How Do You Simplify the Integral in Wald's Quadrupole Formula?

[etotheipi]

To prove (4.4.58) from Wald$$P = \frac{1}{45} \sum_{\mu, \nu = 1}^{n} \left(\frac{d^3 Q_{\mu \nu}}{dt^3} \right)^2$$from starting by writing$$P = - \frac{d}{dt} \int_S t_{a0} dS^a = - \frac{d}{dt} \int_S - \frac{1}{8 \pi} G_{a0}^{(2)} [ \gamma_{cd} ] dS^a = \frac{1}{8\pi} \frac{d}{dt} \int_S \left(R_{a0}^{(2)} - \frac{1}{2} \eta_{a0} R^{(2)} \right) dS^a$$we know that

$$\begin{align*}

R_{a0}^{(2)} = \frac{1}{2} \gamma^{cd} \partial_a \partial_0 \gamma_{cd} &- \gamma^{cd} \partial_c \partial_{(a} \gamma_{0)d} + \frac{1}{4}(\partial_a \gamma_{cd}) \partial_0 \gamma^{cd} + (\partial^d {\gamma^c}_0) \partial_{[d} \gamma_{c]a} \\

&+ \frac{1}{2} \partial_d(\gamma^{dc} \partial_c \gamma_{a0}) - \frac{1}{4}(\partial^c \gamma) \partial_c \gamma_{a0} - (\partial_{d} \gamma^{cd} - \frac{1}{2} \partial^c \gamma) \partial_{(a} \gamma_{0)c}

\end{align*}$$from which it also follows we can write the $- \frac{1}{2} \eta_{a0} R^{(2)}$ term as

$$\begin{align*}

- \frac{1}{2} \eta_{a0} R^{(2)} =
-\frac{1}{4} \eta_{a0} \gamma^{cd} \partial_e \partial^e \gamma_{cd} & + \frac{1}{2} \eta_{a0} \gamma^{cd} \partial_c \partial_{e} {\gamma^{e}}_{d} - \frac{1}{8} \eta_{a0} (\partial_e \gamma_{cd}) \partial^e \gamma^{cd} - \frac{1}{2} \eta_{a0} (\partial^d {\gamma^c}^e) \partial_{[d} \gamma_{c]e} \\

&- \frac{1}{4} \eta_{a0} \partial_d(\gamma^{dc} \partial_c \gamma) + \frac{1}{8} \eta_{a0} (\partial^c \gamma) (\partial_c \gamma) + \frac{1}{2} \eta_{a0} (\partial_{d} \gamma^{cd} - \frac{1}{2} \partial^c \gamma) \partial_{e} {\gamma^{e}}_{c}

\end{align*}
$$where$$\bar{\gamma}_{\mu \nu} = \frac{2}{3R} \frac{d^2 q_{\mu \nu}}{dt^2} = \gamma_{\mu \nu} - \frac{1}{2} \eta_{\mu \nu} \gamma$$How on Earth are you supposed to do that integral?!? Apparently some terms disappear but which ones?

 

[Ambitwistor]

you can approach this problem by using mathematical principles and techniques to simplify the integral. Here are some steps you can follow:

1. Simplify the expressions for $R_{a0}^{(2)}$ and $- \frac{1}{2} \eta_{a0} R^{(2)}$ using the given equations and the definition of $\bar{\gamma}_{\mu \nu}$. This will involve expanding and rearranging terms.

2. Use the fact that $\eta_{a0}$ is a constant to bring it outside the integral. This will allow you to simplify the integrand.

3. Use integration by parts to convert the derivatives in the integrand into simpler expressions. This will also involve using the boundary conditions (i.e. assuming that the surface integral is evaluated at some fixed time).

4. Use the fact that $\gamma_{\mu \nu}$ is symmetric to simplify the integrand further.

5. Use the given equation for $P$ to substitute for $R_{a0}^{(2)}$ and $- \frac{1}{2} \eta_{a0} R^{(2)}$ in the integral. This will give you an expression for $P$ in terms of $\gamma_{\mu \nu}$ and its derivatives.

6. Use the given equation for $P$ again to substitute for $\gamma_{\mu \nu}$ in the expression you obtained in step 5. This will give you an expression for $P$ in terms of $q_{\mu \nu}$ and its derivatives.

7. Evaluate the integral by using the boundary conditions and simplifying the resulting expression. Some terms may cancel out, leaving you with a simpler expression for $P$.

8. Compare the expression you obtained for $P$ with the given equation for $P$ and see if they are equal. If they are, you have successfully proven the equation.

Keep in mind that this is just one possible approach and there may be other ways to simplify the integral. It may also be helpful to consult with a mathematician or a colleague who is familiar with this type of integral. Good luck!