#1 The relativistic doppler effect

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:

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: $$x = \sqrt{1-\beta^2}x^\prime + \beta u$$