Suppose we have a coordinate frame in dimensions, where will typically be 4 for relativistic spacetime (with the 0th coordinate equal to as usual) or 3 for just the spatial part. To simplify our notation, we will use roman characters such as for the three-vector spatial-only part of a four-vector, and use greek characters such as for the entire four-vector (where recall, repeated indices imply summation over e.g. or , hence the distinction as it can be used to de-facto restrict the summation range).

Now suppose that we wish to transform to a new coordinate frame .
At this time we place very few restrictions on this transformation. The
transformation might, therefore, translate, rotate, rescale or otherwise
alter the original coordinate description. As we do this, our *description* of physical quantities expressed in the old coordinates
must systematically change to a description in the new coordinates,
since the actual physical situation being described is not altered by
the change in coordinate frames. All that is altered is our point of
view.

Our first observation might be that it may not be *possible* to
describe our physical quantities in the new frame if the transformation
were completely general. For example, if the dimension of were
different (either larger or smaller than that of ) we might well be
unable to represent some of the physics that involved the missing
coordinate or have a certain degree of arbitrariness associated with a
new coordinate added on. A second possible problem involves regions of
the two coordinate frames that cannot be made to correspond - if there
is a patch of the frame that simply does not map into a
corresponding patch of the frame we cannot expect to correctly
describe any physics that depends on coordinates inside the patch in the
new frame.

These are *not irrelevant mathematical issues* to the physicist. A
perpetual *open question* in physics is whether or not any parts of
it involve *additional variables*. Those variables might just be
``parameters'' that can take on some range of values, or they might be
supported only within spacetime scales that are too small to be directly
observed (leaving us to infer what happens in these microscale
``patches'' from observations made on the macroscale), they may be
macroscopic domains over which frame transformations are singular (think
``black holes'') or they may be actual *extra dimensions* - hidden
variables, if you like - in which interactions and structure can occur
that is only visible to us in our four dimensional spacetime in *projection*. With no *a priori* reason to include or exclude any of
these possibilities, the wise scientist must be prepared to believe or
disbelieve them all and to include them in the ``mix'' of possible
explanations for otherwise difficult to understand phenomena.

However, our purposes here are more humble. We only want to be able to
describe the relatively mundane coordinate transformations that do *not* involve singularities, unmatched patches, or additional or missing
coordinate dimensions. We will therefore require that our coordinate
transformations be *one-to-one* - each point in the spacetime frame
corresponds to one and only one point in the spacetime frame -
and *onto* - no missing or extra patches in the frame. This
suffices to make the transformations *invertible*. There will be
two very general classes of transformation that satisfy these
requirements to consider. In one of them, the new coordinates can be
reached by means of a *parametric transformation* of the original
ones where the parameters can be *continuously varied* from a set of
0 values that describe ``no transformation''. In the other, this is not
the case.

For the *moment*, let's stick to the first kind, and start our
discussion by looking at our friends the coordinates themselves. By
definition, the untransformed coordinates of an inertial reference frame
are *contravariant* vectors. We symbolize contravariant components
(not just 4-vectors - this discussion applies to tensor quantities on
all manifolds on the patch of coordinates that is locally flat around a
point) with *superscript* indices:

(5.12) |

Now let us define a mapping between a point (event) in the frame
and the *same* point described in the frame. in
consists of a set of four scalar numbers, its frame coordinates, and we
need to transform these four numbers into four new numbers in .
From the discussion above, we want this mapping to be a *continuous
function* in both directions. That is:

(5.13) | |||

(5.14) | |||

(5.15) | |||

(5.16) |

and

(5.17) | |||

(5.18) | |||

(5.19) | |||

(5.20) |

have to both exist and be well behaved (continuously differentiable and so on). In the most general case, the coordinates have to be

Given this formal transformation, we can write the following relation
using the chain rule and definition of derivative:

(5.21) | |||

(5.22) | |||

(5.23) | |||

where again, superscripts stand for indices and not powers in this context. We can write this in a tensor-matrix form:

The determinant of the matrix above is called the *Jacobean* of the
transformation and must not be zero (so the transformation is
invertible. This matrix defines the differential transformation between
the coordinates in the and frame, given the invertible maps
defined above. All first rank tensors that transform like the
coordinates, that is to say according to this transformation matrix
linking the two coordinate systems, are said to be *contravariant
vectors* where obviously the coordinate vectors themselves are
contravariant by this construction.

We can *significantly compress* this expression using Einsteinian
summation:

(5.24) |

(5.25) |