Q: meaning \[EmptyUpTriangle]^-1?

[Sam Finn]

I understand \[CapitalDelta][X] means the perturbed X. How should I understand \[CapitalDelta]^{-1}[X]?

Thanks - Sam

[Jose]

Hi,

If you are referring to negative perturbation orders in xPert, we don't support them, and I'm not aware of any definition of such a concept. I was expecting xPert to fail loudly when presented with negative orders, but it seems to return zeros, which is just a random way of failing:

In[1]:= << xAct`xPert`

In[2]:= DefManifold[M, 4, {a, b, c, d}]

In[3]:= DefMetric[-1, g[-a, -b], cd]

In[5]:= DefMetricPerturbation[g, h, eps]

In[6]:= Perturbation[Riccicd[-a, -b], -1] // ExpandPerturbation
Out[6]= 0

Cheers,

Jose.

[Sam Finn]

Hi Jose - 

Thanks for getting back to me. I encountered these terms in an xPert involving a gauge change. I’m assuming from your note the appearance of these is an error that should have been caught. I’ve attached a MWE showing how xPert readily generates them in the hopes that it may be helpful in debugging. 

Again, thanks and best, 

Sam

[Leo Stein]

Hi all,

Sam, I think there are some definitions missing in your notebook: RicciO2 is never defined.

However, here is a simpler MWE. Using the definitions you provided in your notebook before the section "General Gauge Change", the first line below works (of course) whereas the second does not:

GaugeChange[h[LI[1], -a, -b], \[Xi]]
GaugeChange[CD[-c]@h[LI[1], -a, -b], \[Xi]]

Best

Leo 

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Sam Finn

lsf...@gmail.com

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[Sam Finn]

Hi Jose & Leo -

My apologies for an MWE that wasn’t. 

I’ve attached an amended MWE, which I have just now verified works as expected. 

I added Leo’s example to the end. In a second addendum I also added an example where the anomalous result depends on the location of an ExpandPerturbation, either before the gauge change (leading to the anomalous results) or after. 

Leo’s example is, of course, minimal relative to my own; nevertheless, there may some value in recirculating the corrected MWE as it shows the anomalous result arising in the kind of calculation one might actually think to perform. As such it exercise the code in a somewhat different way and be of value in a functional regression testing suite, if such is maintained.

Best, 

Sam