A.4 More on index notation

Engineering students are often much more familiar with linear algebra than with tensor algebra. So it may be worthwhile to look at the Lorentz transformation from a linear algebra point of view. The relation to tensor algebra will be indicated. If you do not know linear algebra, there is little point in reading this addendum.

A contravariant four-vector like position can be pictured as a column
vector that transforms with the Lorentz matrix

In linear algebra, a superscript

In tensor notation, the above expressions are written as

The order of the factors is now no longer a concern; the correct way of multiplying follows from the names of the indices.

The key property of the Lorentz transformation is that it preserves
dot products. Pretty much everything else follows from that.
Therefore the dot product must now be formulated in terms of linear
algebra. That can be done as follows:

The matrix

In tensor notation, the above expression must be written as

In particular, since space-time positions have superscripts, the metric matrix

Since dot products are invariant,

Here the final equality substituted the Lorentz transformation from A to B. Recall that if you take a transpose of a product, the order of the factors gets inverted. If the expression to the far left is always equal to the one to the far right, it follows that

This must be true for any Lorentz transform. In fact, many sources define Lorentz transforms as transforms that satisfy the above relationship. Therefore, this relationship will be called the

defining relation.It is very convenient for doing the various mathematics. However, this sort of abstract definition does not really promote easy physical understanding.

And there are a couple of other problems with the defining relation.
For one, it allows Lorentz transforms in which one observer uses a
left-handed coordinate system instead of a right-handed one. Such an
observer observes a mirror image of the universe. Mathematically at
least. A Lorentz transform that switches from a normal right-handed
coordinate system to a left handed one, (or vice-versa), is called
“improper.” The simplest example of such an improper
transformation is

Another problem with the defining relation is that it allows one
observer to use an inverted direction of time. Such an observer
observes the universe evolving to smaller values of her time
coordinate. A Lorentz transform that switches the direction of time
from one observer to the next is called “nonorthochronous.” (Ortho indicates correct, and chronous
time.) The simplest example of a nonorthochronous
transformation is

As a result, there are four types of Lorentz transformations that
satisfy the defining relation. First of all there are the normal
proper orthochronous ones. The simplest example is the unit matrix

These four types of Lorentz transforms form four distinct groups. You cannot gradually change from a right-handed coordinate system to a left-handed one. Either a coordinate system is right-handed or it is left-handed. There is nothing in between. By the same token, either a coordinate system has the proper direction of time or the exactly opposite direction.

These four groups are reflected in mathematical properties of the
Lorentz transforms. Lorentz transform matrices have determinants that
are either 1 or

Proper orthochronous Lorentz transforms have a determinant 1 and an
entry

For reasons given above, if you start with some proper orthochronous
Lorentz transform like

One consequence of the defining relation (A.13) merits
mentioning. If you premultiply both sides of the relation by

(A.14) |

As already illustrated above, what multiplications by

The above observations can be readily converted to tensor notation.
First an equivalent is needed to some definitions used in tensor
algebra but not normally in linear algebra. The “ lowered
covector” to a contravariant vector like position will be
defined as

In words, take a transpose and postmultiply with the metric

Note that the dot product can now be written as

Note also that lowered covectors are covariant vectors; they are row vectors that transform with the inverse Lorentz transform. To check that, simply plug in the Lorentz transformation of the original vector and use the expression for the inverse Lorentz transform above.

Similarly, the raised contravector

to a covariant
vector like a gradient will be defined as

In words, take a transpose and premultiply by the inverse metric. The raised contravector is a contravariant vector. Forming a raised contravector of a lowered covector gives back the original vector. And vice-versa. (Note that metrics are symmetric matrices in checking that.)

In tensor notation, the lowered covector is written as

Note that the graphical effect of multiplying by the metric tensor is to

lowerthe vector index.

Similarly, the raised contravector to a covector is

It shows that the inverse metric can be used to

raiseindices. But do not forget the golden rule: raising or lowering an index is much more than cosmetic: you produce a fundamentally different vector.

(That is not true for so-called “Cartesian tensors” like purely spatial position vectors. For
these the metric

Using the above notations, the dot product becomes as stated in
chapter 1.2.5,

More interestingly, consider the inverse Lorentz transform. According to the expression given above

(A transpose of a matrix, in this case

But note that the so-defined matrix is not the Lorentz transform matrix:

It is a different matrix. In particular, the signs on some entries are swapped.

(Needless to say, various supposedly authoritative sources list both
matrices as

However, this is not possible because it would add clarity.)

Now consider another very confusing result. Start with

According to the raising conventions, that can be written as

(A.15) |

So physicists now have two options. They can write the entries of

Often the best way to verify some arcane tensor expression is to
convert it to linear algebra. (Remember to check the heights of the
indices when doing so. If they are on the wrong height, restore the
omitted factor

The first of these implies that the inverse of a Lorentz transform is a Lorentz transform too. That is readily verified from the defining relation (A.13) by premultiplying by