Alpha, beta, gamma, delta: Bondis approach

to special relativity

Rajaram Nityananda

December 9, 2010

These notes summarise a very elegant way of introducting the Lorentz trans-formation, due to Hermann Bondi. It starts with the notion that the best wayof assigning space and time coordinates to a distant event is by radar, i.e bytransmitting light signals. It also makes good use of the Doppler effect (hencethe delta in the title). The starting point is the familiar set up of two observerswho choose, for convenience, their origin of space and time to be the event oftheir passing each other, along the one dimension to which they are confined.The event which they are both studying is at a general place on the same line,at a general time, as shown by the standard space-time diagram.

Notice that both observers, traditionally unprimed and primed, (S and S)share the same signals, both outgoing and return. One could imagine that bothof them note the times at which the outgoing (suffix one) and return (suffix two)signals pass them. Then the unprimed co-ordinates of the event are assignedaccording to the basic idea of radar. The transit times for the outgoing andreturn journeys are the same. So the time of the event is the average of thetimes of transmission (t1) and reception (t2). Also, the distance travelled is justthe speed of light, times the travel time which is half the difference of t1 and t2.

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Thus we havex = c(t2 t1)/2, t = (t1 + t2)/2

Now the principle of relativity insists that we must also accept exactly the sameequation between primed quantities

x = c(t2 t

1)/2, t = (t

1+ t

2)/2

These two equations can be turned around to express the two quantitiest1 and t2 in terms of the space time c co-ordinates pf the event we are considering.The relation is

t1 = t x/c, t2 = t + x/c, t

1= t x/c, t

2= t + x/c

Thus t1 and t2 can be interpreted as retarded and advanced times associatedwith the given event and the observer S. The retarded time is familiar in electro-dynamics when S is source of radiation which is whose influence on the event Ehas to be calculated., Since Bondi wrote, the retarded co-ordinate has becomeeven more familiar because of the use of the GPS (global positioning system).Clearly, if your GPS receiver receives a clock signal from a satellite, the readingis precisely retarded time.

The next important remark is that the Doppler effect enables us to relatet1 tot1and t2 to t

2For this purpose, we have to imagine one light signal passing from

S to S at the time that they were coincident. The next one is emitted at t1 andreceived at t

1. The Doppler factor is defined as the ratio of the time intervals

between successive light pulses at reception to that at emission. (Notice thatthe corresponding factor for frequencies as measured by the two observers, themore common way of looking at the Doppler effect, would be 1/. And surely,the same Doppler factor applies when we try to relate t2 a reception event to t

2,

an emission event. So we now get

t1

= t1, t

2= t2/

There is a further assumption of symmetry here, that the Doppler factor goingfrom S to S is the same as that going from S to S. Using the earlier expressionsfor the retarded and advanced times in terms of the co-ordinates of the event,we get the Lorentz transformation in a rather nice form.

t x/c = (t x/c), t + x/c = 1(t + x/c)

Solving for x and t in terms of x and t, we get something looking more like theLorentz transformation, especially if we put

= exp()

We then getx = x cosh() ct sinh()

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ct = x sinh() x cosh()

But where is the relative velocity v of the two frames? Simple, just put x equalto zero, and the ratio of x to t gives

v/c = tanh()

We have followed tradition and denoted the ratio of the speed to that of lightby . The Lorentz factor giving time dilation and length contraction is seen tobe given by cosh(). Since we are expressing everything in terms of , we mightas well have a name for it and fortunately there is one, rapidity. It has thenice property that it adds under Lorentz transformations in the same direction.This could be checked in the usual way by introducing a third frame S andusing identities between hyperbolic functions. But because of s connection tothe Doppler factor we can see the result directly. From Bondis treatment, it isobvious that the Doppler factor is multiplicative, i.e the going from S to S isthe product of the s going from S to S and from S to S. Clearly, the logarithmof , which we called , is going to be additive on these physical grounds.

The relationship between the transverse co-ordinates, y = y, z = z followsin the same way as in the usual textbook treatment - the factor f relating thetransverse co-ordinates could only depend on the relative velocity (and not evenon its sign), so we get f 2 when we go from S to S and come back. so f endsup being one.

With the three dimensional Lorentz transformation in hand, one can do theusual things like Doppler effect and aberration even for rays which make an anglewith the relative motion of the two observers. As a parting shot, let us writethe standard aberration formula in an additive form! To do this, we need tointroduce log(tan(/2) and denote it by (what else?) . The aberration formulanow reads (proof an exercise)

= +

Who would think of adding the rapidity to the logartithm of the tangent of halfthe angle that a ray makes with the x-axis in one frame? But thats the way itworks!

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