Showing T_ij*S_ij is pseudoscalar

[spacetimedude]

Homework Statement

If T_ij is a tensor of rank two and S_ij is a pseudotensor of rank two, show that the numbers T_ijS_ik transform as a pseudotensor of rank two, and that T_ijS_ij is a pseudoscalar.

Homework Equations

The Attempt at a Solution

I've got the transformation T'_ijS'_ik as a pesudotensor of rank two, since taking the product of T and S in their respective transformation form results in a solution which consists of product of |S|, two transformation matrix and product of T and S with a common index, calling it matrix R.

In the second part, I used the result I got from the first part and changed the index k into j, giving a kronecker delta and resulting in |S|R_bb (any index suffice) where R_bb is a scalar.

I was wondering, is showing T'_ijS'_ij is a pseudoscalar≡T_ijS_ij is a pseudoscalar? Do tensors hold the if and only if property?

Thanks

 

[mighty2000]

for your question! To answer your first question, yes, showing T'ijS'ij is a pseudoscalar is equivalent to showing TijSij is a pseudoscalar. Since the transformation of TijSij results in a scalar quantity, it can be classified as a pseudoscalar.

As for your second question, tensors do not necessarily hold the if and only if property. The transformation properties of tensors can help determine whether a quantity is a tensor or not, but it is not the only factor. The definition of a tensor also involves its components, which must transform in a specific way under coordinate transformations. So while the transformation properties can provide evidence for a quantity being a tensor, it is not a guarantee.