Because of relativistic effects, quickly moving objects appear to
shift in color: quickly approaching objects seem bluer, and
quickly departing objects seem redder. Astronomers use this
phenomenon to determine the speed of distant objects. First, they
measure the frequency of the radiation that reaches them. Next,
they infer the frequency of radiation that was originally emitted
by the object. (This surprising feat is possible because each
chemical element has a recognizable light signature that's not
distorted by the doppler effect.) By comparing the emitted
frequency with the observed frequency, astronomers can calculate
the speed of distant objects.
You can use this nomogram to perform the same calculation:
Find the observed frequency of the object on the
lower horizontal axis.
Find the emitted frequency of the object on the
upper horizontal axis.
Draw the line passing through the two points. That line
will cross the vertical axis in exactly one place—
that's the velocity of the object relative to the observer.
Of course, you can also use this nomogram to solve
for any of these variables if you know the other two:
just mark the values of the two variables you know, draw the
line between them, and find where the line intersects the axis
of the unknown variable.
With the visible spectrum overlaid, you can also answer questions such
as “How fast would you have to travel in order for
red objects to appear green?” (Answer: you'd have to
travel towards the red object at about 0.35 times the speed of
light.)
Two different forms of the doppler nomogram
Here's an alternative way to depict the same
relation. This nomogram has the distinction of being both the
first nomogram I ever plotted by hand, and also the first one
which I transformed by hand. The equation for the relativistic
doppler effect is
$$f_o = f_s \sqrt{\frac{1-\beta}{1+\beta}},$$
where \(f_s\) is the frequency emitted by the object, \(f_o\) is
the frequency you observe, and \(\beta\) is your velocity
relative to the object. (Unlike with the nomogram, you must
remember a sign convention that \(\beta\) is positive when
you're moving towards the object and negative when
you're moving away.) Although the equation is distinctly
nonlinear, it is straightfoward to convert into nomographic
form. (You could even use a Z-shaped chart to do it, as I did in
the version shown at the beginning of the article.) First, we
square both sides and move all terms to one side, yielding:
$$\left(\frac{1-\beta}{1+\beta}\right)f_s^2 - f_o^2 = 0.$$
We can convert this into a matrix in standard nomographic form
(I'll explain how I derived it in a later post.)
$$\begin{bmatrix}
f_o^2 & 1 & 1\\
f_s^2 & -1 & 1\\
0 & \frac{1}{\beta} & 1\\
\end{bmatrix}$$
This produces a nomogram like the one shown here. Another valid
choice would have been $$\begin{bmatrix} f_o^2 & -1 & 1\\ 0 &
\beta & 1\\ -f_s^2 & 1 & 1\\ \end{bmatrix}$$
which you can derive most easily by writing the doppler equation
as \((1+\beta)f_o^2 = (1-\beta) f_s^2\) and expanding. In this
form, the nomogram looks instead like the one shown at the
beginning of this article. (It's a Z-shaped nomogram, which has
the general form \( f_1(u) = f_2(v)f_3(w) \)).
#2 A relativistic speedometer
As you approach the speed of light, time dilates and length
contracts along your direction of travel. The relation is
remarkably simple to depict using the unit circle.
This is because the effects of relativity are
trigonometric in nature: for each point on the unit circle, the
y-coordinate corresponds to a velocity, the x-coordinate
coordinate corresponds to the length contraction factor at that
velocity, and the length of the secant line (the line from the
origin to the point on the circle, stopping at x=1) is the time
dilation at that velocity.
Officially, this relation doesn't even require a nomogram, since
the variables are all monotonic functions of each other and could
all three be plotted on a single curve. But on the other hand,
this nomogram is “metrically accurate” — every
quantity is a geometric length in the image.
Here, the orange line indicates how you can use the nomogram: at half
the speed of light, time dilates by a factor of about 1.2, and
length contracts by about a factor of 0.87.
The relations depicted are
$$\ell^\prime/\ell= \sqrt{1-\beta^2}$$
$$t^\prime / t = \frac{1}{\sqrt{1-\beta^2}}$$
Hence we can define an angle \(\theta\) for which
$$\begin{eqnarray*}
\beta &=& \sin{\theta}\\
\ell^\prime/\ell &=& \cos{\theta}\\
t^\prime / t &=& \sec{\theta}.\\
\end{eqnarray*}$$
#3 The Lorentz boost
According to special relativity, observers who are moving at
different speeds will disagree about measurements of length and
duration. This nomogram enables you to measure those differences.
Here, the horizontal axis represents your velocity relative to
another observer, vertical displacement represents your choice
of a length or time measurement in your reference frame, the
blue curve represents the other measurement (i.e. time or
length, respectively) in your reference frame, and the grey
curve represents how it appears to another observer.
How it's made
This is
an oriented
nomogram with transparency for the one-dimensional Lorentz
boost
$$x^\prime = \frac{1}{\sqrt{1-\beta^2}} \cdot x - \frac{\beta}{\sqrt{1-\beta^2}} \cdot u $$
where \(x\) is your length measurement, and \(u\equiv ct\) is your
temporal measurement (converted to units of length). Since the
right-hand side of the equation has two terms of opposite signs,
the two fields overlap, producing a messy design. Instead,
rearrange terms to yield:
$$x = \sqrt{1-\beta^2}x^\prime + \beta u$$
