$\sup_{\bbeta\in\mathbb{B}(r)}\|T_{1\bbeta}\|_2=O_p(K^{1/2}r)$. Again, the key step is to use the results from Lemma <ref>.
For the second term $T_2$, we can use the similar arguments in the proof of the first result to show that  $\EE\|T_2\|^2_2\leq CK\cdot \EE[\eta_i(m^*(\bB(\bX_i))^2]/n=O(K/n)$. The Markov inequality implies $\|T_2\|_2=O_p(K^{1/2}/n^{1/2})$. For the third term $T_3$, after some algebra, we can show that

\|T_3\|_2\leq C\Big(\sup_{\bx\in\mathcal{X}}|K(\bx)-\balpha_1^{*\top}\bh_1(\bx)|+\sup_{\bx\in\mathcal{X}}|L(\bx)-\balpha_2^{*\top}\bh_2(\bx)|\Big)=O_p(K^{-r_h}).

Combining the $L_2$ error bound for $T_{1\bbeta}$, $T_2$ and $T_3$, we obtain the last result. This completes the whole proof.

Under the conditions in Theorem <ref>, it holds that

\|\tilde\bbeta-\bbeta^*\|_2=o_p(1).

Recall that $\bbeta^o$ is the minimizer of $Q(\bbeta)$. We now decompose $Q(\tilde\bbeta)-Q(\bbeta^o)$ as

\begin{equation}\label{eqlemnonrate2}
\end{equation}

In the following, we study the terms I, II and III one by one. For the term I, Lemma <ref> implies
$|Q(\tilde\bbeta)-Q_n(\tilde\bbeta)|\leq \sup_{\bbeta\in\Theta}\Big|Q_n(\bbeta)-Q(\bbeta)\Big|=o_p(1)$.
This shows that $|I|=o_p(1)$ and the same argument yields $|III|=o_p(1)$. For the term II, by the definition of $\tilde\bbeta$, it is easy to see that $II\leq 0$. Thus, combining with (<ref>), we have for any constant $\eta>0$ to be chosen later,  $Q(\tilde\bbeta)-Q(\bbeta^o)<\eta$ with probability tending to one. For any $\epsilon>0$, define $E_\epsilon=\Theta\cap\{\|\bbeta-\bbeta^o\|_2\geq \epsilon\}$. By the uniqueness of $\bbeta^o$, for any $\bbeta\in E_\epsilon$, we have $Q(\bbeta)>Q(\bbeta^o)$. Since $E_\epsilon$ is a compact set, we have $\inf_{\bbeta\in E_\epsilon} Q(\bbeta)>Q(\bbeta^o)$. This implies that for any $\epsilon>0$, there exists $\eta'>0$ such that $Q(\bbeta)>Q(\bbeta^o)+\eta'$ for any $\bbeta\in E_\epsilon$. If $\tilde\bbeta\in E_\epsilon$, then $Q(\bbeta^o)+\eta>Q(\tilde\bbeta)>Q(\bbeta^o)+\eta'$ with probability tending to one. Apparently, this does not holds if we take $\eta<\eta'$. Thus, we have proved that  $\tilde\bbeta\notin E_\epsilon$, that is $\|\tilde\bbeta-\bbeta^o\|_2\leq \epsilon$ for any $\epsilon>0$. Thus, we have $\|\tilde\bbeta-\bbeta^o\|_2=o_p(1)$.

Next, we shall show that $\|\bbeta^o-\bbeta^*\|_2=o_p(1)$. It is easily seen that these together lead to the desired consistency result

$$\|\tilde\bbeta-\bbeta^*\|_2\leq \|\bbeta^o-\bbeta^*\|_2+\|\tilde\bbeta-\bbeta^o\|_2=o_p(1).

To show $\|\bbeta^o-\bbeta^*\|_2=o_p(1)$, we use the similar strategy. That is we want to show that for any constant $\eta>0$,  $Q(\bbeta^*)-Q(\bbeta^o)<\eta$. In the following, we prove that $Q(\bbeta^*)=O(K^{1-2r_b})$. Note that

Q(\bbeta^*)\leq C^2K^{-2r_b}\sum_{j=1}^{K}\EE|\bh_j(\bX)|^2=O(K^{1-2r_b}),

where the first inequality follows from the Cauchy-Schwarz inequality and the last step uses the assumption that $\sup_{\bx\in\mathcal{X}}\|\bh(\bx)\|_2=O(K^{1/2})$.

In addition, it holds that $Q(\bbeta^o)\leq Q(\bbeta^*)=O(K^{1-2r_b})$. As $K\rightarrow\infty$, it yields $Q(\bbeta^*)-Q(\bbeta^o)<\eta$, for any constant $\eta>0$. The same arguments yield $\|\bbeta^o-\bbeta^*\|_2=o_p(1)$. This completes the proof of the consistency result.

Under the conditions in Theorem <ref>, there exists a global minimizer $\tilde\bbeta$ (if $Q_n(\bbeta)$ has multiple minimizers), such that

\begin{equation}\label{eqlemnonrate0}
\|\tilde\bbeta-\bbeta^*\|_2=O_p(K^{1/2}/n^{1/2}+K^{-r_b}).
\end{equation}

We first prove that there exists a local minimizer $\tilde\bDelta$ of $Q_n(\bbeta^*+\bDelta)$, such that $\tilde\Delta\in\mathcal{C}$, where $\mathcal{C}=\{\bDelta\in\RR^K: \|\bDelta\|_2\leq r\}$,  and  $r=C(K^{1/2}/n^{1/2}+K^{-r_b})$ for some constant $C$ large enough. To this end, it suffices to show that

\begin{equation}\label{eqlemnonrate1}
\PP\Big\{\inf_{\bDelta\in\partial\mathcal{C}} Q_n(\bbeta^*+\bDelta)-Q_n(\bbeta^*)>0\Big\}\rightarrow 1, ~~\textrm{as}~n\rightarrow\infty,
\end{equation}

where $\partial\mathcal{C}=\{\bDelta\in\RR^K: \|\bDelta\|_2=r\}$. Applying the mean value theorem to each component of $\bar\bg_{\bbeta^*+\bDelta}(\bT, \bX)$,

\bar\bg_{\bbeta^*+\bDelta}(\bT, \bX)=\bar\bg_{\bbeta^*}(\bT, \bX)+\widetilde\Gb\bDelta,

where $\widetilde\Gb=\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}$ and for notational simplicity we assume there exists a common $\bar\bbeta=v\bbeta^*+(1-v)\tilde\bbeta$ for some $0\leq v\leq 1$ lies between $\bbeta^*$ and $\bbeta^*+\bDelta$ (Rigorously speaking, we need different $\bar\bbeta$ for different component of $\bar\bg_{\bbeta^*+\bDelta}(\bT, \bX)$). Thus, for any $\bDelta\in\partial\mathcal{C}$,

\begin{align}
Q_n(\bbeta^*+\bDelta)-Q_n(\bbeta^*)&=2\bar\bg_{\bbeta^*}(\bT, \bX)\widetilde\Gb\bDelta+\bDelta^\top(\widetilde\Gb^\top\widetilde\Gb)\bDelta\nonumber\\
&\geq-2\|\bar\bg_{\bbeta^*}(\bT, \bX)\|_2\cdot\|\widetilde\Gb\|_2\cdot\|\bDelta\|_2+\|\bDelta\|_2^2\cdot\lambda_{\min}(\widetilde\Gb^\top\widetilde\Gb)\nonumber\\
&\geq-C(K^{1/2}/n^{1/2}+K^{-r_b})\cdot r+C\cdot r^2,\label{eqlemnonrate3}
\end{align}

for some constant $C>0$. In the last step, we first use the results that $\|\bar\bg_{\bbeta^*}(\bT, \bX)\|_2=O_p(K^{1/2}/n^{1/2}+K^{-r_b})$, which is derived by combining Lemma <ref> with the arguments similar to (<ref>) in the proof of Lemma <ref>. In addition, $\|\widetilde\Gb\|_2\leq \|\widetilde\Gb-\Gb^*\|_2+\|\Gb^*\|_2\leq C$, since $\|\Gb^*\|_2$ is bounded by a constant and $\|\widetilde\Gb-\Gb^*\|_2=o_p(1)$ by Lemma  <ref>. By the Weyl inequality and Lemma  <ref>,

\begin{align*}
\lambda_{\min}(\widetilde\Gb^\top\widetilde\Gb)&\geq\lambda_{\min}(\Gb^{*\top}\Gb^*)-\|\widetilde\Gb^\top\widetilde\Gb-\Gb^{*\top}\Gb^*\|_2\\
&\geq C-\|\widetilde\Gb-\Gb^*\|_2\cdot\|\widetilde\Gb\|_2-\|\widetilde\Gb-\Gb^*\|_2\cdot \|\Gb^*\|_2\geq C/2,
\end{align*}

for $n$ sufficiently large. By (<ref>), if $r=C(K^{1/2}/n^{1/2}+K^{-r_b})$ for some constant $C$ large enough, the right hand side is positive for $n$ large enough. This establishes (<ref>). Next, we show that $\tilde\bbeta=\bbeta^*+\tilde\bDelta$ is a global minimizer of $Q_n(\bbeta)$. This is true because the first order condition implies

\Big(\frac{\partial \bar\bg_{\tilde\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)\bar\bg_{\tilde\bbeta}(\bT, \bX)=0, ~~\Longrightarrow~~\bar\bg_{\tilde\bbeta}(\bT, \bX)=0,

provided $\partial \bar\bg_{\tilde\bbeta}(\bT, \bX)/\partial\bbeta$ is invertible. Following the similar arguments by applying the Weyl inequality, $\partial \bar\bg_{\tilde\bbeta}(\bT, \bX)/\partial\bbeta$ is invertible with probability tending to one. Since $\bar\bg_{\tilde\bbeta}(\bT, \bX)=0$, it implies $Q_n(\tilde\bbeta)=0$. Noting that $Q_n(\bbeta)\geq 0$ for any $\bbeta$, we obtain that $\tilde\bbeta$ is indeed a global minimizer of $Q_n(\bbeta)$.

Under the conditions in Theorem <ref>, $\tilde\bbeta$ satisfies the following asymptotic expansion

\begin{equation}\label{lemnonexpansion1}
\tilde\bbeta-\bbeta^*=-\Gb^{-1}\bar\bU+\bDelta_{n},
\end{equation}

where $\bar\bU=\frac{1}{n}\sum_{i=1}^n\bU_i$, $\bU_i=(\bU^\top_{i1},\bU^\top_{i2})^\top$, with

\bU_{i1}=	\left(\frac{T_i}{\pi^*_i}-\frac{1-T_i}{1-\pi^*_i}\right)\bh_1(\bX_i),~~\bU_{i2}= \left(\frac{T_i}{\pi^*_i}-1\right)\bh_2(\bX_i),

\|\bDelta_n\|_2=O_p\Big(K^{1/2}\cdot\Big(\frac{K^{1/2}}{n^{1/2}}+\frac{1}{K^{r_b}}\Big)^2+\sqrt{\frac{K\log K}{n}}\cdot \Big(\frac{K^{1/2}}{n^{1/2}}+\frac{1}{K^{r_b}}\Big)\Big).

Similar to the proof of Lemma <ref>, we apply the mean value theorem to each component of $\bar\bg_{\tilde\bbeta}(\bT, \bX)$,

\bar\bg_{\bbeta^*}(\bT, \bX)+\Big(\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)(\tilde\bbeta-\bbeta^*)=0,

where for notational simplicity we assume there exists a common $\bar\bbeta=v\bbeta^*+(1-v)\tilde\bbeta$ for some $0\leq v\leq 1$ lies between $\bbeta^*$ and $\tilde\bbeta$. After rearrangement, we derive

\begin{align}
\tilde\bbeta-\bbeta^*&=-\Gb^{*-1}\bar\bg_{\bbeta^*}(\bT, \bX)+\Big[\Gb^{*-1}-\Big(\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)^{-1}\Big]\bar\bg_{\bbeta^*}(\bT, \bX)\nonumber\\
\end{align}

\bDelta_{n1}=\Gb^{*-1}[\bar \bU-\bar\bg_{\bbeta^*}(\bT, \bX)], ~~~\bDelta_{n2}=\Big[\Gb^{*-1}-\Big(\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)^{-1}\Big]\bar \bU

\bDelta_{n3}=\Big[\Gb^{*-1}-\Big(\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)^{-1}\Big]\cdot[\bar\bg_{\bbeta^*}(\bT, \bX)-\bar \bU].

We first consider $\bDelta_{n1}$ in (<ref>). Let $\bxi=(\xi_1,...,\xi_n)^\top$, where

$$\xi_i=T_i\Big(\frac{1}{\pi^*_i}-\frac{1}{J^*_i}\Big)-(1-T_i)\Big(\frac{1}{1-\pi^*_i}-\frac{1}{1-J^*_i}\Big), ~~\textrm{for}~ 1\leq i\leq m_1,

$$\xi_i=T_i\Big(\frac{1}{\pi^*_i}-\frac{1}{J^*_i}\Big), ~~\textrm{for}~m_1+1\leq i\leq K.

Let $\Hb=(\bh(X_1),...,\bh(X_n))^\top$ be a $n\times K$ matrix. Then, for some constants $C,C'>0$,

\begin{align}
\|\bDelta_{n1}\|^2_2&= n^{-2}\bxi^\top\Hb\Gb^{*-1}\Gb^{*-1}\Hb^\top\bxi\leq n^{-2}\|\bxi\|^2_2\cdot \|\Hb\Gb^{*-1}\Gb^{*-1}\Hb^\top\|_2\nonumber\\
&\leq Cn^{-1}\|\bxi\|^2_2\cdot \|\Hb^\top\Hb/n\|_2\leq C'n^{-1}\|\bxi\|^2_2,\label{lemnonexpansion3}
\end{align}

where the third step follows from the fact that $\|\Gb^{*-1}\|_2$ is bounded and the last step follows from Lemma <ref> and the maximum eigenvalue of $\EE[\bh(\bX_i)\bh(\bX_i)^\top]$ is bounded. Since $|\partial J(v)/\partial v|$ is upper bounded by a constant for any $v\leq \sup_{\bx\in\mathcal{X}}|m^*(\bx)|$, then there exist some constants $C,C'>0$, suc that for any $m_1+1\leq i\leq K$,

|\xi_i|\leq C|\pi_i^*-J_i^*|\leq C'\sup_{\bx\in\mathcal{X}}|m^*(\bx)-\bbeta^{*\top}\bB(\bx)|\leq C'K^{-r_b}.

Similarly, $|\xi_i|\leq 2C'K^{-r_b}$ for any $1\leq i\leq m_1$. Thus, it yields $n^{-1}\|\bxi\|^2_2=O_p(K^{-2r_b})$. Combining with (<ref>), we conclude that $\|\bDelta_{n1}\|_2=O_p(K^{-r_b})$.

Next, we consider $\bDelta_{n2}$. Since $\|\Gb^{*-1}\|_2$ is bounded, we have

\begin{align*}
\|\bDelta_{n2}\|_2&\leq \|\Gb^{*-1}\|_2 \cdot\Big\|\Big(\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)^{-1}\Big\|_2\cdot \Big\|\Gb^*-\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big\|_2\cdot\|\bar \bU\|_2\\
&\leq C\Big(\|\tilde\bbeta-\bbeta^*\|_2K^{1/2}+\sqrt{\frac{K\log K}{n}}\Big)\cdot \sqrt{\frac{K}{n}},
\end{align*}

where the last step follows from Lemma <ref>.

Finally, we consider $\bDelta_{n3}$. By the same arguments in the control of terms $\bDelta_{n1}$ and $\bDelta_{n2}$, we can prove that

\begin{align*}
\|\bDelta_{n3}\|_2&\leq \Big\|\Gb^{*-1}-\Big(\frac{\partial \bar\bg_{\bar\bbeta}(\bT, \bX)}{\partial\bbeta}\Big)^{-1}\Big\|_2\cdot\|\bar\bg_{\bbeta^*}(\bT, \bX)-\bar \bU\|_2\\
&\leq C\Big(\|\tilde\bbeta-\bbeta^*\|_2K^{1/2}+\sqrt{\frac{K\log K}{n}}\Big)\cdot K^{-r_b}.
\end{align*}

Combining the rates of $\|\bDelta_{n1}\|_2$, $\|\bDelta_{n2}\|_2$ and $\|\bDelta_{n3}\|_2$ with (<ref>), by Lemma <ref>, we obtain

\begin{align*}
\|\tilde\bbeta-\bbeta^*\|_2&\leq \|\Gb^{*-1}\bar\bg_{\bbeta^*}(\bT, \bX)\|_2+\|\bDelta_{n1}\|_2+\|\bDelta_{n2}\|_2+\|\bDelta_{n3}\|_2\\
&\leq C\Big(\frac{K^{1/2}}{n^{1/2}}+\frac{1}{K^{r_b}}\Big)+C'\Big(\|\tilde\bbeta-\bbeta^*\|_2K^{1/2}+\sqrt{\frac{K\log K}{n}}\Big)\cdot \Big(\frac{K^{1/2}}{n^{1/2}}+\frac{1}{K^{r_b}}\Big),
\end{align*}

for some constants $C,C'>0$.
Therefore, (<ref>) holds with $\bDelta_n=\bDelta_{n1}+\bDelta_{n2}+\bDelta_{n3}$, where

\|\bDelta_n\|_2=O_p\Big(K^{1/2}\cdot\Big(\frac{K^{1/2}}{n^{1/2}}+\frac{1}{K^{r_b}}\Big)^2+\sqrt{\frac{K\log K}{n}}\cdot \Big(\frac{K^{1/2}}{n^{1/2}}+\frac{1}{K^{r_b}}\Big)\Big).

This completes the proof.

We now consider the following decomposition of $\tilde\mu_{\tilde\bbeta}-\mu$,

\begin{align*}
\tilde\mu_{\tilde\bbeta}-\mu&=\frac{1}{n}\sum_{i=1}^n\Big[\frac{T_i(Y_i(1)-K(\bX_i)-L(\bX_i))}{\tilde J_i}-\frac{(1-T_i)(Y_i(0)-K(\bX_i))}{1-\tilde J_i}\Big]\\
&~~~~~~~~+\frac{1}{n}\sum_{i=1}^n\Big(\frac{T_i}{\tilde J_i}-\frac{1-T_i}{1-\tilde J_i}\Big)K(\bX_i)+\frac{1}{n}\sum_{i=1}^n\Big(\frac{T_i}{\tilde J_i}-1\Big)L(\bX_i)+\frac{1}{n}\sum_{i=1}^n L(\bX_i)-\mu\\
&=\frac{1}{n}\sum_{i=1}^n\Big[\frac{T_i(Y_i(1)-K(\bX_i)-L(\bX_i))}{\tilde J_i}-\frac{(1-T_i)(Y_i(0)-K(\bX_i))}{1-\tilde J_i}\Big]\\
&~~~~~~~~+\frac{1}{n}\sum_{i=1}^n\Big(\frac{T_i}{\tilde J_i}-\frac{1-T_i}{1-\tilde J_i}\Big)\Delta_K(\bX_i)+\frac{1}{n}\sum_{i=1}^n\Big(\frac{T_i}{\tilde J_i}-1\Big)\Delta_L(\bX_i)+\frac{1}{n}\sum_{i=1}^n L(\bX_i)-\mu,
\end{align*}

where $\tilde J_i=J(\tilde\bbeta^\top\bB(X_i))$, $\Delta_K(\bX_i)=K(\bX_i)-\balpha_1^{*\top}\bh_1(\bX_i)$ and $\Delta_L(\bX_i)=L(\bX_i)-\balpha_2^{*\top}\bh_2(\bX_i)$. Here, the second equality holds by the definition of $\tilde\bbeta$. Thus, we have

\begin{align*}
\tilde\mu_{\tilde\bbeta}-\mu&=\frac{1}{n}\sum_{i=1}^n S_i+R_0+R_1+R_2+R_3
\end{align*}

R_0=\frac{1}{n}\sum_{i=1}^n \frac{T_i(Y_i(1)-K(\bX_i)-L(\bX_i))}{\tilde J_i\pi^*_i}(\pi^*_i-\tilde J_i),

R_1=\frac{1}{n}\sum_{i=1}^n \frac{(1-T_i)(Y_i(0)-K(\bX_i))}{(1-\tilde J_i)(1-\pi^*_i)}(\pi^*_i-\tilde J_i),

R_2=\frac{1}{n}\sum_{i=1}^n\Big(\frac{T_i}{\tilde J_i}-\frac{1-T_i}{1-\tilde J_i}\Big)\Delta_K(\bX_i),
~~R_3=\frac{1}{n}\sum_{i=1}^n\Big(\frac{T_i}{\tilde J_i}-1\Big)\Delta_L(\bX_i).

In the following, we will show that $R_j=o_p(n^{-1/2})$ for $0\leq j\leq 3$. Thus, the asymptotic normality of $n^{1/2}(\tilde\mu_{\tilde\bbeta}-\mu)$ follows from the previous decomposition. In addition, $S_i$ agrees with the efficient score function for estimating $\mu$ <cit.>. Thus, the proposed estimator $\tilde\mu_{\tilde\bbeta}$ is also semiparametrically efficient.

Now, we first focus on $R_0$. Consider the following empirical process $\GG_n(f_0)=n^{1/2}(\PP_n-\PP)f_0(T, Y(1), \bX)$, where $\PP_n$ stands for the empirical measure and $\PP$ stands for the expectation, and

f_0(T, Y(1), \bX)=\frac{T(Y(1)-K(\bX)-L(\bX))}{J(m(\bX))\pi^*(\bX)}[\pi^*(\bX)-J(m(\bX))].

By Lemma <ref>, we can easily show that

\begin{align*}
\sup_{\bx\in\mathcal{X}}&|J(\tilde\bbeta^\top\bB(\bx))-\pi^*(\bx)|\lesssim \sup_{\bx\in\mathcal{X}}|\tilde\bbeta^\top\bB(\bx)-\bbeta^{*\top}\bB(\bx)|\\
\end{align*}

For notational simplicity, we denote $\|f\|_\infty=\sup_{\bx\in\mathcal{X}}|f(\bx)|$. Define the set of functions $\mathcal{F}=\{f_0: \|m-m^*\|_\infty\leq \delta\}$, where $\delta=C(K/n^{1/2}+K^{1/2-r_b})$ for some constant $C>0$.
By the strong ignorability of the treatment assignment, we have that $\PP f_0(T, Y(1), \bX)=0$. By the Markov inequality and the maximal inequality in Corollary 19.35 of <cit.>,

n^{1/2}R_0\leq \sup_{f_0\in\mathcal{F}}\GG_n(f_0)\lesssim \EE \sup_{f_0\in\mathcal{F}}\GG_n(f_0)\lesssim J_{[~]}(\|F_0\|_{P,2}, \mathcal{F}, L_2(P)),

where $J_{[~]}(\|F_0\|_{P,2}, \mathcal{F}, L_2(P))$ is the bracketing integral, and $F_0$ is the envelop function. Since $J$ is bounded away from 0, we have $|f_0(T, Y(1), \bX)|\lesssim \delta |Y(1)-K(\bX)-L(\bX)|:=F_0$. Then $\|F_0\|_{P,2}\leq \delta \{\EE|Y(1)|^2\}^{1/2}\lesssim \delta$. Next, we consider $N_{[~]}(\epsilon, \mathcal{F}, L_2(P))$. Define $\mathcal{F}_0=\{f_0: \|m-m^*\|_\infty\leq C\}$ for some constant $C>0$. Thus, it is easily seen that $\log N_{[~]}(\epsilon, \mathcal{F}, L_2(P))\lesssim \log N_{[~]}(\epsilon, \mathcal{F}_0\delta, L_2(P))=\log N_{[~]}(\epsilon/\delta, \mathcal{F}_0, L_2(P))\lesssim \log N_{[~]}(\epsilon/\delta, \mathcal{M}, L_2(P))\lesssim (\delta/\epsilon)^{1/k_1}$, where we use the fact that $J$ is bounded away from 0 and $J$ is Lipschitz. The last step follows from the assumption on the bracketing number of $\mathcal{M}$. Then

\begin{align*}
J_{[~]}(\|F_0\|_{P,2}, \mathcal{F}, L_2(P))\lesssim \int_0^\delta \sqrt{\log N_{[~]}(\epsilon, \mathcal{F}, L_2(P))}d\epsilon \lesssim \int_0^\delta (\delta/\epsilon)^{1/(2k_1)}d \epsilon,
\end{align*}

which goes to 0, as $\delta\rightarrow 0$, because $2k_1>1$ by assumption and thus the integral converges. Thus, this shows that $n^{1/2}R_0=o_p(1)$. By the similar argument, we can show that $n^{1/2}R_1=o_p(1)$.

Next, we consider $R_2$. Define the following empirical process $\GG_n(f_2)=n^{1/2}(\PP_n-\PP)f_2(T, \bX)$, where

f_2(T, \bX)=\frac{T-J(m(\bX))}{J(m(\bX))(1-J(m(\bX)))}\Delta_K(\bX).

By the assumption on the approximation property of the basis functions, we have $\|\Delta_K\|_\infty\lesssim K^{-r_h}$. In addition,

\begin{align*}
\|J(\tilde\bbeta^\top\bB(\bX))-\pi^*(\bX)\|_{P,2}&\leq \|J(\tilde\bbeta^\top\bB(\bX))-J(\bbeta^{*\top}\bB(\bX))\|_{P,2}+\|J(\bbeta^{*\top}\bB(\bX))-\pi^*(\bX)\|_{P,2}\\
&\lesssim \|\tilde\bbeta^\top\bB(\bX)-\bbeta^{*\top}\bB(\bX)\|_{P,2}+\sup_{\bx\in\mathcal{X}}|m^*(\bx)-\bbeta^{*\top}\bB(\bx)|\\
\end{align*}

where the last step follows from Lemma <ref>.

Define the set of functions $\mathcal{F}=\{f_2: \|m-m^*\|_{P,2}\leq \delta_1, \|\Delta\|_\infty\leq \delta_2\}$, where $\delta_1=C(K^{1/2}/n^{1/2}+K^{-r_b})$ and $\delta_2=CK^{-r_h}$ for some constant $C>0$. Thus,

n^{1/2}R_2\leq \sup_{f_2\in\mathcal{F}}\GG_n(f_2)+n^{1/2}\sup_{f_2\in\mathcal{F}}\PP f_2.

We first consider the second term $n^{1/2}\sup_{f_2\in\mathcal{F}}\PP f_2$. Let $\mathcal{G}_1=\{m\in\mathcal{M}:  \|m-m^*\|_{P,2}\leq \delta_1\}$ and $\mathcal{G}_2=\{\Delta\in\mathcal{H}-\balpha_1^{*\top}\bh_1:  \|\Delta\|_{\infty}\leq \delta_2\}$.
By the definition of the propensity score and Cauchy inequality,

\begin{align*}
n^{1/2}\sup_{f_2\in\mathcal{F}}\PP f_2&=n^{1/2}\sup_{m\in\mathcal{G}_1, \Delta\in\mathcal{G}_2}\EE \frac{\pi^*(\bX)-J(m(\bX))}{J(m(\bX))(1-J(m(\bX)))}\Delta(\bX)\\
&\lesssim n^{1/2}\sup_{m\in\mathcal{G}_1}\|\pi^*-J(m)\|_{P,2}\sup_{\Delta\in\mathcal{G}_2}\|\Delta\|_{P,2}\\
&\lesssim n^{1/2}\delta_1\delta_2\lesssim n^{1/2}(K^{1/2}/n^{1/2}+K^{-r_b})K^{-r_h}=o(1),
\end{align*}

where the last step follows from $r_h>1/2$ and the scaling assumption $n^{1/2}\lesssim K^{r_b+r_h}$ in this theorem. Next, we need to control the maximum of the empirical process $\sup_{f_2\in\mathcal{F}}\GG_n(f_2)$. Following the similar argument to that for $R_0$, we only need to upper bound the bracketing integral $J_{[~]}(\|F_2\|_{P,2}, \mathcal{F}, L_2(P))$. Since $J$ is bounded away from 0 and 1, we can set the envelop function to be $F_2:=C\delta_2$ for some constant $C>0$ and thus $\|F_2\|_{P,2}\lesssim \delta_2$. Define $\mathcal{F}_0=\{f_2: \|m-m^*\|_{P,2}\leq C, \|\Delta\|_{P,2}\leq 1\}$ for some constant $C>0$, $\mathcal{G}_{10}=\{m\in\mathcal{M}+m^*:  \|m\|_{P,2}\leq C\}$ and $\mathcal{G}_{20}=\{\Delta\in\mathcal{H}-\balpha_1^{*\top}\bh_1:  \|\Delta\|_{P,2}\leq 1\}$. Similarly, we have

\begin{align*}
\log N_{[~]}(\epsilon, \mathcal{F}, L_2(P))&\lesssim \log N_{[~]}(\epsilon/\delta_2, \mathcal{F}_0, L_2(P))\\
&\lesssim \log N_{[~]}(\epsilon/\delta_2, \mathcal{G}_{10}, L_2(P))+\log N_{[~]}(\epsilon/\delta_2, \mathcal{G}_{20}, L_2(P))\\
&\lesssim \log N_{[~]}(\epsilon/\delta_2, \mathcal{M}, L_2(P))+\log N_{[~]}(\epsilon/\delta_2, \mathcal{H}, L_2(P))\\
&\lesssim (\delta_2/\epsilon)^{1/k_1}+(\delta_2/\epsilon)^{1/k_2},
\end{align*}

where the second step follows from the boundness assumption on $J$ and its Lipschitz property, the third step is due to $\mathcal{G}_{10}-m^*\subset \mathcal{M}$ and $\mathcal{G}_{20}+\balpha_1^{*\top}\bh_1\subset \mathcal{H}$ and the last step is by the bracketing number condition in our assumption.  Since $2k_1>1$ and $2k_2>1$, it is easily seen that the bracketing integral $J_{[~]}(\|F_2\|_{P,2}, \mathcal{F}, L_2(P))=o(1)$. This shows that $\sup_{f_2\in\mathcal{F}}\GG_n(f_2)=o_p(1)$.  Thus, we conclude that $n^{1/2}R_2=o_p(1)$. By the similar argument, we can show that $n^{1/2}R_3=o_p(1)$. This completes the whole proof.

§ DISCUSSION ON THE RESULTS IN SECTION <REF>

Under the conditions in Theorem <ref>, it is well known that the convergence rate for estimating $K(\bx)$ (and also $L(\bx)$, $\psi^*(\bx)$) in the $L_2(P)$ norm (i.e, $\int (\hat K(\bx)-K(\bx))^2P(d\bx)$) is $O_p(\kappa^{-2r_h}+\kappa/n)$; see <cit.>. Thus, the optimal choice of $\kappa$ that minimizes the rate is $\kappa\asymp n^{1/(2r_h+1)}$. Assume that $r_b=r_h$. With $\kappa\asymp n^{1/(2r_h+1)}$, the conditions $\kappa=o(n^{1/3})$ and $n^{\frac{1}{2(r_b+r_h)}}=o(\kappa)$ always hold as long as $r_h>1$. Recall that from the previous discussion $r_h=s/d$, where $s$ is the smoothness parameter and $d$ is the dimension of $\bX$. Thus under very mild conditions $s>d$, we do not need to under-smooth the estimator.

By the proof of Theorem <ref>, we find that when
$\kappa=o(n^{1/(2r_b+1)})$ and $\kappa=o(n^{1/(2r_h+1)})$ hold, the asymptotic
bias of the estimator $\tilde{\mu}_{\tilde\bbeta}$ is of order
$O_p(K^{-(r_b+r_h)})$, which is the product of the approximation
errors for $\psi^*(\bx)$ and $K(\bx)$ (also $L(\bx)$). Thus, to make
the bias of the estimator $\tilde{\mu}_{\tilde\bbeta}$
asymptotically ignorable, we can require either $r_b$ or $r_h$
sufficiently large (not necessarily both). This phenomenon can be
viewed as the double robustness property in the nonparametric
context, which holds for the kernel based doubly robust estimator
<cit.> and the targeted maximum likelihood
estimator <cit.>.  In addition, our estimator
has smaller asymptotic bias than the usual nonparametric method. For
simplicity, assume $r_b=r_h=r$. The asymptotic bias of the IPTW
estimator in <cit.> is generally of order
$O_p(\kappa^{-r})$, whereas our estimator has a smaller bias of order

§ ESTIMATION OF ATT

We consider the estimation of the average treatment effect for the treated (ATT)

\tau^*=\EE(Y_i(1)-Y_i(0)|T_i=1).

$\tau^*_1=\EE(Y_i(1)\mid T_i=1)$ and $\tau^*_0=\EE(Y_i(0)\mid
T_i=1)$. By the law of total probability,

\tau^*_1=\EE(T_iY_i(1)\mid T_i=1)=\EE(T_iY_i(1))/\PP(T_i=1).

Thus, a simple estimator of $\tau^*_1$ is

\hat\tau_1=\frac{\sum_{i=1}^n T_iY_i}{\sum_{i=1}^n T_i}.

To estimate $\tau_0^*$, we notice that

\begin{align*}
\tau_0^*&=\EE[\EE(Y_i(0)\mid T_i=1,\bX_i)\mid T_i=1]=\EE[\EE(Y_i(0)\mid \bX_i)\mid T_i=1]\\
&= \frac{\EE[T_i  \EE(Y_i(0)\mid \bX_i)]} {\PP(T_i=1)}
%\int \EE(Y_i(0)\mid \bX_i)\frac{\PP(T_i=1\mid\bX_i)f(\bX_i)}{\PP(T_i=1)}d\bX_i
=\frac{\EE(\pi(\bbeta^{*\top}\bX_i)\EE(Y_i(0)\mid \bX_i))}{\PP(T_i=1)}\\
\end{align*}

Similar to the bias and variance calculation for the ATE, we can estimate $\bbeta$ by the solving the following estimating equations

{n}^{-1}\sum_{i=1}^n \left(T_i-\frac{(1-T_i)\pi(\bbeta^\top\bX)}{1-\pi(\bbeta^\top\bX)}\right)\fb(\bX)=0.

Then, we set $\hat\pi_i=\pi(\hat\bbeta^\top\bX_i)$ and estimate $\tau_0$ by

\hat\tau_0=\frac{\sum_{i=1}^n(1-T_i)\hat r_iY_i}{\sum_{i=1}^n (1-T_i) \hat r_i},

where $\hat r_i=\hat\pi_i/(1-\hat\pi_i)$. The final estimator of
the ATT is $\hat\tau=\hat\tau_1-\hat\tau_0$. Similar to the proof of the main results on ATE, we can show that when both models are correct, $n^{1/2}(\hat{\tau}-\tau^*)\rightarrow_d N(0,W)$, where

W \ = \ p^{-2}\EE\left[\pi^*\EE(\epsilon_1^2\mid \bX)+\frac{\pi^{*2}}{1-\pi_i^*}\EE(\epsilon_0^2\mid \bX)+\pi^*(L(\bX_i)-\tau^*)^2\right].

Here, $\epsilon_0=Y(0)-K(\bX)$, $\epsilon_1=Y(1)-K(\bX)-L(\bX)$ and $p=\PP(Y=1)$

§ DERIVATION OF (<REF>) AND  (<REF>)

In this appendix, we only provide a sketch of the proof of (<ref>) and  (<ref>), because the detail is very similar to the proof of Theorem <ref>. Recall that as in Section <ref>, $\bbeta^o$ which satisfies $\EE(\bar\bg_{\bbeta^o}(\bT,\bX))=0$ is the limiting value of $\hat\bbeta$ as in Lemma <ref>.  In addition, denote $K^o(\bX_i)=\balpha^{*T}\bh_1(\bX_i)+\delta \bA_1\bh_1(\bX_i)$ and $L^o(\bX_i)=\bgamma^{*T}\bh_2(\bX_i)+\delta \bA_2\bh_2(\bX_i)$, where the vectors $\bA_1$ and $\bA_2$ are to be determined. We have the following decomposition

\begin{align*}
\hat\mu_{\hat\bbeta}-\mu&=\frac{1}{n}\sum_{i=1}^n\Big[\frac{T_i}{\pi_{\beta^o}(\bX_i)}\{Y_i(1)-K^o(\bX_i)-L^o(\bX_i)\}-\frac{1-T_i}{1-\pi_{\beta^o}(\bX_i)}\{Y_i(0)-K^o(\bX_i)\}+L^o(\bX_i)-\mu\Big]\\
&+\frac{1}{n}\sum_{i=1}^n \Big\{\frac{T_i}{\pi_{\hat\beta}(\bX_i)}-\frac{T_i}{\pi_{\beta^o}(\bX_i)}\Big\}\{Y_i(1)-K^o(\bX_i)-L^o(\bX_i)\}\\
\end{align*}

We first consider $I_2$. The mean value theorem implies

I_2=-\frac{1}{n}\sum_{i=1}^n \frac{T_i}{\pi^2_{\tilde\beta}(\bX_i)}\frac{\partial\pi_{\tilde\beta}(\bX_i)}{\partial\beta}\{Y_i(1)-K^o(\bX_i)-L^o(\bX_i)\} (\hat\beta-\beta^o),

where $\tilde\beta$ is an intermediate value between $\hat\beta$ and $\beta^o$. Under Assumptions similar to  <ref>, the dominated convergence theorem implies

-\frac{1}{n}\sum_{i=1}^n \frac{T_i}{\pi^2_{\tilde\beta}(\bX_i)}\frac{\partial\pi_{\tilde\beta}(\bX_i)}{\partial\beta}\{Y_i(1)-K^o(\bX_i)-L^o(\bX_i)\}=O_p(\delta).

Similar to Lemma <ref>, we can show that $\hat\beta-\beta^o=O_p(n^{-1/2})$. The Slutsky theorem yields $I_2=O_p(\delta n^{-1/2})$. The same argument implies that $I_3=O_p(\delta n^{-1/2})$.  Finally, we focus on $I_1$. Note that

\begin{align*}
\Delta_i&=\Big\{\frac{T_i}{\pi_{\beta^o}(\bX_i)}-\frac{T_i}{\pi(\bX_i)}\Big\}\{Y_i(1)-K^o(\bX_i)-L^o(\bX_i)\}\\
\end{align*}

The central limit theorem implies $n^{1/2}(\frac{1}{n}\sum_{i=1}^n\Delta_i-\EE\Delta_i)/sd(\Delta_i)\rightarrow N(0,1)$. In order to derive the order of $\frac{1}{n}\sum_{i=1}^n\Delta_i$, it suffices to compute the $\EE(\Delta_i)$ and $sd(\Delta_i)$. As in the derivation of (<ref>), after some algebra, we similarly obtain

\beta^o-\beta^*=\xi \Tb^{-1}\Mb+O(\xi^2),

\Mb=\left( \begin{array}{c}
\EE(\frac{1}{1-\pi_{\beta^*}(\bX_i)}u_i^*\bh_1(\bX_i)) \\
\EE(u_i^*\bh_2(\bX_i) \end{array} \right)

and $\Tb=[\EE(\frac{1}{\pi_{\beta^*}(\bX_i)(1-\pi_{\beta^*}(\bX_i))}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\bh_1^T(\bX_i)),\EE(\frac{1}{\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\bh_2^T(\bX_i))]^T$.

Denote $\tilde r_1(\bX_i)=r_1(\bX_i)-\bA_1\bh_1(\bX_i)$ and $\tilde r_2(\bX_i)=r_2(\bX_i)-\bA_2\bh_2(\bX_i)$. Note that

\begin{align*}
\EE(\Delta_i)&=\EE\Big\{\frac{\pi(\bX_i)}{\pi_{\beta^o}(\bX_i)}\delta(\tilde r_1(\bX_i)+\tilde r_2(\bX_i))-\frac{1-\pi(\bX_i)}{1-\pi_{\beta^o}(\bX_i)}\delta\tilde r_1(\bX_i)-\delta\tilde r_2(\bX_i)\Big\}\\
&=\EE\Big\{\{1+\xi u_i^*-\frac{1}{\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}(\beta^o-\beta^*)\}\delta(\tilde r_1(\bX_i)+\tilde r_2(\bX_i))\\
&-\{1-\frac{\pi_{\beta^*}(\bX_i)}{1-\pi_{\beta^*}(\bX_i)}\xi u_i^*+\frac{1}{1-\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}(\beta^o-\beta^*)\}\delta\tilde r_1(\bX_i)-\delta\tilde r_2(\bX_i))\Big\}+O(\xi^2\delta)\\
&=\xi\delta\EE\Big[\{\frac{1}{1-\pi_{\beta^*}(\bX_i)} u_i^*-\frac{1}{(1-\pi_{\beta^*}(\bX_i))\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\}\tilde r_1(\bX_i)\Big]\\
&+\xi\delta\EE\Big[\{ u_i^*-\frac{1}{\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\}\tilde r_2(\bX_i)\Big]+O(\xi^2\delta).
\end{align*}

Assume that at least one entry of $\EE\Big[\{\frac{1}{1-\pi_{\beta^*}(\bX_i)} u_i^*-\frac{1}{(1-\pi_{\beta^*}(\bX_i))\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\}\bh_1(\bX_i)\Big]$ is nonzero. Then, there exists $\bA_1$ such that

\begin{align*}
&\bA_1\EE\Big[\{\frac{1}{1-\pi_{\beta^*}(\bX_i)} u_i^*-\frac{1}{(1-\pi_{\beta^*}(\bX_i))\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\}\bh_1(\bX_i)\Big]\\
&=\EE\Big[\{\frac{1}{1-\pi_{\beta^*}(\bX_i)} u_i^*-\frac{1}{(1-\pi_{\beta^*}(\bX_i))\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\} r_1(\bX_i)\Big],
\end{align*}

which implies

\EE\Big[\{\frac{1}{1-\pi_{\beta^*}(\bX_i)} u_i^*-\frac{1}{(1-\pi_{\beta^*}(\bX_i))\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\}\tilde r_1(\bX_i)\Big]=0.

Similarly, by choosing a proper $\bA_2$, we have

\EE\Big[\{ u_i^*-\frac{1}{\pi_{\beta^*}(\bX_i)}\frac{\partial\pi_{\beta^*}(\bX_i)}{\partial\beta}\Tb^{-1}\Mb\}\tilde r_2(\bX_i)\Big]=0.

As a result, we obtain $\EE(\Delta_i)=O(\xi^2\delta)$. Finally, after some tedious calculation, we can show that $sd(\Delta_i)=O(\xi+\delta)$. This implies $\frac{1}{n}\sum_{i=1}^n\Delta_i=O_p(\xi^2\delta+\xi n^{-1/2}+\delta n^{-1/2})$. This completes the proof of (<ref>). The proof of (<ref>) follows from the similar argument and we omit the details.

§ ASYMPTOTIC VARIANCE FORMULAS USED FOR SIMULATIONS

In this appendix, we present the asymptotic variance formulas used for constructing the 95% confidence intervals for calculating the coverage probabilities in the simulations in Section <ref>. In particular, for a generic estimator $\hat\mu$, the 95% confidence interval is $(\hat\mu-1.96*\hat \sigma, \hat\mu+1.96*\hat \sigma)$, where $\hat\sigma^2$ is the estimate of the asymptotic variance of $\sqrt{n}(\hat\mu-\mu)$.

For the True estimator, the asymptotic variance formula is similar to the one given in Section <ref> and is as follows:

\begin{eqnarray*}
& \Sigma_{\mu_0}  =   \Var\bigl(\mu_{\bbeta_0}(T_i, Y_i, \bX_i) \bigr) \ = \ \mathbb{E}\biggl(\frac{Y_i(1)^2}{\pi_{\bbeta_0}(\bX_i)} + \frac{Y_i(0)^2}{1-\pi_{\bbeta_0}(\bX_i)}-(\mathbb{E}(Y_i(1))-\mathbb{E}(Y_i(0)))^2\biggr).
\end{eqnarray*}

For the GLM estimator, the asymptotic variance formula is as follows:

\begin{eqnarray*}
& \Sigma_{\textrm{GLM}}  =   \Sigma_{\mu_0} - \bH_y^\top \bI^{-1} \bH_y
\end{eqnarray*}

where $\Sigma_{\mu_0}$ is defined like before, $\bI$ is the Fisher Information Matrix, and

\begin{eqnarray*}
\bH_y  & = &  -\EE\left(\frac{K(\bX_i)+(1-\pi_{\bbeta_0}(\bX_i))L(\bX_i)}{\pi_{\bbeta_0}(\bX_i) (1-\pi_{\bbeta_0}(\bX_i))}\cdot \frac{\partial \pi_{\bbeta_0}(\bX_i)}{\partial\bbeta}\right).
\end{eqnarray*}

Since the second term is positive definite, $ \Sigma_{GLM} < \Sigma_{\mu} $ and thus the variance decreases.

The GAM estimator achieves the semiparametric efficiency bound <cit.> and so we can use $V_{\textrm{opt}}$ given in (<ref>) as the asymptotic variance formula. The CBPS estimator has the following asymptotic variance formula:

\begin{align*}
\Sigma_{\textrm{CBPS}} ~~ = ~~ \Sigma_{\mu_0} ~~ + ~~ & \bH_y^\top(\bH_{\fb}^\top\bOmega^{-1}\bH_{\fb})^{-1}\bH_y \\
- ~~ &2\bH_y^\top(\bH_{\fb}^\top\bOmega^{-1}\bH_{\fb})^{-1}\bH_{\fb}^\top\bOmega^{-1}\Cov(\mu_{\bbeta_0}(T_i, Y_i, \bX_i),\bg_{\bbeta_0}(T_i, \bX_i))
\end{align*}

where $\Sigma_{\mu_0}$ and $\bH_y$ are defined like before, and we have:

\begin{eqnarray*}
\bH_{\fb}  & = &  -\EE \left(\frac{\fb(\bX_i)}{\pi_{\bbeta_0}(\bX_i)(1-\pi_{\bbeta_0}(\bX_i))}\left(\frac{\partial \pi_{\bbeta_0}(\bX_i)}{\partial \bbeta}\right)^{\top}\right)\\
\bOmega & = & \Var(\bg_{\bbeta_0}(T_i, \bX_i))\\
\bg_{\bbeta_0}(T_i,\bX_i) & =  & \ \left(\frac{T_i}{\pi_{\bbeta_0}(\bX_i)}-\frac{1-T_i}{1-\pi_{\bbeta_0}(\bX_i)} \right)\fb(\bX_i)\\
\mu_{\bbeta_0}(T_i, Y_i, \bX_i) & =  & \frac{T_iY_i}{\pi_{\bbeta_0}(\bX_i)}-\frac{(1-T_i)Y_i}{1-\pi_{\bbeta_0}(\bX_i)}.
\end{eqnarray*}

The asymptotic variance for the DR estimator is automatically computed in the R package drtmle
and the confidence interval was constructed accordingly.

Finally, we note that when we estimate the asymptotic variances, we simply replace the quantities $\pi_{\beta_0}$ and $K(\bX)$ and $L(\bX)$ with their estimates and replace the expectation with the sample average. To save space, we do not repeat the formulas of the estimated variances.


[1,4,5]Richard Jiarui Tong

[2,5]Cassie Chen Cao

1]IEEE Artificial Intelligent Standards Committee, IEEE, NJ, USA

2]University of Sheffield, Sheffield, UK

3]National Institute of Education, Singapore

4]Macao University of Science and Technology, Macao

5]Carnegie Learning, PA, USA

6]University of Memphis, TN, USA

7]Carnegie Mellon University, PA, USA

8]University of Rochester, NY, USA

9]Stanford University, CA, USA

10]Beijing Normal University, Beijing, China

# NEOLAF, an LLM-powered neural-symbolic cognitive architecture

<EMAIL_ADDRESS><EMAIL_ADDRESS>Timothy Xueqian Lee
<EMAIL_ADDRESS>Guodong Zhao<EMAIL_ADDRESS>Ray Wan  Feiyue Wang
Xiangen Hu<EMAIL_ADDRESS>Robin Schmucker<EMAIL_ADDRESS>Jinsheng Pan
<EMAIL_ADDRESS>Julian Quevedo<EMAIL_ADDRESS>Yu Lu
<EMAIL_ADDRESS>[  [  [  [  [  [  [  [  [  [

###### Abstract

This paper presents the Never Ending Open Learning Adaptive Framework
(NEOLAF), an integrated neural-symbolic cognitive architecture that models and
constructs intelligent agents. The NEOLAF framework is a superior approach to
constructing intelligent agents than both the pure connectionist and pure
symbolic approaches due to its explainability, incremental learning,
efficiency, collaborative and distributed learning, human-in-the-loop
enablement, and self-improvement. The paper further presents a compelling
experiment where a NEOLAF agent, built as a problem-solving agent, is fed with
complex math problems from the open-source MATH dataset. The results
demonstrate NEOLAF’s superior learning capability and its potential to
revolutionize the field of cognitive architectures and self-improving adaptive
instructional systems.

###### keywords:

cognitive architecture, agent, LLM, neural-symbolic

##  1 Purpose

This paper presents the Never Ending Open Learning Adaptive Framework
(NEOLAF), which is an integrated neural-symbolic cognitive architecture. It
can be used to model and construct intelligent agents, such as self-improving
intelligent tutor agents in an adaptive instructional system environment. The
design is inspired by the human cognitive development process, especially the
human learning process. The authors propose that the NEOLAF framework is a
superior approach to constructing intelligent agents than both the pure
connectionist approach, such as ChatGPT and the Yann Lecun Approach [1] and
the pure symbolic approach such as SOAR [2] and ACT-R [3], for several
reasons, including its explainability, incremental learning, efficiency,
collaborative and distributed learning, human-in-the-loop enablement[4],
avoidance of hallucinations[5], and self-improvement.

##  2 Methodology

###  2.1 NEOLAF Agent

NEOLAF learning agents are instantiated from a DNA-like starter kit (the
”innate or system-0 layer”) leveraging pre-trained large language models
(LLMs) as the foundation for their reasoning and cognition. The agent employs
and develops its cognitive capabilities at two levels akin to human cognition.
The first is a system-1 “fast” process relying on zero-shot LLM response, and
the second a “slow” system-2 process employing explicit reasoning and external
services.[6] NEOLAF agents utilize natural language as the primary
communication interface and are self-learning, developing new cognitive
capabilities by acquiring and applying grounded knowledge gradually through
experience.

###  2.2 KSTAR framework

KSTAR (knowledge, situation, task, action, result) is both a representation of
the knowledge-experience duality and a process to acquire knowledge through
experience by planning, doing, and learning in a continuous, recursive, and
multitasking manner.

The KSTAR representation breaks down any encounter into its constituent parts:
Situation, Task, Action, and Result:

  1. 1.

Identify the Situation: Start by describing the specific situation or context
in which the experience occurred.

  2. 2.

Define the Task: Determine the objective or goal that needs to be accomplished
within the given situation. In KSTAR, the task may be further decomposed into
subtasks and the implied co-tasks of planning, forecasting, and grounding.

  3. 3.

Plan and Execute the Actions: Identify the specific actions taken to address
the task at hand. Describe the steps, decisions, or strategies employed to
accomplish the goal. The actions can be further represented by the subject-
verb-objectives structure (agent-skill-constraints) to describe both the
planned and actual action steps.

  4. 4.

Forecast and Evaluate the Result: Forecast the outcome or result of the
actions taken as part of the plan and assess whether the objective of the main
task and co-tasks have been successfully achieved by grounding to truth and
reality and consider the consequences, impact, and any feedback or reactions
received as a result of the actions.

The NEOLAF agent uses LLM capability to reason from both explicit Knowledge
accumulated incrementally from prior encounters and in context learning as
well as implicit knowledge encoded in LLM to guide its interactions. Knowledge
reflects comparisons of expected results given prior knowledge, the situation,
task, and planned actions, against actual outcomes after the action is
executed.

  * •

Expected Result: Before taking any action, NEOLAF agents generate an
anticipated outcome based on prior knowledge, the situation, and task.

  * •

Planned Action: NEOLAF agents formulate a plan or strategy to achieve the
expected result. This involves making decisions, setting goals, and
determining the steps to be taken.

  * •

Actual Result: After executing the planned action, NEOLAF agents observe,
evaluate, and encode the outcome that actually occurs.

###  2.3 Memory

Memory in NEOLAF agents are reflected in two ways.

  * •

Explicit memory: Encoded prior knowledge reflecting agents’ KSTAR process for
each encounter.

  * •

Implicit Memory: Offline knowledge injection is used for model fine-tuning and
knowledge embedding plug-ins to improve the system-1 zero-shot performance.
This is analogous to memory consolidation during human sleep.

###  2.4 Related Work

Our approach utilizes chain of thought reasoning to conduct planning and
learning tasks in KSTAR process using recent advances in LLM for reasoning[7]
[8] [9] [10] [11] [12], RL[13], multitask learning[14], and planning[15].

##  3 Result

###  3.1 Initial Implementation

While the NEOLAF architecture is intended to serve as a domain-general
architecture, initial implementation and evaluation are ongoing for a math
problem-solving agent. The NEOLAF agent will be instantiated and trained on
math problems from the open-source MATH dataset. We hypothesise that the
NEOLAF agent’s continuous learning process will allow it to develop reasoning
capabilities that will enable it to perform well on problems beyond its
training set and achieve problem-solving capabilities rivalling leading LLMs
(e.g. GPT-4).

In our initial experiment, we will evaluate the performance of four different
AI setups for performance: 1) ChatGPT 3.5, 2) ChatGPT 4.0, 3) A ‘Chain of
Thought’ model with WolframAlpha and other plugins, and 4) a NEOLAF Agent, on
metrics such as answer correctness, response time, and explanation quality. 50
test cases from AIME and USAMO Math Competitions will be utilized as a test
set. These challenging problems would require novel approaches that diverge
substantially from the simpler problems found in the MATH dataset, providing a
challenging test of the capabilities hypothesized to emerge from the NEOLAF
continuous learning process.

###  3.2 Application to Education

We intend for NEOLAF to serve as a cognitive architecture underlying functions
of an agent-based learning environment (Open Learning Adaptive Framework -
OLAF) involving 3 types of agents - learners, human teachers, and AI agents.
OLAF will combine a learning management approach from XAPI CMI5 to allow
teacher and student interaction and using an autonomous agent framework
similar to the Stanford Interactive Simulacra[16].

##  4 Conclusions

The key elements of the NEOLAF architecture include the integration of
system-1 LLM capabilities with system-2 explicit reasoning and external
services, problem-solving within encounters using the KSTAR representation,
and learning and memory through encoding prior experiences using the KSTAR
representation and model fine-tuning.

We contend that the combination of these elements, drawing inspiration from
human cognition and combining symbolic and connectionist approaches to AI,
will overcome some of the key challenges of either approach. We expect that
NEOLAF agents, powered by a local LLM, will display highly competitive
performance in a diverse array of tasks through continual learning from
experience. This would constitute a lightweight and continually improving AI
model, improving on current leading LLMs that are expensive to train and
maintain.

Beyond the math problem-solving experiment and learning settings in OLAF, we
hope to introduce multimodal reasoning[17] to the agent framework and build a
co-habitat ecosystem called BotLand, designed as a complex adaptive multi-
agent environment where both intelligent agents and humans can interact and
evolve socially. We hypothesize that the potential of using complex adaptive
systems (CAS) and evolution to achieve artificial generalized intelligence
(AGI) is better than the alternative routes.

## References

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# Dynamical spacetimes in conformal gravity

Hongsheng Zhang ${}^{1,2,3~{}}$111Electronic address<EMAIL_ADDRESS>Yi Zhang 4, Xin-Zhou Li 2 1 School of Physics and Technology, University of
Jinan, West Road of Nan Xinzhuang 336, Jinan, Shandong 250022, China
2 Center for Astrophysics, Shanghai Normal University, 100 Guilin Road,
Shanghai 200234, China
3 State Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing, 100190, China
4 College of Science, Chongqing University of Posts and Telecommunications,
Chongqing 400065, China

###### Abstract

The conformal gravity remarkably boosts our prehension of gravity theories. We
find a series of dynamical solutions in the $W^{2}$-conformal gravity,
including generalized Schwarzschild-Friedmann-Robertson-Walker (GSFRW),
charged generalized Schwarzschild-Friedmann-Robertson-Walker (CGSFRW),
especially rotating Friedmann-Robertson-Walker (RFRW), charged rotating
Friedmann-Robertson-Walker (CRFRW), and a dynamical cylindrically symmetric
solutions. The RFRW, CRFRW and the dynamical cylindrically symmetric solutions
are never found in the Einstein gravity and modified gravities. The GSFRW and
CGSFRW solutions take different forms from the corresponding solutions in the
Einstein gravity.

conformal gravity; Kerr-FRW metric

###### pacs:

04.20.-q, 04.70.-s

††preprint: arXiv:

##  I Introduction

In 1918, Weyl proposed the purely infinitesimal geometry to unify gravity and
electromagnetism wey , the only two known interactions that time. Though
Weyl’s original suggestion to unify gravity and electromagnetic fields fails,
the essential idea of the Weyl theory, i.e., the conformal symmetry, becomes
more and more important in modern physics, for a modern review see Scholz olz
. The massless particles are conformally invariant. In sufficient high energy
region, classically all the fields in the standard model are conformally
invariant. Based on these facts, the conformal field theory is extensively
applied in string theory, statistical physics, and condensed matter physics.
However, gravity theory makes a significant exception. The Einstein gravity is
not conformally invariant no matter how high the energy is. Imposing the
condition of conformal invariance, one finds an interesting pure geometric
gravity,

$S_{W}=\int d^{4}x\sqrt{-g}~{}W_{abcd}W^{abcd},$ (1)

in which $W_{abcd}$ denotes the Weyl tensor and $\alpha$ is a constant.

The $W^{2}$-conformal gravity has been investigated in many aspects, for a
review see Mannheim man1 . For spherical case, the solution of conformal
gravity is helpful to interpret the surplus rotation velocity of the celestial
objects far from the center of galaxies man2 , which is assumed to be the
effects of dark matters in general relativity. The conformal gravity is subtle
for cosmology. For an exact Friedmann-Robertson-Walker (FRW) universe, the
Weyl tensor always vanishes. An arbitrary scale factor is a solution of a void
universe. The merit is that we do not need any exotic matter to explain the
acceleration of the universe. The shortcoming is that an arbitrary scale
factor makes the theory lose the ability of prediction. Fortunately the
perturbations of matters in the universe will yield a non-zero Weyl tensor,
and thus can present non-trivial constraints on the dynamical behaviours of
the universe man3 . A data fitting of the cosmology in conformal gravity with
perturbations has been done by Yang et al yang .

The boundary term of conformal gravity in the scenario of holography is
presented by Grumiller et al gru . The mass of any spacetime which is
conformal to the Minkowski spacetime, for example an FRW universe, should be
zero, which makes the concept of gravity mass in conformal gravity does not
coincide with intuition des . The conformal gravity is power-counting
renormalizable. Thus, it is treated as a proper candidate for quantum gravity
theory. Recently ’t Hooft presents an interesting argument how the requirement
of renormalization leads to the conformal gravity from the Einstein gravity
hoo . All the vacuum solutions of the Einstein gravity are the vacuum
solutions of the conformal gravity. The conformal gravity has more vacuum
solutions, which do not satisfy the Einstein equation. We call such solutions
“non-trivial solutions” in conformal gravity. Maldacena shows that we can pick
out the solutions of the Einstein equations from the solutions of conformal
gravity by imposing a special boundary condition mal . Liu et al find several
interesting non-trivial solutions in conformal gravity liu . Meng and Zhao
find the C-metric solution in conformal gravity zhao . The cylindrical
solution in conformal gravity is studied by Said et al said . We shall
investigate the dynamical cylindrical solutions in this paper.

A general conformally invariant gravity can be constructed by introducing an
extra dilaton field. Narlikar et al showed that a wide class of singularities
can be removed by choosing a “physical frame” in the conformal gravity with a
dilaton field nar  in cosmology. In black hole physics, the conformal
invariance is also helpful to remove the spacetime singularities, see recent
works by Bambi et al bambi1 , bambi2 , and Modesto et al modesto .

In this paper, we work in the frame of geometric conformal gravity (1). This
paper is organized as follows. In section 2 we study the GSFRW solution and
the CGSFRW solution. Similar solutions have been studied in the Einstein
gravity. Here they appear in different form with vanishing energy-stress. In
section 3 we obtain the RFRW and CRFRW solutions, which never found in the
Einstein gravity and modified gravities. In section 4, we present special
dynamical cylindrical solutions. They are non-trivial solutions in conformal
gravity. In section 5, we conclude this paper.

##  II GSFRW and CGSFRW spacetimes in conformal gravity

To find an exact solution of a mass point in a dynamical universe is an old
problem. In 1933, McVittie found a solution which can describe the
gravitational field of a mass point in an FRW universe mcv . Recently, Gao et
al found the higher dimensional McVittie solution and charged FRW solution in
the Einstein gravity gao . Here, we try to find the corresponding solutions in
conformal gravity.

###  II.1 GSFRW

We will show that the McVittie metric and Gao’s charged FRW metric are not
solutions of conformal gravity. Hence it is a non-trivial work to find the
GSFRW and CGSFRW in conformal gravity.

The McVitte solution reads,

$ds^{2}=-\frac{\left(\sqrt{a(t)}-\frac{M}{2r\sqrt{a(t)}}\right)^{2}}{\left(\frac{M}{2r\sqrt{a(t)}}+\sqrt{a(t)}\right)^{2}}dt^{2}+\left(\frac{M}{2r\sqrt{a(t)}}+\sqrt{a(t)}\right)^{4}\left(dr^{2}+r^{2}d\theta^{2}+r^{2}\text{sin}^{2}\theta
d\phi^{2}\right),$ (2)

where $M$ is the Schwarzschild mass parameter, $a(t)$ denotes the scale
factor, and $t,~{}r,~{}\theta,~{}\phi$ are the isotropic spherical
coordinates. The action of the conformal gravity with matter reads,

$S=\int d^{4}x\sqrt{-g}~{}\left(-\frac{\alpha}{16\pi}W_{abcd}W^{abcd}+{\cal
L}_{m}\right),$ (3)

where ${\cal L}_{m}$ denotes the Lagrangian of the matter fields. The
corresponding equation of field reads,

$-4\alpha C_{ab}=8\pi T_{ab},$ (4)

where $T_{ab}$ is the stress-energy corresponding the ${\cal L}_{m}$, and
$C_{ab}$ is the Bach tensor,

$C_{ab}=(\nabla^{c}\nabla^{d}+\frac{1}{2}R^{cd})W_{acbd}.$ (5)

Substituting the McVittie metric (2) into the field equation (4) and assuming
the stress energy is in perfect fluid form, we derive

$\rho=-T_{0}^{0}=0,$ (6) $8\pi p_{r}=8\pi T_{1}^{1}=\frac{16\alpha
Mr^{3}a\left(-\dot{a}^{2}+a\ddot{a}\right)}{(M-ra)(M+ra)^{5}},$ (7) $8\pi
p_{\theta}=8\pi p_{\phi}=8\pi T_{2}^{2}=8\pi T_{3}^{3}=-\frac{8\alpha
Mr^{3}a\left(-\dot{a}^{2}+a\ddot{a}\right)}{(M-ra)(M+ra)^{5}}.$ (8)

It is clear that $p_{r}\neq p_{\theta}$, and thus the universe is anisotropic.
This form of pressure violates the fundamental motivation of McVittie to find
his solution. Furthermore, a universe with zero density but finite pressure is
very difficult to understand. This leads to the conclusion that the McVittie
solution is not the SFRW solution in conformal gravity. From the other point
of view, the stress energy tensor of the matters in conformal gravity must be
traceless. It is clear that the McVittie metric does not satisfy this
requirement. Thus, it is not a solution of conformal gravity.

Now we try to find the GSFRW solution in conformal gravity. Both for the
Schwarzschild and FRW solutions the stress-energy vanish in conformal gravity.
This condition determines the asymptotic behaviour of the metric. Generally,
the metric ansatz can be written as,

$ds^{2}=-f(r)F(t,r)dt^{2}+B^{2}(t,r)\left(f^{-1}(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\text{sin}^{2}\theta
d\phi^{2}\right).$ (9)

Here we write the metric in a Schwarzschild-like form rather than the
isotropic form. The convenience of such coordinates will be clear later. The
corresponding field equation of the metric (9) is very involved. We cast it
into the Appendix A. It is difficult to find a general solution for $f$, $F$
and $B$. Here we study an explicit and physically more important case for $F$
and $B$,

$F=F(r),~{}~{}~{}B=B(t).$ (10)

Then we obtain the field equations as follows,

$-8\pi\rho=-\frac{\alpha}{24r^{4}B^{4}F^{4}}\left(49r^{4}f^{2}F^{\prime
4}-2r^{3}fFF^{\prime
2}\left(47rf^{\prime}F^{\prime}+f\left(22F^{\prime}+58rF^{\prime\prime}\right)\right)+r^{2}F^{2}\left(25r^{2}f^{\prime
2}F^{\prime 2}\right.\right.$
$\left.+4rfF^{\prime}\left(13rF^{\prime}f^{\prime\prime}+f^{\prime}\left(31F^{\prime}+41rF^{\prime\prime}\right)\right)+4f^{2}\left(-5F^{\prime
2}+9r^{2}F^{\prime\prime
2}+2rF^{\prime}\left(13F^{\prime\prime}+6rF^{\prime\prime\prime}\right)\right)\right)$
$+4F^{4}\left(-4+4f^{2}+4r^{2}f^{\prime 2}+r^{4}f^{\prime\prime
2}-2r^{3}f^{\prime}\left(2f^{\prime\prime}+rf^{\prime\prime\prime}\right)-4rf\left(2f^{\prime}+r\left(-f^{\prime\prime}+3rf^{\prime\prime\prime}+r^{2}f^{\prime\prime\prime\prime}\right)\right)\right)$
$\left.-4r^{3}F^{3}\left(5f^{\prime
2}\left(2F^{\prime}+rF^{\prime\prime}\right)+16ff^{\prime}\left(3F^{\prime\prime}+rF^{\prime\prime\prime}\right)+2f\left(7rf^{\prime\prime}F^{\prime\prime}+F^{\prime}\left(14f^{\prime\prime}+3rf^{\prime\prime\prime}\right)+8fF^{\prime\prime\prime}+2rfF^{\prime\prime\prime\prime}\right)\right)\right)$
(11) $8\pi
p_{r}=\frac{\alpha}{24r^{4}B^{4}F^{4}}\left(16F^{4}-16f^{2}F^{4}+32rfF^{4}f^{\prime}-16r^{2}F^{4}f^{\prime
2}+32rf^{2}F^{3}F^{\prime}-80r^{2}fF^{3}f^{\prime}F^{\prime}+40r^{3}F^{3}f^{\prime
2}F^{\prime}-4r^{2}f^{2}F^{2}F^{\prime 2}\right.$
$+52r^{3}fF^{2}f^{\prime}F^{\prime 2}-25r^{4}F^{2}f^{\prime 2}F^{\prime
2}-20r^{3}f^{2}FF^{\prime 3}-2r^{4}fFf^{\prime}F^{\prime
3}+7r^{4}f^{2}F^{\prime
4}-16r^{2}fF^{4}f^{\prime\prime}+16r^{3}F^{4}f^{\prime}f^{\prime\prime}-16r^{3}fF^{3}F^{\prime}f^{\prime\prime}$
$+16r^{4}fF^{2}F^{\prime 2}f^{\prime\prime}-4r^{4}F^{4}f^{\prime\prime
2}-32r^{2}f^{2}F^{3}F^{\prime\prime}-16r^{3}fF^{3}f^{\prime}F^{\prime\prime}+20r^{4}F^{3}f^{\prime
2}F^{\prime\prime}+48r^{3}f^{2}F^{2}F^{\prime}F^{\prime\prime}-8r^{4}fF^{2}f^{\prime}F^{\prime}F^{\prime\prime}$
$-12r^{4}f^{2}FF^{\prime
2}F^{\prime\prime}-8r^{4}fF^{3}f^{\prime\prime}F^{\prime\prime}-4r^{4}f^{2}F^{2}F^{\prime\prime
2}-16r^{3}fF^{4}f^{\prime\prime\prime}+8r^{4}F^{4}f^{\prime}f^{\prime\prime\prime}+8r^{4}fF^{3}F^{\prime}f^{\prime\prime\prime}-16r^{3}f^{2}F^{3}F^{\prime\prime\prime}$
$\left.+8r^{4}fF^{3}f^{\prime}F^{\prime\prime\prime}+8r^{4}f^{2}F^{2}F^{\prime}F^{\prime\prime\prime}\right)$
(12) $8\pi p_{\theta}=8\pi
p_{\phi}=\frac{\alpha}{24r^{4}B^{4}F^{4}}\left(-16F^{4}+16f^{2}F^{4}-32rfF^{4}f^{\prime}+16r^{2}F^{4}f^{\prime
2}-16rf^{2}F^{3}F^{\prime}+40r^{2}fF^{3}f^{\prime}F^{\prime}-40r^{3}F^{3}f^{\prime
2}F^{\prime}\right.$ $-8r^{2}f^{2}F^{2}F^{\prime
2}+36r^{3}fF^{2}f^{\prime}F^{\prime 2}+25r^{4}F^{2}f^{\prime 2}F^{\prime
2}-12r^{3}f^{2}FF^{\prime 3}-46r^{4}fFf^{\prime}F^{\prime
3}+21r^{4}f^{2}F^{\prime
4}+16r^{2}fF^{4}f^{\prime\prime}-16r^{3}F^{4}f^{\prime}f^{\prime\prime}$
$-48r^{3}fF^{3}F^{\prime}f^{\prime\prime}+18r^{4}fF^{2}F^{\prime
2}f^{\prime\prime}+4r^{4}F^{4}f^{\prime\prime
2}+16r^{2}f^{2}F^{3}F^{\prime\prime}-88r^{3}fF^{3}f^{\prime}F^{\prime\prime}-20r^{4}F^{3}f^{\prime
2}F^{\prime\prime}+28r^{3}f^{2}F^{2}F^{\prime}F^{\prime\prime}$
$+86r^{4}fF^{2}f^{\prime}F^{\prime}F^{\prime\prime}-52r^{4}f^{2}FF^{\prime
2}F^{\prime\prime}-24r^{4}fF^{3}f^{\prime\prime}F^{\prime\prime}+20r^{4}f^{2}F^{2}F^{\prime\prime
2}-16r^{3}fF^{4}f^{\prime\prime\prime}-8r^{4}F^{4}f^{\prime}f^{\prime\prime\prime}-16r^{4}fF^{3}F^{\prime}f^{\prime\prime\prime}$
$\left.-24r^{3}f^{2}F^{3}F^{\prime\prime\prime}-36r^{4}fF^{3}f^{\prime}F^{\prime\prime\prime}+20r^{4}f^{2}F^{2}F^{\prime}F^{\prime\prime\prime}-8r^{4}fF^{4}f^{\prime\prime\prime\prime}-8r^{4}f^{2}F^{3}F^{\prime\prime\prime\prime}\right)$
(13)

To obtain a reasonable solution, we further make two assumptions. The first
one is the density is independent of $r$,

$\rho=\rho(t).$ (14)

This requirement is inspired by the McVittie solution in the Einstein gravity.
The second is the universe is isotropic. We should be cautious when using this
condition. When we say that the pressure is isotropic we usually require that
the pressure is measured in the isotropic coordinates. But we are working in
the Schwarzschild-like coordinates. Now we prove that the pressures in
different directions are not changed by coordinates transformation in the
following form,

$\bar{t}=t,~{}~{}~{}\bar{r}=\bar{r}(r),~{}~{}~{}\bar{\theta}=\theta,~{}~{}~{}\bar{\phi}=\phi.$
(15)

In the bar coordinates, the pressure in the $\bar{r}$-direction reads,

$\bar{T}^{1}_{1}=T^{\mu}_{\nu}\frac{\partial\bar{x}^{1}}{\partial
x^{\mu}}\frac{\partial{x}^{\nu}}{\partial\bar{x}^{1}}=T^{1}_{1}\frac{\partial\bar{r}}{\partial
r}\frac{\partial{r}}{\partial\bar{r}}=T^{1}_{1}.$ (16)

Similarly, we can demonstrate that $p_{\theta}$ and $p_{\phi}$ are coordinate-
independent under the transformation of (15). We can obtain the isotropic
coordinates through coordinates transformation as a special case of (15). Thus
we prove that the isotropic condition is the same in Schwarzschild-like
coordinates as in the isotropic ones,

$p_{r}=p_{\theta}=p_{\phi}.$ (17)

Since the Bach tensor $C_{ab}$ is traceless, the stress energy $T_{ab}$ in (4)
is traceless, which requires

$-\rho+p_{r}+p_{\theta}+p_{\phi}=0.$ (18)

Solving the equations (14) and (17), we obtain,

$F={\rm constant},$ (19)

and

$f=\frac{h}{r}+i+jr+kr^{2},$ (20)

where $h,~{}i,~{}j,~{}k$ are integration constants, and

$i^{2}-3hj=1.$ (21)

$B(t)$ is an arbitrary function of $t$. Without loss of generality we set
$F=1$. In this solution the stress energy vanishes,

$\rho=0,~{}p_{r}=p_{\theta}=p_{\phi}=0,$ (22)

which is the simplest case satisfying the condition (18). The stress energy
satisfies the required asymptotic behaviours of the GSFRW solution. An empty
universe is not completely nonsense. As an important toy model, the Milne
universe has been discussed for many years mil . Recently, Nielsen et al show
that the dynamics of the universe may be exactly like an empty universe nie .
Furthermore, the perturbation of matters in the universe will yield a non-zero
stress-energy in conformal gravity.

On one hand, from the above discussions the conformal invariance imposes so
harsh constraint on the matter distributions in the universe that it seems
only a vanishing stress energy is permitted for isotropic distributions. On
the other hand, it offers extra freedoms in the dynamics of the universe. In
an empty FRW universe, the scale factor can be an arbitrary function of the
cosmic time. We see that the GSFRW metric inherits this property. In fact, we
can solve the metric ansatz (9) through the other road by taking full
advantages of the conformal invariance. Here we present a sketch of this
method. First, we set

$F(t,r)=P(r)B^{2}(t,r).$ (23)

And then the metric is equal to the following metric in conformal gravity,

$ds^{2}=-f(r)P(r)dt^{2}+\left(f^{-1}(r)dr^{2}+r^{2}d\theta^{2}+r^{2}\text{sin}^{2}\theta
d\phi^{2}\right).$ (24)

By using a similar procedure in bir , one can prove $P(r)=1$. Then restoring
the scale factor $B$, and making a time rescaling, one can also arrive at the
previous GSFRW metric.

Now we discuss some special cases of the above solution. When $f$ degenerates
to the Schwarzschild form, we have $j=k=0,~{}i=1$, and thus the solution
becomes,

$ds^{2}=-fdt^{2}+B^{2}(f^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}).$ (25)

In the isotropic coordinates, it reads,

$ds^{2}=-\frac{\left(1-\frac{M}{2r}\right)^{2}}{\left(1+\frac{M}{2r}\right)^{2}}dt^{2}+B^{2}\left(1+\frac{M}{2r}\right)^{4}\left(dr^{2}+r^{2}d\theta^{2}+r^{2}\text{sin}^{2}\theta
d\phi^{2}\right).$ (26)

Comparing with (2), one finds that it is still different from the McVitte
solution. As we have mentioned, the McVitte metric is not a solution of the
conformal gravity. So it is impossible to recover the McVitte metric from our
solution.

Now we make a short discussion about the isotropic coordinates. Generally, the
isotropic coordinate is derived by

$\rho_{i}=\int\frac{~{}dr}{\sqrt{f}}.$ (27)

It has no analytical result for a general $f$ in (20), which implies that we
cannot obtain the analytical result if we work in the isotropic coordinates
from the beginning. That is the reason why we use Schwarzschild-like
coordinates in (9). The next important case is the static limit. When
$B=$constant, one recovers the solution found in bir .

For a fixed $r$, the GSFRW metric describes an expanding/contracting two-
sphere, which can be treated as a three-dimensional FRW universe. However, for
a variable $r$, it does not become a four-dimensional FRW universe when
$r\to\infty$ except the case $j=k=0$. So a proper name for this solution is
generalized FRW solution.

###  II.2 CGSFRW

Because the charged FRW solution in [13] admits the McVitte solution as a
special case, it is not a solution of conformal gravity either. We try to find
the CGSFRW solution in conformal gravity. When we turn on the (sourced) $U(1)$
gauge field, the matter lagrangian in (3) becomes

${\cal L}_{m}={\cal\bar{L}}-\frac{1}{4}{\cal F}_{ab}{\cal F}^{ab}-A_{a}J^{a},$
(28)

where ${\cal F}=dA$ denotes the $U(1)$ gauge field, $J$ denotes the
electromagnetic current, and ${\cal\bar{L}}$ represents the contribution of
other matters.

Similar to the previous discussions, we obtain a solution under the metric
ansatz (9) and (10),

$F={\rm constant},$ (29) $f=\frac{h}{r}+i+jr+kr^{2},$ (30)

where $h,~{}i,~{}j,~{}k$ are constants, and the stress energy corresponding to
$\cal\bar{L}$ vanishes. The stress energy for this metric reads,

$8\pi\rho=-8\pi T_{0}^{0}=-\frac{2\alpha(1-i^{2}+3hj)}{3r^{4}B^{4}},$ (31)
$8\pi p_{r}=8\pi T_{1}^{1}=\frac{2\alpha(1-i^{2}+3hj)}{3r^{4}B^{4}},$ (32)
$8\pi p_{\theta}=8\pi p_{\phi}=8\pi T_{2}^{2}=8\pi
T_{3}^{3}=-\frac{2\alpha(1-i^{2}+3hj)}{3r^{4}B^{4}}.$ (33)

One can check that the stress energy is traceless,

$T^{\mu}_{\mu}=0.$ (34)

The potential of the U(1) field is,

$A=\frac{q(t)}{r}dt-v\cos\theta d\phi.$ (35)

Here $q(t),~{}v$ are the electric and the magnetic charge of the point mass at
$r=0$, respectively. The stress energy of the U(1) gauge field reads,

$8\pi
T^{0}_{0}(U(1))=-\frac{\left(v^{2}+B^{2}q^{2}\right)}{r^{4}B^{4}},~{}T^{1}_{1}(U(1))=-T^{2}_{2}(U(1))=-T^{3}_{3}(U(1))=T^{0}_{0}(U(1)).$
(36)

Then the field equation requires

$Bq=K,$ (37)

and

$K^{2}=\frac{2\alpha(i^{2}-1-3hj)}{3}-v^{2}.$ (38)

The total charge in a 2-surface with radius $r$ is derived by the integration
of the Hodge dual of the field strength,

$Q=\frac{1}{4\pi}\int*{\cal F}=K,$ (39)

which is independent on $r$. This is the property of a point charge. One sees
that the total charge is time independent, which is different from $q(t)$.
This point also can be seen from the electric current reads,

$J=\frac{1}{r^{2}fB}(q\dot{B}+B\dot{q})dr=0$ (40)

Thus the charge conservation law requires the central charge to be a constant.
The potential (35) becomes

$A=\frac{K}{Br}dt-v\cos\theta d\phi.$ (41)

At the first sight, this is a very natural, even obvious result. The electric
potential decreases to $1/B$ when the universe expand by a factor $B$ for a
constant charge. In fact, it is a highly non-trivial result, since the stress
energy must increase or decrease synchronously to satisfy the field equation.
In fact this point cannot be accomplished in the Einstein gravity gao , in
which a variable charge, and thus a non-zero current, are required.

At the limit $B$=constant, such that $q$=constant, and the magnetic charge $v$
vanishes, the previous result in bir  recovers.

The metric cannot come back to the Reissner-Nordstrom metric with any special
choices of parameters. It really describes an asymptotic FRW spacetime with a
point charge. So we name it CGSFRW metric in conformal gravity.

##  III RFRW and CRFRW spacetimes in conformal gravity

To describe a rotating black hole in an FRW universe is a long-waited
achievement. Unfortunately, the Kerr-FRW metric has not been found in the
Einstein gravity up to now. And to our knowledge, it has never been found in
any modified gravity theory. By using the special structure of the conformal
gravity, we successfully find the RFRW and CRFRW metrics.

To derive the Kerr solution in the Einstein gravity is a very complicated
work. The field equation becomes extremely complicated when one considers a
rotating solution which is asymptotic to the FRW metric. In this section we
just present our results. One can verify that the following metric solves the
field equation of the conformal gravity with a $U(1)$ gauge field,

$ds^{2}=B(t)^{2}\left(r^{2}+a^{2}\cos^{2}\theta\right)\left(\frac{1}{a^{2}+r^{2}-2r\left(M+cr^{2}\right)}dr^{2}+\frac{1}{1+b\cos\theta}d\theta^{2}\right)$
$+\frac{1+b\cos\theta}{r^{2}+a^{2}\cos^{2}\theta}\sin^{2}\theta\left(adt-\left(a^{2}+r^{2}\right)B(t)d\varphi\right)^{2}-\frac{a^{2}+r^{2}-2r\left(M+cr^{2}\right)}{r^{2}+a^{2}\cos^{2}\theta^{2}}\left(dt-
aB(t)\sin^{2}\theta d\varphi\right)^{2}.$ (42)

Here, $a$ denotes the angular momentum per mass, $M$ is a mass parameter,
$B(t)$ is a function of time, which reduces to the scale factor of the
Schwarzschild-FRW universe in Schwarzschild-like coordinate system when
$a=0,~{}b=0,~{}c=0$, and $b$ and $c$ are two extra parameters which have no
correspondence in the Einstein gravity. It is easy to check that it comes back
to the Kerr metric when $b=c=0$ and $B$ is a constant. The stress energy of
the solution (42) reads,

$-8\pi\rho=8\pi
T^{0}_{0}=\frac{8\alpha\left(b^{2}+4cM\right)\left(3a^{2}+2r^{2}-a^{2}\cos
2\theta\right)}{B^{4}\left(a^{2}+2r^{2}+a^{2}\cos 2\theta\right)^{3}}$ (43)

and

$8\pi p_{1}=8\pi
T_{1}^{1}=\frac{2\alpha\left(b^{2}+4cM\right)}{B^{4}\left(r^{2}+a^{2}\cos^{2}\theta\right)^{2}},$
(44) $p_{2}=-p_{1}=T^{2}_{2},p_{3}=T_{3}^{3}=-T^{0}_{0}.$ (45)

One sees that the components of the stress energy satisfy,

$T_{3}^{3}+T^{0}_{0}=0;~{}T_{1}^{1}+T_{2}^{2}=0.$ (46)

Thus it satisfies the requirement of the stress energy in conformal gravity.
The potential of the $U(1)$ gauge field reads,

$A=\frac{Kr}{\rho B}(dt-a\sin^{2}\theta Bd\varphi)+\frac{v\cos\theta}{\rho
B}(adt-B(r^{2}+a^{2})d\varphi),$ (47)

whose energy stress reads,

$8\pi T^{0}_{0}=-\frac{4\left(K^{2}+v^{2}\right)\left(3a^{2}+2r^{2}-a^{2}\cos
2\theta\right)}{B^{4}\left(a^{2}+2r^{2}+a^{2}\cos 2\theta\right)^{3}}$ (48)
$8\pi
T^{1}_{1}=-\frac{\left(K^{2}+v^{2}\right)}{B^{4}\left(r^{2}+a^{2}\cos^{2}\theta\right)^{2}},$
(49) $T_{2}^{2}=-T^{1}_{1},~{}~{}~{}T^{3}_{3}=-T^{0}_{0}.$ (50)

The field equation requires,

$(K^{2}+v^{2})=-2\alpha(b^{2}+4cM).$ (51)

Similar to the demonstrations in the above context, one can show that $K$ is
the total electric charge and $v$ is the total magnetic charge in the
universe. Also, this point can be confirmed by the fact that the
electromagnetic currents vanish. Thus generally, it describes a charged
rotating asymptotic FRW universe in conformal gravity. When $b^{2}+4cM=0$, we
obtain a neutral universe. And furthermore, we obtain a Kerr-FRW universe if
we set $b=0,~{}c=0$. The Kerr-FRW universe is never obtained in the Einstein
gravity, nor in any modified gravity theory.

##  IV dynamical cylindrically symmetric solution

The general line element of a cylindrically symmetric solution can be written
as,

$ds^{2}=-H_{0}dt^{2}+H_{1}dz^{2}+H_{2}dr^{2}+H_{3}d\phi^{2},$ (52)

where $H_{0},~{}H_{1},~{}H_{2},~{}H_{3}$ are functions of $r$ and $t$. The
cylindrically symmetric solutions are widely investigated in the Einstein
gravity. But, the dynamical solutions are seldom studied. We shall study some
cylindrically dynamical solutions in conformal gravity.

A class of static cylindrically symmetric solution has been found in said ,

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}dz^{2}+r^{2}\beta d\phi^{2},$ (53)

in which the constant $\beta$ denotes the possible conical singularity, and

$f=\frac{h}{r}+i+jr+kr^{2}.$ (54)

This looks very similar to the spherical solution in the conformal gravity.
The different points are discussed as follows. The first one is that the
subspace ($z,~{}\phi$) is flat, while the subspace ($\theta,~{}\phi$) is
curved in the spherical case. The second one is that the stress energy
vanishes, while a stress energy of a U(1) gauge field appears if one does not
impose any constraints among the constants $h,~{}i,~{}j,$ and $k$ in the case
of the spherical solution. Actually, (53) also can be treated as a topological
solution corresponding to the spherical solution when $\beta=1$, like the
topological Schwarzschild solution in the Einstein gravity.

Then we discuss the dynamical case. Mimicking the spherical case, we derive
the following dynamical solution in conformal gravity

$ds^{2}=-f(r)dt^{2}+B(t)^{2}(f^{-1}(r)dr^{2}+r^{2}dz^{2}+r^{2}\beta
d\phi^{2}).$ (55)

The stress energy in this solution is zero. So it is a vacuum solution which
describes an expanding/contracting cylindrically symmetric space.

When $i=j=0$, (54) becomes the Melvin solution mel . We find the other type of
dynamical solution of with dynamical subspace of ($z,~{}\phi$) belonging to
the Melvin class,

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+B^{2}(r^{2}dz^{2}+r^{2}\beta d\phi^{2}),$
(56)

in which $i=j=0,$ $hk=0$ and $B=t^{3/4}$. This solution describes a
cylindrically symmetric with an expanding subspace ($z,~{}\phi$).

##  V conclusions and discussions

The conformal gravity takes a special status in modified gravity theories.
Classically, the standard model of particle physics is conformally invariant
at high energy region. If the quantum effects are considered, a conformal
anomaly appears modesto . In classical level, if the putative unified theory
including gravity and gauge interactions exists, the gravity may also be
conformally invariant. However, classically the Einstein gravity is not
conformally invariant. A pure geometric Lagrangian of conformally invariant
gravity in four dimensional spacetime is the contraction of the Weyl tensor.
We study some dynamical solutions in such a conformal gravity.

All the vacuum solutions of the Einstein gravity are the solutions of
conformal gravity. A sourced solution in the Einstein gravity may not solve
the conformal gravity. For example, a sourced FRW universe is not a solution
in conformal gravity. The FRW metric, as a solution, is a vacuum solution in
conformal gravity. We show that the Schwarzschild-FRW metric (the McVittie
metric) is not a solution of conformal gravity. In this article we find the
GSFRW solution in conformal gravity. And then, we extend it to the charged
case.

To describe a rotating black hole inhabited in an FRW universe is an extremely
difficult problem in gravity theory. Before this research, there is no exact
solution to describe a rotating black hole in an FRW universe in the Einstein
gravity and in modified gravity theories. We find a class of rotating solution
in an FRW universe in conformal gravity. Both the neutral and charged cases
are considered. Finally, we investigate the dynamical cylindrically symmetric
solutions. There are several other methods to obtain the exact solutions
besides directly solving the field equation. A noticeable recent approach is
the thermodynamic method self1 . Also, there are several solutions with
different symmetries in the Einstein gravity, for example the plane symmetric
solutions self2 . How to extend these method and solutions to the case of the
conformal gravity is the future topic.

Acknowledgments. The authors thank the anonymous referee for her/his valuable
comments. This work is supported by the Program for Professor of Special
Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning,
National Education Foundation of China under grant No. 200931271104, Shanghai
Key Laboratory of Particle Physics and Cosmology under grant No.11DZ2230700,
and National Natural Science Foundation of China under Grant Nos. 11075106,
11275128 and 11105004.

Appendix A

The field equations under the ansatz (9) read,

$-8\pi\rho=\frac{\alpha}{24r^{4}B^{6}F^{4}}\left(135r^{4}f^{2}F^{4}B^{\prime
4}-10r^{3}BfF^{3}B^{\prime
2}\left(15rFf^{\prime}B^{\prime}+2f\left(rB^{\prime}F^{\prime}+F\left(11B^{\prime}+13rB^{\prime\prime}\right)\right)\right)\right.$
$+5r^{2}B^{2}F^{2}\left(3r^{2}F^{2}f^{\prime 2}B^{\prime
2}+2rfFB^{\prime}\left(rf^{\prime}B^{\prime}F^{\prime}+F\left(6rf^{\prime\prime}B^{\prime}+f^{\prime}\left(26B^{\prime}+22rB^{\prime\prime}\right)\right)\right)\right.$
$\left.\left.\left.+2f^{2}\left(-3r^{2}B^{\prime 2}F^{\prime
2}+2rFB^{\prime}\left(2rF^{\prime}B^{\prime\prime}+B^{\prime}\left(5F^{\prime}+rF^{\prime\prime}\right)\right)\right.\right.+2F^{2}\left(B^{\prime
2}+3r^{2}B^{\prime\prime
2}+2rB^{\prime}\left(7B^{\prime\prime}+2rB^{\prime\prime\prime}\right)\right)\right)\right)$
$-2r^{2}B^{3}F\left(18r^{2}f^{2}B^{\prime}F^{\prime
3}-rfFF^{\prime}\left(23rf^{\prime}B^{\prime}F^{\prime}+2f\left(7rF^{\prime}B^{\prime\prime}+B^{\prime}\left(19F^{\prime}+14rF^{\prime\prime}\right)\right)\right)\right.$
$+F^{2}\left(-5r^{2}f^{\prime
2}B^{\prime}F^{\prime}+2rf\left(2rf^{\prime\prime}B^{\prime}F^{\prime}+f^{\prime}\left(7rF^{\prime}B^{\prime\prime}+B^{\prime}\left(34F^{\prime}+7rF^{\prime\prime}\right)\right)\right)+4f^{2}\left(r\left(3rB^{\prime\prime}F^{\prime\prime}+F^{\prime}\left(11B^{\prime\prime}+2rB^{\prime\prime\prime}\right)\right)\right.\right.$
$+B^{\prime}\left.\left.\left(5F^{\prime}+r\left(11F^{\prime\prime}+2rF^{\prime\prime\prime}\right)\right)\right)\right)+2rF^{3}\left(5f^{\prime
2}\left(2B^{\prime}+rB^{\prime\prime}\right)+16ff^{\prime}\left(3B^{\prime\prime}+rB^{\prime\prime\prime}\right)\right.$
$+2f\left.\left.\left(3rf^{\prime\prime\prime}B^{\prime}+7f^{\prime\prime}\left(2B^{\prime}+rB^{\prime\prime}\right)+8fB^{\prime\prime\prime}+2rfB^{\prime\prime\prime\prime}\right)\right)\right)$
$+B^{4}\left(-4F^{4}\left(-4+4f^{2}+4r^{2}f^{\prime 2}+r^{4}f^{\prime\prime
2}-2r^{3}f^{\prime}\left(2f^{\prime\prime}+rf^{\prime\prime\prime}\right)-4rf\left(2f^{\prime}+r\left(-f^{\prime\prime}+3rf^{\prime\prime\prime}+r^{2}f^{\prime\prime\prime\prime}\right)\right)\right)\right.$
$-49r^{4}f^{2}F^{\prime 4}+2r^{3}fFF^{\prime
2}\left(47rf^{\prime}F^{\prime}+f\left(22F^{\prime}+58rF^{\prime\prime}\right)\right)-r^{2}F^{2}\left(25r^{2}f^{\prime
2}F^{\prime
2}+4rfF^{\prime}\left(13rf^{\prime\prime}F^{\prime}+f^{\prime}\left(31F^{\prime}+41rF^{\prime\prime}\right)\right)\right.$
$+4f^{2}\left.\left(-5F^{\prime 2}+9r^{2}F^{\prime\prime
2}+2rF^{\prime}\left(13F^{\prime\prime}+6rF^{\prime\prime\prime}\right)\right)\right)+4r^{3}F^{3}\left(5f^{\prime
2}\left(2F^{\prime}+rF^{\prime\prime}\right)+16ff^{\prime}\left(3F^{\prime\prime}+rF^{\prime\prime\prime}\right)\right.$
$+2f\left.\left.\left.\left(3rf^{\prime\prime\prime}F^{\prime}+7f^{\prime\prime}\left(2F^{\prime}+rF^{\prime\prime}\right)+8fF^{\prime\prime\prime}+2rfF^{\prime\prime\prime\prime}\right)\right)\right)\right),$
(57)

$8\pi
p_{r}=\frac{\alpha}{24r^{4}B^{6}fF^{5}}\left(16B^{4}fF^{5}-16B^{4}f^{3}F^{5}+32rB^{4}f^{2}F^{5}f^{\prime}-16r^{2}B^{4}fF^{5}f^{\prime
2}-16r^{2}B^{4}f^{2}F^{5}f^{\prime\prime}+16r^{3}B^{4}fF^{5}f^{\prime}f^{\prime\prime}\right.$
$-4r^{4}B^{4}fF^{5}f^{\prime\prime
2}-16r^{3}B^{4}f^{2}F^{5}f^{\prime\prime\prime}+8r^{4}B^{4}fF^{5}f^{\prime}f^{\prime\prime\prime}-32rB^{3}f^{3}F^{5}B^{\prime}+80r^{2}B^{3}f^{2}F^{5}f^{\prime}B^{\prime}-40r^{3}B^{3}fF^{5}f^{\prime
2}B^{\prime}+16r^{3}B^{3}f^{2}F^{5}f^{\prime\prime}B^{\prime}$
$-8r^{4}B^{3}f^{2}F^{5}f^{\prime\prime\prime}B^{\prime}-68r^{2}B^{2}f^{3}F^{5}B^{\prime
2}+20r^{3}B^{2}f^{2}F^{5}f^{\prime}B^{\prime 2}+15r^{4}B^{2}fF^{5}f^{\prime
2}B^{\prime 2}+20r^{3}Bf^{3}F^{5}B^{\prime
3}-30r^{4}Bf^{2}F^{5}f^{\prime}B^{\prime 3}$ $+15r^{4}f^{3}F^{5}B^{\prime
4}+32rB^{4}f^{3}F^{4}F^{\prime}-80r^{2}B^{4}f^{2}F^{4}f^{\prime}F^{\prime}+40r^{3}B^{4}fF^{4}f^{\prime
2}F^{\prime}-16r^{3}B^{4}f^{2}F^{4}f^{\prime\prime}F^{\prime}+8r^{4}B^{4}f^{2}F^{4}f^{\prime\prime\prime}F^{\prime}$
$+72r^{2}B^{3}f^{3}F^{4}B^{\prime}F^{\prime}-72r^{3}B^{3}f^{2}F^{4}f^{\prime}B^{\prime}F^{\prime}+10r^{4}B^{3}fF^{4}f^{\prime
2}B^{\prime}F^{\prime}-16r^{4}B^{3}f^{2}F^{4}f^{\prime\prime}B^{\prime}F^{\prime}+36r^{3}B^{2}f^{3}F^{4}B^{\prime
2}F^{\prime}$ $+10r^{4}B^{2}f^{2}F^{4}f^{\prime}B^{\prime
2}F^{\prime}-20r^{4}Bf^{3}F^{4}B^{\prime
3}F^{\prime}-4r^{2}B^{4}f^{3}F^{3}F^{\prime
2}+52r^{3}B^{4}f^{2}F^{3}f^{\prime}F^{\prime 2}-25r^{4}B^{4}fF^{3}f^{\prime
2}F^{\prime 2}$ $+16r^{4}B^{4}f^{2}F^{3}f^{\prime\prime}F^{\prime
2}-36r^{3}B^{3}f^{3}F^{3}B^{\prime}F^{\prime
2}+22r^{4}B^{3}f^{2}F^{3}f^{\prime}B^{\prime}F^{\prime
2}+2r^{4}B^{2}f^{3}F^{3}B^{\prime 2}F^{\prime
2}-20r^{3}B^{4}f^{3}F^{2}F^{\prime 3}$
$-2r^{4}B^{4}f^{2}F^{2}f^{\prime}F^{\prime
3}-4r^{4}B^{3}f^{3}F^{2}B^{\prime}F^{\prime 3}+7r^{4}B^{4}f^{3}FF^{\prime
4}+32r^{2}B^{3}f^{3}F^{5}B^{\prime\prime}+16r^{3}B^{3}f^{2}F^{5}f^{\prime}B^{\prime\prime}-20r^{4}B^{3}fF^{5}f^{\prime
2}B^{\prime\prime}$
$+8r^{4}B^{3}f^{2}F^{5}f^{\prime\prime}B^{\prime\prime}-48r^{3}B^{2}f^{3}F^{5}B^{\prime}B^{\prime\prime}+40r^{4}B^{2}f^{2}F^{5}f^{\prime}B^{\prime}B^{\prime\prime}-20r^{4}Bf^{3}F^{5}B^{\prime
2}B^{\prime\prime}-16r^{4}B^{3}f^{2}F^{4}f^{\prime}F^{\prime}B^{\prime\prime}$
$+32r^{4}B^{2}f^{3}F^{4}B^{\prime}F^{\prime}B^{\prime\prime}-12r^{4}B^{3}f^{3}F^{3}F^{\prime
2}B^{\prime\prime}-4r^{4}B^{2}f^{3}F^{5}B^{\prime\prime
2}-32r^{2}B^{4}f^{3}F^{4}F^{\prime\prime}-16r^{3}B^{4}f^{2}F^{4}f^{\prime}F^{\prime\prime}$
$+20r^{4}B^{4}fF^{4}f^{\prime
2}F^{\prime\prime}-8r^{4}B^{4}f^{2}F^{4}f^{\prime\prime}F^{\prime\prime}-16r^{4}B^{3}f^{2}F^{4}f^{\prime}B^{\prime}F^{\prime\prime}-4r^{4}B^{2}f^{3}F^{4}B^{\prime
2}F^{\prime\prime}+48r^{3}B^{4}f^{3}F^{3}F^{\prime}F^{\prime\prime}$
$-8r^{4}B^{4}f^{2}F^{3}f^{\prime}F^{\prime}F^{\prime\prime}+16r^{4}B^{3}f^{3}F^{3}B^{\prime}F^{\prime}F^{\prime\prime}-12r^{4}B^{4}f^{3}F^{2}F^{\prime
2}F^{\prime\prime}+8r^{4}B^{3}f^{3}F^{4}B^{\prime\prime}F^{\prime\prime}-4r^{4}B^{4}f^{3}F^{3}F^{\prime\prime
2}$
$+16r^{3}B^{3}f^{3}F^{5}B^{\prime\prime\prime}-8r^{4}B^{3}f^{2}F^{5}f^{\prime}B^{\prime\prime\prime}+8r^{4}B^{2}f^{3}F^{5}B^{\prime}B^{\prime\prime\prime}-8r^{4}B^{3}f^{3}F^{4}F^{\prime}B^{\prime\prime\prime}-16r^{3}B^{4}f^{3}F^{4}F^{\prime\prime\prime}$
$+8r^{4}B^{4}f^{2}F^{4}f^{\prime}F^{\prime\prime\prime}-8r^{4}B^{3}f^{3}F^{4}B^{\prime}F^{\prime\prime\prime}+8r^{4}B^{4}f^{3}F^{3}F^{\prime}F^{\prime\prime\prime}-24r^{3}B^{2}fF^{4}B^{\prime}\dot{B}^{2}+36r^{4}B^{2}F^{4}f^{\prime}B^{\prime}\dot{B}^{2}$
$-120r^{4}BfF^{4}B^{\prime
2}\dot{B}^{2}+24r^{4}B^{2}fF^{3}B^{\prime}F^{\prime}\dot{B}^{2}+24r^{4}B^{2}fF^{4}B^{\prime\prime}\dot{B}^{2}-8r^{3}B^{3}fF^{3}B^{\prime}\dot{B}\dot{F}+12r^{4}B^{3}F^{3}f^{\prime}B^{\prime}\dot{B}\dot{F}$
$-24r^{4}B^{2}fF^{3}B^{\prime
2}\dot{B}\dot{F}-8r^{3}B^{4}fF^{2}F^{\prime}\dot{B}\dot{F}+12r^{4}B^{4}F^{2}f^{\prime}F^{\prime}\dot{B}\dot{F}+32r^{4}B^{3}fF^{2}B^{\prime}F^{\prime}\dot{B}\dot{F}-8r^{4}B^{4}fFF^{\prime
2}\dot{B}\dot{F}$
$+8r^{4}B^{3}fF^{3}B^{\prime\prime}\dot{B}\dot{F}+8r^{4}B^{4}fF^{2}F^{\prime\prime}\dot{B}\dot{F}+40r^{3}B^{5}fFF^{\prime}\dot{F}^{2}-60r^{4}B^{5}Ff^{\prime}F^{\prime}\dot{F}^{2}+40r^{4}B^{4}fFB^{\prime}F^{\prime}\dot{F}^{2}$
$+56r^{4}B^{5}fF^{\prime
2}\dot{F}^{2}-40r^{4}B^{5}fFF^{\prime\prime}\dot{F}^{2}+24r^{3}B^{3}fF^{4}\dot{B}\dot{B}^{\prime}-36r^{4}B^{3}F^{4}f^{\prime}\dot{B}\dot{B}^{\prime}+168r^{4}B^{2}fF^{4}B^{\prime}\dot{B}\dot{B}^{\prime}$
$-24r^{4}B^{3}fF^{3}F^{\prime}\dot{B}\dot{B}^{\prime}+8r^{3}B^{4}fF^{3}\dot{F}\dot{B}^{\prime}-12r^{4}B^{4}F^{3}f^{\prime}\dot{F}\dot{B}^{\prime}+24r^{4}B^{3}fF^{3}B^{\prime}\dot{F}\dot{B}^{\prime}$
$-40r^{4}B^{4}fF^{2}F^{\prime}\dot{F}\dot{B}^{\prime}-48r^{4}B^{3}fF^{4}\dot{B}^{\prime
2}+8r^{3}B^{4}fF^{3}\dot{B}\dot{F}^{\prime}-12r^{4}B^{4}F^{3}f^{\prime}\dot{B}\dot{F}^{\prime}-24r^{4}B^{3}fF^{3}B^{\prime}\dot{B}\dot{F}^{\prime}$
$+8r^{4}B^{4}fF^{2}F^{\prime}\dot{B}\dot{F}^{\prime}-40r^{3}B^{5}fF^{2}\dot{F}\dot{F}^{\prime}+60r^{4}B^{5}F^{2}f^{\prime}\dot{F}\dot{F}^{\prime}-40r^{4}B^{4}fF^{2}B^{\prime}\dot{F}\dot{F}^{\prime}$
$-72r^{4}B^{5}fFF^{\prime}\dot{F}\dot{F}^{\prime}+32r^{4}B^{4}fF^{3}\dot{B}^{\prime}\dot{F}^{\prime}+16r^{4}B^{5}fF^{2}\dot{F}^{\prime
2}-24r^{4}B^{3}fF^{4}\dot{B}\dot{B}^{\prime\prime}$
$-8r^{4}B^{4}fF^{3}\dot{F}\dot{B}^{\prime\prime}-8r^{4}B^{4}fF^{3}\dot{B}\dot{F}^{\prime\prime}+40r^{4}B^{5}fF^{2}\dot{F}\dot{F}^{\prime\prime}+16r^{3}B^{3}fF^{4}B^{\prime}\ddot{B}-24r^{4}B^{3}F^{4}f^{\prime}B^{\prime}\ddot{B}$
$+48r^{4}B^{2}fF^{4}B^{\prime
2}\ddot{B}-16r^{4}B^{3}fF^{3}B^{\prime}F^{\prime}\ddot{B}-16r^{4}B^{3}fF^{4}B^{\prime\prime}\ddot{B}-16r^{3}B^{5}fF^{2}F^{\prime}\ddot{F}+24r^{4}B^{5}F^{2}f^{\prime}F^{\prime}\ddot{F}$
$-16r^{4}B^{4}fF^{2}B^{\prime}F^{\prime}\ddot{F}-16r^{4}B^{5}fFF^{\prime
2}\ddot{F}+16r^{4}B^{5}fF^{2}F^{\prime\prime}\ddot{F}-16r^{3}B^{4}fF^{4}\ddot{B}^{\prime}+24r^{4}B^{4}F^{4}f^{\prime}\ddot{B}^{\prime}-48r^{4}B^{3}fF^{4}B^{\prime}\ddot{B}^{\prime}$
$+16r^{4}B^{4}fF^{3}F^{\prime}\ddot{B}^{\prime}+16r^{3}B^{5}fF^{3}\ddot{F}^{\prime}-24r^{4}B^{5}F^{3}f^{\prime}\ddot{F}^{\prime}+16r^{4}B^{4}fF^{3}B^{\prime}\ddot{F}^{\prime}+16r^{4}B^{5}fF^{2}F^{\prime}\ddot{F}^{\prime}+16r^{4}B^{4}fF^{4}\ddot{B}^{\prime\prime}$
$\left.-16r^{4}B^{5}fF^{3}\ddot{F}^{\prime\prime}\right),$ (58)

$8\pi p_{\theta}=8\pi
p_{\phi}=\frac{\alpha}{24r^{4}B^{6}fF^{5}}\left(-16B^{4}fF^{5}+16B^{4}f^{3}F^{5}-32rB^{4}f^{2}F^{5}f^{\prime}+16r^{2}B^{4}fF^{5}f^{\prime
2}+16r^{2}B^{4}f^{2}F^{5}f^{\prime\prime}\right.$
$-16r^{3}B^{4}fF^{5}f^{\prime}f^{\prime\prime}+4r^{4}B^{4}fF^{5}f^{\prime\prime
2}-16r^{3}B^{4}f^{2}F^{5}f^{\prime\prime\prime}-8r^{4}B^{4}fF^{5}f^{\prime}f^{\prime\prime\prime}-8r^{4}B^{4}f^{2}F^{5}f^{\prime\prime\prime\prime}+16rB^{3}f^{3}F^{5}B^{\prime}$
$-40r^{2}B^{3}f^{2}F^{5}f^{\prime}B^{\prime}+40r^{3}B^{3}fF^{5}f^{\prime
2}B^{\prime}+48r^{3}B^{3}f^{2}F^{5}f^{\prime\prime}B^{\prime}+16r^{4}B^{3}f^{2}F^{5}f^{\prime\prime\prime}B^{\prime}+24r^{2}B^{2}f^{3}F^{5}B^{\prime
2}$ $-140r^{3}B^{2}f^{2}F^{5}f^{\prime}B^{\prime
2}-15r^{4}B^{2}fF^{5}f^{\prime 2}B^{\prime
2}-30r^{4}B^{2}f^{2}F^{5}f^{\prime\prime}B^{\prime
2}+100r^{3}Bf^{3}F^{5}B^{\prime 3}+90r^{4}Bf^{2}F^{5}f^{\prime}B^{\prime
3}-75r^{4}f^{3}F^{5}B^{\prime 4}$
$-16rB^{4}f^{3}F^{4}F^{\prime}+40r^{2}B^{4}f^{2}F^{4}f^{\prime}F^{\prime}-40r^{3}B^{4}fF^{4}f^{\prime
2}F^{\prime}-48r^{3}B^{4}f^{2}F^{4}f^{\prime\prime}F^{\prime}-16r^{4}B^{4}f^{2}F^{4}f^{\prime\prime\prime}F^{\prime}-16r^{2}B^{3}f^{3}F^{4}B^{\prime}F^{\prime}$
$+104r^{3}B^{3}f^{2}F^{4}f^{\prime}B^{\prime}F^{\prime}-10r^{4}B^{3}fF^{4}f^{\prime
2}B^{\prime}F^{\prime}+12r^{4}B^{3}f^{2}F^{4}f^{\prime\prime}B^{\prime}F^{\prime}-68r^{3}B^{2}f^{3}F^{4}B^{\prime
2}F^{\prime}-10r^{4}B^{2}f^{2}F^{4}f^{\prime}B^{\prime 2}F^{\prime}$
$+20r^{4}Bf^{3}F^{4}B^{\prime 3}F^{\prime}-8r^{2}B^{4}f^{3}F^{3}F^{\prime
2}+36r^{3}B^{4}f^{2}F^{3}f^{\prime}F^{\prime 2}+25r^{4}B^{4}fF^{3}f^{\prime
2}F^{\prime 2}+18r^{4}B^{4}f^{2}F^{3}f^{\prime\prime}F^{\prime 2}$
$-20r^{3}B^{3}f^{3}F^{3}B^{\prime}F^{\prime
2}-34r^{4}B^{3}f^{2}F^{3}f^{\prime}B^{\prime}F^{\prime
2}+14r^{4}B^{2}f^{3}F^{3}B^{\prime 2}F^{\prime
2}-12r^{3}B^{4}f^{3}F^{2}F^{\prime
3}-46r^{4}B^{4}f^{2}F^{2}f^{\prime}F^{\prime 3}$
$+20r^{4}B^{3}f^{3}F^{2}B^{\prime}F^{\prime 3}+21r^{4}B^{4}f^{3}FF^{\prime
4}-16r^{2}B^{3}f^{3}F^{5}B^{\prime\prime}+88r^{3}B^{3}f^{2}F^{5}f^{\prime}B^{\prime\prime}+20r^{4}B^{3}fF^{5}f^{\prime
2}B^{\prime\prime}+24r^{4}B^{3}f^{2}F^{5}f^{\prime\prime}B^{\prime\prime}$
$-116r^{3}B^{2}f^{3}F^{5}B^{\prime}B^{\prime\prime}-130r^{4}B^{2}f^{2}F^{5}f^{\prime}B^{\prime}B^{\prime\prime}+140r^{4}Bf^{3}F^{5}B^{\prime
2}B^{\prime\prime}+44r^{3}B^{3}f^{3}F^{4}F^{\prime}B^{\prime\prime}+22r^{4}B^{3}f^{2}F^{4}f^{\prime}F^{\prime}B^{\prime\prime}$
$-36r^{4}B^{2}f^{3}F^{4}B^{\prime}F^{\prime}B^{\prime\prime}-8r^{4}B^{3}f^{3}F^{3}F^{\prime
2}B^{\prime\prime}-28r^{4}B^{2}f^{3}F^{5}B^{\prime\prime
2}+16r^{2}B^{4}f^{3}F^{4}F^{\prime\prime}-88r^{3}B^{4}f^{2}F^{4}f^{\prime}F^{\prime\prime}$
$-20r^{4}B^{4}fF^{4}f^{\prime
2}F^{\prime\prime}-24r^{4}B^{4}f^{2}F^{4}f^{\prime\prime}F^{\prime\prime}+44r^{3}B^{3}f^{3}F^{4}B^{\prime}F^{\prime\prime}+22r^{4}B^{3}f^{2}F^{4}f^{\prime}B^{\prime}F^{\prime\prime}-8r^{4}B^{2}f^{3}F^{4}B^{\prime
2}F^{\prime\prime}$
$+28r^{3}B^{4}f^{3}F^{3}F^{\prime}F^{\prime\prime}+86r^{4}B^{4}f^{2}F^{3}f^{\prime}F^{\prime}F^{\prime\prime}-36r^{4}B^{3}f^{3}F^{3}B^{\prime}F^{\prime}F^{\prime\prime}-52r^{4}B^{4}f^{3}F^{2}F^{\prime
2}F^{\prime\prime}+8r^{4}B^{3}f^{3}F^{4}B^{\prime\prime}F^{\prime\prime}$
$+20r^{4}B^{4}f^{3}F^{3}F^{\prime\prime
2}+24r^{3}B^{3}f^{3}F^{5}B^{\prime\prime\prime}+36r^{4}B^{3}f^{2}F^{5}f^{\prime}B^{\prime\prime\prime}-44r^{4}B^{2}f^{3}F^{5}B^{\prime}B^{\prime\prime\prime}+12r^{4}B^{3}f^{3}F^{4}F^{\prime}B^{\prime\prime\prime}$
$-24r^{3}B^{4}f^{3}F^{4}F^{\prime\prime\prime}-36r^{4}B^{4}f^{2}F^{4}f^{\prime}F^{\prime\prime\prime}+12r^{4}B^{3}f^{3}F^{4}B^{\prime}F^{\prime\prime\prime}+20r^{4}B^{4}f^{3}F^{3}F^{\prime}F^{\prime\prime\prime}+8r^{4}B^{3}f^{3}F^{5}B^{\prime\prime\prime\prime}-8r^{4}B^{4}f^{3}F^{4}F^{\prime\prime\prime\prime}$
$+12r^{3}B^{2}fF^{4}B^{\prime}\dot{B}^{2}-18r^{4}B^{2}F^{4}f^{\prime}B^{\prime}\dot{B}^{2}+60r^{4}BfF^{4}B^{\prime
2}\dot{B}^{2}-12r^{4}B^{2}fF^{3}B^{\prime}F^{\prime}\dot{B}^{2}-12r^{4}B^{2}fF^{4}B^{\prime\prime}\dot{B}^{2}$
$+4r^{3}B^{3}fF^{3}B^{\prime}\dot{B}\dot{F}-6r^{4}B^{3}F^{3}f^{\prime}B^{\prime}\dot{B}\dot{F}+12r^{4}B^{2}fF^{3}B^{\prime
2}\dot{B}\dot{F}+4r^{3}B^{4}fF^{2}F^{\prime}\dot{B}\dot{F}-6r^{4}B^{4}F^{2}f^{\prime}F^{\prime}\dot{B}\dot{F}$
$-16r^{4}B^{3}fF^{2}B^{\prime}F^{\prime}\dot{B}\dot{F}+4r^{4}B^{4}fFF^{\prime
2}\dot{B}\dot{F}-4r^{4}B^{3}fF^{3}B^{\prime\prime}\dot{B}\dot{F}-4r^{4}B^{4}fF^{2}F^{\prime\prime}\dot{B}\dot{F}-20r^{3}B^{5}fFF^{\prime}\dot{F}^{2}$
$+30r^{4}B^{5}Ff^{\prime}F^{\prime}\dot{F}^{2}-20r^{4}B^{4}fFB^{\prime}F^{\prime}\dot{F}^{2}-28r^{4}B^{5}fF^{\prime
2}\dot{F}^{2}+20r^{4}B^{5}fFF^{\prime\prime}\dot{F}^{2}-12r^{3}B^{3}fF^{4}\dot{B}\dot{B}^{\prime}+18r^{4}B^{3}F^{4}f^{\prime}\dot{B}\dot{B}^{\prime}$
$-84r^{4}B^{2}fF^{4}B^{\prime}\dot{B}\dot{B}^{\prime}+12r^{4}B^{3}fF^{3}F^{\prime}\dot{B}\dot{B}^{\prime}-4r^{3}B^{4}fF^{3}\dot{F}\dot{B}^{\prime}+6r^{4}B^{4}F^{3}f^{\prime}\dot{F}\dot{B}^{\prime}-12r^{4}B^{3}fF^{3}B^{\prime}\dot{F}\dot{B}^{\prime}$
$+20r^{4}B^{4}fF^{2}F^{\prime}\dot{F}\dot{B}^{\prime}+24r^{4}B^{3}fF^{4}\dot{B}^{\prime
2}-4r^{3}B^{4}fF^{3}\dot{B}\dot{F}^{\prime}+6r^{4}B^{4}F^{3}f^{\prime}\dot{B}\dot{F}^{\prime}+12r^{4}B^{3}fF^{3}B^{\prime}\dot{B}\dot{F}^{\prime}$
$-4r^{4}B^{4}fF^{2}F^{\prime}\dot{B}\dot{F}^{\prime}+20r^{3}B^{5}fF^{2}\dot{F}\dot{F}^{\prime}-30r^{4}B^{5}F^{2}f^{\prime}\dot{F}\dot{F}^{\prime}+20r^{4}B^{4}fF^{2}B^{\prime}\dot{F}\dot{F}^{\prime}+36r^{4}B^{5}fFF^{\prime}\dot{F}\dot{F}^{\prime}$
$-16r^{4}B^{4}fF^{3}\dot{B}^{\prime}\dot{F}^{\prime}-8r^{4}B^{5}fF^{2}\dot{F}^{\prime
2}+12r^{4}B^{3}fF^{4}\dot{B}\dot{B}^{\prime\prime}+4r^{4}B^{4}fF^{3}\dot{F}\dot{B}^{\prime\prime}+4r^{4}B^{4}fF^{3}\dot{B}\dot{F}^{\prime\prime}$
$-20r^{4}B^{5}fF^{2}\dot{F}\dot{F}^{\prime\prime}-8r^{3}B^{3}fF^{4}B^{\prime}\ddot{B}+12r^{4}B^{3}F^{4}f^{\prime}B^{\prime}\ddot{B}-24r^{4}B^{2}fF^{4}B^{\prime
2}\ddot{B}+8r^{4}B^{3}fF^{3}B^{\prime}F^{\prime}\ddot{B}+8r^{4}B^{3}fF^{4}B^{\prime\prime}\ddot{B}$
$+8r^{3}B^{5}fF^{2}F^{\prime}\ddot{F}-12r^{4}B^{5}F^{2}f^{\prime}F^{\prime}\ddot{F}+8r^{4}B^{4}fF^{2}B^{\prime}F^{\prime}\ddot{F}+8r^{4}B^{5}fFF^{\prime
2}\ddot{F}-8r^{4}B^{5}fF^{2}F^{\prime\prime}\ddot{F}+8r^{3}B^{4}fF^{4}\ddot{B}^{\prime}$
$-12r^{4}B^{4}F^{4}f^{\prime}\ddot{B}^{\prime}+24r^{4}B^{3}fF^{4}B^{\prime}\ddot{B}^{\prime}-8r^{4}B^{4}fF^{3}F^{\prime}\ddot{B}^{\prime}-8r^{3}B^{5}fF^{3}\ddot{F}^{\prime}+12r^{4}B^{5}F^{3}f^{\prime}\ddot{F}^{\prime}-8r^{4}B^{4}fF^{3}B^{\prime}\ddot{F}^{\prime}$
$\left.-8r^{4}B^{5}fF^{2}F^{\prime}\ddot{F}^{\prime}-8r^{4}B^{4}fF^{4}\ddot{B}^{\prime\prime}+8r^{4}B^{5}fF^{3}\ddot{F}^{\prime\prime}\right),$
(59)

$0=T^{1}_{0}=\frac{\alpha}{6r^{2}B^{6}F^{4}}\left(-8B^{2}f^{2}F^{4}B^{\prime}\dot{B}+16rB^{2}fF^{4}f^{\prime}B^{\prime}\dot{B}-3r^{2}B^{2}F^{4}f^{\prime
2}B^{\prime}\dot{B}+6r^{2}B^{2}fF^{4}f^{\prime\prime}B^{\prime}\dot{B}\right.$
$-30rBf^{2}F^{4}B^{\prime 2}\dot{B}-15r^{2}BfF^{4}f^{\prime}B^{\prime
2}\dot{B}+30r^{2}f^{2}F^{4}B^{\prime
3}\dot{B}+14rB^{2}f^{2}F^{3}B^{\prime}F^{\prime}\dot{B}-r^{2}B^{2}fF^{3}f^{\prime}B^{\prime}F^{\prime}\dot{B}$
$-6r^{2}B^{2}f^{2}F^{2}B^{\prime}F^{\prime
2}\dot{B}+8rB^{2}f^{2}F^{4}B^{\prime\prime}\dot{B}+8r^{2}B^{2}fF^{4}f^{\prime}B^{\prime\prime}\dot{B}-30r^{2}Bf^{2}F^{4}B^{\prime}B^{\prime\prime}\dot{B}+2r^{2}B^{2}f^{2}F^{3}F^{\prime}B^{\prime\prime}\dot{B}$
$+4r^{2}B^{2}f^{2}F^{3}B^{\prime}F^{\prime\prime}\dot{B}+4r^{2}B^{2}f^{2}F^{4}B^{\prime\prime\prime}\dot{B}+8B^{4}f^{2}F^{2}F^{\prime}\dot{F}-16rB^{4}fF^{2}f^{\prime}F^{\prime}\dot{F}+3r^{2}B^{4}F^{2}f^{\prime
2}F^{\prime}\dot{F}$
$-6r^{2}B^{4}fF^{2}f^{\prime\prime}F^{\prime}\dot{F}+14rB^{3}f^{2}F^{2}B^{\prime}F^{\prime}\dot{F}-r^{2}B^{3}fF^{2}f^{\prime}B^{\prime}F^{\prime}\dot{F}-2r^{2}B^{2}f^{2}F^{2}B^{\prime
2}F^{\prime}\dot{F}+2rB^{4}f^{2}FF^{\prime 2}\dot{F}$
$+17r^{2}B^{4}fFf^{\prime}F^{\prime
2}\dot{F}-8r^{2}B^{3}f^{2}FB^{\prime}F^{\prime
2}\dot{F}-14r^{2}B^{4}f^{2}F^{\prime
3}\dot{F}+4r^{2}B^{3}f^{2}F^{2}F^{\prime}B^{\prime\prime}\dot{F}-8rB^{4}f^{2}F^{2}F^{\prime\prime}\dot{F}$
$-8r^{2}B^{4}fF^{2}f^{\prime}F^{\prime\prime}\dot{F}+2r^{2}B^{3}f^{2}F^{2}B^{\prime}F^{\prime\prime}\dot{F}+18r^{2}B^{4}f^{2}FF^{\prime}F^{\prime\prime}\dot{F}-4r^{2}B^{4}f^{2}F^{2}F^{\prime\prime\prime}\dot{F}+8B^{3}f^{2}F^{4}\dot{B}^{\prime}$
$-16rB^{3}fF^{4}f^{\prime}\dot{B}^{\prime}+3r^{2}B^{3}F^{4}f^{\prime
2}\dot{B}^{\prime}-6r^{2}B^{3}fF^{4}f^{\prime\prime}\dot{B}^{\prime}+30rB^{2}f^{2}F^{4}B^{\prime}\dot{B}^{\prime}+15r^{2}B^{2}fF^{4}f^{\prime}B^{\prime}\dot{B}^{\prime}$
$-30r^{2}Bf^{2}F^{4}B^{\prime
2}\dot{B}^{\prime}-14rB^{3}f^{2}F^{3}F^{\prime}\dot{B}^{\prime}+r^{2}B^{3}fF^{3}f^{\prime}F^{\prime}\dot{B}^{\prime}+6r^{2}B^{3}f^{2}F^{2}F^{\prime
2}\dot{B}^{\prime}+16r^{2}B^{2}f^{2}F^{4}B^{\prime\prime}\dot{B}^{\prime}$
$-4r^{2}B^{3}f^{2}F^{3}F^{\prime\prime}\dot{B}^{\prime}-8B^{4}f^{2}F^{3}\dot{F}^{\prime}+16rB^{4}fF^{3}f^{\prime}\dot{F}^{\prime}-3r^{2}B^{4}F^{3}f^{\prime
2}\dot{F}^{\prime}+6r^{2}B^{4}fF^{3}f^{\prime\prime}\dot{F}^{\prime}-14rB^{3}f^{2}F^{3}B^{\prime}\dot{F}^{\prime}$
$+r^{2}B^{3}fF^{3}f^{\prime}B^{\prime}\dot{F}^{\prime}+2r^{2}B^{2}f^{2}F^{3}B^{\prime
2}\dot{F}^{\prime}-2rB^{4}f^{2}F^{2}F^{\prime}\dot{F}^{\prime}-17r^{2}B^{4}fF^{2}f^{\prime}F^{\prime}\dot{F}^{\prime}+8r^{2}B^{3}f^{2}F^{2}B^{\prime}F^{\prime}\dot{F}^{\prime}$
$+14r^{2}B^{4}f^{2}FF^{\prime
2}\dot{F}^{\prime}-4r^{2}B^{3}f^{2}F^{3}B^{\prime\prime}\dot{F}^{\prime}-8r^{2}B^{4}f^{2}F^{2}F^{\prime\prime}\dot{F}^{\prime}-8rB^{3}f^{2}F^{4}\dot{B}^{\prime\prime}-8r^{2}B^{3}fF^{4}f^{\prime}\dot{B}^{\prime\prime}$
$+14r^{2}B^{2}f^{2}F^{4}B^{\prime}\dot{B}^{\prime\prime}-2r^{2}B^{3}f^{2}F^{3}F^{\prime}\dot{B}^{\prime\prime}+8rB^{4}f^{2}F^{3}\dot{F}^{\prime\prime}+8r^{2}B^{4}fF^{3}f^{\prime}\dot{F}^{\prime\prime}-2r^{2}B^{3}f^{2}F^{3}B^{\prime}\dot{F}^{\prime\prime}$
$\left.-10r^{2}B^{4}f^{2}F^{2}F^{\prime}\dot{F}^{\prime\prime}-4r^{2}B^{3}f^{2}F^{4}\dot{B}^{\prime\prime\prime}+4r^{2}B^{4}f^{2}F^{3}\dot{F}^{\prime\prime\prime}\right).$
(60)

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