April 2nd, 2021

A note regarding the inverse of a Lorentz transformation and its representation in Einstein notation

Originally published at 狗和留美者不得入内. You can comment here or there.

I can remember the velocity addition formula corresponding to Lorentz transformations along a single dimension, [公式] , off the top of my head. It might mistakenly give the impression that the Lorentz group in general is abelian. It is not for four dimension. A Lorentz boost along one direction and one along another direction do not commute. The Lorentz boost matrix is given by

[公式]

Moreover, [公式] tells us that to obtain the inverse, we flip the sign of the entries in the boost matrix such that one index corresponds to time coordinate and the other index corresponds to a space coordinate. This is given formulaically by

[公式] wherein [公式] is the matrix corresponding to Minkowski metric. In Einstein notation, we denote [公式] with [公式] and raising the lower index and then lowering the upper index in it gives

[公式] Raising an index corresponds to applying Minkowski matrix from the left, in which case for each entry in the result, we iterate along the entries of some column of the Lorentz matrix, which is along the row index. Lowering an index then corresponds to applying Minkowski matrix from the right, which demonstrates the equivalence of [公式] and [公式], with [公式] . [公式] equates to[公式]which if appearing in expression involving Lorentz transformations of tensors of higher rank conveniently cancels out to the identity matrix. Letting [公式] , we obtain

[公式] where [公式] is the matrix corresponding to the transform of the contravariant four-vector for a given Lorentz transformation.

On the completely antisymmetric unit rank 4 tensor (or pseudotensor) over spacetime coordinates

Originally published at 狗和留美者不得入内. You can comment here or there.

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 [公式] are special in the sense that their components are the same in all coordinate systems (This was demonstrated via [公式] in [2]). We want that the completely antisymmetric unit tensor of fourth rank, [公式] , which changes sign under interchange of any pair of indices, has the same property. We set

[公式] Now, how do we lower all the indices?

[公式] In the above, we notice that because [公式] is diagonal, the only non-zero term in the sum across [公式] permutations is given by [公式] , thus three negatives and one positive gives a sign reversal ( [公式] ). Moreover,

[公式] We notice that for any [公式] ,

In the above, we viewed the contravariant rank four tensor [公式] 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

[公式] wherein we simply applied the same Lorentz transform to each of the indices. In [公式] , if [公式] , then we have

[公式] 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 [公式] .

Moreover, by [公式] of [3] we have

[公式] from which follows that

[公式] namely invariance under change of coordinates per some Lorentz rotation.

Let [公式] denote a parity inversion transformation, one which flips one or three of the coordinate axes. In this case, the values of

[公式] would be signed flipped relative to [公式] and thus if [公式] 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 [公式] is a pseudotensor, which is a quantity that behaves the same in all coordinate transformations not reducible to rotations.

The products [公式] form a four-tensor of rank [公式] . One observes how in the case of parity inversion, it is done twice here, thereby no change. Relatedly, for any permutation [公式] ,

[公式] We now determine the value of [公式] . We note that because [公式] , [公式] . In doing so, we first take [公式] to be some permutation of [公式]. As for [公式] , 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 [公式] with respect to [公式] . This feature is provided to us by the permutation based definition of determinant. We have accordingly

[公式]

Contracting the last index, we get

[公式] We note that in the above, [公式] and if the two sets of three indices (one covariant, one contravariant in the [公式] ) 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 [公式] to emphasize that we are summing across all [公式] . However, for fixed [公式] , there is only only one value of [公式] that yields a non-zero result.

We next set [公式] and compute [公式] . We notice that for this tensor to be non-zero, [公式] is a must. Upon fixing [公式] , there are two choices for the value of [公式] . Thus,

[公式]

Similarly, we obtain

[公式] and [公式] upon further contraction.

We furthermore observe that

[公式] where [公式] is the determinant formed from the quantities [公式] .

The fact that the components of the four-tensor [公式] are unchanged under rotations of a four-dimensional coordinate system, and that the components of the three-tensor [公式] 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 [公式] antisymmetric tensor to [公式] th tensor power of any rotation results in the permutation based formula for determinant to be applied to the rotation matrix, which has determinant [公式] .

References