Constant tensor

[ehki...@gmail.com]

Hi,

Is there an option in DefTensor to define a tensor as constant with respect to all types of derivatives?

Best regards,
Ehsan

[Jose]

Hi,

No, there is no such option in xTensor. I don't think the concept of "constant tensor with respect to all derivatives" is geometrically possible. Imagine a vector field that is constant with respect to two arbitrary covariant derivatives. Then that means that the vector field is zero when contracted with arbitrary Christoffel tensors, which I think makes the vector field zero. Perhaps only the delta tensors are constant with respect to all derivatives.

Having said that, we have the xTensor command FirstDerQ, which gives True on all types of derivatives, so you can do something like this, if you are sure it makes sense in your computation:

   << xAct`xTensor`

   DefManifold[M, 4, {a, b, c, d, e}]

   DefTensor[v[a], M]

   v /: der_?FirstDerQ[v[_]] := 0

And then every derivative of v will give zero.

   In[]:= PD[-a][v[b]]

   Out[]= 0

Note also that DefTensor does not complain with something like this:

   DefTensor[w[a], {}]

meaning that w[a] is a field with no manifold or parameter dependencies. This is a strange situation, similar in spirit to what you are asking for, but again I don't think it is self-consistent, and should be used only with much care, or not at all.

Cheers,

Jose.

[ehki...@gmail.com]

Thank you for your response!

Actually, I am dealing with objects that are not tensors. Instead, they are some constants in global transformations.

Best regards,
Ehsan