Foundation lesson

Matrices and linear maps

A linear map makes each output a sum of the inputs, each multiplied by a fixed number, and a matrix is the table of those multipliers. The change from one frame to another at 0.6c is such a map. Writing it as a matrix is a later aid: the 1905 paper writes it as two equations.

Written for this edition, not translated from Einstein. Editorial review pending.

How can a table of four numbers turn one frame's coordinates into another's?

Some rules turn inputs into outputs by multiplying and adding, and nothing else: $x' = p\,x + q\,t$ and $t' = r\,x + s\,t$ , with the same four numbers p, q, r and s for every input. Such a rule is a linear map. Set the four numbers in a table, one row per output, and the table is its matrix. Each row says how much of each input goes into that output.

\begin{aligned} x' &= p\,x + q\,t \\ t' &= r\,x + s\,t \end{aligned}

x prime equals p x plus q t, and t prime equals r x plus s t.

The change from a frame at rest to one moving at v along x is a linear map: $x' = \gamma(x - vt)$ and $t' = \gamma(t - vx/c^2)$ , with $\gamma = 1/\sqrt{1 - v^2/c^2}$ . The relativity paper prints this factor as β; here it is written γ, the modern letter, and the ratio v/c is written out. At v = 0.6c, γ = 1.25 and γv = 0.75c, so the table of multipliers is:

\left(\begin{aligned} 1.25 &\quad -0.75c \\ -0.75/c &\quad 1.25 \end{aligned}\right)

A two by two table: first row 1.25 and minus 0.75 c; second row minus 0.75 over c and 1.25.

Two numbers read off the table describe what the map does. The determinant, ps − qr, is the factor by which it scales areas in the plane of x and t; for this map it is $1.25^2 - 0.75^2 = 1$ , so areas are kept. The eigenvectors are the directions the map only stretches. For this map they are the two light lines, $x = ct$ and $x = -ct$ , stretched by 0.5 and by 2: a flash sent either way still moves at c in the new frame.

The older Galilean rule, $x' = x - vt$ and $t' = t$ , is also linear, with the table $p = 1$ , $q = -v$ , $r = 0$ , $s = 1$ , and its determinant is also 1. What separates the two is the light lines. The Galilean map sends $x = ct$ to $x' = (c - v)t$ : the flash now moves at c − v, not c.

Writing the transformation as a matrix, with determinants and eigenvectors, is a later aid, tied to the geometry Minkowski gave the theory in 1908. §3 of the paper writes the transformation as equations and derives them from light signals and clocks; the table adds a way of reading them, not a premise.

Worked example: Two simultaneous events, seen from 0.6c

Two events happen at the same time, 10 light-seconds apart along x: $\Delta x = 10$ light-seconds and $\Delta t = 0$ .
The first row gives $\Delta x' = 1.25 \times 10 + (-0.75c) \times 0 = 12.5$ light-seconds.
The second row gives $\Delta t' = (-0.75/c) \times 10$ light-seconds $+\ 1.25 \times 0$ . Ten light-seconds divided by c is 10 seconds, so $\Delta t' = -7.5$ s.
In the moving frame the events are not simultaneous: the one farther along x happened 7.5 seconds earlier.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: a change of frame as a table of multipliers

Choose the speed of the new frame as a fraction of c, and which rule to use. The table turns four events, measured in light-seconds and seconds, into the new frame's numbers.

Speed of the new frame, v/c

Type a speed and press Enter, or drag the slider from −0.9 to 0.9. A minus sign means the frame moves the other way.

x′ = 1.25 x − 0.75 t
t′ = −0.75 x + 1.25 t

γ = 1.25. Determinant 1. A flash from the origin still moves at c in the new frame, and the two light lines are stretched by 0.5 and 2.

Four events, before and after

The origin: (x, t) = (0, 0) becomes (x′, t′) = (0, 0).
Same moment, 10 light-seconds away: (x, t) = (10, 0) becomes (x′, t′) = (12.5, −7.5).
Same place, 10 seconds later: (x, t) = (0, 10) becomes (x′, t′) = (−7.5, 12.5).
Where a flash from the origin is after 10 s: (x, t) = (10, 10) becomes (x′, t′) = (5, 5).

The first two events happen at the same moment in the old frame. In the new frame the one 10 light-seconds away happens 7.5 seconds earlier than the one at the origin.

What it shows, in words

Each row of the table makes one new coordinate from the old ones by multiplying and adding. For the relativistic rule at 0.6c the factor γ is 1.25, the determinant is 1, and a flash of light still moves at c in the new frame; two events that were simultaneous are 7.5 seconds apart. The Galilean rule also has determinant 1 and keeps simultaneous events simultaneous, but it changes the speed of the flash. Writing the change of frame as a table is a later aid: the 1905 paper writes it as two equations.

Where this lesson stops

A matrix is a table of multipliers applied to inputs. This lesson stops at two inputs and two outputs.

If you came here from a passage, Back returns you to the exact place you left.

All foundation lessons · Read as Markdown