Lecture 6
We discussed the algebra of Lorentz transformations.
Particularly, because ct and 
mix into each other under Lorentz
transformations of the reference frame, neither can be considered a
well-defined, independent attribute of an event. This is entirely
analogous to the fact that spatial coordinates mix into each under
under rotations -- so that x or y is not a robust characteristic of a
particle, but depends on the axes chosen.
We know how to handle this interdependence in the case of spatial
rotations: the position is a vector which changes under
rotation: goes to 
, where R is a rotation matrix. The
rotation leaves the norm 
invariant. Physical laws need not
involve only scalars (or invariants) under rotation. They can instead
be vector equations, like 
-- as long as both sides
transform the same way under rotations, the equation remains true no
matter what axes we rotate our system to.
We generalize this structure to the Lorentz transformations. Since ct
and x,y,z rotate into each other, we regard them as components of a
vector (ct, x, y,z) (really a column vector in our notation, but ascii
is limited). We call this column vector 
, where 
denotes
the time component, and 
ranges over the spatial
components. Under a Lorentz boost the vector is multiplied by a
Lorentz transformation matrix 
, whose form was discussed in
class. The specific form of insures that the norm 
-- the proper distance -- is preserved.
We then discussed how to express such a norm in terms of the vector
. We defined the metric tensor 
, as the diagonal
matrix with diagonal entries +1,-1,-1,-1. We then wrote the norm as
the sum 
, explicitly doing the
sum to see how it reproduced . We discussed
how this metric tensor was a natural extension of the metric tensors
introduced on flat space, which describe how small variations in given
coordinates correspond to small changes in physical position. The only
new subtlety is the signature: the fact that the time eigenvalue has
opposite sign from the spatial eigenvalues.
We then introduced some notation, the entire purpose of which is to
always keep proper track of all the minus signs while doing a minimum
of writing. These notations were: 1) the Einstein implicit summation
convention, where any index which is repeated, once as a subscript and
once as a superscript, is understood to be summed over all its allowed
values. 2) The raising and lowering conventions. We define the
operation of lowering an index as follows: 
is defined to be
(with of course an implicit sum on the 
index, so
that this corresponds to matrix multiplication of by the matrix
). This associates with , a vector, a different kind of
object, called a covector. Because this notation allows us to write
the norm as 
, we see that the product of a covector and
vector is invariant. Thus under Lorentz transformations, when the
vector is multiplied by the matrix , the covector must be
multiplied by 
-- so that the product remains unchanged.
We saw that the effect on components of lowering to , by
matrix multiplying by , is just to multiply all the spatial
components by a minus sign.
We discussed how the role played by vectors and covectors is analogous to that played by column vectors and row vectors in flat space. To get the norm of a column vector, we must multiply it by its transpose, a row vector. Every column vector determines a unique partner among the row vectors, and the two combine to give the vector's magnitude. Here we have the complication of the Minkowski signature. To get the magnitude of a four-vector, which we represent as a column vector, we must multiply it by its covector -- however that covector is not just its transpose, but the matrix g multiplied times the four-vector, *then* transposed. The effect of this complication is to introduce a sign on every spatial component.
We then discussed raising indices: going from a covector back to its
partner vector. To do this we must use the matrix 
, which for
flat Minkowki space with its (+1,-1,-1,-1) diagonal metric, is just
the same as g. We denote the matrix in general by 
;
then 
, and proved that taking a vector to its
covector and back reproduced the original vector.
Finally, we talked about the Lorentz transformation properties of
and introduced tensors with arbitrary numbers of upper and
lower indices.
-- KB