Polychromatic Derivation of the Lorentz Transformations

In my lectures in Oxford on the Philosophy of Physics during the 1960s and 1970s I used to present a version of the derivation of the Lorentz Transformations put forward by Milne and Whitrow, in which I used two colours to distinguish, more easily than with the normal dashes, between a Greek using green and a Russian using red. I incorporated it in my A Treatise on Time and Space, but the publisher reckoned that the distinction between red and green would be lost on those who were colour blind, and urged me to change green to Green. This proved difficult, as there was no word in Greek for `Green', and four words in Russian, all with different senses, and all irregular. I put forward another, monochromatic, version in Spacetime and Electromagnetism with the two communicators conversing in French and German. With the advent of the Internet it seems worth making the bones of my original presentation available in a form that can be downloaded onto transparencies. My computer cannot manage Russian or Greek yet, but the colours are enough to make the beginner aware of the different frames of reference in issue. In this text I have emphasized the universality of the speed of light, c2 = c2 by using black throughout, writing c2; numerals, brackets, + and - are likewise the same for both correspondents, but sometimes it seemed to make the argument more perspicuous to have them in the colour of the person carrying out the argument. There are three separate files, showing the letters, the derivation, and the full version of the Lorentz transformations, in larger type. It should be possible to make transparencies from these for an overhead projector. An outline presentation, based on the transparencies alone, followed a week later by a fuller discussion of the argument and objections to it, may prove effective in encouraging pupils to wrestle with Lorentz.

Whitrow and Milne1 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)

Date t1 Home (0, 0, 0)

Dear Green,
hope this finds you as it leaves me.
Yours ever,
Red

To which Green replies:

Date t2 Home (0, 0, 0)

Dear Red,
Thank you for your letter, dated `t1', which has just arrived.
Yours ever,
Green

Red then answers Green's Ietter:

Date t3 Home (0, 0, 0)

Dear Green,
Thank you for your letter, dated `t2', which has just arrived.
Yours ever,
Red

To which Green again replies:

Date t4 Home (0, 0, 0)

Dear Red,
Thank you for your letter, dated `t3', which has just arrived.
Yours ever,
Green

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)

t2 = 1/2 ( t1 + t3 ).

It is reasonable to adopt a comparable rule for assigning distances, as we do when using Radar. Red will assign a distance l2 to Green at t2, where

l2 = 1/2 c (t1 - t3) .

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

t3 = 1/2 (t2 + t4)
l3 = 1/2 c ( t2 - t4 ).

Consider what rules Red and Green must adopt if, being in communication with one another, each is to translate the other's references into his own system. They can iron out differences in their zeros very easily. For, since they are moving at a non-zero velocity with respect to each other, there was or will be a date when, if the velocity had been uniform throughout, they would have been or will be in the same place; and this date provides a natural and symmetrical starting point of the time reckoning of each. Let them both agree to measure time from then: and let us assume that this has already been done, so that
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
. t2 = f( t1 ) and t3 = g(t2 ), where f(w) and g(w) are strictly monotonic increasing functions of w, which correlate not only the particular dates t1 and t2 , and t2 and t3 , but any dates when a message is dispatched by one and received by the other. More carefully:
(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

t = f(t)


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

t = g(t)

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,

k-1(h(h(k(x)))u) = g(f(x)).

Red and Green agree to recalibrate their clocks, so that henceforward, according to their new scheme of time reckoning, it will be true that a message dispatched by Red at t by Red's new time reckoning will be received by Green at t by Green's new time reckoning, where t = h(t); and a message dispatched at t by Green's new time reckoning will be received by Red at t by Red's new time reckoning, where t = h(t): and h( ) is the same in either colour.

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:

I
Radar Rule

(i) t2 = 1/2 ( t1 + t3 )
(ii) l2 = 1/2 c (t1 - t3 ).

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

II
Radio Rule

(i) t = h(t)
(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 t2 = Q(t2), 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 t2 in terms of t1 and t3 , and the radio rule to express t1 and t3 in terms of t2 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 t2

By radar rule I (i) t2 = 1/2 ( t1 + t3)

By radio rule II (i) t1 = h-1( t2)

By radio rule II (i) t3 = h( t2)

Therefore

Q( t2) = t2 = 1/2 (h-1 (t2) + h(t2)). (1)

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 t1 and t3 with l3, 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 l2 again in terms of t2, namely l2 = vt2. 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),

t2 = l2/v = 1/2 c (t3 - t1)/v

and hence, as before,

Q( t2) = 1/2 c (h(t2) - h-1 (t2))/v

therefore

vQ( t2)/c = 1/2 (h(t2) - h-1 (t2)) (2)

We can now solve for h( ) and Q( ). It is convenient to solve for h( ) first.
Adding (1) and (2)

Q( t2) + vQ( t2)/c = h(t2)
(1 + v/c) Q( t2) = h(t2) (3)

Subtracting (2) from (1)

Q( t2) - vQ( t2)/c = h-1 (t2)

(1 - v/c) Q( t2) = h-1 (t2) (4)

Dividing (3) by (4)

(1 + v/c) / (1 - v/c) = h(t2) / h-1 (t2) (5)

This has to hold for any and every value of t2. It would therefore have to hold for w, where w = h(t2).
Substituting w for t2, we have

(1 + v/c) / (1 - v/c) = h(w)/ h-1(w) = h(h(t2)) / h-1 (h(t2)) = h(h(t2)) / t2 (6)
h(h(t2)) = t2 ((1 + v/c) / (1 - v/c)) (7)

It is easy to see that, taking square roots,

h(t2) = t2 \/ (1 + v/c) / (1 - v/c)

is a solution; and, indeed, it can be shown to be the only one.
From it we can obtain h-1 (t2), and so Q(t2).

h-1 (t2) = t2 \/( (1 - v/c) / (1 + v/c))

Thus, by (1)

2 Q( t2 ) / t2 = 1 / \/(1-v2 /c2).

This, then, is the way in which Red must correlate the date t2 which he ascribes by the radar rule to Green's receipt of his message, with the date t2 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 t2 and t2 in this way if he is to have both the radar and the radio rule without inconsistency.

Q(t2) is a special case of

L(iv) t = (1 - v2/ c2)-1/2 (t - vx/ c2 ).

We have x = 0, because Green sends all his messages from Home, (0,0,0). We have therefore worked out the transformation for the origin of Green's frame of reference (or coordinate system), not for a general point (x,y,z,t), but it is easy to generalise.

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; c2 = c2 = c2.
(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.

The Lorentz Transformations

(for larger version, click here)

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 v2 = v2, both are written in black, like c and c2; but some of the other brackets and symbols are in local colours.

Red's Rules

(i) Red into Green (x,y,z,t) = L(x,y,z,t)
Russian-Greek dictionary published in Russia

L(i) x = (1 - v2/ c2) -1/2 (x + vt)
L(ii) y = y
L(iii) z = z
L(iv) t = (1 - v2/ c2) -1/2 (t + vx/ c2)
v = -v

(ii) Green into Red (x,y,z,t) = N(x,y,z,t)
Greek-Russian dictionary published in Russia

N(i) x = (1 - v2/ c2) -1/2 (x - vt)
N(ii) y = y
N(iii) z = z
N(iv) t = (1 - v2/ c2) -1/2 (t - vx/ c2)
v = -v
LN = I

Green's Rules

(i) Red into Green (x,y,z,t) = M (x,y,z,t)
Russian-Greek lexicon published in Greece

M(i) x = (1 - v2/ c2) -1/2 (x - vt)
M(ii) y = y
M(iii) z = z
M(iv) t = (1 - v2/ c2) -1/2 (t - vx/ c2)
v = -v

(ii) Green into Red (x,y,z,t) = K(x,y,z,t)
Greek-Russian lexicon published in Greece

K(i) x = (1 - v2/ c2) -1/2 (x + vt)
K(ii) y = y
K(iii) z = z
K(iv) t = (1 - v2/ c2) -1/2 (t + vx/ c2)
v = -v
MK = I

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)

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1. G.J.Whitrow, The Natural Philosophy of Time, Edinburgh, 1961, ch.3, section 8, pp.171-173; 2nd ed. Oxford, 1980, ch.5, esp. Sections 5.2-5.4, pp.230--253; and E.A.Milne, Modern Cosmology and the Christian Idea of God, Oxford, 1952, chs.3 and 4; and Kinematic Relativity, Oxford, 1948, ch.2, esp. Section 24. My exposition is heavily indebted to C.W.Kilmister, Special Theory of Relativity, Oxford, 1970, ch.2, pp.14--19, and to G.Stephenson and C.W.Kilmister, Special Relativity for Physicists, London, 1958, ch.1, Section 7, pp.16--19 and C.W.Kilmister, The Environment in Modern Physics, London, 1965, ch.4, pp.46--53. Click here to return to home page

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