text:
\section{ \label{sec:Intro} Introduction}

One of the important advances made by Landau when he developed Fermi-liquid theory in the mid-1950's~\cite{Landau56,*Landau57} was the introduction of a kinetic equation that governed the time dependence of the quasiparticle distribution function.  The solution of this kinetic equation yields both the propagation speeds and the attenuation of the collective excitations, among much additional information.  In a recent paper,~\cite{LAM_PRB2013}  we solved Landau's linearized kinetic equation for two-dimensional Fermi liquids with arbitrary polarization, and obtained both analytic expressions and numerically computed results for the propagation speed and attenuation of zero sound and first sound.  The method that we used was the approach pioneered by Khalatnikov and Abrikosov~\cite{Khalatnikov1958,*AK1959}  (AK). This is the classic method that is described in the standard reviews of Landau's Fermi-liquid theory: for example,  Pines and Nozi\`{e}res~\cite{PinesNoz1966} and Baym and Pethick.~\cite{BP1991}   In fact there exists a second approach for solving the kinetic equation that yields results that are similar to but not exactly the same as that of the AK theory but seems to be relatively unknown in the Fermi-liquid literature.  This second approach can be found in the 1989 paper of Sanchez-Castro, Bedell, and Wiegers\cite{CBW1989} (SBW),  and its application to zero sound was discussed briefly in Refs.~\onlinecite{AM2011} and \onlinecite{LAM_PRB2012}.  In this note, we wish to compare in detail the analytic and numerical predictions of the AK and SBW approaches for solving the kinetic equation for the propagation speeds and attenuation of the collective excitations covering both the collisionless and hydrodynamic regimes. To the best of our knowledge such a comparison has not yet been made. In particular, we wish to derive expressions for the propagation speeds and attenuation of both zero sound and first sound as functions of density and polarization.  The attenuation will be determined in the relaxation time approximation for the collision integral.

In Sec.~\ref{sec:candatt} we shall briefly review the two approaches for solving the kinetic equation.  We shall then use the SBW approach to calculate analytic expressions for zero sound and first sound propagation speeds and attenuation at arbitrary polarization.  These results will be compared and contrasted with our previous analytic results~\cite{LAM_PRB2013} obtained using the AK approach.  Finally, we derive the behavior of the the sound speeds and attenuation in the weak-coupling and strong-coupling limits for both approximations. In Sec.~\ref{sec:he3}, using Landau parameters~\cite{AM2011,LAM_PRB2012,LAM_JLTP2012}  and  quasiparticle lifetimes~\cite{LAM_PRB2013} determined previously, we shall compute sound speeds and attenuation for the system of  thin \he3 films at arbitrary polarization for both approximations, and compare their results.  In Sec.~\ref{sec:Conclusion}, the Conclusion, we shall discuss among other matters the effects on the attenuation and first-sound speeds of using a state-dependent relaxation time approximation for the collision integral. Finally, we shall point out that there are notable advantages of the SBW method especially for systems at non-zero polarization, and also in the zero-sound limit.

\section{\label{sec:candatt}Sound speed and attenuation}

We examine a system of $N = N_{\ua} + N_{\da}$, spin-up and spin-down fermions in a box of area $L^{2}$.  The particles have bare mass $m$, and interact with a typical two-body potential $V(r)$ that is assumed to depend only on the scalar distance between the particles. The energy $E\{n_{\pvec,\sigma}\}$ is a functional of the quasiparticle distribution function.   The particles fill two Fermi seas up to Fermi momenta $\kup$ and  $\kdn$, and we introduce the convention that the spin-down Fermi sea will always be the minority Fermi sea in the case of nonzero polarization. The term \textit{polarization} denotes the magnetization per particle which will be denoted by $\pol$, thus $\pol \equiv M/N = \left( N_{\ua} - N_{\da}\right) / N$.  The system is assumed to be at some finite but low temperature $T$ in the sense that $T << T_{\text{F} \da}$.   The derivation of the dispersion relations, and attenuation of the collective excitations proceeds as in three-dimensions beginning with Landau's linearized kinetic equation:~\cite{BP1991}
\begin{equation} \label{eq:linker}
\frac{\delta  }{\delta t}\delta n_{\pvec,\sigma}(\rvec, t) + \mathbf{v}_{\pvec, \sigma} \cdot \mathbf{\nabla}_{\rvec} \delta n_{\pvec,\sigma} (\rvec, t) - \mathbf{\nabla}_{\pvec}  n_{\pvec,\sigma}^{0} (\rvec, t)  \sum_{\pvec',\sigma'} f_{\pvec \pvec'}^{\sigma \sigma'} \mathbf{\nabla}_{\rvec} \delta n_{\pvec,\sigma} (\rvec, t) = I[ n_{\pvec,\sigma}] \,,
\end{equation}
where $\mathbf{v}_{\pvec, \sigma} \equiv \nabla_{\pvec} \epsilon_{\pvec, \sigma}$ is the Fermi velocity for spin state $\sigma$, $ f_{\pvec \pvec'}^{\sigma \sigma'} = {\delta^{2} E} / {\delta n_{\pvec, \sigma} \delta n_{\pvec^{\prime}, \sigma^{\prime}}}$ are the Landau parameters, and $I[n_{\pvec,\sigma}]$ is the collision integral.   For small oscillations we can assume traveling wave solutions for the excitations: $\delta n_{\pvec,\sigma}(\rvec, t) = \delta n_{\pvec,\sigma} ({\qvec,\omega}) \exp{(i(\qvec \cdot \rvec - \omega t))}$.  The kinetic equation (\ref{eq:linker}) reduces to
\begin{align} \label{eq:linkep}
&\left( q \vfs \cos{\theta} - \omega  \right) \nu_{\sigma}(\theta) + \left( q \cos{\theta} \right) \sum_{\pvec' \sigma'}  f_{\pvec \pvec'}^{\sigma \sigma'}  \delta(\epsilon_{\text{F}}^{\sigma'} - \epsilon_{\pvec' \sigma'}) \vfsp \nu_{\sigma'}(\theta')  \nonumber \\
&= -\frac{1}{i \tau_{\sigma}} \left[  \nu_{\sigma}(\theta) - \langle \nu_{\sigma}(\theta) \rangle - 2 \langle  \nu_{\sigma}(\theta) \cos{\theta} \rangle \cos{\theta} \right] \,,
\end{align}
where $\nu_{\sigma}(\theta)$ is the  Fermi surface distortion for spin state $\sigma$ introduced in the usual way:
\begin{equation}
\delta n_{\pvec,\sigma} (\qvec, \omega) = - \delta(\epsilon_{\text{F}}^{\sigma} - \epsilon_{\pvec \sigma}) \vfs \nu_{\sigma}(\theta) \,,
\end{equation}
and $\theta$ is the angle between $\pvec$ and $\qvec$.  In addition, we have written the collision integral in the relaxation time approximation:
\begin{equation}
\label{eq:CollInt}
I[n_{\pvec, \sigma}] = -\frac{1}{\tau_{\sigma}} \left[ \delta n_{\pvec, \sigma} - \langle \delta n_{\pvec, \sigma} \rangle - 2 \langle \delta n_{\pvec, \sigma} \cos{\theta} \rangle \cos{\theta} \right] \,.
\end{equation}
The second and third terms are added to ensure conservation of particle number, energy, and momentum.\cite{Khalatnikov1958,*AK1959}  The angular brackets are angular averages for two-dimensions: $\langle \dots \rangle \equiv (1/2 \pi) \int_{0}^{2 \pi} \dots d\theta $. Expressions for the quasiparticle lifetimes $\tau_{\sigma}$ were derived in Ref.~\onlinecite{LAM_PRB2013}, and will be briefly discussed below.  There will be additional discussion of the form of Eq.~(\ref{eq:CollInt}) in the Conclusion.

Eq. (\ref{eq:linkep}) can be simplified by introducing Fourier decompositions for the angle-dependent quantities:
\begin{subequations}
\begin{gather}
\nu_{\sigma}(\theta)  = \sum_{\ell = 0}^{\infty} \alpha_{\ell} \nu_{\ell}^{\sigma} \cos{(\ell \theta_{\pvec \qvec})}  \equiv \sum_{\ell = 0}^{\infty} \alpha_{\ell} \nu_{\ell}^{\sigma} T_{\ell}(\cos{\theta_{\pvec \qvec}})  \,, \\
f_{\pvec \pvec'}^{\sigma \sigma'}  = \sum_{\ell = 0}^{\infty} \alpha_{\ell} f_{\ell}^{\sigma \sigma'} \cos{(\ell \theta_{\pvec \pvec'})}  \equiv \sum_{\ell = 0}^{\infty} \alpha_{\ell}  f_{\ell}^{\sigma \sigma'}  T_{\ell}(\cos{(\theta_{\pvec \pvec'})}) \,.
\end{gather}
\end{subequations}
The constants $\alpha_{\ell}$ are defined by
\begin{equation}
\alpha_{\ell} =
\begin{cases}
	1& \text{if $\ell = 0$},\\
	2& \text{if $\ell \ge 1$}.
\end{cases}
\end{equation}
The quantities $T_{\ell}(\cos{\theta}) \equiv \cos{(\ell \theta)}$ are Chebyshev polynomials of the first kind,~\cite{A&S} and were introduced for convenience in Ref.~\onlinecite{LAM_PRB2012}. It was pointed out that integrals over $\theta$ from $0$ to $2 \pi$ can be replaced with integrals over $x \equiv \cos{\theta}$ from $-1$ to $+1$  by simply introducing the weight function $w(x) = 1/\sqrt{1 - x^2}$ and multiplying by a factor of $2$.  This is valid whenever the function involved is real, even in $\theta$, and periodic in $\theta$ with a period of $2 \pi$, which is the case for all  functions needed in this work.  There is no calculational advantage of this second representation. However, when using these variables, expressions in two dimensions become very similar to the familiar expressions in three dimensions by simply substituting Chebyshev polynomials for the usual Legendre polynomials.
After performing the indicated integrations we find
\begin{align} \label{eq:keq}
(s^{\sigma} - \cos{\theta}) &\sum_{\ell =0}^{\infty} \alpha_{\ell}  \nu_{\ell}^{\sigma} T_{\ell} (\cos{\theta} ) \nonumber \\
&- \cos{\theta} \sum_{\sigma'} N^{\sigma'}_{0} \left(\dfrac{\vfsp}{\vfs} \right) \sum_{\ell= 0}^{\infty} \alpha_{\ell} f_{\ell}^{\sigma \sigma'}  \nu_{\ell}^{\sigma'} T_{\ell} (\cos{\theta} ) \nonumber \\
&=  \frac{1}{i \tau_{\sigma} q \vfs } \left(  \nu_{\sigma}(\theta) - \nu_{0}^{\sigma} - 2\nu_{1}^{\sigma} \cos{\theta} \right)  \,,
\end{align}
where  $s^{\sigma} \equiv c/\vfs$, $c \equiv \omega /q $ is the complex valued speed of sound, and $\Ns$ is the \textit{single} spin-state density of states:
\begin{align} \label{eq:N0s}
\Ns &\equiv   \frac{m^{*}_{\sigma}L^{2}}{2 \pi \hbar^{2}} \,.
\end{align}
In Eq. (\ref{eq:keq}) the wave vector $q$ is a manifestly complex variable with real and imaginary parts defined by
\begin{equation}
q = q_{1} + i q_{2}  \,,
\end{equation}
where $q_{1}$ and $q_{2}$ are both real.   In the following,  we will obtain for the SBW approximation analytic expressions for the dimensionless speed of sound $s \equiv \omega /( q_{1} v_{\text{F}})$ and the attenuation $\operatorname{Im} (q) \equiv q_{2}$ at arbitrary polarization.  In the limits of zero polarization and full polarization we obtain simple analytic results which can be compared  with those previously obtained with the AK approximation.~\cite{LAM_PRB2013}  The question of whether $s$ is zero sound or first sound will depend on whether $\omega \tau \gg 1$ or $\omega \tau \ll 1$, respectively. Expressions for the quasiparticle lifetimes have been derived in Ref.~\onlinecite{LAM_PRB2013}.

For the AK approach one divides both sides of (\ref{eq:keq}) by $(s^{\sigma} - \cos{\theta})$ and then take moments with respect to the  $T_{\ell} (\cos{ \theta} )$, the angular functions in two dimensions.   We emphasize that despite the presence of the denominator all integrals can be evaluated exactly and analytically.~\cite{LAM_PRB2013}  In practice assuming that the series is truncated after the $\ell =1$ term one obtains a $4 \times 4$ secular equation of substantial complexity  (see Eq. (2.32) in Ref.~\onlinecite{LAM_PRB2013}).  In Ref.~\onlinecite{LAM_PRB2012} it was shown that truncation after the $\ell = 1$ contribution yields accurate sound speeds for a system with \he3  valued Landau parameters.  For the SBW approach one omits the first step of division by $(s^{\sigma} - \cos{\theta})$, and immediately takes moments of (\ref{eq:keq}) with respect to the  $T_{\ell} (\cos{\theta} )$ for $\ell = 0, 1, 2$.   We note that the relevant matrix elements can be found in Ref.~\onlinecite{AM2011}.

There are numerous ways to truncate the AK and SBW sets of linear equations. In a theory capable of yielding both zero sound and spin-zero sound we need to retain deformation coefficients $\nu_{\ell}^{\sigma}$ corresponding to $\ell = 0, 1$.  For the AK set of equations we set $f_{\ell}^{\sigma \sigma'} = 0$ for $\ell \ge 2$.  The deformation parameters $\nu_{\ell}^{\sigma}$ for $\ell \ge 2$ only couple to that Landau parameter with the same value of $\ell$. Thus, with this truncation, no deformation parameter with $\ell \ge 2$ will appear in the truncated AK equations. An example of this can be found in Eqs.~(2.31) in Ref.~\onlinecite{LAM_PRB2013}.

For the SBW equations the procedure differs slightly because the angular integrals couple the deformation parameters differently. In the following we show the first three moments for the SBW equations:
\begin{subequations} \label{eq:kineqtrunc}
\begin{alignat}{2}
&\ell=0  &s^{\sigma} \nu_{0}^{ \sigma} &=  \nu_{1}^{ \sigma} + \sum_{\sigma'} \left(\dfrac{\vfsp}{\vfs} \right) N^{\sigma'}_{0} f_{1}^{\sigma \sigma'} \nu_{1}^{ \sigma'} \,, \label{eq:u0}\\
&\ell=1   &s^{\sigma} \nu_{1}^{ \sigma} &=  \hlf (\nu_{0}^{ \sigma} + \nu_{2}^{ \sigma})   + \hlf\sum_{\sigma'} \left(\dfrac{\vfsp}{\vfs} \right) N^{\sigma'}_{0} f_{0}^{\sigma \sigma'} \nu_{0}^{ \sigma'}  \,, \label{eq:u1} \\
&\ell=2 \quad  &\left(s^{\sigma} - \frac{ 1}{i \tau_{\sigma} q \vfs } \right) \nu_{2 \sigma} &= \hlf (\nu_{1}^{ \sigma} + \nu_{3}^{ \sigma}) + \hlf \sum_{\sigma'} \left(\dfrac{\vfsp}{\vfs} \right) N^{\sigma'}_{0} f_{1}^{\sigma \sigma'} \nu_{1}^{ \sigma'}  \,. \label{eq:u2}
\end{alignat}
\end{subequations}
In these equations we have applied the same restriction on the Landau parameters that we used for AK: $f_{\ell}^{\sigma \sigma'} = 0$ for $\ell \ge 2$.

For the SBW set of equations we need to include  the $\ell = 2$ moment.  This is because for the SBW equations particle number conservation and momentum conservation remove the attenuation contribution to the $\ell =0$ and $\ell = 1$ moments.  Thus the $\ell =2$ moment is the lowest order contribution for attenuation to the SBW equations. Further, for SBW, the angular integral for the $\nu_{\ell}^{ \sigma}$ moment couples it to $\nu_{\ell - 1}^{ \sigma}$ and $\nu_{\ell + 1}^{ \sigma}$. Thus the deformation parameter $\nu_{3}^{ \sigma}$ is present in (\ref{eq:u2}).

The value of $\nu_{3}^{ \sigma}$ can be fixed by the following argument. We can proceed in the simplest approximation: zero polarization and $f_{\ell}^{\sigma \sigma'} = 0$ for $\ell \ge 1$, with no loss of generality. In this limit the exact solution to the kinetic equation can be written:
\begin{equation} \label{eq:nu0theta}
\nu (\theta) \sim \dfrac{\cos{\theta}}{s - \cos{\theta}} \,,
\end{equation}
where we ignore a system dependent constant. In the same limit we can compute $\nu_{3}$. Using Eq.~(3.3a) in Ref.~\onlinecite{AM2011}, and Eqs.(3.23, 3.24, 3.25) in Ref.~\onlinecite{LAM_PRB2012}, we find:
\begin{equation}
\dfrac{\nu_{3}}{s \nu_{0}} = \dfrac{\Omega_{3,0}}{s \Omega_{0, 0}} = z_{0}^{2} = \dfrac{1}{(s + \sqrt{s^{2} - 1})^{2}} \,,
\end{equation}
where the angular integrals $\Omega_{m, n}$ and the parameter $z_{0}$ are defined in Refs.~\onlinecite{AM2011,LAM_PRB2012}.  This ratio equals one in the weak coupling limit ($s \rightarrow 1$), zero in the strong coupling limit ($s \rightarrow \infty$), and it drops monotonically as the interaction strength $f_{0}$ increases. Thus, one can replace $\nu_{3}^{\sigma}$ by $s_{\sigma} \nu_{0}^{\sigma}$ in Eq.~(\ref{eq:u2}), and then Eq.(\ref{eq:u2}) becomes
\begin{equation} \label{eq:nu20}
\nu_{2}^{\sigma} = \nu_{0}^{\sigma} \,.
\end{equation}
This replacement is inaccurate as one approaches the strong-coupling limit.  However, by inspection of Eq.~(\ref{eq:nu0theta}) one can see that in the strong-coupling limit  all of the $\nu_{\ell}$'s vanish except for $\ell =1$, and therefore this inaccuracy is irrelevant.  In Sec.~\ref{ssec:Asym} we shall show that this change to Eq.~(\ref{eq:u2}) gives both the weak and strong-coupling limits correctly.  We note that because  \he3 Landau parameters are not small, \he3 sound speeds are in fact insensitive to the value used for the parameter $\nu_{3}$, and that SBW in particular simply set $\nu_{3} = 0$.

Substituting (\ref{eq:nu20}) into (\ref{eq:u1}) one finds the following eigenvalue equation:
\begin{equation}
\sum_{\sigma''}
\left[ \left( \dfrac{c}{\vfs} \right)^{2} \delta_{\sigma \sigma''} - A_{\sigma \sigma''}\right] \eta_{0 \sigma''} = 0 \,,
\end{equation}
where
\begin{equation} \label{eq:Assp}
A_{\sigma \sigma''} = A_{\sigma \sigma''}' + i A_{\sigma \sigma''}'',
\end{equation}
and we have defined
\begin{subequations} \label{eq:A}
\begin{align}
A_{\sigma \sigma''}' &\equiv \sum_{\sigma'} \left( \delta_{\sigma \sigma'} + \Nsp f_{1}^{\sigma \sigma'}  \right) \hlf \left[  \left(\dfrac{ 2 + \frac{1}{(\omega \tau_{\sigma'})^{2}}}{ 1 + \frac{1}{(\omega \tau_{\sigma'})^{2}}}  \right) \delta_{\sigma' \sigma''} +  N_{0}^{\sigma''} f_{0}^{\sigma' \sigma''} \right] \left(\dfrac{\vfsp}{\vfs} \right) \,,  \\
A_{\sigma \sigma''}'' &\equiv  - \sum_{\sigma'} \left( \delta_{\sigma \sigma'} + \Nsp f_{1}^{\sigma \sigma'}  \right)   \hlf  \left( \dfrac{ \frac{1}{(\omega \tau_{\sigma'})}}{1 + \frac{1}{(\omega \tau_{\sigma'})^{2}}} \right) \delta_{\sigma' \sigma''}  \left(\dfrac{\vfsp}{\vfs} \right) \,.
\end{align}
\end{subequations}
The eigenfunctions are defined by $ \eta_{0 \sigma} \equiv \left(\dfrac{\vfs}{c} \right) \nu_{0 \sigma} $.  The secular determinant is $2 \times 2$ and the eigenvalues are the roots of a quadratic equation:
\begin{equation} \label{eq:csqpm}
c_{\pm}^{2} = \hlf \left( v_{\text{F} \da}^{2} A_{\da \da} +  v_{\text{F} \ua}^{2} A_{\ua \ua}\right) \pm \hlf \sqrt{\left( v_{\text{F} \da}^{2} A_{\da \da} +  v_{\text{F} \ua}^{2} A_{\ua \ua}\right)^{2} - 4 v_{\text{F} \da}^{2}  v_{\text{F} \ua}^{2} \left( A_{\da \da} A_{\ua \ua}   - A_{\ua \da}  A_{\da \ua}\right)} \,.
\end{equation}
The propagation speed and attenuation can be determined from the real and imaginary parts of $c_{\pm}^{2}$. If we compare this quadratic equation with our previous equivalent AK result Eq. (2.32) in Ref.~\onlinecite{LAM_PRB2013}: it is clear that the numerical solution of  (2.14) is much easier, and yet gives very similar results as will be discussed in detail in the next section.

 In Eq.~(\ref{eq:csqpm}) and below we will have need of the explicit components of $A_{\sigma \sigma''}$.  First the real parts:
\begin{subequations} \label{eq:Areal}
\begin{align}
A_{\ua \ua}' &=  \left( 1 + N_{0}^{\ua}  f_{1}^{\ua \ua}  \right) \hlf \left[  \left(\dfrac{ 2 + \frac{1}{(\omega \tau_{\ua})^{2}}}{ 1 + \frac{1}{(\omega \tau_{\ua})^{2}}}  \right) +  N_{0}^{\ua} f_{0}^{\ua \ua} \right] + \hlf \left(N_{0}^{\da} f_{1}^{\ua \da}\right) \left(N_{0}^{\ua} f_{0}^{\da \ua} \right) \dfrac{\vf^{\da}}{\vf^{\ua}} \,,  \\
A_{\ua \da}' &=  \hlf \left(N_{0}^{\da}  f_{1}^{\ua \da}  \right) \left[  \left(\dfrac{ 2 + \frac{1}{(\omega \tau_{\da})^{2}}}{ 1 + \frac{1}{(\omega \tau_{\da})^{2}}}  \right) +  N_{0}^{\da} f_{0}^{\da \da} \right] \dfrac{\vf^{\da}}{\vf^{\ua}} + \hlf \left(1 + N_{0}^{\ua} f_{1}^{\ua \ua}\right) N_{0}^{\da} f_{0}^{\ua \da}   \,.
\end{align}
\end{subequations}
For $A_{\da \da}'$ and $A_{\da \ua}'$ reverse the spins in $A_{\ua \ua}'$ and $A_{\ua \da}'$, respectively.
Next, the imaginary parts:
\begin{subequations} \label{eq:Aimag}
\begin{align}
A_{\ua \ua}'' &=  -\hlf \left( 1 + N_{0}^{\ua}  f_{1}^{\ua \ua}  \right)   \left(\dfrac{ \frac{1}{(\omega \tau_{\ua})}}{ 1 + \frac{1}{(\omega \tau_{\ua})^{2}}}  \right)  \,,  \\
A_{\ua \da}'' &=  -\hlf \left(N_{0}^{\da}  f_{1}^{\ua \da}  \right)  \left(\dfrac{ \frac{1}{(\omega \tau_{\da})}}{ 1 + \frac{1}{(\omega \tau_{\da})^{2}}}  \right)  \dfrac{\vf^{\da}}{\vf^{\ua}}    \,.
\end{align}
\end{subequations}
For $A_{\da \da}''$ and $A_{\da \ua}''$ reverse the spins in $A_{\ua \ua}''$ and $A_{\ua \da}''$, respectively.

In the small attenuation regime $q_{2}/q{1} \ll 1$ we can substitute (\ref{eq:Assp}) into (\ref{eq:csqpm}) and expand to first-order in the $A_{\sigma \sigma''}''$'s:
\begin{align} \label{eq:csq}
c^{2} = (c^{\prime})^{ 2} &+ i \Biggl[ \hlf \left(  v_{\text{F} \ua}^{2} A_{\ua \ua}'' + v_{\text{F} \da}^{2} A_{\da \da}''  \right) \Biggr. \nonumber \\
 &+ \left\{ \hlf \left( v_{\text{F} \da}^{2} A_{\da \da}' +  v_{\text{F} \ua}^{2} A_{\ua \ua}' \right) \left( v_{\text{F} \da}^{2} A_{\da \da}'' +  v_{\text{F} \ua}^{2} A_{\ua \ua}'' \right) \right. \nonumber \\
 &-  \Biggl. v_{\text{F} \da}^{2}  v_{\text{F} \ua}^{2} \left[ A_{\ua \ua}' A_{\da \da}'' +  A_{\da \da}' A_{\ua \ua}''   - A_{\ua \da}'  A_{\da \ua}'' - A_{\da \ua}' A_{\ua \da}''  \right] \Bigr\} \nonumber \\
 &\times \Biggl. \dfrac{1}{\sqrt{\left( v_{\text{F} \ua}^{2} A_{\ua \ua}' +  v_{\text{F} \da}^{2} A_{\da \da}' \right)^{2} - 4 v_{\text{F} \da}^{2}  v_{\text{F} \ua}^{2} \left( A_{\ua \ua}'  A_{\da \da}'   - A_{\ua \da}'  A_{\da \ua}' \right)}} \Biggr] \,.
\end{align}
In this expression, $ (c^{\prime})^{ 2}$ is the real part of the complex speed squared. That is, it is Eq.~(\ref{eq:csqpm}) with $A_{\sigma \sigma'}$ replaced by $A_{\sigma \sigma'}'$.  We have omitted the spin-zero-sound (minus) root because spin-zero-sound does not propagate in \he3 thin films.~\cite{LAM_PRB2012} It is simple to recover that solution by replacing the plus sign by a minus sign in front of the first curly bracket in Eq.~(\ref{eq:csq}), and also before the radical in (\ref{eq:csqpm}). We have defined $c^{\prime} \equiv \omega/q_{1}$ and in lowest order of $q_{2}/q_{1}$ the attenuation can be found from:
\begin{equation} \label{eq:csqrealimag}
c^{2} = (c^{\prime})^{2} \left( 1 - 2i \frac{q_{2}}{q_{1}}  \right) \,.
\end{equation}
Eqs. (\ref{eq:csq}) and (\ref{eq:csqrealimag}) are an analytic expression for the complex speed of sound in the SBW approximation.  In the following we shall use Eq.(\ref{eq:csq}) for obtaining simple analytic expressions  in terms of the Landau parameters in the zero-sound and first-sound limits.  For the numerical work that is discussed in Sec.~\ref{sec:he3}, we use Eq.~(\ref{eq:csqpm}) directly.

\subsection{ \label{ssec:zero} Zero sound}

In the zero sound limit, we let $\omega \tau_{\sigma} \rightarrow \infty$. Thus, from Eq.~(\ref{eq:A}) we find:
\begin{equation} \label{eq:A0ssp}
A_{\sigma \sigma''}^{(0)} = A_{\sigma \sigma''}^{(0) \prime} + i A_{\sigma \sigma''}^{(0) \prime \prime} \,,
\end{equation}
where to lowest order in $1/\omega \tau_{\sigma}$:
\begin{align} \label{eq:A0sspRe}
 A_{\sigma \sigma''}^{(0) \prime} &= \sum_{\sigma'} \left( \delta_{\sigma \sigma'} + \Nsp f_{1}^{\sigma \sigma'}  \right) \left[ \delta_{\sigma' \sigma''} + \hlf N_{0}^{\sigma''} f_{0}^{\sigma' \sigma''} \right] \left(\dfrac{\vfsp}{v_{\text{F}}^{\sigma}} \right) \,, \\
\label{eq:A0sspIm}
A_{\sigma \sigma''}^{(0) \prime \prime} &=  -\hlf \sum_{\sigma'} \left( \delta_{\sigma \sigma'} + \Nsp f_{1}^{\sigma \sigma'}  \right)  \left(\dfrac{\vfsp}{\vfs} \right) \frac{1}{\omega \tau_{\sigma'}} \delta_{\sigma' \sigma''} \,.
\end{align}
Substituting Eq.~(\ref{eq:A0sspRe}) into Eq.~(\ref{eq:csqpm}), and substituting Eqs.~(\ref{eq:A0sspRe}) and \ref{eq:A0sspIm}) into Eq.~(\ref{eq:csq}) and comparing with Eq.~(\ref{eq:csqrealimag}), one obtains analytic expressions for the speed and attenuation of zero sound, respectively. Comparing with the analogous expressions previously obtained by the AK method, Eqs.~(2.37)--(2.41) in Ref.~\onlinecite{LAM_PRB2013}, the SBW results have a much simpler mathematical structure. This will be seen more clearly when the two limiting cases, zero and full polarization, are discussed below. Despite the fact that the SBW approximation leads to very different \textit{looking} expressions for the speed and attenuation of zero sound, we shall show in Sec.~\ref{sec:he3} below that SBW yields numerical results that are very close to the AK approximation.

\subsubsection{\label{ssec:z0p0}Zero polarization}

In the zero polarization limit $ v_{\text{F}} = v_{\text{F} \ua} =  v_{\text{F} \da}$, and  Eq.~(\ref{eq:csq}) becomes
\begin{equation} \label{eq:csqp0}
s^{2} = s_{0}^{2} +   i \left( A_{\ua \ua}^{(0) \prime \prime}  +  A_{\ua \da}^{(0) \prime \prime} \right)   \,,
\end{equation}
where the zero-sound dimensionless propagation speed $s_{0} = \omega/(\vf q_{1})$ is given by :
\begin{equation}
s_{0}^{2} =  A_{\ua \ua}^{(0) \prime} + A_{\da \ua}^{(0) \prime} \,.
\end{equation}
Using Eq.~(\ref{eq:csq}) we find
\begin{equation}
-2ic_{0}^{2} \dfrac{q_{2}}{q_{1}} = i \left( A_{\ua \ua}^{(0) \prime \prime} + A_{\ua \da}^{(0) \prime \prime} \right) v_{\text{F}}^{2} = -i \hlf \dfrac{1}{ \omega \tau} \left(  1 +  F_{1}^{s} \right) v_{\text{F}}^{2}\,,
\end{equation}
where:
\begin{align}
A_{\ua \ua}^{(0) \prime} &=  (1 + \hlf F_{0}^{\ua \ua}) (1 + F_{1}^{\ua \ua}) + \hlf F_{1}^{\ua \da} F_{0}^{\da \ua} \nonumber \,, \\
A_{\ua \da}^{(0) \prime} &=  (1 + \hlf F_{0}^{\da \da}) F_{1}^{\ua \da} + \hlf (1 + F_{1}^{\ua \ua}) F_{0}^{\ua \da} \nonumber \,, \\
A_{\ua \ua}^{(0) \prime \prime} &= -\hlf \left( \dfrac{1}{ \omega \tau}  \right) (1 + F_{1}^{\ua \ua}) \nonumber \,, \\
A_{\ua \da}^{(0) \prime \prime} &= -\hlf \left( \dfrac{1}{ \omega \tau}  \right) F_{1}^{\ua \da} \,.
\end{align}
For convenience we have introduced dimensionless Landau parameters defined by
\begin{equation}
F_{\ell}^{\sigma \sigma'} \equiv \Nsp f_{\ell}^{\sigma \sigma'} \,.
\end{equation}
Note that these are defined with the single spin-state density of states, and they are not symmetric in the indices at non-zero polarization.  The symmetric and antisymmetric Landau parameters used in the zero-polarization limit are defined as usual by
\begin{align} \label{eq:symmF}
2 \Nsp f_{\ell}^{\sigma \sigma'} = {F}_{\ell}^{s} + \sigma  \sigma'  {F}_{\ell}^{a}  \,,
\end{align}
where for this definition we associate $\sigma (\ua) = +1$, and $\sigma (\da) = -1$. The Landau parameters that appear in Eq.~(\ref{eq:symmF}) are the usual  parameters scaled with the \textit{two} spin-state density of states. Two spin-state Landau parameters are only used in the zero-polarization case.

Thus the  speed and attenuation become:
\begin{subequations} \label{eq:zsp0sbw}
\begin{align}
s_{0}^{2}   &=    ( 1 + \hlf F_{0}^{s}  ) \left(1 + F_{1}^{s}   \right)  \,, \\
\dfrac{q_{2}}{q_{1}} &= \frth   \dfrac{ \left( 1 +  F_{1}^{s} \right)}{s_{0}^{2}} \dfrac{1}{\omega \tau} =  \dfrac{1}{4 ( 1 + \hlf F_{0}^{s}  )} \dfrac{1}{\omega \tau} \,.
\end{align}
\end{subequations}
The expressions for spin-zero-sound can be obtained by replacing the symmetric Landau parameters with antisymmetric Landau parameters.

The analogous results for the zero-sound speed and attenuation that one obtains from the AK approach~\cite{LAM_PRB2013} are:
\begin{subequations} \label{eq:zsp0ka}
\begin{align}
g(s_{0}) &= \frac{1 + \Fos}{(1 + \Fos)\Fzs + 2s_{0}^{2}\Fos} \,, \label{eq:c0P0} \\
\frac{q_{2}}{q_{1}}&= \frac{1}{\omega \tau} \Bigg[1 -  \frac{(1 + \Fos)(1 + g(s_{0})) + 2 s_{0}^{2} (1 - \Fos) g(s_{0})}{\frac{\displaystyle (1 + \Fos)}{\displaystyle g(s_{0})} \frac{\displaystyle (1 + g(s_{0}))}{\displaystyle s_{0}^{2} - 1} - 4 s_{0}^{2} \Fos g(s_{0})}  \Bigg] \,. \label{eq:xiprime}
\end{align}
\end{subequations}
The important function $g(s)$ is defined by:
\begin{equation} \label{eq:g}
g(s) \equiv \dfrac{s}{\sqrt{s^{2} - 1}} - 1 \,.
\end{equation}

\subsubsection{Full Polarization}

In the limit of full polarization, we set $ v_{\text{F}} =  v_{\text{F} \ua} \text{ and }  v_{\text{F} \da} = 0$.  Then Eq.~(\ref{eq:csq}) becomes
\begin{equation} \label{eq:csqp1}
c^{2} = c_{0}^{2} +  i A_{\ua \ua}^{(0) \prime \prime}  v_{\text{F} \ua}^{2}\,,
\end{equation}
where the zero-sound propagation speed is $c_{0}^{2} =  A_{\ua \ua}^{0}  v_{\text{F} \ua}^{2}$.  In terms of Landau parameters the dimensionless zero-sound speed $s_{\ua}$  and attenuation are given by:
\begin{subequations} \label{eq:zsp1sbw}
\begin{align}
s_{0} ^{2} &= (1 + \hlf F_{0}^{\ua \ua}) (1 + F_{1}^{\ua \ua})  \,, \\
\dfrac{q_{2}}{q_{1}} &=  \dfrac{1}{4 (1 + \hlf F_{0}^{\ua \ua})} \left( \dfrac{1}{\omega \tau_{\ua}}  \right) \,.
\end{align}
\end{subequations}
The analogous results using the AK approach yield:
\begin{subequations} \label{eq:zsp1ka}
\begin{align} \label{eq:c0P1}
g(s_{\ua}) &=    \frac{1 + F_{1}^{\ua \ua}  }{F_{0}^{\ua \ua} (1 + F_{1}^{\ua \ua}) + 2 s_{\ua}^{2} F_{1}^{\ua \ua}} \,, \\
\dfrac{q_{2}}{ q_{1}} &= \Big[\frac{c_{\tau \ua}^{0} + F_{1}^{\ua \ua} c_{\tau \ua}^{1} }{ F_{0}^{\ua \ua} \hu +   F_{1}^{\ua \ua} c_{q \ua}^{1}  } \Big] \left( \dfrac{1}{\omega \tau_{\ua}}  \right)  \,.
\end{align}
\end{subequations}
The parameters in the attenuation were introduced for convenience in Ref.~\onlinecite{LAM_PRB2013} and are defined by:
\begin{subequations}
\begin{align}
\hu &\equiv s_{\ua} \Big( \frac{\partial \gu}{\partial s_{\ua}} \Big)_{\xi' = 0} \,, \\
\xu' &\equiv \frac{1}{\omega \tau_{\ua}} -  \frac{q_{2}}{q_{1}} \,, \\
 c_{\tau, \sigma}^{0}  &\equiv 1 + (2 s_{\sigma}^{2} + 1) g(s_{\sigma}) + F_{0}^{\sigma \sigma} h_{\sigma} \,, \\
 c_{\tau, \sigma}^{1}  &\equiv 1 + 2 s_{\sigma}^{2} (g(s_{\sigma})  + h_{\sigma}) + g(s_{\sigma})  + F_{0}^{\sigma \sigma} h_{\sigma} \,, \\
 c_{q, \sigma}^{1}  &\equiv 2 s_{\sigma}^{2} (2 g(s_{\sigma})  + h_{\sigma}) + F_{0}^{\sigma \sigma} h_{\sigma}  \,.
\end{align}
\end{subequations}

\subsection{First sound}

In the hydrodynamic regime $\omega \tau \ll 1$.  Using the same notation as for zero sound, we have from Eq.~(\ref{eq:A}):
\begin{equation} \label{eq:A1ssp}
A_{\sigma \sigma''}^{(1)} = A_{\sigma \sigma''}^{(1) \prime} + i A_{\sigma \sigma''}^{(1) \prime \prime} \,,
\end{equation}
where to lowest order in $\omega \tau_{\sigma}$:
\begin{subequations} \label{eq:FsApApp}
\begin{align}
 A_{\sigma \sigma''}^{(1) \prime} &= \hlf \sum_{\sigma'} \left( \delta_{\sigma \sigma'} + \Nsp f_{1}^{\sigma \sigma'}  \right)  \left[ \delta_{\sigma' \sigma''} + N_{0}^{\sigma''} f_{0}^{\sigma' \sigma''} \right] \left(\dfrac{\vfsp}{v_{\text{F}}^{\sigma}} \right)  \label{eq:A1real} \,, \\
A_{\sigma \sigma''}^{(1) \prime \prime} &=  -\hlf \sum_{\sigma'} \left( \delta_{\sigma \sigma'} + \Nsp f_{1}^{\sigma \sigma'}  \right)  \left(\dfrac{\vfsp}{\vfs} \right) \left( \omega \tau_{\sigma'} \right) \delta_{\sigma' \sigma''} \,.
\end{align}
\end{subequations}
The general expression for the square of the complex first sound speed is the same as Eq.~(\ref{eq:csq}) with the $0$ superscripts replaced by $1$'s.

For the AK approach, it was shown in Eq.~(2.42) of Ref.~\onlinecite{LAM_PRB2013} that the first sound speed could be written:
\begin{equation} \label{eq:c1P}
\begin{split}
&\Big[  2  (\dfrac{c_{1}}{v_{\text{F}}^{\ua}})^{2} - (1 + F_{0}^{\ua \ua})  (1 + F_{1}^{\ua \ua}) \Big] \Big[  2  (\dfrac{c_{1}}{v_{\text{F}}^{\da}})^{2} - (1 + F_{0}^{\da \da})  (1 + F_{1}^{\da \da}) \Big]  \\
- &(F_{0}^{\ua \da})^{2} \Big[  (1 + F_{1}^{\ua \ua})  (1 + F_{1}^{\da \da})  -  (F_{1}^{\ua \da})^{2}\Big] - (F_{1}^{\ua \da})^{2}  (1 + F_{0}^{\ua \ua})  (1 + F_{0}^{\da \da})  \\
- &4 F^{\ua \da}_{0} F^{\ua \da}_{1} \dfrac{c_{1}}{v_{\text{F}}^{\ua}} \dfrac{c_{1}}{v_{\text{F}}^{\da}} = 0 \,.
\end{split}
\end{equation}
It was surprising to discover that after some algebra (\ref{eq:c1P}) was found to be identical to Eq.~(\ref{eq:csq}) together with Eq.~(\ref{eq:A1real}).  Thus the SBW and AK methods yield identical results for the first-sound speeds at all polarizations. As will be seen below this equality is not carried over to the attenuation.

\subsubsection{Zero polarization}

At zero polarization $s^{2} = s_{1}^{2} + i (A_{\ua \ua}^{(1)  \prime \prime} + A_{\ua \da}^{(1)  \prime \prime})$. Thus, we find for the dimensionless first-sound propagation speed $s_{1} \equiv \omega / (q_{1} \vf)$, and the attenuation:
\begin{subequations} \label{eq:fsp0sbw}
\begin{align}
s_{1}^{2} &=  A_{\ua \ua}^{(1)  \prime} + A_{\ua \da}^{(1)  \prime} =\hlf \left(1 +  F_{0}^{s}\right) \left(1 + F_{1}^{s}   \right) \,, \\
\dfrac{q_{2}}{q_{1}} &=    \dfrac{\omega \tau }{2 ( 1 +  F_{0}^{s} )} \label{eq:q2SBWP0}\,.
\end{align}
\end{subequations}
Using AK theory, we find:
\begin{subequations} \label{eq:fsp0ka}
\begin{align}
\label{eq:c1P0}
s_{1}^{2} &= \hlf (1 + \Fzs) (1 + \Fos)  \,, \\
\label{eq:q2AKP0}
\dfrac{q_{2}}{q_{1}} &= \frac{\omega \tau}{4 (1 + \Fzs)} \,.
\end{align}
\end{subequations}

\subsubsection{Full polarization}

In the limit of full polarization, we set $ v_{\text{F}} =  v_{\text{F} \ua} \text{ and }  v_{\text{F} \da} = 0$.  Thus we obtain $c^{2} = c_{1}^{2} +  i A_{\ua \ua}^{(1) \prime \prime}  v_{\text{F} \ua}^{2}$, and therefore
\begin{subequations} \label{eq:fsp1sbw}
\begin{align}
s_{1} ^{2} &=  \hlf (1 + F_{0}^{\ua \ua}) (1 + F_{1}^{\ua \ua}) \,, \\
\dfrac{q_{2}}{q_{1}} &=  \hlf \dfrac{\omega \tau_{\ua}}{(1 + F_{0}^{\ua \ua})} \label{eq:q2SBWP1} \,.
\end{align}
\end{subequations}
In the AK approach we found:
\begin{subequations} \label{eq:fsp1ka}
\begin{align}
s_{1}^{2} &= \frac{1}{2}  \left( 1+ F_{0}^{\ua \ua} \right) \left( 1+ F_{1}^{\ua \ua} \right)  \,, \\
\dfrac{q_{2}}{ q_{1}} &= \frac{ \omega \tau_{\ua}}{4 ( 1+ F_{0}^{\ua \ua} ) } \label{eq:q2AKP1}  \,.
\end{align}
\end{subequations}
Thus, in the zero and full polarization limits the SBW and AK attenuation differ by a factor of two.

We can obtain additional information concerning first sound from the thermodynamic derivation of the compressibility $\kappa_{T}$ that can be found in Ref.~\onlinecite{AM2011}.  This yields an essentially exact zero-temperature expression for the first sound speed as a function of polarization:
\begin{equation} \label{eq:kappainv}
c_{1}^{2} =  \frac{2 \pi \hbar^{2}}{\overline{n} m^{2}} \biggl[ \left( \frac{m}{m^{*}_{\ua}}\overline{n}_{\ua}^{2}  + \frac{m}{m^{*}_{\da}}\overline{n}_{\da}^{2} \right)  + \tilde{F}_{0}^{\ua \, \ua} \overline{n}_{\ua}^{2}
			+ 2 \tilde{F}_{0}^{\ua \, \da} \overline{n}_{\ua} \overline{n}_{\da}
                        	+   \tilde{F}_{0}^{\da \, \da}  \overline{n}_{\da}^{2} \biggr] \,.
\end{equation}
Here $\overline{n}_{\sigma}$ is the areal density in the $\sigma^{\text{th}}$ Fermi sea, $\overline{n} =  \overline{n}_{\ua} + \overline{n}_{\da}$, and $\tilde{F}_{0}^{\sigma \, \sigma'}$ is a dimensionless Landau parameter defined with the bare mass rather than the effective mass. We can rewrite this in a form that is more useful in the present work:
\begin{align} \label{eq:c1sqthermo}
c_{1}^{2}  &= \frac{\left( 1 + \Pee \right)}{4} \Big[ {v_{\text{F}}^{\ua}}^{2} \left( \frac{m^{*}_{\ua}}{m} \right)  [1 + F_{0}^{\ua \ua} ] +  {v_{\text{F}}^{\da}}^{2} \left( \frac{m^{*}_{\da}}{m} \right)  [F_{0}^{\ua \da} ] \Big] \nonumber \\
                 &+ \frac{\left( 1 - \Pee \right)}{4} \Big[  {v_{\text{F}}^{\da}}^{2} \left( \frac{m^{*}_{\da}}{m} \right)  [1 + F_{0}^{\da \da} ] +  {v_{\text{F}}^{\ua}}^{2} \left( \frac{m^{*}_{\ua}}{m} \right)  [F_{0}^{\da \ua} ] \Big] \,.
\end{align}
By inspection of (\ref{eq:c1sqthermo}), in the zero-polarization limit this expression becomes:
\begin{equation}
s_{1}^{2} = \hlf \dfrac{m^{\ast}}{m} (1 + F_{0}^{s}) = \hlf (1 + F_{0}^{s}) (1 + F_{1}^{s}) \,,
\end{equation}
and in the full polarization limit:
\begin{equation}
s_{1} ^{2} =  \hlf (1 + F_{0}^{\ua \ua}) (1 + F_{1}^{\ua \ua}) \,.
\end{equation}
In the following, these will be referred to as the \textit{thermodynamic} results.

Thus, both the SBW and AK results in the zero and full polarization limits,  (\ref{eq:fsp0sbw}) and (\ref{eq:fsp0ka}) respectively, are in agreement with the thermodynamic first-sound speeds.   However, as will be seen in Sec.~\ref{sec:he3} there is a slight disagreement at finite polarization $0 < \Pee < 1$ between the thermodynamic sound speed and the SBW/AK results.

\subsection{\label{ssec:Asym} Asymptotic behavior of zero sound and first sound}

In this section we shall compare the SBW and AK sound speeds and attenuations in the strong and weak-coupling asymptotic limits.  In the strong-coupling limit:  $F_{0} \gg 1$ and $F_{\ell} = 0 \text{ for } \ell \ge 1$.   In the weak-coupling limit, all Landau parameters vanish. Note that for Tables~\ref{tab:zss},~\ref{tab:fss}, and \ref{tab:zsw} $F_{0}^{s}$ is defined as usual with a two spin-state density of states whereas $F_{0}^{\ua \ua}$ is defined with a single spin-state density of states.

The results for the strong-coupling limit are gathered together in Tables~\ref{tab:zss} and \ref{tab:fss}. These come from analyzing Eqs.~(\ref{eq:zsp0sbw}), (\ref{eq:zsp0ka}), ({\ref{eq:zsp1sbw}), ({\ref{eq:zsp1ka}),     (\ref{eq:fsp0sbw}), (\ref{eq:fsp0ka}), ({\ref{eq:fsp1sbw}), and (\ref{eq:fsp1ka}). The SBW and AK approaches are in exact agreement as far as the sound speeds are concerned. There is a disagreement of $O(1)$ between the two approaches in the coefficients of the attenuation.
 \begin{table}[h]
\caption{\label{tab:zss} The expressions for the  speed and attenuation of  zero sound in the strong-coupling limit $F_{0} \gg 1$, and $F_{\ell} = 0$ for $\ell \ge 1$ .  }
\begin{ruledtabular}
\begin{tabular}{cccc}
Polarization ($\Pee$) & Approach & Speed squared ($s_{0}^{2}$) & Attenuation ($q_{2}/q_{1}$) \\
\hline
0  & SBW & $\hlf F_{0}^{s}$ & $\frac{1}{2 F_{0}^{s}} \frac{1}{\omega \tau}$ \\
0  & AK   & $\hlf F_{0}^{s}$ & $\frac{5}{2}\frac{1}{F_{0}^{s}} \frac{1}{\omega \tau}$ \\
1 &  SBW & $ \hlf F_{0}^{\ua \ua}$ & $\frac{1}{2 F_{0}^{\ua \ua}} \frac{1}{\omega \tau_{\ua}}$  \\
1 & AK & $ \hlf F_{0}^{\ua \ua}$ & $\frac{5}{2}\frac{1}{F_{0}^{\ua \ua }} \frac{1}{\omega \tau_{\ua}}$
\end{tabular}
\end{ruledtabular}
\end{table}
 \begin{table}[h]
\caption{\label{tab:fss} The expressions for the  speed and attenuation of  first sound in the strong-coupling limit $F_{0} \gg 1$, and $F_{\ell} = 0$ for $\ell \ge 1$ .   }
\begin{ruledtabular}
\begin{tabular}{cccc}
Polarization ($\Pee$) & Approach & Speed squared ($s_{1}^{2}$) & Attenuation ($q_{2}/q_{1}$) \\
\hline
0  & SBW  & $\hlf F_{0}^{s}$ & $ \frac{\omega \tau}{2 F_{0}^{s}}$  \\
0  & AK  & $\hlf F_{0}^{s}$ & $ \frac{\omega \tau}{4 F_{0}^{s}}$  \\
1 &  SBW & $\hlf F_{0}^{\ua \ua}$ & $ \frac{\omega \tau_{\ua}}{2 F_{0}^{\ua \ua}}$  \\
1 & AK & $\hlf F_{0}^{\ua \ua}$ & $ \frac{\omega \tau_{\ua}}{4 F_{0}^{\ua \ua}}$
\end{tabular}
\end{ruledtabular}
\end{table}

In the weak-coupling limit, both approaches have the correct ideal Fermi gas  limiting value (at all polarizations).  In Table~\ref{tab:zsw} we point out that the SBW and AK methods approach the limiting values in different ways. The SBW method approaches 1 linearly in the Landau parameter whereas the AK method approaches 1 quadratically. As with the attenuation in the strong-coupling limit, the SBW and AK approaches have slightly different coefficients of the $\omega \tau$ term.
 \begin{table}[h]
\caption{\label{tab:zsw} The expressions for the  speed and attenuation of  zero sound in the weak-coupling limit $F_{0} \rightarrow 0$, and $F_{\ell} = 0$ for $\ell \ge 1$ . }
\begin{ruledtabular}
\begin{tabular}{cccc}
Polarization ($\Pee$) & Approach & Speed squared ($s_{0}^{2}$) & Attenuation ($q_{2}/q_{1}$) \\
\hline
0  & SBW & $1 + O(F_{0}^{s})$ & $ \frth \frac{1}{\omega \tau}$ \\
0  & AK   & $1 + O((F_{0}^{s})^{2})$ & $\frac{1}{\omega \tau}$ \\
1 &  SBW & $1 + O(F_{0}^{\ua \ua})$ & $\frth  \frac{1}{\omega \tau_{\ua}}$  \\
1 & AK & $1 + O((F_{0}^{\ua \ua})^{2})$ & $\frac{1}{\omega \tau_{\ua}}$
\end{tabular}
\end{ruledtabular}
\end{table}
 \begin{table}[h]
\caption{\label{tab:fsw} The expressions for the  speed and attenuation of  first sound in the weak-coupling limit $F_{0} \rightarrow 0$, and $F_{\ell} = 0$ for $\ell \ge 1$ .   }
\begin{ruledtabular}
\begin{tabular}{cccc}
Polarization ($\Pee$) & Approach & Speed squared ($s_{1}^{2}$) & Attenuation ($q_{2}/q_{1}$) \\
\hline
0  & SBW  & $\hlf $ & $ \hlf \omega \tau$  \\
0  & AK  & $\hlf $ & $ \frth \omega \tau$  \\
1 &  SBW & $\hlf$ & $ \hlf \omega \tau_{\ua}$  \\
1 & AK & $\hlf $ & $ \frth \omega \tau_{\ua}$
\end{tabular}
\end{ruledtabular}
\end{table}

\section{ \label{sec:he3} Application to $^{3}$H\lowercase{e} thin films}

In this section we shall compare the results of the SBW and AK approximations as a function of polarization and temperature using Landau parameters that were determined by fitting experimental data for  \he3.    In Refs. \onlinecite{AM2011,LAM_PRB2012} the methods for obtaining these parameters are described in detail.   In very brief  summary, we fit experimental data  for the heat capacity effective mass and the spin susceptibility, obtained for thin \he3 films on graphite substrates.  These fits determine values for the $s$-wave and $p$-wave effective $T$-matrix components at zero polarization.    We then substitute these into expressions for the Landau parameters that are valid to quadratic order in the $T$-matrix components  at arbitrary polarization. The derivations are fairly involved and we refer the reader to the original references for the details.

In Figs.~\ref{fig:ZsSpeeds}  and  \ref{fig:ZsAtt} we show the zero-sound speed and attenuation at an areal density $\overline{n} = 0.0132$~\AA$^{-2}$ as a function of polarization. We chose  $T = 10$~mK and $\omega = 1500$~MHz  to make sure that the condition for a well-defined zero-sound mode  $\omega \tau \gg 1$ is satisfied at all polarizations. The SBW results are calculated using Eq.~(\ref{eq:csqpm}) directly, while the AK results are obtained using Eq.~(2.32) in Ref.~\onlinecite{LAM_PRB2013}. We note that since we are well into the zero-sound limit, the analytic expressions (\ref{eq:csq}), (\ref{eq:A0sspRe}) and (\ref{eq:A0sspIm}) in the SBW method, and Eqs.~(2.37)-(2.41) in Ref.~\onlinecite{LAM_PRB2013} in the AK method can be used as accurate approximations to the numerical solutions.

Fig.~\ref{fig:ZsSpeeds} shows that the SBW zero-sound speeds are slightly larger than those of AK and the difference is approximately constant for all polarizations ($\sim 5$ m/s at this density). This behavior was noted previously in Ref.~\onlinecite{LAM_PRB2012} (Fig.~9). In Fig.~\ref{fig:ZsAtt} for $0.2 \lesssim \Pee \lesssim 0.9$ the values of the attenuation for both models are fairly close and fairly flat.  Essentially the two models are in reasonable agreement for the zero-sound attenuation.
\begin{figure}[ht]
\includegraphics[]{ZsSpeeds}
\caption{(Color online) Comparison of zero-sound speeds as a function of polarization  for the SBW approach  and the AK approach at an areal density $\overline{n} = 0.0132$~\AA$^{-2}$.  The SBW speeds are a fairly constant $\sim 5$~m/s faster than the AK. We note that the two approaches give different values in the limits of zero and full polarization.
\label{fig:ZsSpeeds}}
\vspace{1truein}
\end{figure}
\begin{figure}[ht]
\includegraphics[]{ZsAtt}
\caption{(Color online) Comparison of zero-sound attenuation as a function of polarization  for the SBW approach  and the AK approach at an areal density $\overline{n} = 0.0132$~\AA$^{-2}$, frequency $\omega = 1500$~MHz, and temperature $T = 10$~mK.   The results from both methods are very similar. They predict maximum attenuation at zero polarization, and minimum attenuation at full polarization as is to be expected from a simple quasiparticle scattering picture.
\label{fig:ZsAtt}}
\vspace{1truein}
\end{figure}

In Figs.~\ref{fig:FsSpeeds} and \ref{fig:FsAtt} we show the first-sound speed and attenuation at an areal density $\overline{n} = 0.0132$~\AA$^{-2}$ as a function of polarization. We chose  $T = 10$~mK and $\omega = 500$~kHz  to ensure that the condition for a well-defined first-sound mode  $\omega \tau \ll 1$ is satisfied at all polarizations.  Again the SBW results are calculated using Eq.~(\ref{eq:csqpm}), and the AK results are obtained using Eq.~(2.32) in Ref.~\onlinecite{LAM_PRB2013}. In the first-sound limit Eq.~(\ref{eq:c1P}) can be used to evaluate the speed in both models. The attenuation can be evaluated using analytic expressions (\ref{eq:csq}) combined with (\ref{eq:FsApApp}) in the SBW method, or Ref.~\onlinecite{LAM_PRB2013} Eqs.~(2.44)-(2.46) for the AK method.

We note that in Fig.~\ref{fig:FsSpeeds}, for the first-sound speeds, we also show the \textit{thermodynamic} results from Eq.~(\ref{eq:c1sqthermo}).  As pointed out above: the results for SBW and AK are identical, and at  zero and full polarization SBW/AK agree with the thermodynamic results. However for in-between polarizations $0 < \Pee < 1$ there is a slight disagreement between SBW/AK and Eq.~(\ref{eq:c1sqthermo}).  The \textit{maximum} disagreement, less than 2\%, is certainly less than the uncertainty associated with the values of our Landau parameters. In any case, the possible sources for this disagreement will be discussed in Sec.~\ref{sec:Conclusion}.

The first-sound attenuation calculated by the two models have the same basic behavior: they are minimum at zero polarization and maximum at full polarization.  The factor of two difference in the SBW and AK results can be seen in comparing Eqs.~(\ref{eq:q2SBWP0}) and (\ref{eq:q2AKP0}), and also (\ref{eq:q2SBWP1}) with (\ref{eq:q2AKP1}).

The qualitative reason for the observed polarization dependence of the  attenuation for both zero sound and first sound can be understood as the consequence of a simple quasiparticle scattering argument.  As shown in Fig.~7 of Ref.~\onlinecite{LAM_PRB2013} the majority spin quasiparticle lifetime increases with increasing polarization. This is mainly due to the decreasing phase space for $s$-wave scattering. Roughly speaking we expect the zero-sound attenuation $\sim 1/ (\omega \tau)$, and thus we expect that zero-sound attenuation should decrease with increasing polarization. Similarly, first-sound attenuation $\sim \omega \tau$, and thus we should expect first-sound attenuation to increase with increasing polarization. Finally, we note that the limiting cases $\Pee \rightarrow 0$ and $\Pee \rightarrow 1$ have technical issues that are discussed in detail in Ref.~\onlinecite{LAM_PRB2013}. We believe that the abrupt behavior seen in these limits in Fig~\ref{fig:ZsAtt} and especially Fig.~\ref{fig:FsAtt} is artificial.
\begin{figure}[ht]
\includegraphics[]{FsSpeeds}
\caption{(Color online) Comparison of first-sound speeds as a function of polarization  for the SBW approach,  the AK approach, and the exact zero temperature thermodynamic result from Eq.~(\ref{eq:kappainv}).  The areal density is $\overline{n} = 0.0132$~\AA$^{-2}$ and temperature $T = 10$~mK.  The AK and SBW results are in agreement for all polarizations.  The zero-temperature  thermodynamic result Eq.~(\ref{eq:kappainv}) is slightly smaller in magnitude than the AK and SBW.    polarization.
\label{fig:FsSpeeds}}
\vspace{1truein}
\end{figure}
\begin{figure}[ht]
\includegraphics[]{FsAtt}
\caption{(Color online) Comparison of first-sound attenuation as a function of polarization  for the SBW approach  and the AK approach at an areal density $\overline{n} = 0.0132$~\AA$^{-2}$, frequency $\omega = 1500$~MHz, and temperature $T = 10$~mK.   The results from both methods are fairly similar. The abrupt behavior in the regions close to zero polarization and full polarization is probably artificial as discussed in Ref.~\onlinecite{LAM_PRB2013}.
\label{fig:FsAtt}}
\vspace{1truein}
\end{figure}

In Figs.~\ref{fig:GP00}, \ref{fig:GP05}, \ref{fig:GP10} we show the transition from zero sound to first sound at $\Pee  = 0, 0.5, 1.0$, respectively. For the SBW method the speed and attenuation are obtained by numerically solving Eq.~(\ref{eq:csqpm}), whereas for the AK method they are calculated by numerically solving Eq.~(2.32) in Ref.~\onlinecite{LAM_PRB2013}. We let the temperature increase gradually so that the magnitude of $\omega \tau$ moves smoothly from  $\omega \tau \gg 1$  to $\omega \tau \ll 1$, thus the system transitions continuously from a collisionless zero-sound regime to a hydrodynamic normal sound regime.

In each of these figures the left hand ordinates have their spacing magnified in order to show  clearly the difference in the zero-sound speeds. The small differences between the two methods  decreases as the system moves towards the hydrodynamic regime, and completely vanishes at the first sound limit. The right hand ordinates show the variations of the attenuation over the transition region. The SBW magnitudes are consistently greater but nonetheless still on the same order as the AK ones.   For simplicity we call the \textit{transition temperature} from zero sound to first sound as the temperature that occurs at the peak of the attenuation curve $\omega \tau \approx 1$.  For each polarization  the frequency $\omega$ was adjusted so that the transition temperature is  $\lesssim 10$~mK.  We need to ensure both that the transition temperature is not too low as to not be experimentally accessible, and yet is low enough so that at the highest temperatures needed (in the hydrodynamic regime) the system is still degenerate.  One important feature that these figures show is that for a given frequency, density and polarization, both SBW and AK have their transitions occur at almost the same temperature.
\begin{figure}[ht]
\includegraphics[]{GP00}
\caption{(Color online) Sound speed and attenuation as functions of temperature or $\omega \tau$ at zero polarization $\Pee = 0$, density $\overline{n} = 0.0132$~\AA$^{-2}$, and frequency $\omega = 60$~MHz in the region of transition from zero sound to first sound. The SBW and AK approaches yield slightly different results for zero sound but are identical for first sound. The attenuation structures have the same form with different peak heights at the transition.
\label{fig:GP00}}
\vspace{1truein}
\end{figure}
\begin{figure}[ht]
\includegraphics[]{GP05}
\caption{(Color online) Sound speed and attenuation as functions of temperature or $\omega \tau$ at polarization $\Pee = 0.5$, density $\overline{n} = 0.0132$~\AA$^{-2}$, and frequency $\omega = 5.5$~MHz in the region of transition from zero sound to first sound. The SBW and AK approaches yield slightly different results for zero sound but are identical for first sound. The attenuation structures have the same form with different peak heights at the transition.
\label{fig:GP05}}
\vspace{1truein}
\end{figure}\begin{figure}[ht]
\includegraphics[]{GP10}
\caption{(Color online) Sound speed and attenuation as functions of temperature or $\omega \tau$ at full polarization $\Pee = 1.0$, density $\overline{n} = 0.0132$~\AA$^{-2}$, and frequency $\omega = 10$~MHz in the region of transition from zero sound to first sound. The SBW and AK approaches yield slightly different results for zero sound but are identical for first sound. The attenuation structures have the same form with different peak heights at the transition.
\label{fig:GP10}}
\vspace{1truein}
\end{figure}

\section{Conclusion} \label{sec:Conclusion}
In this manuscript we have explored the solution of Landau's linearized kinetic equation by the method introduced by Sanchez-Castro, Bedell, and Wiegers~\cite{CBW1989} (SBW), and compared it with the solution by using the standard Khalatnikov and Abrikosov~\cite{Khalatnikov1958,*AK1959} (AK) approach.  Both methods rewrite the kinetic equation as an infinite sum of coupled algebraic equations by utilizing a Fourier expansion of all relevant quantities. It is assumed that if both methods retained all terms,  they would yield exact solutions of the kinetic equation. However, for practical solutions both methods need to be truncated, and so one can ask whether the approximate solutions are close to one another, and also whether they are close to exact results.

There are two fundamental issues that need to be addressed concerning the form and solution of the kinetic equation (\ref{eq:linkep}).  The first concerns the collision integral Eq.~(\ref{eq:CollInt}).  The third term in parentheses was identified as being introduced in order to ensure conservation of momentum.  We note that this is an approximate form for a model with state-dependent lifetimes.  That is, if one requires that total momentum is conserved then the total momentum needs to be computed by summing over the contributions of each constituent. This can easily be done but it leads to an expression in which there are factors of $1 / \tau_{\ua}$ and $1 / \tau_{\da}$ that appear in a numerator and denominator in addition to the overall factor of $1 / \tau_{\ua}$ or $1 / \tau_{\da}$.  This leads to terms that are nonlinear in the collision frequencies. We note that this problem is introduced because of our use of state-dependent lifetimes. If the lifetimes were state independent then the factors of $1 / \tau$ in the numerator and denominator would cancel out. Our choice of (\ref{eq:CollInt}) is essentially equivalent to requiring conservation of momentum for each separate component. In that sense our choice is a sufficient condition for conservation of total momentum.

The second issue is an important observation made by Troian and Mermin~\cite{TroianMermin1985} concerning the requirement that the collision integral must vanish when the quasiparticle distribution function is in local equilibrium. In order to accomplish this one needs to include the contribution to the quasiparticle energies of the quasiparticle-quasiparticle interaction in the local equilibrium state. Troian and Mermin showed that this inclusion changed the form of the conservation of momentum term by replacing a scalar term by a matrix. They showed that even with a state independent lifetime this change would alter the value of the first sound speed, and bring it into agreement with the exact thermodynamic result. They also pointed out that in the limit of zero polarization and full polarization this change makes no difference since the matrix becomes a $1 \times 1$.  It is possible that we are seeing this effect in the slight difference between the SBW/AK first-sound speeds  and the thermodynamic first-sound speeds for $0 < \Pee < 1$ that we pointed out in Fig.~\ref{fig:FsSpeeds}.   Unfortunately, the Troian-Mermin cure for this problem would involve an analysis similar to the one discussed above concerning conservation of momentum. For \textit{state dependent lifetimes} this would introduce higher order terms in $1/\tau$ than simply linear. We do note that the differences between the SBW/AK and the thermodynamic first-sound speeds are very small: a \textit{maximum} difference of less than 2\% .

In summary we have examined the SBW approach to solving the kinetic equation for complex sound speeds, and compared it with the well-known AK method.   We have calculated analytic expressions for the propagation speeds and attenuation, and also numerical results using Landau parameters appropriate for \he 3 adsorbed on graphite substrates.  For zero-sound speed and attenuation, and first-sound attenuation, the SBW method yields significantly simpler expressions.  Nevertheless, the numerical speeds and attenuation predicted by both methods are very close.  We have noted that in the case of first-sound propagation speeds where SBW and AK yield identical expressions, these results are not in agreement with the thermodynamic results. In the previous section we discussed the possibility that this behavior is due to overlooking of the quasiparticle-quasiparticle interaction contribution to the quasiparticle energy in the distribution function.  In conclusion, we find that the SBW method requires far less algebraic effort than AK to derive analytic expressions for the propagation speeds and attenuation of sound from Landau's kinetic equation, yet nevertheless produces numerical results almost as accurate as the AK method. This is especially true for the zero-sound case at finite polarization $0 < \pol < 1$. Thus, at finite polarization we recommend SBW method as a very attractive alternative to the classic AK.

\newpage

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

Several scenarios in which the quantum gravity scale or the string
scale might be $O(\textrm{TeV})$ have been proposed~\cite{ref1}.
In these scenarios, our space is the 3-brane in large extra
dimensions. These brane world scenarios attract
much attention because of their great significance, namely, the
possibility of observing the quantum gravity effects in future-planned
accelerators.
In the far above regime of the Planck energy, the characteristic
scales of particle interactions are given by the gravitational radius of
the system and the classical description is expected to become valid.
In this energy scale, the black hole production in the high-energy
particle collision occurs. The phenomenology of the black hole production
in accelerators was first discussed in~\cite{ref2}, and subsequently a great amount of
papers concerning this theme have appeared.

One of the necessary investigations is to analyze the black hole formation
in the high-energy particle system using the higher-dimensional general relativity.
In the author's previous paper with Nambu~\cite{YN03},
we numerically calculated the apparent horizon in the grazing
collision of high-energy particles. Because the apparent horizon formation
is the sufficient condition for the black hole formation, our analysis
provides the lowerbound of the cross section of the black hole production.
In this analysis, we used the metric of the massless point particle,
which is obtained by boosting the Schwarzschild black hole to the speed of light,
fixing the energy $p$~\cite{AS71}.   However, there are various
characters of the particle that might affect its
gravitational field, namely, the electric charge, the color charge,
the spin, the wave feature (which was recently pointed out
in~\cite{GR04}), and the length of the string.
Because one should be able to obtain the metric of high-energy
objects to investigate these effects,  the development of
such methods is important.

In this paper, we would like to propose the (probably) easiest method
for deriving the metric of the objects which is boosted to the speed of light.
Because using a simple example is useful for this purpose, we consider the lightlike limit
of the boosted Kerr black holes in higher-dimensional spacetimes.
The higher-dimensional Kerr black hole solution was given by Myers and Perry~\cite{MP86}.
Although the higher-dimensional Kerr black hole can have more than one
Kerr parameter, we restrict our attention to the
one with only one Kerr parameter $a$ for simplicity.
In this case, the singularity is a one-dimensional rotating ring.
We consider two cases: the boost in the parallel
direction to the spin (i.e., the velocity vector is orthogonal
to the plane with the ring singularity) and the boost in the transverse direction to the spin
(i.e., the velocity vector is parallel to the plane with the ring singularity).
In the boost, we fix the energy $p$ and the Kerr parameter $a$.

In the four-dimensional case, the lightlike limit of the boosted
Kerr black hole fixing the energy $p$ and the Kerr parameter $a$
has attracted a lot of attentions.
Loust\'{o} and S\'{a}nchez considered the lightlike boost of the
Kerr black hole in the parallel direction to the spin~\cite{LS92}. They just boosted
the metric and succeeded in taking the lightlike limit.
But because they expanded the metric in the calculation,
it is applicable only at the distant region from the black hole and
the physical interpretation is difficult.
Further, their method contained ambiguity as pointed out
in~\cite{BN, BH03}. Balasin and Nachbagauer proposed an
alternative method~\cite{BN}. They calculated the distributional energy-momentum
tensor of the Kerr black hole and boosted it to the speed of light
in arbitrary directions.
Then they solved the Einstein's equation with this boosted energy-momentum tensor.
Barrab\`{e}s and Hogan proposed a somewhat easier method~\cite{BH03}, in which they
boost the Riemann tensor of the Kerr black hole to the speed of light
and construct the spacetime so that it has the same curvature as the
boosted one. There is also a related work by Burinskii and Magli~\cite{BM00}.

We choose the simplest method here: we just boost the
metric without considering the energy-momentum tensor and the
Riemann tensor. In this sense, our method is
similar to that of Loust\'{o} and S\'{a}nchez, but we do not expand
the metric. Using this method, we can analytically
derive the lightlike Kerr metric in the higher-dimensional spacetimes.
Although we reconfirm the ambiguity of this method in the four-dimensional case,
we find that there is no such ambiguity in the higher-dimensional cases.
Our analysis indicates that our method would be applicable
for a wide class of solutions and shows the condition for deriving the analytic
metric. It also indicates that we would be able to
calculate the lightlike metric in the general cases at least numerically.

Although our motivation is mainly mathematical, we also would like
to point out two possible implications of our results. First,
the Kerr black hole might provide the model of the
gravitational field of a particle with spin. Although the spin angular momentum
is different from the orbital angular momentum, there are several discussions
about the relation between the spinning particle and the Kerr black hole~\cite{ref3}.
Second, in the string theory, there are several discussions that
some black hole solutions should be regarded as the elementary particles~\cite{ref4},
or the Kerr black hole might be a classical model for the gravitational field
of some string~\cite{Nis95, NR02}.
In the last section of this paper, we give brief discussions about
the implications of our results from the viewpoint of
these contexts and the brane world scenarios.

This paper is organized as follows. In Sec. II, we analyze the boost
of the Kerr black hole in the parallel direction to the spin.
We propose the general strategy to obtain the lightlike limit by
just boosting the metric and construct the analytic metric of the lightlike Kerr black hole.
Then we show that there is no ambiguity in this method in the higher-dimensional
cases and discuss the properties of
the resulting metric.
In Sec. III, we repeat the similar analysis and discussions
in the case of the boost in the transverse direction to the spin.
Sec. IV is devoted to summary and discussion.
The discussion includes the advantage and the limitation in our method in general cases.
We also discuss the possible implications of our results, namely,
the effect of spin and the string length on the black hole formation
in the collision of the high-energy particles.

\section{Lightlike boost in the parallel direction to the spin}

In this section, we consider the lightlike limit of the Kerr black hole with
one Kerr parameter $a$
that is boosted in the parallel direction
to the spin for total spacetime dimension number $D\ge 5$.
We introduce the Kerr-Schild coordinate $(\bar{t}, \bar{x}_1, \bar{x}_2, \bar{z}, \bar{y}_i)$
in which the metric becomes
\begin{multline}
ds^2=-d\bar{t}^2+d\bar{x}_1^2+d\bar{x}_2^2+d\bar{z}^2+\sum_{i}d\bar{y}_i^2
+\frac{\mu\bar{r}^{7-D}}{\bar{r}^4+a^2\left(\bar{z}^2+Y^2\right)}\\
\times
\left[d\bar{t}
+\frac{\bar{r}}{\bar{r}^2+a^2}\left(\bar{x}_1d\bar{x}_1+\bar{x}_2d\bar{x}_2\right)
+\frac{a}{\bar{r}^2+a^2}\left(\bar{x}_1d\bar{x}_2-\bar{x}_2d\bar{x}_1\right)
+\frac{\bar{z}d\bar{z}+\sum_i\bar{y}_id\bar{y}_i}{\bar{r}}\right]^2,
\end{multline}
where $i=1,...,D-4$, and $Y$ and $\bar{r}$ are defined via
\begin{align}
&~~~~~~~~Y^2=\sum_{i} \bar{y}_i^2,\\
&\frac{X^2}{\bar{r}^2+a^2}+\frac{\bar{z}^2+Y^2}{\bar{r}^2}=1,
\label{parallel_rbar}
\end{align}
with $X^2=\bar{x}_1^2+\bar{x}_2^2$.
The total mass (the Arnowitt-Deser-Misner mass) $M$ and
the angular momentum $J$ are given by
\begin{align}
 M&=\frac{(D-2)\Omega_{D-2}}{16\pi G_D}\mu,\\
 J&=\frac{-2}{D-2}Ma,
\end{align}
where $G_D$ and $\Omega_{D-2}$ denote the gravitational constant of the $D$-dimensional
spacetime and the $(D-2)$-dimensional area of the unit sphere, respectively.
The singularity is rotating in the $(\bar{x}_1, \bar{x}_2)$-plane. In the boost,
we fix the energy $p=\gamma M$ and the Kerr parameter $a$. Here, $\gamma$ is
the usual $\gamma$ factor given by $\gamma\equiv 1/\sqrt{1-v^2}$
with the velocity of the Kerr black hole $v$. For convenience,
we introduce $P=\gamma \mu$. Now we consider the Lorentz boost
in the $\bar{z}$ direction:
\begin{align}
\bar{t}&=\gamma(t-vz),\\
\bar{z}&=\gamma(-vt+z),\\
\bar{x}_j&=x_j,\\
\bar{y}_i&=y_i,
\end{align}
where $j=1,2$. The boosted metric becomes
\begin{equation}
ds^2=ds_F^2+
\frac{\mu\bar{r}^{7-D}}{\bar{r}^4+a^2(\bar{z}^2+Y^2)}
\left(d\bar{t}^2+\frac{\bar{z}^2d\bar{z}^2}{\bar{r}^2}-\frac{2\bar{z}}{\bar{r}}
d\bar{t}d\bar{z}\right)+O(\gamma^{-1}),
\end{equation}
where $ds_F^2=-dt^2+\sum_jdx_j^2+dz^2+\sum_idy_i^2$.
In the limit $v\to 1$, this is reduced to
\begin{align}
&~~~~~~ds^2=ds_F^2+\lim_{v\to 1}f\left(\gamma(z-vt), X, Y\right)(dz-dt)^2,
\label{parallel_limit_f}\\
&f\left(\gamma(z-vt), X, Y\right)=\frac{\gamma P \bar{r}^{7-D}}{\bar{r}^4+a^2(\bar{z}^2+Y^2)}
\left(1+\frac{\bar{z}^2}{\bar{r}^2}-\frac{2\bar{z}}{\bar{r}}\right).
\label{parallel_f}
\end{align}
The strategy for obtaining this limit is as follows. We find the
primitive function $\tilde{\Phi}$ such that
\begin{equation}
\frac{\partial}{\partial z}\tilde{\Phi}\left(\gamma(z-vt), X, Y\right)=
f\left(\gamma(z-vt), X, Y\right).
\end{equation}
Then the limit can be obtained by
\begin{equation}
\lim_{v\to 1}f=\frac{\partial}{\partial z}\left\{\lim_{v\to 1}\tilde{\Phi}
\left(\gamma(z-vt), X, Y\right)\right\}.
\end{equation}
As we will see later, $\tilde{\Phi}$ is reduced to
\begin{equation}
\lim_{v\to 1}\tilde{\Phi}\left(\gamma(z-vt), X, Y\right)=(1/2)\textrm{sgn}(z-t)\Phi(X,Y)
+\bar{\Phi}\left(\gamma(z-t),X,Y\right),
\label{para_primitive}
\end{equation}
where $\textrm{sgn}(z-t)$ is determined by
$\textrm{sgn}(z-t)\equiv 2\theta(z-t)-1$ with the Heviside step function $\theta(z-t)$.
Hence we obtain
\begin{equation}
\lim_{v\to 1}f=\Phi(X,Y)\delta(z-t)
+\frac{\partial}{\partial z}\bar{\Phi}\left(\gamma(z-t),X,Y\right).
\end{equation}
The key point for writing down the analytic form of the lightlike metric
is whether we can obtain this primitive function $\tilde{\Phi}$ or not.
Note that $\Phi(X,Y)$ comes from the even part of $f$ with respect to $\bar{z}$,
and the odd part of $f$ contributes only to $\bar{\Phi}(\gamma(z-t),X,Y)$. Because we
can show that the primitive function
of the odd part becomes a finite value independent of $z$
in the limit $v\to 1$ for all $D\ge 4$,
it does not affect the result. Hence we omit the term $2\bar{z}/\bar{r}$ in eq.~\eqref{parallel_f} hereafter.

Now we construct the lightlike Kerr metric by finding the primitive function $\tilde{\Phi}$
in the higher-dimensional cases.
Using eq.~\eqref{parallel_rbar}, we find the following relations:
\begin{align}
&\frac{\partial\bar{r}}{\partial z}=\gamma\frac{\bar{r}^2+a^2}{\bar{r}}
\left(\bar{r}^2+a^2-\frac{X^2a^2}{\bar{r}^2+a^2}\right)^{-1}\gamma(z-vt),
\\
&\gamma(z-vt)=\textrm{sgn}(z-vt)\sqrt{\bar{r}^2-X^2-Y^2+\frac{a^2X^2}{\bar{r}^2+a^2}}.
\end{align}
Using these relations, we derive
\begin{equation}
f=\lim_{v\to 1}\textrm{sgn}(z-vt)
\frac{P\bar{r}^{6-D}}{\sqrt{(\bar{r}^2+a^2)(\bar{r}^2-\alpha)(\bar{r}^2-\beta)}}
\left(2-\frac{X^2}{\bar{r}^2+a^2}-\frac{Y^2}{\bar{r}^2}\right)\frac{\partial\bar{r}}{\partial z},
\end{equation}
where $\alpha$ and $\beta$ are given by
\begin{align}
\alpha&=\frac12\left[X^2+Y^2-a^2+\sqrt{(X^2+Y^2-a^2)^2+4a^2Y^2}\right],\\
\beta&=\frac12\left[X^2+Y^2-a^2-\sqrt{(X^2+Y^2-a^2)^2+4a^2Y^2}\right].
\end{align}
Now we introduce $R=\bar{r}^2$. Because $\bar{z}=0$ corresponds to $R=\alpha$,
we derive
\begin{equation}
\tilde{\Phi}/P=(1/2)\textrm{sgn}(z-vt)\left[2I^{(D)}_1(R)-X^2I^{(D)}_2(R)-Y^2I^{(D)}_3(R)\right],
\end{equation}
where
\begin{align}
&I^{(D)}_1(R)=\int_{\alpha}^{R}\frac{R^{(5-D)/2}}{\sqrt{(R+a^2)(R-\alpha)(R-\beta)}}dR,
\label{para_I1}\\
&I^{(D)}_2(R)=\int_{\alpha}^{R}\frac{R^{(5-D)/2}}{(R+a^2)\sqrt{(R+a^2)(R-\alpha)(R-\beta)}}dR,
\label{para_I2}\\
&I^{(D)}_3(R)=\int_{\alpha}^{R}\frac{R^{(3-D)/2}}{\sqrt{(R+a^2)(R-\alpha)(R-\beta)}}dR.
\label{para_I3}
\end{align}
Because taking the limit $v\to 1$ for $t\neq z$ corresponds to $R\to \infty$
and each integral is finite at $R=\infty$ for $D\ge 5$,
we derive
\begin{align}
&\Phi(X,Y)=2I^{(D)}_1(\infty)-X^2I^{(D)}_2(\infty)-Y^2I^{(D)}_3(\infty),
\label{parallel_F}\\
&\bar{\Phi}\left(\gamma(z-t),X,Y\right)=0.
\end{align}
The function $\Phi(X,Y)$ is expressed by the elliptic integrals, although
we do not show explicitly here because it is quite complicated.
The primitive functions $I_i^{(D)}(R)$ necessary for this calculation
are given in Appendix A. In summary, the metric of the lightlike Kerr black hole
has the following form:
\begin{equation}
ds^2=ds_F^2+\Phi(X, Y)\delta(z-t)(dz-dt)^2.
\end{equation}
The spacetime is flat except at $z=t$. The delta function indicates
that the two coordinates in the regions $z>t$ and $z<t$
are discontinuously connected and
there is the distributional Riemann tensor at $z=t$, which
we call the gravitational shock wave. The function $\Phi(X,Y)$ characterizes
the structure of this shock wave and we call it the potential of the shock wave
hereafter.

Before looking at the behavior of the potential $\Phi$ for $D\ge 5$, we would like
to see what occurs in our method in the $D=4$ case.
In this case, we should be careful in the following two points. First,
there is no $y_i$ direction and we need not consider $I_3^{(4)}$.
Because of this absence of $y_i$, there are two possible cases for
$\bar{z}$=0: the $R=X^2-a^2$ case and the $R=0$ case.
Depending on these choices, we should adopt $[X^2-a^2, R]$ and $[0, R]$
as the integral regions of $I_i^{(4)}$, respectively. It turns out later that
they lead to different results, which
 represent the solutions in the $X>|a|$ region and in the $X<|a|$ region, respectively.
Second, because $I_1^{(4)}$ logarithmically diverges at $R\to\infty$,
we should consider the $\bar{\Phi}$ part of the primitive function~\eqref{para_primitive}.
If we take the choice such that $\bar{z}=0$ corresponds to $R=X^2-a^2$,
$I_1^{(4)}$ and $I_2^{(4)}$ are given by
\begin{align}
I_1^{(4)}(R)&=\log\left|2R+2a^2-X^2+2\sqrt{(R+a^2)(R+a^2-X^2)}\right|-\log X^2
\nonumber \\
&\to \log\left|4\gamma^2(z-t)^2\right|-\log X^2~~(v\to 1),
\label{I1(4)_1}\\
I_2^{(4)}(R)&=\frac{2}{X^2}\sqrt{\frac{R+a^2-X^2}{R+a^2}}\to
\frac{2}{X^2}~~(v\to 1),
\end{align}
and we find
\begin{equation}
\lim_{v\to 1}f=2P\left[\frac{1}{|z-t|}+(-2\log |X|-1)\delta(z-t)\right].
\label{parallel_4D_result1}
\end{equation}
On the other hand, adopting the other choice in which $\bar{z}=0$ is equivalent
to $R=0$, $I_1^{(4)}$ and $I_2^{(4)}$ become
\begin{align}
I_1^{(4)}(R)&=\log\left|2R+2a^2-X^2+2\sqrt{(R+a^2)(R+a^2-X^2)}\right|
-\log\left|2a^2-X^2+2\sqrt{a^2(a^2-X^2)}\right|\nonumber\\
&\to \log\left|4\gamma^2(z-t)^2\right|
-\log\left|2a^2-X^2+2\sqrt{a^2(a^2-X^2)}\right|~~(v\to 1),\\
I_2^{(4)}(R)&=\frac{2}{X^2}\left(\sqrt{\frac{R+a^2-X^2}{R+a^2}}-
\sqrt{\frac{a^2-X^2}{a^2}}\right)
\to \frac{2}{X^2}\left(1-
\sqrt{\frac{a^2-X^2}{a^2}}\right)~~(v\to 1),
\end{align}
which lead to
\begin{equation}
\lim_{v\to 1}f=2P\left\{\frac{1}{|z-t|}+
\left[-2\log\left(|a|+\sqrt{a^2-X^2}\right)-1+\frac{\sqrt{a^2-X^2}}{|a|}\right]\delta(z-t)
\right\}.
\label{parallel_4D_result2}
\end{equation}
The latter result~\eqref{parallel_4D_result2} is defined only in the region $|X|<|a|$.
Because the former result~\eqref{parallel_4D_result1} has the same value
at $|X|=|a|$ as that of eq.~\eqref{parallel_4D_result2}, $\lim_{v\to 1}f$
can be extended to the region $|X|>|a|$ using eq.~\eqref{parallel_4D_result1}.
Since we can eliminate the $1/|z-t|$ term using the proper coordinate
transformation, we find that this result is the same as the previous result of~\cite{BN}
using the relation $P=2G_4p$.

At this point, we should point out the ambiguity of this method for $D=4$.
For example, we can consider the following form instead of eq.~\eqref{I1(4)_1}:
\begin{equation}
I_1^{(4)}(R)=\log\left|\left[2R+2a^2-X^2+2\sqrt{(R+a^2)(R+a^2-X^2)}\right]/e^{g(X)}\right|
+g(X)-
\log X^2
\end{equation}
with arbitrary function $g(X)$. This leads to
\begin{equation}
\lim_{v\to 1}f=2P\left[\frac{1}{|z-t|}+\left(g(X)-2\log |X|-1\right)\delta(z-t)\right].
\end{equation}
instead of eq.~\eqref{parallel_4D_result1}.
This ambiguity is the reason why the previous authors have developed the
various methods other than just boosting the metric~\cite{BN, BH03,BM00}.
In fact, there would not exist the method to determine this factor $g(X)$
rigorously without checking the consistency with the boosted energy-momentum tensor
or the boosted Riemann tensor.
Hence the above calculation is just the formal derivation
of the lightlike Kerr black hole metric. However,
we would like to point out that there
is no ambiguity for the higher-dimensional cases, because the primitive function
$\tilde{\Phi}$ is finite in the limit $v\to 1$. Hence in the higher-dimensional cases,
it is sufficient to consider the boost of the metric and
we need not calculate the energy-momentum tensor or the Riemann tensor
of this system.

\begin{figure}[tb]
\centering
{\includegraphics[width=0.35\textwidth]{Para_Poten_D5_light.eps}\hspace{5mm}
\includegraphics[width=0.35\textwidth]{Para_Poten_D6_light.eps}\\
\vspace{5mm}
\includegraphics[width=0.35\textwidth]{Para_Poten_D7_light.eps}\hspace{5mm}
\includegraphics[width=0.35\textwidth]{Para_Poten_D8_light.eps}}
\caption{The behavior of the function $\Phi$ for $D=5,...,8$
in the parallel case. The unit of $X$
and $Y$ is $a$, and the unit of $\Phi$ is $P|a|^{D-4}$. In the
case of $D=5$, $\Phi$ diverges at $Y=0, X=\pm a$, which indicates that the
singularity is a one-dimensional ring. In other cases, $\Phi$ diverges
at $Y=0, -|a|<X<|a|$. This implies that the singularity is a two-dimensional disk.}
\end{figure}

Now we discuss the properties of the potential $\Phi(X, Y)$
in the higher-dimensional cases.
Figure 1 shows the behavior of $\Phi(X,Y)$ for $D=5,...,8$.
In the $D=5$ case, the potential $\Phi$ logarithmically diverges
at $X=\pm a$. Hence the shock wave contains a one-dimensional
ring-shaped singularity.
 For $D\ge 6$,
the potential $\Phi$ diverges at $Y=0, -|a|\le X\le |a|$. Hence the singularity
is a two-dimensional disk in these cases.
This is interpreted as follows: For $D=5$,
the black hole horizon does not exist in the resulting spacetime because
we fixed $a$ in the boosting. The singularity becomes naked and
the potential $\Phi$ diverges at the singularity. While for $D\ge 6$,
the horizon exists for arbitrary $M$ and $a$.
Hence the resulting gravitational shock wave contains an extremely
oblate horizon, on which the potential $\Phi$ diverges.
The function $\Phi$ behaves like
$\log Y$ for $D=6$ and $1/Y^{D-6}$ for $D\ge 7$. One can understand this
potential behavior using the Newtonian analogy. The Newtonian
potential of $n$-dimensional matter distribution in $N$-dimensional
space is given by $\sim 1/r^{N-n-2}$ for $N-n>2$ and $\sim -\log r$
for $N-n=2$. Because the shock wave is
$(D-2)$ dimensional, the behavior of the potential $\Phi$
and the dimensionality of the singularity have the same relation
as the Newtonian gravity.
Hence, we see that these results are consistent with our intuition
and we expect that also in general cases we
can guess the behavior of $\Phi(X,Y)$ to a certain extent,
even if the calculation of the primitive function of $f$ cannot be proceeded.

\section{Lightlike boost in the transverse direction of the spin}

In this section, we consider the lightlike limit of the boosted Kerr black hole in the transverse
direction to the spin. In this case, we use a coordinate system
$(\bar{t}, \bar{x}, \bar{y}, \bar{z}_i)$ in which the metric of
the Kerr black hole becomes
\begin{equation}
ds^2=-d\bar{t}^2+d\bar{x}^2+d\bar{y}^2+\sum_id\bar{z}_i^2
+\frac{\mu\bar{r}^{7-D}}{\bar{r}^4+a^2Z^2}
\left[d\bar{t}
+\frac{\bar{r}\left(\bar{x}d\bar{x}+\bar{y}d\bar{y}\right)}{\bar{r}^2+a^2}
+\frac{a\left(\bar{x}d\bar{y}-\bar{y}d\bar{x}\right)}{\bar{r}^2+a^2}
+\frac{\sum_i\bar{z}_id\bar{z}_i}{\bar{r}}\right]^2,
\end{equation}
where $i=1,...,D-3$, and $Z$ and $\bar{r}$ are defined by
\begin{align}
&~~~~Z^2=\sum_i \bar{z}_i^2,\\
&\frac{\bar{x}^2+\bar{y}^2}{\bar{r}^2+a^2}+\frac{Z^2}{\bar{r}^2}=1.
\label{transverse_rbar}
\end{align}
The singularity is rotating in the $(\bar{x}, \bar{y})$-plane and we will consider
the boost in the $\bar{x}$ direction. Again
we fix the energy $p=\gamma M$ and the Kerr parameter $a$ in the boost
and use the notation $P=\gamma\mu$. By the Lorentz transformation
\begin{align}
\bar{t}&=\gamma(t-vx),\\
\bar{x}&=\gamma(-vt+x),\\
\bar{y}&=y,\\
\bar{z}_i&=z_i,
\end{align}
the metric becomes
\begin{equation}
ds^2=ds_F^2+
\frac{\mu\bar{r}^{7-D}}{\bar{r}^4+a^2Z^2}
\left(d\bar{t}^2+\frac{\bar{r}^2\bar{x}^2+a^2{y}^2-2a\bar{r}\bar{x}{y}}{(\bar{r}^2+a^2)^2}d\bar{x}^2
+\frac{2(\bar{r}\bar{x}-a{y})}{\bar{r}^2+a^2}
d\bar{t}d\bar{x}\right)+O(\gamma^{-1}),
\end{equation}
where $ds_F^2=-dt^2+dx^2+dy^2+\sum_idz_i^2$. In the limit $v\to 1$, this is reduced to
\begin{align}
&~~~~~~~~ds^2=ds_F^2+\lim_{v\to 1}f\left(\gamma(x-vt), y, Z\right)(dt-dx)^2,
\label{transverse_limit_f}\\
&f\left(\gamma(x-vt), y, Z\right)=\frac{\gamma P \bar{r}^{7-D}}{\bar{r}^4+a^2Z^2}
\left(2-\frac{Z^2+(y-a)^2}{\bar{r}^2+a^2}+\frac{2a^2y^2}{\left(\bar{r}^2+a^2\right)^2}\right),
\label{transverse_f}
\end{align}
where we have omitted the
odd terms concerning $\bar{x}$ with the same reason as the parallel case.

The method for obtaining this limit is the same as the parallel case. We find the
primitive function $\tilde{\Phi}$ and take the limit as follows:
\begin{equation}
\lim_{v\to 1}f\left(\gamma(x-vt), y, Z\right)
=\frac{\partial}{\partial x}\left\{\lim_{v\to 1}\tilde{\Phi}\left(\gamma(x-vt), y, Z\right)\right\}.
\end{equation}
Again $\tilde{\Phi}$ is reduced to
\begin{equation}
\lim_{v\to 1}\tilde{\Phi}\left(\gamma(x-vt), y, Z\right)=(1/2)\textrm{sgn}(x-t)\Phi(y,Z)+\bar{\Phi}\left(\gamma(x-t),y,Z\right),
\end{equation}
and we obtain
\begin{equation}
\lim_{v\to 1}f=\Phi(y,Z)\delta(x-t)
+\frac{\partial}{\partial x}\bar{\Phi}\left(\gamma(x-t),y,Z\right).
\end{equation}

Now we find the primitive function $\tilde{\Phi}$.
Using eq.~\eqref{transverse_rbar}, we find the following relations
\begin{align}
&~~~~~~~~~~~\frac{\partial\bar{r}}{\partial x}=\gamma\frac{\bar{r}^3}{\bar{r}^4+a^2Z^2}
\gamma(x-vt),
\\
&\gamma(x-vt)=\textrm{sgn}(x-vt)\sqrt{\bar{r}^2+a^2-y^2-Z^2+\frac{a^2Z^2}{\bar{r}^2}},
\end{align}
with which we derive
\begin{equation}
f=\lim_{v\to 1}\textrm{sgn}(x-vt)
\frac{P\bar{r}^{5-D}}{\sqrt{(\bar{r}^2-\alpha)(\bar{r}^2-\beta)}}
\left(2-\frac{Z^2+(y-a)^2}{\bar{r}^2+a^2}+\frac{2a^2y^2}{(\bar{r}^2+a^2)^2}\right)\frac{\partial\bar{r}}{\partial x},
\end{equation}
where $\alpha$ and $\beta$ are given by
\begin{align}
\alpha&=\frac12\left[y^2+Z^2-a^2+\sqrt{(y^2+Z^2-a^2)^2+4a^2Z^2}\right],\\
\beta&=\frac12\left[y^2+Z^2-a^2-\sqrt{(y^2+Z^2-a^2)^2+4a^2Z^2}\right].
\end{align}
Because $\bar{x}=0$ corresponds to $R\equiv \bar{r}^2=\alpha$,
we obtain
\begin{equation}
\tilde{\Phi}/P=(1/2)\textrm{sgn}(x-vt)\left[2I^{(D)}_1(R)-\left(Z^2+(y-a)^2\right)I^{(D)}_2(R)+2a^2y^2I^{(D)}_3(R)\right],
\end{equation}
where
\begin{align}
&I^{(D)}_1(R)=\int_{\alpha}^{R}\frac{R^{2-D/2}}{\sqrt{(R-\alpha)(R-\beta)}}dR,
\label{trans_I1}\\
&I^{(D)}_2(R)=\int_{\alpha}^{R}\frac{R^{2-D/2}}{(R+a^2)\sqrt{(R-\alpha)(R-\beta)}}dR,
\label{trans_I2}\\
&I^{(D)}_3(R)=\int_{\alpha}^{R}\frac{R^{2-D/2}}{(R+a^2)^2\sqrt{(R-\alpha)(R-\beta)}}dR.
\label{trans_I3}
\end{align}
Because taking the limit $v\to 1$ for $t\neq z$ corresponds to $R\to \infty$
and each integral is finite at $R=\infty$ for $D\ge 5$,
we find
\begin{align}
&\Phi(X,Y)=2I^{(D)}_1(\infty)-\left[Z^2+(y-a)^2\right]I^{(D)}_2(\infty)+2a^2y^2I^{(D)}_3(\infty),
\label{transverse_F}\\
&\bar{\Phi}\left(\gamma(x-t),X,Y\right)=0.
\end{align}
The function $\Phi$ can be expressed in terms of the elementary functions
for even $D$ and in terms of the elliptic integrals for odd $D$.
The formulas necessary for this calculation are given in Appendix B.

Now we consider the $D=4$ case before looking at the behavior of $\Phi$.
In this case, $I_1^{(4)}$ is given by
\begin{align}
I_1^{(4)}(R)&=\log\left[2R-\alpha-\beta+2\sqrt{(R-\alpha)(R-\beta)}\right]
-\log\left(\alpha-\beta\right)
\nonumber\\
&\to \log\left|4\gamma^2(x-t)^2\right|-\log\left(\alpha-\beta\right)
~(v\to 1),
\label{transverse_I1(4)}
\end{align}
and we show $I_2^{(4)}(R)$ and $I_3^{(4)}(R)$ in Appendix B.
The limit $\lim_{v\to 1}f$ becomes
\begin{equation}
\lim_{v\to 1}f/P=\frac{2}{|x-t|}+\left\{
-2\log(\alpha-\beta)
-\left[z^2+(y-a)^2\right]I_2^{(4)}(\infty)
+2a^2y^2I_3^{(4)}(\infty)
\right\}\delta(x-t).
\end{equation}
Although we should
calculate two cases $ay<0$ and $ay>0$ separately
because $I_2(\infty)$ and $I_3(\infty)$ contain $|ay|$,
they give the same result:
\begin{equation}
\lim_{v\to 1}f=2P\left\{\frac{1}{|x-t|}-\left[\log\left((y-a)^2+z^2\right)+1\right]
\delta(x-t)\right\}.
\end{equation}
This coincides with the previous results of~\cite{BN, BH03}. Again we should mention that
there remains the ambiguity in the form of eq.~\eqref{transverse_I1(4)},
because we can substitue the arbitrary function $g(y,z)$
as follows:
\begin{equation}
I_1^{(4)}(R)=\log\left\{\left[2R-\alpha-\beta+2\sqrt{(R-\alpha)(R-\beta)}\right]/e^{g(y, z)}\right\}
+g(y, z)-\log\left(\alpha-\beta\right).
\end{equation}
Hence we are required to check the consistency with the
boosted energy-momentum tensor or the boosted Riemann tennsor
and thus the above derivation is just the formal one.
For the higher-dimensional cases, there is no such ambiguity
because all integrals \eqref{trans_I1}, \eqref{trans_I2} and \eqref{trans_I3}
are finite for $R\to \infty$.

\begin{figure}[tb]
\centering
{\includegraphics[width=0.35\textwidth]{PotentD5_light.eps}\hspace{5mm}
\includegraphics[width=0.35\textwidth]{PotentD6_light.eps}\\
\vspace{5mm}
\includegraphics[width=0.35\textwidth]{PotentD7_light.eps}\hspace{5mm}
\includegraphics[width=0.35\textwidth]{PotentD8_light.eps}}
\caption{The behavior of the function $\Phi$ for $D=5,...,8$
in the transverse case. The unit of $y$
and $Z$ is $a$, and the unit of $\Phi$ is $P|a|^{D-4}$.
For all $D$,
$\Phi$ diverges at $Z=0, -a\le y\le a$, which indicates that the
singularity is a one-dimensional segment.}
\end{figure}

Figure 2 shows the potential $\Phi$ of the shock wave
for $D=5,...,8$.
For all $D\ge 5$, the potential diverges at $Z=0, -a\le y\le a$.
Hence in the transverse case,
the singularity has the shape of a one-dimensional segment for all $D\ge 5$.
This is interpreted as follows. In the boost, the ring singularity
becomes naked for $D=5$ and the horizon becomes extremely oblate for $D\ge 6.$
This singularity or the horizon is infinitely Lorentz-contracted in the
direction of the motion. This leads to the segmental shape of the singularity
in the gravitational shock wave.
The behavior
of the potential $\Phi$
around the singularity is $\sim -\log |Z|$ for $D=5$,
and $\sim 1/|Z|^{D-5}$ for $D\ge 6$.
Similarly to the parallel case, we see that the relation between the
potential behavior near the singularity and the dimensionality
of the singularity is the same as the Newtonian gravity.
The behavior of the potential is not symmetric to the $y=0$ plane.
This is the manifestation of the fact that the lightlike Kerr black hole
carries the angular momentum $J=-2ap/(D-2)$. The rotating singularity has
the velocity in the direction of the boost at $(y,Z)=(a,0)$ and
it has the velocity in the converse direction of the boost at $(y,Z)=(-a,0)$.
Correspondingly, the divergence of the potential at $(y,Z)=(a,0)$ is stronger than
the one at $(y,Z)=(-a,0)$.

\section{Summary and discussion}

In this paper, we have developed the method for taking the lightlike limit of the boosted
Kerr black holes in the higher-dimensional spacetimes. We analyzed two cases,
the boosts in the parallel direction and in the transverse direction to the spin.
Our method is quite simple: just boosting the metric without
considering the energy-momentum tensor or the Riemann tensor.
The problem is reduced to taking the limit $v\to 1$ of the function $f$ of
eqs.~\eqref{parallel_limit_f} and \eqref{transverse_limit_f}.
We solved this problem by introducing the primitive function $\tilde{\Phi}$ of $f$,
taking the limit $v\to 1$ of $\tilde{\Phi}$, and then differentiating it. Because
the primitive function $\tilde{\Phi}$ is finite for the higher-dimensional cases,
there is no ambiguity which occurred in the four-dimensional case.
Hence, we consider that the boost of the metric is the simplest
and easiest method in the higher-dimensional cases.

We discuss whether this method works for the general systems.
Also in the general systems, the lightlike limit of the metric would have
the form like eq.~\eqref{parallel_limit_f} or eq.~\eqref{transverse_limit_f}.
The problem is whether we can construct the primitive function of $f$.
If not, we cannot analytically derive the metric of the lightlike system.
This is the limitation in our method. But even if we cannot find the primitive function of $f$,
there remains the possibility that we can numerically construct the potential $\Phi$.
Namely, if $\Phi$ is reduced to the form like eqs.~\eqref{para_I1},
\eqref{para_I2}, \eqref{para_I3} and \eqref{parallel_F},
it is easy to calculate the potential $\Phi$ at each point by numerical integrals.
Hence, our analysis shows the direction of the calculation at which we should
aim in general cases.  We also would like to mention that we can guess
the resulting potential $\Phi$ to some extent even if we fail the above
directions, because the results are quite consistent with our intuition.
We can intuitively imagine the shape of the infinitely Lorentz-contracted
source and guess the shape of the potential $\Phi$ by the Newtonian analog.

Now we discuss the possible implications of our results in the context
of the brane world scenario. There are several discussions that
the Kerr black hole might provide the gravitational field of
the elementary particle with spin~\cite{ref3}, especially
motivated by the fact that the Kerr-Newman black hole has
the same gyromagnetic ratio as that of the electron. We consider that this
scenario is unlikely, because the electron in this model has the characteristic
scale $a$, whose value is similar to its Compton wave length.
Furthermore, we do not have the criterion for determining $a$ in the
case of a massless spinning particle.
However, we can expect that the spin has an effect on the gravitational
field of the elementary particle, although the spin angular momentum
and the orbital angular momentum are different. This expectation
comes from the fact that the sum of the spin and orbital angular momentum
is the conserved quantity.  It is natural that the conserved quantity affects the
spacetime structure. This expectation is strengthened
by the following observation. If the black hole forms in the head-on collision
of spinning particles,  the produced black hole would have the angular momentum
that corresponds to the sum of two spins.
In order that the black hole has the angular momentum,
the gravitational field of each particle should not be the same as that of the massless
point particle given in \cite{AS71}.

Because the lightlike Kerr black hole has the quite different gravitational field
from that of the massless point particle, we can also expect that the spin of a particle
has a large effect on its gravitational field. Hence it would affect the condition
of the black hole formation in the high-energy particle collisions.
To clarify this, the investigation of the Einstsein-Dirac system would be useful.
As the first step, the classical field treatment would be appropriate.
The formalism of this system was  recently developed by Finster~\cite{Fin98}, and
Finster {\it et al.} constructed the spin singlet state
of two fermions~\cite{FSY99}.
The generalization to the spin eigenstate of one particle and
its lightlike boost would be an interesting next problem.

Now we turn to the discussion about another possible implication, the relation
between the Kerr black holes and the strings.
In the string theory, there are several discussions that some
black holes should be regarded as the elementary particles~\cite{ref4}.
Our formalism would also be applicable to these black hole solutions.
Furthermore, Nishino~\cite{Nis95} and Nishino and Rajpoot~\cite{NR02}
discussed the fact that the Kerr black hole might be the classical model of the gravitational
field around the closed string,
simply because the Kerr black hole has the distributional energy-momentum tensor
on its singularity. Hence, our boosted Kerr solution might have an implication
for the black hole formation in the collision of high-energy strings with characteristic
length scale $a$.

Investigatiton of the properties of the
black hole formation in the collision of two lightlike Kerr strings requires
the analysis of the apparent horizon, which we have not done.
But using the fact that each incoming Kerr string has a singularity with the
characteristic scale $a$, we can discuss the condition for the black hole formation
as follows. For some impact parameter $b$,
we write the horizon shape in the case of the two-point-particle collision which
has been calculated in \cite{YN03}. Then we draw on it the shape of the singularity
of the lightlike Kerr strings with Kerr parameter $a$.
If the singularity crosses the horizon,
the apparent horizon would not form for such $b$ and $a$.
Under this assumption, we can calculate the maximal impact
parameter of the black hole production as a function of $a$.

\begin{figure}[tb]
\centering
{\includegraphics[width=0.45\textwidth]{Spin_amax.eps}}
\caption{The expected relation between $a$ and the maximal impact parameter
$b_{\rm max}$ for the black hole production in the collision of two
lightlike Kerr strings boosted in the parrallel direction to the spin.
The unit of axis is $r_h(2p)$.
We see that $b_{\rm max}=0$ in the case of $a\gtrsim r_h(2p)$ for all $D$.}
\end{figure}

Figure 3 shows the expected maximal impact parameter $b_{\rm max}$
for the black hole production
in the collision of two Kerr strings which are boosted
in the direction of the spin.
If the Kerr parameter $a$ (i.e., the length of the string) is small compared
to the gravitational radius of the system energy $r_h(2p)$,
it does not affect the black hole formation.
But $b_{\rm max}$ decreases with the increase in $a$, and
becomes zero at $a\sim r_h(2p)$.
We can write a similar figure in the case of two lightlike Kerr strings
which are boosted in the transverse direction to the spin.
Hence, the condition for $b_{\rm max}\neq 0$
is written as $a\lesssim r_h(2p)$.
If the Planck energy is $O({\rm TeV})$ and the string length $a$ is
similar to the Planck length, the gravitational radius for the two particle system
becomes $r_h(2p)\sim a$ if the incoming energy is (few) TeV.
Hence the string length might have the effect such that it prevents the black hole formation.

We should mention that the above discussion is not rigorous, because
we do not know the validity of the assumption we have used.
We can also have the totally different expectation, because
in the higher-dimensional spacetime, the arbitarily long black hole can form
as clarified by Ida and Nakao~\cite{IN02}. The higher-dimensional effect might
lead to the formation of the long apparent horizon in the high-energy
string collision depending on the shape of two strings.
If this is true, the cross section for the black hole
production might be enhanced. Furthermore, we can also
expect the formation of the black ring,
which is the higher-dimensional black hole whose
horizon topology is $S^1\times S^{D-3}$. (See \cite{ER02}
for the five-dimensional solution.)
If the shape of two strings at the instant of the collision is similar to the ring,
the ring-shaped apparent horizon that surrounds the two strings
might form. Although the author and Nambu discussed
one possible method to produce the black rings in particle systems in the previous paper~\cite{YN04},
it required more than two particles. If the black ring forms due to the
effect of the string length, it would be the more interesting phenomena.
To clarify the effect of the string length on the black hole formation
in the high-energy string collisions
would be the interesting problem to be tackled as the next step.

\acknowledgments

The author thanks Ken-ichi Nakao, Yasusada Nambu and Akira Tomimatsu for helpful discussions.
I also thank Kin-ya Oda, Daisuke Ida and Yasunari Kurita for their encouragement.
This work is partly supported by the grant-in-aid for the Nagoya
University 21st Century COE Program (ORIUM).