Whitrow and Milne^{1}
developed a thought-provoking
derivation of the Lorentz transformation from Einstein's
"Radar Rule", granted certain other assumptions,
e.g. that
electromagnetic radiation not only is reflected back, so
as to locate distant events in a given frame of
reference, but can be received and understood, so as to
be a means of communication between different observers
representing different inertial frames of reference,
whereby their different ways of referring to dates,
places and events, can be harmonized.

Suppose Red and Green exchange a series of messages thus:

(for larger version of the letters, click here)

_{1}',
which has just arrived.

_{2}',
which has just arrived.

To which Green again replies:

_{3}', which has just arrived.

From this exchange of messages, Red knows when he sent his first message, when he received one back, and also knows the date on which Green says he, Green, received the message according to his, Green's, reckoning, Similarly Green knows when he sent his first answer, when he received an answer acknowledging it and also knows the date on which Red says he, Red, received the message according to his, Red's, reckoning.

But neither Red nor Green are prepared to accept the other's *ipse dixit* as conclusive evidence for dates. But I am prepared to accept a colleague's word for it that he is truthfully reporting the date according to his own country's dating system, which differs from country to country.
Nevertheless, I cannot accept another man's dating system as my own. I live in Britain and go by British time, and I want to know when, by British time, the man received and answered my letter. Now what can I say? I know that he cannot have received my letter, by British time, before I sent it; nor could he have answered it after I had received his answer. If I sent the letter in January and received an answer in June, it must have reached him some time between January and June inclusive. Any date between that of dispatch and that of receipt of the answer is possible. Einstein took the mid date.
He said it was a convention, but it is not just that,
for there are *arguments*, mostly of symmetry and parity of esteem for assuming that the date of the arrival of a message to be exactly halfway between the date of its dispatch and the date of the receipt of its acknowledgement:

(for outline version of the derivation, click here)

Similarly Green will adopt corresponding Radar Rules for assigning dates and distances to Red:

0= 0.

We cannot assume that the unit will be so easily synchronized as the zero. All we know is that messages dispatched later will arrive later, so that Green's time reckoning for receipt of messages must be a strictly monotonic increasing function of Red's time reckoning for the dispatch of messages, and *vice versa; i.e*.
t

(i) If t is the date in Red's time reckoning when he dispatches a message to Green, and t is the date in Green's time reckoning when he receives the message, then

and (ii) if t is the date in Green's time reckoning when he dispatches a message to Red, and t is the date in Red's time reckoning when he receives the message, then

Given a sufficient number of messages Red and Blue can both discover what the functions f(w) and g(w) are, more or less. Each of them will be correlating the numbers given as dates by his own time reckoning with the numbers reported in the messages being the date of dispatch according to the time
reckoning of the other. In general f(w) and g(w) will not be the same function. It would obviously be more symmetrical if they were. And this is therefore what we set about achieving by altering the measures---recalibrating---either or both of the time scales of Red and Green.
We seek as it were a "functional square root", a function h( ) such that for any variable w, h(h(w)) = g(f(w)).
We cannot do this directly, but we can find functions h( ) and k( ), such that for any variable x,

By thus recalibrating their clocks, Red and Green have gone as far as possible in achieving perfect symmetry between the one and the other, and eschewing egocentricity on the part of either. They have revised their systems of time reckoning so that now, even on the dates supplied by Green, Red will not be able to say My messages travel faster than yours do, Green. Nor Green to say My messages travel faster than yours do, Red.

We now have developed two rules for ascribing dates at a distance: the Radar Rule, which applies to objects as well as persons; and what we may now call the Radio Rule, which applies only to other observers, who can communicate their own time reckoning in answering messages, as well as merely reflecting them. We thus have, essentially, two rules:

Radar Rule

the Radar Rule, giving the date and distance, assigned by Red, to a distant event at Green's home.

Radio Rule

(ii) t = h(t)

The radio rule, depending on a functional square root, correlating the date of dispatch, according to the time reckoning of the dispatcher, with the date of receipt, according to the time reckoning of the receiver.

How is Red to correlate t,
and t?
He needs to find a function Q( ),
such that
t_{2} =
Q(t_{2}),
not only for the particular cases first considered,
but on all occasions---i.e. for all dates
t and
t
that are to be correlated by Red's transformation rule for translating red into green. What we do, essentially, is to use the radar rule to express
t_{2}
in terms of
t_{1} and
t_{3} ,
and the radio rule to express
t_{1}
and
t_{3}
in terms of
t_{2}
We then can work out what Q( )
must be if the radar and the radio rules are both to hold good for any and every
t_{2}

By radar rule I (i)
t_{2}
= 1/2 (
t_{1}
+
t_{3})

By radio rule II (i)
t_{1} =
h^{-1}(
t_{2})

By radio rule II (i)
t_{3}
= h(
t_{2})

Therefore

Since we do not know in general what
h( ) might be, this is not enough for us to be able to say what Q( ) is.
We therefore need to use the second, main, radar rule, which correlates
t_{1}
and
t_{3}
with
l_{3},
the distance.
This is where we use the fact that Red is allowed to assume that Green is moving with *uniform* velocity with respect to him. Together with the convention that Red and Green have each agreed to set their zero dates at the time of their coincidence, it enables us to express
l_{2}
again in terms of
t_{2},
namely
l_{2} =
vt_{2}.
We should note that it is not necessary that Red and Green should have coincided in time past, and actually have been in uniform motion ever since. All that is necessary is that they should be in uniform motion while they recalibrate their clocks and that each should set his zero to the date when, according to his calculation of the other's velocity, the other would have been, or (if the motion is towards each other) would be, coincident. This is enough. And of course we could derive our result, at the cost of somewhat heavy algebra, without making this simplifying assumption about the zero.

With this assumption we have, using radar rule I (ii),

Adding (1) and (2)

This, then, is the way in which Red must correlate the date
t_{2}
which he ascribes by the radar rule to Green's receipt of his message, with the date
t_{2}
which he agrees, in virtue of the radio rule, Green should assign to the same
event of Green's receipt of Red's message. Red must correlate
t_{2}
and
t_{2}
in this way if he is to have both the radar and the radio rule without inconsistency.

_{2})
is a special case of

The key to the argument is that each person is *both* an *observer and *an *object*.
He both is the centre of his own world, and occupies some position in the other's world. We have not one but two frames of reference, and every point of the one, including the origin, is a point of the other, and *vice versa*.
Red can not only refer to Green as an object, but tell Green what he, Red, is saying about him, Green, and be told by Green, as an observer, how he, Green, would describe the same events. Red then has to learn how to translate Green's own account of himself, Green, as an observer, into his, Red's, account of Green as an object; and *vice versa*.
The radar rule enables Red to talk about Green: the radio rule enables him to talk with Green about the same things. And the Lorentz transformations are generated from our making these two rules compatible.
Assumptions

(i) Non-egocentricity:
My messages travel neither faster nor slower than anybody else's; c^{2} =
c^{2} =
c^{2}.

(ii) Egocentricity:
My frame of reference is at rest

(iii) Isotropy of space: Speed of communication not different in different directions

(iv) No bouncing: Speed of communication independent of source.

We are also assuming that time is one-dimensional, Archimedean and non-cyclic.

There are four versions, each consisting of four equations for the 3+1 dimensions of Minkowski spacetime X,Y,Z,T.
There are four versions, because there are two directions of translation and two possible frames of reference. We have
a Russian-Greek dictionary published in Russia,
a Greek-Russian dictionary published in Russia,
a Greek-Russian lexicon published in Greece, and
a Russian-Greek lexicon published in Greece.
Since v^{2} =
v^{2}, both are written in black, like c and c^{2}; but some of the other brackets and symbols are in local colours.

There are criticisms and discussions of this argument in
R.B. Angel, *Relativity: The Theory and its Philosophy*,
Pergamon, Oxford, 1980, ch.4, pp.110ff.,
J.R. Lucas, *A Treatise on Time and Space*, London, 1973,
Section 45, pp.225-227, and
J.R. Lucas and P.E. Hodgson, *Spacetime and Electromagnetism*, Oxford, 1990,
Section 4.5 pp.143-147.
(for larger version of the letters, click here)

(for outline version of the derivation, click here)

(for larger version of the Lorentz Transformations, click here)

Click here to return to home page

1. G.J.Whitrow,

Click here to return to bibliography

Click here to go to PSIgate, which is a free online catalogue of high quality Internet resources in the physical sciences. Resources are selected, catalogued and indexed by researchers and other specialists in their respective fields.