# The k-calculus

### Keywords

Bondi, Lorentz, k-calc, k-calculus, Minkowskian geometry, relativity,anti-relativity

This is a method of deriving the **Lorentz transform equations** was invented by **Hermann Bondi**. It uses space time diagrams like this

in which lines at 45° to the axes represent rays of light. The two lines A and B are the "world lines" of two observers. Observer A is stationary and his world line is vertical. Observer B is motion and her world line slopes to the right indicating that she is moving away from A. She looks back at A and observes two events which are separated by a time T as measured by observer A. However, the time between them as observed by B is greater by a factor which we call k. Hence the name k-calculus.

My diagram (fig 2) represents the space time diagram drawn from the point of view of observer B and shows A observing two events separated by a time T as measured by observer B.

The description of the k-calculus I am using is by Ray D'Inverno who says in his book that he spent his years as a research student at King's College London in the era of Bondi. Ray D'Inverno does not include the diagram of fig 2, but makes an addition to his diagram giving fig 3. He then simplifies this by moving the top pair of parallel lines down until they meet the other pair on B's world line and then moves the world lines of A and B together so that they now meet at the first event. This reduces the length of the lower of each pair of parallel lines to zero and they disappear giving the diagram in fig 4.

The mathematical fiddle lies in the geometry of fig 3. If we look at fig 2, we find that if B sees the time interval as T, then A will see the same time interval as kT. If B measured the time interval as kT then A would observe this to be k^{2}T. In his diagram (fig 4) D'Inverno inserts this result, but the geometry of his diagram no longer supports the result. The result which is written in fig 4 fits the geometry of similar triangles.

In fig 5, the triangles OAB and OBC are similar. That means that

Multiplying both sides by we get

This is precisely the relationship claimed in k-calculus, but the triangles in fig 4 are not similar.

The problem comes from the fact that it is impossible to draw a space time diagram which gives equal status to all observers. We might draw a diagram in which A and B have equal status, but it would always be possible to draw a vertical line and call it observer C. Whoever is represented by the vertical line is the privileged observer because their world line is draw at 45° to the lines representing light rays. Figs 2 and 4 both put A in the position of privilege contrary to the fundermental assertion that all observers have equal status. To give both observers equal status, we must assert that figs 1 and 2 are equally valid. but how do we then combine them? It is impossible to do so in a way which gives equal status and allows us to derive the required result.

When we apply the assumptions of special relativity to fig 5 we require that BAC = OBA=45° and that ABC=90°. This can only be so if OC is parallel to OB which makes a nonsense of the diagram.

The k-calculus is a fiddle.

## Note

In asking Ray D'Inverno's permission to use his book as the example of k-calculus, he was kind enough to point out that the arguments I have given are invalid. Chapter 2 of his book should be read through to the end and then reread in the light of the fact that the geometry of space and time is **Minkowskian**. Euclidean geometry should not be used in interpreting the diagrams because their geometry is Minkowskian.
The essence of my argument is that in mathematics, we move from things which have already been proven to prove new things. We cannot start with the proposition that the normal rules of geometry do not apply and then prove that they do not apply. We have an example in the geometry of navigation on the spherical surface of the earth. We know that we cannot take a curved surface and flatten it without distortion, so we work in three dimensions intersecting the curved surface of the earth with flat planes on which we can draw triangles. All the geometry which we do is Euclidean and it is only in the final result that we come up with triangles drawn on the curved surface of the earth whose angles do not add up to 180 degrees.

Special Relativity

Relativity and the nature of time

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© Copyright Bruce Harvey 1997.