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Tensors and Relativity

The question is attached along with relevant course notes. Only part ii needs to be answered.

Not everything with indices is a tensor. If theta and A^u transform as a scalar and a contravariant vector under general coordinate transformations, respectively, then show that:

(See attached formula)

Is a tensor, but that:

(See attached formula)

Is not. (Hint: apply the transformation laws for theta, A^u, and (see attached), to determine how the RHS transforms in each case.)

ii) Let G^uv be a symmetric tensor function under general coordinate transformations, i.e. g^uv = g^vu, and let h_uv be a tensor that satisfies

(See attachment)

In a particular set of coordinates.

a) Is the above relation satisfied in all sets of coordinates? Why or why not?
b) By multiplying each side of the above relation by (see attached), show that (see attached) is also symmetric.
c) By differentiating each side of the above relation with respect to X^alpha, show that (see attached)
Warning: never use the same dummy index more than twice in a given index. Otherwise the summation convention becomes inconsistent. For example (A^uB_u)^2


Solution Preview


You have correctly presented the argument that a contraction of tensors is a tensor. Then because in some particular coordinate system the contraction is equal to delta^{mu}_{nu} the question is then simply how the Kronecker delta tensor transforms under general coordinate transforms. It turns out that the tensor stays invariant, but your argument for this isn't fully conclusive. A compication here is that you can't now assume that Kronecker delta in a different coordinate system delta^{mu'}_{nu'} is the usual Kronecker delta i.e. it is equal to 1 if mu' = mu' and 0 otherwise, this is precisely what we want to prove. So, you do apply the transformation rule. Let's use the notation:

dx^{mu'}/dx^{nu} = ...

Solution Summary

We explain the problem and the solution in detail. Particular attention is paid to pointing out what you need to prove here.