## Lorentz Transforms

This is taken from a Mathcad file, saved in RTF, loaded into Word97 which generated the original html file. It has since been edited with HTMLEdit on an Acorn RiscPC. We apologise for the PC quality of the maths.

There are two different ways of representing the time dimension. We can also choose units so that the speed of light is 1 leading to a simplification of the terms Here I have used c = 1 and real time.

[L] = where =

giving = =

(Mathcad responds to the use of the hyphen key by placing brackets around the expression, so x' cannot be used)

For v = the matrix transform = =

The transform for v is

=

=

Those with Mathcad 6.0 can download the Mathcad file and verify that

=

So we are confident that this is indeed a Lorentz transform.

If we now consider three observers in inertial reference frames S_0, S_1 and S_2 such that at some moment, their axes are coincident and that they zero their clocks at that moment. If the velocity of S_1 in S_0 is u along the x axis and the velocity of S_2 in S_1 is the vector v defined above, we can apply first a Lorentz from S_0 to S_1 and then a Transform from S_1 TO S_2.

=

=

=

It can be seen that the second column remains unchanged by the post multiplication. If the Lorentz group exists then we should be able to equate this to a single tranform:

=

But this requires us to equate the terms of the second columns

=
=
=

We note that these equations are independent of u and conclude that no solution is possible which will be valid for all u less than c. Therefore, we are unable to show that there is a Lorentz transform from S_0 to S_2.

However, the condition that = is satisfied.

This can be verified by doing the sums on a separate Mathcad sheet. (It has a habit of filling its memory and crashing if too much is done in one sheet)

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© Copyright Bruce Harvey 1997.