CLASSROOM
Classroom
In this section of Resonance, we invite readers to pose questions likely to be raised in a
classroom situation. We may suggest strategies for dealing with them, or invite responses,
or both. “Classroom” is equally a forum for raising broader issues and sharing personal
experiences and viewpoints on matters related to teaching and learning science.
Giorgio Goldoni
Civic Planetarium
“Francesco Martino”
Modena, Italy
Rapidity: The Physical Meaning of the Hyperbolic Angle in
Special Relativity
In Special Relativity the Galilean law of addition of velocities is
replaced by a law of composition formally identical to the
addition formula of the hyperbolic tangent. So a simple additive
law can be obtained by inverting the hyperbolic tangent and
passing to the hyperbolic angle, called rapidity. This article
intends to demonstrate that the use of the hyperbolic angle is not
a mere mathematical trick but that rapidity admits a simple
physical meaning.
Slope and Angle
Let us imagine a strange civilization where mathematicians know
the concept of slope but ignore that of angle. Those mathematicians
have discovered that, given three straight lines r, s and s', if s has
slope k with respect to r and s' has slope k' with respect to s then the
slope K of s' with respect to r is given by the formula
K
Keywords
Hyperbolic tangent, hyperbolic
angle, rapidity, inertial observer,
beta-velocity.
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k  k
1  kk 
(1)
For example, if k = 1/2 and k' =1 then K = 3 (Figure 1).
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One day, a mathematician named Angle, while studying a complicated
power series, obtains a new function, that he calls ‘tangent’ and
writes ‘tan’. Angle easily shows that tangent is invertible in the
interval ]–/2, /2[ . He then discovers an incredible connection
between the tangent and the slope. Namely, if you pass from the
slope k to the new quantity  = tan–1 k you can replace the
complicated formula (1) with a simple additive one. In other words,
if = tan–1 k, '= tan–1 k' and  = tan–1 K, then
=  +'.
(2)
It goes without saying that the new quantity was named after Mr
Angle! His problem now is searching for the geometrical meaning
of the angle. His starting point is the fact that for small values k of
Figure 1. Slopes do not add.
A slope 1/2 followed by a
slope 1 gives a slope 3 and
not 3/2 !
the slope, the approximate equality  = tan–1 k  k holds.
Angle’s main idea is to express an angle as the sum of small angles,
where a small angle is one for which   k. Wanting to fix an
appropriate small unit and being his 45th birthday, he considers
1/45 of the angle corresponding to the slope k = 1, that is
 = (tan–1 1)/45 = 0.78540/45 = 0.01745. He is very happy to see
that tan 0.01745 = 0.01745 and calls the unit for angles with the
name of degree.
So, by very definition, to slope 1 there corresponds an angle of 45
degrees. Moreover, while composing two slopes 1, the formula (1)
misses sense because one obtains perpendicular straight lines; Angle
discovers that the angle between perpendicular lines exists and is 90
degrees!
But his real understanding of the geometrical meaning of angle
clearly emerges only when he has the idea to build a simple analogue
computer for measuring it! For this purpose Angle draws on a film
a counter-clockwise oriented pencil of straight lines each with slope
0.01745 with respect to the previous one. To measure directly the
angle corresponding to the slope between an ordered pair of lines
drawn on a paper sheet, he simply overlaps the film on to the paper
sheet with the centre of the pencil on the intersection point of the two
lines and with one line of the pencil on the first of the two lines. Now
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CLASSROOM
he must merely count the lines of the pencil from the first to the
second line on the sheet and take a positive or negative sign
depending on the orientation and obtaining a result between –90
degrees and 90 degrees. Angle then improves his device using a
semicircle with only a notch corresponding to each line of the pencil.
So Angle, on his 45th birthday, discovered the protractor!
Obviously, identifying two lines forming an angle  = 0.01745
with two lines with relative slope k= 0.01745 leads to a little error.
For example, with this kind of protractor, to an angle = 45 degrees
there corresponds a slope k= 0.99984, but Angle can decrease the
error at will by passing to an opportune submultiple of degree.
Velocity and Rapidity
In special relativity one learns that, given two inertial observers O
and O', if O' moves at speed vR with respect to O in a given direction
then a particle A moving in the same direction with speed v' with
respect to O' has in the frame of O a speed given by the formula
v
vR  v ,
v v
1  R2
c
(3)
where c is the speed of light.
Passing to the beta velocities, where = v/c and the speed of light is
1, formula (3) takes the simpler form

R    ,
1  R 
(4)
formally identical to the addition formula for the hyperbolic tangent.
This suggests that we introduce the new quantity = tanh–1, called
rapidity, for which a simple additive law holds
 = R +'.
(5)
As we can easily see from the graph of the hyperbolic tangent, which
is an increasing function with horizontal asymptotes at y = ± 1 and
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Figure 2. From the graph of
the hyperbolic tangent it is
easily seen that, for small
values, velocity is nearly
equal to rapidity and that
to the speed of light there
corresponds an infinite rapidity.
slope 1 at the origin, rapidity is nearly equal to velocity for small
velocities and is infinite for the speed of light (Figure 2).
At this point the analogy with the previous problem is obvious:
physicists, who knew velocity before rapidity, are in the same
position as those mathematicians who knew slope before angle!
Is it possible to find a simple physical meaning for rapidity or does
it only consist in a mathematical trick? Once again, the key argument
is that for a small velocity  and the corresponding rapidity  the
approximate equality  = tanh–1  holds.
This time let us take as a unit for rapidity the 45th part of hyperbolic
angle corresponding to a velocity =1/2, that is = (tanh–1 ½ )/45
= 0.54931/45 = 0.01221 that we call ‘rap’. So 1 rap = 0.01221 =
3662 km/h. It can be considered indistinguishable from velocity
= 0.01221 because tanh 0.01221 = 0.01221.
But what is the equivalent of the protractor in this case? Let us
imagine that there are n inertial reference frames, each moving with
the same velocity  with respect to the previous one and in the
same direction. To find the rapidity of a particle moving with
velocity  with respect to the first reference frame, again in the same
direction, one must simply count the number n of frames necessary
to reach the one which is moving together with the particle: the
rapidity  corresponding to the velocity  of the particle is then n
rap!
Here again we introduce an error because of the approximate
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Figure 3. In this way it is
impossible for the ‘n’th
little man to reach the
speed of light; that is, to
the speed of light there
corresponds an infinite
rapidity.
equality tanh–1  , but we can reduce it at will by choosing
an adequately small submultiple of rap.
We can imagine (Figure 3) a little man walking on the floor while
pulling a very long cart. On the cart there is a second little man
pulling a second long cart and so on. Every little man walks with
Box 1.
The measure of angle t (in radians) may be alternatively defined as twice the area of the corresponding sector
in the unit circle x2 + y2 = 1, as shown in Figure A. This definition has the advantage that it can be immediately
transferred to the unit hyperbola x2 – y2 = 1 for giving the notion of hyperbolic angle of measure t (Figure B).
Now we may define the hyperbolic tangent of the hyperbolic angle t in the same way as in the circular case. The
main difference is that the hyperbolic tangent is a bounded and non-periodic function.
A
90
B
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velocity  with respect to the previous cart, except the first one
who walks with velocity  with respect to the floor. Let us now
consider a rocket moving with velocity  with respect to the floor
and in the same direction of the little men. If the nth little man moves
together with the rocket then the rapidity of the rocket is n times the
velocity .
Conclusion
In Galilean relativity one could have chosen to define velocity in a
different but equivalent manner. For example, one could define the
unit of speed by means of a ball falling from a height of 1 centimetre
on an inclined plane. By very definition, a particle moves at speed
two if it moves together with a ball falling from an inclined plane like
the first that in turn moves at unitary speed, and so on. But following
Einstein’s special relativity, this way of measuring velocity operates
correctly only for small velocities and, moreover, the speed of light
is infinite! We are not measuring velocity, but rapidity!
Suggested Reading
[1] J McMahon, Hyperbolic Functions, John Wiley and Sons, New York,
1906.
[2] V G Shervatov, Hyperbolic Functions, DC Heath and Company, Boston,
1963 (Translated and adapted from the second Russian edition, 1958).
[3] E F Taylor and J A Wheeler, Spacetime Physics, WH Freeman and Co.,
San Francisco, 1963.
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