MakeRule(s) for Transverse-Traceless Gauge

[Alex Cassem]

Hello everyone (I am new).

I just started to learn xAct (with xPert, xTras, etc), and I am trying to implement the transverse-traceless condition on perturbations to the equations of motion. Here is my setup:

DefManifold[M, 4, IndexRange[a, q]]

DefMetricPerturbation /. Options@DefMetric

DefMetric[-1, metric[-a, -b], CD, PrintAs -> "g", CurvatureRelations -> True]

Then, I define the Lagrangian and compute the equations of motion, and then expand on some background:

LEH = RicciScalarCD[];
eom = CollectTensors@VarL[metric[-a, -b], CD][LEH]

lineareom = ExpandBackground[eom] // CollectTensors

I then define the perturbed metric and send it to a flat background

DefTensor[H[-a, -b], M, Symmetric[{-a, -b}]]

Eomlinearized =lineareom /. Perturbationmetric[LI[1], inds__] :> H[inds] //ToCanonical // ToFlat

What I am hoping this has done so far is: find Einstein's eom, linearize first on g_{\mu\nu} = \overline{g}_{\mu\nu} + h_{\mu\nu}, but then the //ToFlat puts me on Minkowski $\eta_{\mu\nu}$ (when I do this I find the correct linearized eom so I am assuming this is correct, please correct me if I am dead wrong). 

But, now I want to implement the transverse-traceless gauge. The traceless part is simple, just do:

Eomlinearized /. {H[_, -_] -> 0, H[-_, _] -> 0}

But I would like to be able to do this with MakeRule.

I tried using MakeRule for the transverse part of it but the following line fails:

MakeRule[{CD[-a][H[a, b]], 0}, PatternIndices -> All, MetricOn -> All]

I should only have -(1/2)  (CD[-c][CD[b][H[a, c]]]) leftover after transverse-traceless has worked, but I do not.

Any suggestions?

Thanks.

-Alex Cassem