{equation}
\begin{split}
 ds^2&=\frac{(1-u_2^2-v^{2}_{3})^{3/2}}
{\sqrt{1-u_2^2}\sqrt{1-v^{2}_{3}}
	(u_2v_{3}+\sqrt{1-u_2^2}
	\sqrt{1-v^{2}_{3}})^2}du_2dv_3\\
&-(1-u_2^2-v^{2}_{3}
)\left(\frac{\sqrt{1-u_2^2}+v_{3}}
{\sqrt{1-u_2^2}-v_{3}}\right)
\left(\frac{\sqrt{1-v^{2}_{3}}+u_2}
{\sqrt{1-v^{2}_{3}}-u_2}\right)dx^2\\
&-(1-u_2^2-v^{2}_{3})
\left(\frac{\sqrt{1-u_2^2}-v_{3}}
{\sqrt{1-u_2^2}+v_{3}}\right)
\left(\frac{\sqrt{1-v^{2}_{3}}-u_2}
{\sqrt{1-v^{2}_{3}}+u_2}\right)dy^2.
\end{split}
\end{equation}
It has been checked that, as for the Sandwich waves, this is an exact solution. The
singularity in Region IV is given by $u_2^2+v_3^2=1$. This boundary is, again
asymmetric. Here, writing $T_3=\beta^2T_2$, the apex occurs at
$t=T\sqrt{\beta+\beta^{-1}}/2,~z=T(\beta-\beta^{-1})/2\sqrt{\beta+\beta^{-1}}$.
Again, as the asymmetry is increased the time of the apex increases, and the
least value, when there is symmetry, is $t=T/\sqrt{2}$
The original diagram given by Szekeres allowed for Regions IV and V to extend
to infinity, but physically this does make sense as the plane waves are for
infinite planes, and there can be no part of the spacetime {\it after} the
collision ``outside'' it. As such the Universe must end at $T_2$ on the left,
$T_3$ on the right and at the curvature singularity in between. It has been
argued \cite{kq,kq1,jbg} that the limits of the left and right regions
are topological singularities. Since all the cases of colliding plane waves,
including unequal strengths, lead to curvature singularities, it might be
wondered if, as for gravitational collapse, it is not only on account of some
special symmetry but the generic collision of gravitational waves might lead
to singularities. Since gravitational waves {\it must} be colliding in the
Universe, this seems unlikely to be the case.

\section{Conclusion and Discussion}

In this paper, we used two different values of the parameter for the two
waves in the Szekeres solution, and ``doomsday'' could not be identified
quite so simply, in terms of the two parameters, but it was nevertheless
well-defined. More precisely, It turns out to be the fourth root of a linear
combination of the squares of the two parameters, and is space and time
dependent. The ``strength'' of the waves was shown to vary as the inverse fourth power of their width, or duration. The spacetime was probed using the
$e\psi N$ formalism. It was verified that the metric satisfies the vacuum
Einstein field equations, and so this is a new exact solution in the sense
that the Szekeres solution is!

Letting the pulse widths of the sandwich waves tend to zero, while increasing
their strengths, in the limit the extended Szekeres solution yielded a
generalization of the Khan-Penrose colliding impulsive plane gravitational
wave solution to unequal strengths. This has the same qualitative features
that the extended Szekres solution has, and is also a new exact solution! It
may be worth mentioning that there is no derivation of the Khan-Penrose in
the literature. As such, this work provides the required derivation, by 
taking $T_2=T_3$.

Penrose points out \cite{Roger} that as both waves will focus each other in
the transverse directions, and are infinite in extent, this would cause the
curvature singularity to form. Essentially, the spacetime must disappear when
the focussing is complete. {\it That is the nature of ``doomsday''.} This
argument would support the previous argument for the left side and right side
singularities being topological. Penrose also conjectures that for
attenuating gravitational waves there will be no curvature singularities
formed. He further drew attention to the fact that the Khan-Penrose spacetime
could be generalized by applying a Lorentz transformation in the
$z$-direction. It would be very interesting to see if that procedure yields
the same generalization as was arrived at here.

It would be of interest to see if the exact solutions can be extended further
to incorporate a cosmological constant for the colliding plane waves
\cite{bh}. (Of course, it would no longer be an exact solution of the vacuum
equations but yield an Einstein space.) It would be worth seeing, also,
whether an exact solution can be constructed for colliding {\it cylindrical}
waves, and trying to incorporate the cosmological constant for it. For these
waves, the amplitude gives one measure of the strength, while the frequency
gives another. As such it is worthwhile to try to develop the solution for
colliding gravitational cylindrical waves of unequal {\it frequency}.

\section*{Acknowledgments}

We are most grateful to Prof. Sir Roger Penrose (NL) for extremely useful and
insightful comments on an earlier draft of this paper in his usual kind way. One of us (K. Q. Abbasi) is grateful to Prof. Ibrar Hussain for his useful discussions and guidance on the work.

\section*{Declarations}
\section*{Ethical Approval}
We would like to highlight that our study involves neither human participants nor animals. Therefore, there is no potential for ethical conflicts or harm to either humans or animals in this study.
\section*{Competing Interests}
In this study, there are no competing interests to declare. The authors, affiliated institutions, or any other related bodies do not have any financial or personal relationships that could be perceived as potentially influencing the research or its publication. We affirm that there are no conflicts of interest to disclose.
\section*{Authors' Contributions}
Both authors contributed equally to the research and preparation of this manuscript.
\section*{Funding}
In this study, there was no funding received or financial support provided for the research. The authors conducted the study independently without any external funding sources.
\section*{Availability of Data and Materials }
Not applicable.
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\end{document}
