symmetries of eplisonmetric are not recognized when indices are mixed

[Masoud Shokri]

Dear all,

It sometimes happens that when I define a rule with epislonmetric, the result is a mixture of up and down indices. For instance,

$

\epsilon^{\beta }{}_{\lambda \sigma \delta } E^{\lambda } B^{\sigma } u_{\alpha } u^{\delta }

$

Despite the the obvious fact that this example is zero, it does not disappear on canonicalization unless I fix the indices in a dirty way. What is the best way to do this?

Best regards

Masoud

[Jose]

Hi,

Why do you say this is obviously zero ? Did you mean to write u_{\beta} instead of u_{\alpha} ? I'm going to assume that's the case.

In general, it is much better (and safer) to send full xAct code, for reproducibility. In this case:

<< xAct`xTensor`

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

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

DefTensor[{u[a], v[a], w[a]}, M]

In[5]:= epsilong[a, -b, -c, -d] v[b] w[c] u[d] u[-a] // ToCanonical

Out[5]= 0

Cheers,

Jose.

[Masoud Shokri]

Dear Jose,

As you noticed, I copied the wrong expression. The correct one is

$

- \epsilon_{\alpha \lambda \sigma }{}^{\delta } E^{\beta } B^{\lambda } u^{\sigma } u_{\delta }

$

And this is the expression in input form

-(epsilonmetric[-a, -l, -o, p]*fourelectric[b]*fourmagnetic[l]*
    fourvelocity[o]*fourvelocity[-p]) // ToCanonical

Since I was not able to reproduce the problem, I am sending my file with this email. The problem appears in line 323.

Best regards

Masoud

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Best Regards
Masoud Shokri
http://www.ipm.ac.ir/personalinfo.jsp?PeopleCode=IP1900030
--------------------------
Post-Doctoral Research Fellow, School of Particles and Accelerator
IPM
Tehran, Iran

[Jose]

Hi,

This notebook contains the line

   SetOptions[ToCanonical, UseMetricOnVBundle -> None]

which instructs ToCanonical not to use the metric. Hence if the indices of a tensor are not placed in their original positions there will not be proper canonicalization. Restoring the default setting UseMetricOnVBundle -> All fixes the issue.

It's always much better to send a small notebook or minimal collection of inputs that reproduce the problem, instead of a large notebook where it takes considerable time to decide what is important or not for the problem at hand.

Cheers,

Jose.