Over the past month, I re-derived much of special relativity and its relation to electromagnetism in four-vector form. I thought perhaps I was ready to move on to general relativity. I did learn a bit about it, including writing a piece on Christoffel symbols; however, from the combined experience of reading Sean Carroll’s lecture notes and L&L Classical Theory of Fields, surfing relevant Wikipedia pages, and trying to re-derive the theory of differential forms on manifolds on my own (with mixed success), it became somewhat clear that my foundation on tensors and differential forms is not solid enough. I was quickly a bit lost reading the first chapter on general relativity especially when they went into transformations of rank four tensors in relation to determinants that referred to section 6, the title of which is “four-vectors”. So I will, here, go through that section in detail, at least the parts of it which I do not already know pretty well.
The tensors are special in the sense that their components are the same in all coordinate systems (This was demonstrated via in ). We want that the completely antisymmetric unit tensor of fourth rank, , which changes sign under interchange of any pair of indices, has the same property. We set
Now, how do we lower all the indices?
In the above, we notice that because is diagonal, the only non-zero term in the sum across permutations is given by , thus three negatives and one positive gives a sign reversal ( ). Moreover,
We notice that for any ,
In the above, we viewed the contravariant rank four tensor as a linear map of a covariant 4-vector to a scalar, and this should hold in any coordinate system. If we Lorentz transform this four tensor, the result is
wherein we simply applied the same Lorentz transform to each of the indices. In , if , then we have
We note that the Lorentz group actually has four connected components (parity inversion flips the sign of the determinant, and one can then partition again corresponding to time reversal). When we speak of Lorentz group, we actually mean the proper orthochronous subgroup of the Lorentz group, which is its own connected component. Since interchanging two rows in a matrix results in a flip of sign of the determinant, the above result generalizes to .
Moreover, by of  we have
from which follows that
namely invariance under change of coordinates per some Lorentz rotation.
Let denote a parity inversion transformation, one which flips one or three of the coordinate axes. In this case, the values of
would be signed flipped relative to and thus if were strictly a tensor, it would not be the same in all coordinate systems which preserve the Minkowski metric. If we enforce that condition already shown impossible for a tensor, then is a pseudotensor, which is a quantity that behaves the same in all coordinate transformations not reducible to rotations.
The products form a four-tensor of rank . One observes how in the case of parity inversion, it is done twice here, thereby no change. Relatedly, for any permutation ,
We now determine the value of . We note that because , . In doing so, we first take to be some permutation of . As for , for the corresponding tensor to be non-zero, it must be a permutation. The sign of the value is determined by the relative sign of with respect to . This feature is provided to us by the permutation based definition of determinant. We have accordingly
Contracting the last index, we get
We note that in the above, and if the two sets of three indices (one covariant, one contravariant in the ) are not equal, we must have some row all zero and some column all zero, thereby the minor determinant must also be zero, thus the reduction. Moreover, we did not explicitly write to emphasize that we are summing across all . However, for fixed , there is only only one value of that yields a non-zero result.
We next set and compute . We notice that for this tensor to be non-zero, is a must. Upon fixing , there are two choices for the value of . Thus,
Similarly, we obtain
and upon further contraction.
We furthermore observe that
where is the determinant formed from the quantities .
The fact that the components of the four-tensor are unchanged under rotations of a four-dimensional coordinate system, and that the components of the three-tensor are unchanged by rotations of the space axes are special cases of a general rule: any completely antisymmetric tensor of rank equal to the number of dimensions of the space in which it is defined is invariant under rotations of the coordinate system in the space. This general rule holds because:
- Antisymmetric implies zero if the indices are not all equal, restricting to permutations.
- Antisymmetric with respect to permutation swap implies a single absolute value for all non-zero entries.
- Application of this rank antisymmetric tensor to th tensor power of any rotation results in the permutation based formula for determinant to be applied to the rotation matrix, which has determinant .