Published definitions of rapidity

[DrGreg]

Rapidity of a particle with speed v can be defined as either

c tanh^-1(v/c).....(1)​

or

tanh^-1(v/c).....(2)​

The difference is that (2) is a dimensionless hyperbolic angle whereas (1) has the same dimensions as speed (and is nearly equal to speed when small).

Equivalent definitions are

$$\frac{c}{2}\log_e \frac{1+v/c}{1-v/c}$$.....(3)
$$\frac{1}{2}\log_e \frac{1+v/c}{1-v/c}$$.....(4)​

I'd like to know which published authors use which of these definitions. I'm really not interested in authors using geometrised units where c = 1, because then (1)=(2) and (3)=(4).

The small number of books I possesses all use (2) or (4), but the Physics FAQ uses (1).

If you can, I'd like you to reply with a reference of the form

(4) Rindler, W. (2006), Relativity: Special, General, and Cosmological, Oxford University Press, Oxford, ISBN 978-0-19-856732-5, p.53.

 

[matheinste]

Hello DrGreg,

I have a quite large collection of books on Relativity. A quick random glance at the index of the first few reveals no entry for rapidity. The first use I came across was Eddington A. -The Mathematical Theory of Relativity - Cambridge University Press - London - 1923 (No ISBN) - P22 where form 2 is used. I will be quite happy, should you wish, to add to any references given by other respondents and to refer to those books/authors who do not have the term in their indexes when I have more time.

Matheinste.

 

[matheinste]

All of the following use version 2 but most of them in non-inverse trig form. Some do not appear in main text but in exercises. If these are of any use I can carry on as I am
about half way through.

D’Inverno R (1998) : Introducing Einstein’s Relativity, Oxford University Press, New York, ISBN 0-19-859688-3. Page 40

Ferrearo R. (2007) : Einstein’s Space-time, Springer, ISBN 978-0387-69946-2 Page 111.

Forshaw J. et al. (2009) : Dynamics and Relativity, Wiley. ISBN 978-0-470-01459-2
Page 251.

Johns O. (2005) : Analytic Mechanics for Relativity and Quantum Mechanics, Oxford University Press. ISBN 0-19-856726-X. Page 408.

Lasenby A. et al. (2006) : General Relativity, Cambridge University Press, New York. ISBN 958-0-521-82951-9 Pages 18 and 19.

Rindler W. (1966) : Special Relativity, Oliver and Boyd, Edinburgh. Pages 38 and 77.

As an afterthought, some of the better known authors who have books on Relativity which do not contain the term, at least not in the index, are Moeller, Schutz, Dirac, French, Hartle, Tolman, MacDonald, Kilmister, Ohanian, Wald, Weinberg and others.

Matheinste.

 

[George Jones]

I'd like to know which published authors use which of these definitions. I'm really not interested in authors using geometrised units where c = 1, because then (1)=(2) and (3)=(4).

I don't know if I've ever seen definition (1). I have a fairly large collection of relativity books. I'm out of town for the next nine days, so I can't check them, but matheinste has mentioned many of them.

the Physics FAQ uses (1).

Wow! Made by a young Terence Tao!

 

[DrGreg]

Thanks very much, matheinste, George and someone else who PMd me. All the evidence seems to suggest that almost everyone uses the dimensionless (2) or (4), and that the velocity-dimensioned (1) or (3) are rare. Which pretty much answers my question.

I don't really need any more references, but I'd still be interested if anyone finds any (1)s or (3)s (or equations that are mathematically equivalent, of course), but it looks like that will be unlikely.

 

[robphy]

For completeness... (and to give [often overlooked or forgotten] credit to AA Robb)...
I believe it was AA Robb (1911) who first used the term "rapidity"

A.A. Robb, Optical Geometry of Motion: A New View of the Theory of Relativity (1911), p. 8 [pdf]
(http://books.google.com/books?id=XGBDAAAAIAAJ&pg=PA9&lpg=PA9")

RAPIDITY OF A PARTICLE WITH RESPECT TO A SYSTEM.

We propose now to define what we shall call the rapidity
of a particle with respect to a system of permanent configuration.

If v be the absolute velocity of the particle with respect
to the system, then the inverse hyperbolic tangent of v will
be spoken of as the rapidity.

Thus if $\omega$ be the rapidity,
$$v = \tanh \omega.$$
As $\omega$ increases from 0 to $\infty$ , $v$ increases from 0 to 1.

For small values of $\omega$ we have, practically, velocity is
equal to rapidity, but we shall see later that, for large
values, it is the rapidity and not the velocity which follows
the additive law.

Presumably, in the last paragraph, Robb means (in his terminology) "absolute velocity" [i.e. the dimensionless velocity, $\beta=v/c$].

(In the quote, I boldfaced words which were originally italicized [since all quoted text at PF is italicized].)

 

[HallsofIvy]

Rapidity of a particle with speed v can be defined as either

c tanh^-1(v/c).....(1)​

or

tanh^-1(v/c).....(2)​

The difference is that (2) is a dimensionless hyperbolic angle whereas (1) has the same dimensions as speed (and is nearly equal to speed when small).

Equivalent definitions are

$$\frac{c}{2}\log_e \frac{1+v/c}{1-v/c}$$.....(3)
$$\frac{1}{2}\log_e \frac{1+v/c}{1-v/c}$$.....(4)​

I'd like to know which published authors use which of these definitions. I'm really not interested in authors using geometrised units where c = 1, because then (1)=(2) and (3)=(4).

I am puzzled that you are "not interested" in authors that use specific units. That's like saying "I am not interested in authors that use cm/s rather than m/s". What reason do you have for that?

The small number of books I possesses all use (2) or (4), but the Physics FAQ uses (1).

If you can, I'd like you to reply with a reference of the form

(4) Rindler, W. (2006), Relativity: Special, General, and Cosmological, Oxford University Press, Oxford, ISBN 978-0-19-856732-5, p.53.

 

[DrGreg]

I am puzzled that you are "not interested" in authors that use specific units. That's like saying "I am not interested in authors that use cm/s rather than m/s". What reason do you have for that?

Maybe I didn't make it clear that I want to find out if there is a consensus as to whether rapidity is dimensionless or has dimensions of velocity. Anyone using c=1 has already opted out of that debate; for such authors the question is meaningless.