The ADM energy of gravitational waves? I have been looking for books about this question for several days. However, almost all books use Landau–Lifshitz pseudotensor to calculate the energy of gravitational waves.  And they said the result of Gravitational Waves' energy doesn't depend on the kinds of pseudotensor. So, I want to try to use another way to calculate the energy of gravitational waves, such as the ADM Energy.
First of all, we let $$ g_{ab}=\eta_{ab}+\gamma_{ab} \,.$$
Then use the linear Einstein's Field Equation,$$ R_{ab}=0 ~~\Rightarrow~~ \Box^2\gamma_{ab}=0 \,.$$
For a plane wave propagating along the $x^3 $-axis,we know that the only components of $ \gamma_{\mu\nu} $ that are different from zero are $$ \gamma_{11}=-\gamma_{22},\gamma_{12}=\gamma_{21} $$
So, $$ \gamma_{jj}=\gamma_{11}+\gamma_{22}+\gamma_{33}=0 $$
Consider the ADM Energy $$ E=\frac{c^4}{16 \pi G} \lim_{r\to\infty} \iint_{S_{r}}\hspace{-34.5px}\subset\!\supset \left(\partial_{j}h_{ij}-\partial_{i}h_{jj}\right) \, \mathrm{d}S^i$$
Some calculation about $ h_{ab} $,$$ h_{ab}=g_{ab} \mp n_{a}n_{b} ~~\Rightarrow~~ h_{ij}=g_{ij}=\eta_{ij}+\gamma_{ij} $$
$$ h_{jj}=\eta_{jj}+\gamma_{jj}=\eta_{jj}= \text{const} $$
$$ \partial_{1}h_{ij}=\partial_{2}h_{ij}=0 $$
Finally, we have $$ E=0 $$
The result is certainly wrong, but where is the mistake? I have been thinking for a long time, but I don't get anything. 
 A: OK, maybe I get where my mistake is.
It's very important that $ T^{(1)}_{ab}=G^{(1)}_{ab}=0 $ but $ T_{ab}\not=0 $.
From this PDF, we can learn $$ T^{03}=-\frac{1}{16\pi}\left[\left(\frac{\partial h_{11}}{\partial t}\right)^2+\left(\frac{\partial h_{12}}{\partial t}\right)^2 \right]
\,,$$
noting that $c=1$ and $G=1.$  Because $$ \sqrt{-g}(T^{\mu\nu}+t^{\mu\nu})=E=0 $$
So $$t^{03}=\frac{1}{16\pi}\left[\left(\frac{\partial h_{11}}{\partial t}\right)^2+\left(\frac{\partial h_{12}}{\partial t}\right)^2\right] \,.$$
It is the same as the result of Landau–Lifshitz pseudotensor.
I hope my answer is correct.
A: I am not sure whether it is true that the ADM energy depends on rest mass. But if it is true, an object with zero rest mass generate no gravity such as plane electromagnetic wave. The property of plane gravitational wave is analogous to plane electromagnetic wave, so it should has zero rest mass as well. You can apply it to gravitational solitons (a wave packet which travels at the speed of light) to see whether it is true.
If a gravitational wave packet has non zero rest mass, it should be regarded as "geon". I know there are some solution of them, but whether they are stable are not sure.
