xAct rule for antisymmetrization

Question

using the xAct I'm trying to make a rule that the antisymmetrization in three indices of an expression goes to zero. Namely I have the killing vector

$$ \xi^a $$

defined using DefTensor[\[xi][a],M,KillingVectorOf -> metric] with $M$ being the manifold. Now I would like to create a rule such that applied to an expression it acts as

$$ \xi_{[a} \nabla_{b} \xi_{d]}=0. $$ I can easily define a new tensor with such symmetries, but I have trouble defining such a rule. For example when I try

MakeRule[{(\[Xi][-d] cd[-b]@\[Xi][-a] - \[Xi][-b] cd[-d]@\[Xi][-a] + \\[Xi][-a] cd[-d]@\[Xi][-b]), 0}, UseSymmetries -> True, 

MetricOn -> All]

I end up getting a "There is more than one term on the LHS of the rule" error. Any advice on how to implement such symmetries for expressions?

Answer 1

As pointed out in a comment above, the key here is to add a condition on the indices. For example, with the setup

<< xAct`xTensor`

DefManifold[M, 4, {a, b, c, d, e, f}]

DefTensor[X[a], M]

DefMetric[-1, g[-a, -b], cd]

construct this sum, which you want to be zero:

zero = 6 Antisymmetrize[X[-a] cd[-b][X[-d]], {-a, -b, -d}]

And now construct the rule you need with something like this, and note the use of OrderedQ so that the rule only applies to one of the six terms of the sum:

rule = With[{lhs = First[zero], rhs = - Rest[zero]}, MakeRule[{lhs, rhs, OrderedQ[{a, b, d}]}]]

Check the rule:

In[]:= zero /. rule
Out[]= 0