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 . A
covariant four-vector like the gradient of a scalar function can be
pictured as a row vector that transforms with the inverse Lorentz
In tensor notation, the above expressions are written as
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:
In tensor notation, the above expression must be written as
Since dot products are invariant,
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 . That is called the “parity transformation.” Its effect is to flip over all spatial position vectors. (If you make a picture of it, you can see that inverting the directions of the , , and axes of a right-handed coordinate system produces a left-handed system.) To see that satisfies the defining relation above, note that is symmetric, , and its own inverse, .
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 . That transformation is called “time-reversal.” Its effect is to simply replace the time by . It satisfies the defining relation for the same reasons as the parity transformation.
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 , corresponding to the case that the observers A and B are identical. Second, there are the improper ones like that switch the handedness of the coordinate system. Third there are the nonorthochronous ones like that switch the correct direction of time. And fourth, there are improper nonorthochronous transforms, like , that switch both the handedness and the direction of time.
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 1. That is easily seen from taking determinants of both sides of the defining equation (A.13), splitting the left determinant in its three separate factors. Also, Lorentz transforms have values of the entry that are either greater or equal to 1 or less or equal to 1. That is readily seen from writing out the entry of (A.13).
Proper orthochronous Lorentz transforms have a determinant 1 and an entry greater or equal to 1. That can readily be checked for the simplest example . More generally, it can easily be checked that is the time dilatation factor for events that happen right in the hands of observer A. That is the physical reason that must always be greater or equal to 1. Transforms that have less or equal to 1 flip over the correct direction of time. So they are nonorthochronous. Transforms that switch over the handedness of the coordinate system produce a negative determinant. But so do nonorthochronous transforms. If a transform flips over both handedness and the direction of time, it has a time dilatation less or equal to 1 but a positive determinant.
For reasons given above, if you start with some proper orthochronous Lorentz transform like and gradually change it, it stays proper and orthochronous. But in addition its determinant stays 1 and its time-dilatation entry stays greater or equal to 1. The reasons are essentially the same as before. You cannot gradually change from a value of 1 or above to a value of 1 or below if there is nothing in between.
One consequence of the defining relation (A.13) merits
mentioning. If you premultiply both sides of the relation by
, you immediately see that
As already illustrated above, what multiplications by do is flip over the sign of some entries. So to find an inverse of a Lorentz transform, just flip over the right entries. To be precise, flip over the entries in which one index is 0 and the other is not.
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
Note that the dot product can now be written as
raised contravector to a covariant
vector like a gradient will be defined as
In tensor notation, the lowered covector is written as
lowerthe vector index.
Similarly, the raised contravector to a covector is
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 is the unit matrix. Then raising or lowering an index has no real effect. By the way, the unit matrix is in tensor notation . That is called the Kronecker delta. Its entries are 1 if the two indices are equal and 0 otherwise.)
Using the above notations, the dot product becomes as stated in
(Needless to say, various supposedly authoritative sources list both
matrices as for that exquisite final bit of
confusion. It is apparently not easy to get subscripts and
superscripts straight if you use some horrible product like MS Word.
Of course, the simple answer would be to use a place holder in the
empty position that indicates whether or not the index has been raised
or lowered. For example:
Now consider another very confusing result. Start with
So physicists now have two options. They can write the entries of in the understandable form . Or they can use the confusing, error-prone form . So what do you think they all do? If you guessed option (b), you are making real progress in your study of modern physics.
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 or .) Some
additional results that are useful in this context are