Verifying Simple Quadrupole Field Not in Lorenz Gauge?

[CharlesJQuarra]

I'm having trouble reproducing some of the results regarding gravitational waves in the Wald's General Relativity

In section 4.4 of gravitational radiation, eq.4.4.49 shows the far-field generated by a variable mass quadrupole:

$$ \gamma_{i j}(t,r)=\frac{2}{3R} \frac{d^2 q_{i j}}{dt^2} \bigg|_{t'=t-R/c} $$

I want to verify that this simple field satisfies the Lorenz gauge (eq.4.4.25)

$$\partial_{i} \gamma_{i j}=0 $$
I wrote the $q_{i j}$ for a simple rotating binary
$\ddot{q}_{i j} =$
\begin{bmatrix}
2 \omega^2 \cos{2\omega(t-R/c)} & - 2 \omega^2 \sin{2\omega(t-R/c)} & 0 \\
- 2 \omega^2 \sin{2\omega(t-R/c)} & - 2 \omega^2 \cos{2\omega(t-R/c)} & 0 \\
0 & 0 & 0
\end{bmatrix}

then, I wrote $R=\big|(x,y,z)\big|$$\partial_{i} \gamma_{i 3}$ trivially cancels, but when I compute the other components of the divergence I get

$$ \partial_{i} \gamma_{i 1} = \frac{2 \omega^2([-c x+2 y R \omega]\cos{2\omega(t-R/c)}+[c y+2 x R \omega]\sin{2\omega(t-R/c)})}{c R^3} $$

$$ \partial_{i} \gamma_{i 2} = \frac{2 \omega^2([c y+2 x R \omega]\cos{2\omega(t-R/c)}+[c x-2 y R \omega]\sin{2\omega(t-R/c)})}{c R^3} $$

Which as you might have noticed, are not zero in general

Given that the Lorenz gauge is used everywhere one wants to study gravitational wave propagation, it seems unexpected that the far-field of a simple binary quadrupole is not automatically in the Lorenz gauge

Question: I want to understand what is wrong here, if there is anything wrong. Am I wrong/naive in expecting this simple physical system field to be in the Lorenz gauge? Is there a simple transformation that can be applied to this field in order to be manifestly in harmonic coordinates?

 

[Ambitwistor]

Thank you for bringing this issue to our attention. I understand your frustration with not being able to reproduce the results regarding gravitational waves in Wald's General Relativity. it is important to thoroughly investigate and understand any discrepancies in our findings.

After reviewing your calculations, I do not see any errors in your approach. However, I believe the issue lies in the assumption that the far-field generated by a variable mass quadrupole automatically satisfies the Lorenz gauge. While this may be true in some cases, it is not a general rule.

In fact, the Lorenz gauge is a gauge condition that is imposed on the metric tensor, not the gravitational wave itself. It ensures that the equations of motion for the metric tensor are consistent with the wave equation for gravitational waves. Therefore, it is possible for the far-field to not automatically satisfy the Lorenz gauge.

I suggest looking into the general transformation for a metric tensor to be in harmonic coordinates, which is the equivalent of the Lorenz gauge in General Relativity. This transformation involves a coordinate transformation and a metric transformation. By applying this transformation to your far-field, you should be able to obtain a metric tensor that is manifestly in harmonic coordinates.

I hope this helps in your understanding of the issue. Please let me know if you have any further questions or if you need any assistance in applying the transformation.