epsilong with two identical indexes is not zero

[Nail Khusnutdinov]

Hi!

I defined 3D manifold DefManifold[M3, 3, {i, k, l, j}] and metric DefMetric[-1, g[-i, -k], CD, {"|", "\[EmptyDownTriangle]"}]. The antisymmetric tensor is automatically defined epsilong[-i,-k,-l]. But why epsilong[i,i,k] is not zero? Is there a specific simplification operator?

Nail

[Thomas Bäckdahl]

Hi!

Use
ToCanonical

Regards
Thomas

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[Nail Khusnutdinov]

Nothing are changed. epsilon[i,i,k]//ToCanonical gives the same result, not zero.

четверг, 20 июня 2024 г. в 02:21:42 UTC-3, Thomas Bäckdahl:

[Jose]

Hi,

Abstract indices can never be repeated in xAct tensors. The only exception are 1-dimensional indices. Something like epsilong[i, i, k] is not valid syntax.

    << xAct`xCoba`

    DefManifold[M, 3, {i, j, k, l}]

    DefTensor[eps[i, j, k], M, Antisymmetric[{1, 2, 3}]]

    In[]:= eps[i, i, j] // Validate
    Validate::repeated: Found indices with the same name i.

What is needed here is a basis in which to address individual components of the tensor:

    DefBasis[B, TangentM, {1, 2, 3}]

Then something like this is zero:

   In[]:= eps[{1, B}, {1, B}, {k, B}] // ToCanonical

   Out[]= 0

Note that currently we don't support eps[{i, B}, {i, B}, {k, B}] as valid either, but I guess this could be made to be zero.

Cheers,

Jose.

[Nail Khusnutdinov]

Thanks a lot for the explanation.

четверг, 20 июня 2024 г. в 17:26:35 UTC-3, Jose: