§2 · lengths and measurement events · Special relativity: from clock operations to electrodynamics

§2 · lengths and measurement events

A length is a specified pair of events

Which endpoint measurements define a moving rod’s length?

A ruler travelling with the rod gives its rest length. An observer the rod moves past has to note where both ends are at the same moment of that observer’s time; marking one end now and the other a little later adds the distance the rod moved in between.

Differences between two events transform by the same map as the events themselves. So a pair with $\Delta t = 0$ and $\Delta x \ne 0$ has $\Delta t' = -\gamma v\,\Delta x/c^2$ , which is not zero for any $v \ne 0$ : what one frame calls simultaneous, the other does not.

\Delta t=0:\quad\Delta t'=-\gamma\frac{v\Delta x}{c^2}

Events simultaneous in the unprimed frame have a primed time difference of minus gamma v delta x over c squared.

In the simultaneity laboratory you pick the two end events yourself and see which frame calls them simultaneous. A pair chosen in the wrong frame gives a larger separation, and that does not mean the rod got longer.

Return to which end marks measure a moving rod. (included in this file)

Letters: the rod has length l when measured at rest. Moving with speed v along the X axis toward increasing x, its length measured in the stationary system is $r_{AB}$ , where A and B are its ends; V is the speed of light. The γ, c and primes of the formula below are this explanation's: the paper's §2 argues without the transformation, which it derives only in §3.

The two principles, as §2 defines them. First: the laws by which the states of physical systems change do not depend on which of two coordinate systems in uniform translation relative to each other they are referred to. Second: every light ray moves in the stationary system with the definite speed V, whether it was emitted by a body at rest or a body in motion, where speed means light path divided by time interval in the sense of §1.
Operation a: an observer moving with the rod lays a measuring rod along it, exactly as if rod, observer and measuring rod were at rest. By the principle of relativity this gives l, which the paper calls the length of the rod in the moving system.
Operation b: an observer at rest uses the stationary system's clocks, set by §1's rule, to find where the two ends of the rod are at one given time t, and measures the distance between those two points. That is the length of the moving rod in the stationary system, $r_{AB}$ .
The kinematics then in general use tacitly assumed that a and b give the same length. The paper sets out to determine b from its two principles, and finds that it differs from l.
Now put a clock at each end, A and B, in step with the stationary system's clocks, and let observers moving with the clocks apply §1's rule. A ray leaves A at $t_A$ , is reflected at B at $t_B$ , and arrives back at A at $t'_A$ .
Outward, the light chases the receding end B, so the gap closes at $V-v$ : $t_B-t_A=r_{AB}/(V-v)$ . Coming back, the light meets A moving toward it, so the gap closes at $V+v$ : $t'_A-t_B=r_{AB}/(V+v)$ .
Worked case: v = 0.6V and $r_{AB}$ = 1 light-second. Out: 1/0.4 = 2.5 seconds. Back: 1/1.6 = 0.625 seconds. The two are unequal, so observers moving with the rod would find the clocks not in step, while observers in the stationary system call them in step.
So simultaneity has no absolute meaning: two events that are simultaneous seen from one coordinate system are not simultaneous seen from a system moving relative to it.
To measure a moving rod, therefore, mark both ends at the same time in your own frame. Mark one end, then the other 1 second later while the rod moves at 0.6c, and you add $0.6\times3\times10^{8}=1.8\times10^{8}$ metres, the distance it moved in between.
Borrowing §3's transformation puts a number on the disagreement. Two events with no time difference in one frame and a separation Δx along the motion have, in the other frame, the time difference below.

\Delta t=0:\quad\Delta t'=-\gamma\frac{v\Delta x}{c^2}

Events simultaneous in the unprimed frame have a primed time difference of minus gamma v delta x over c squared.

Worked: Δx = 1 light-second, v = 0.6c, γ = 1.25: $\Delta t'=-1.25\times0.6\times1$ = −0.75 seconds. The minus sign says that, in the moving frame, the event farther along x happens first.
A measured length is not what a photograph shows. A camera records light that arrives at the same moment, and that light left the two ends at different times.

Return to which end marks measure a moving rod. (included in this file)

Simultaneous in one system, not in the other

Modern teaching equation, not the equation as printed.

\Delta t' = -\frac{\gamma\,v\,\Delta x}{c^{2}}

Events simultaneous in the unprimed frame have a primed time difference of minus gamma v delta x over c squared.

Two events at the same time in the stationary system but at different places are, in the moving system, separated in time by this amount. It is the Lorentz map applied to a pair of events with no time between them.

Assumptions

The rod moves at constant velocity along the direction of motion and keeps its rest length.
Each frame reads time on its own synchronized clocks.

Limits

The measured length is not what a photograph of the moving rod would show.
Choosing a different pair of end events for each frame is the point of the measurement, not a correction to it.

Foundations included in this file

Reading a graph

How does a curve show the relationship between two physical quantities?

Walk for a minute and note how far you have gone every ten seconds: 0 m at the start, 14 m after ten seconds, 28 m after twenty, and so on. The first drawing below puts time along the bottom and distance up the side, with a dot for each pair. That is a graph: each dot is one moment, read across for the time and up for the distance.

The dots of a steady walk fall on a straight line, and the steeper the line, the faster the walk. This one climbs 14 metres for every 10 seconds: 1.4 metres per second. A flat line would mean standing still, and a line that bends upward would mean speeding up.

The papers draw other pairs the same way. Along the bottom goes what you choose or wait for, such as a position or a time; up the side goes what you then find, such as how many particles sit there. A bell-shaped curve, high in the middle and falling away on both sides, says that most particles are near the centre and fewer are farther out.

Worked example: Four particles that average to zero

Four particles end at −3, −1, +1 and +3 micrometres from where they started.
The second drawing puts those positions along the bottom, with a bar one particle tall above each.
The bars stand two on each side of zero, at the same distances out. The picture is symmetric, so the average position is 0.
Yet no bar stands at zero: every particle moved. The graph shows at a glance what the average hides.

Where this lesson stops

Before reading a curve, read its two labels and their units, and ask whether its height is a count, a density or a running total.

This lesson builds on

A sign records direction

Return to the chapter

A sign records direction

How can movement add up to zero?

On the ruler below, the starting point is zero. A final mark three units to the right has displacement +3; one three units to the left has displacement −3. Adding the signed displacements gives zero. Adding the distances from the start gives six.

Displacement compares where something ends with where it started, whatever it did in between. A walker who goes three units right and comes back has displacement zero after walking six, which is the lower pair of arrows in the drawing.

Worked example: Two walkers, opposite directions, average zero

Two walkers both finish one unit from their start, one at −1 and the other at +1. The signed average is zero. The average distance is one.

Where this lesson stops

The sign names a direction relative to a chosen axis; it does not mean a negative distance.

This lesson builds on

Adding and averaging

Return to the chapter

Adding and averaging

What does an average keep, and what does it lose?

Four jars hold 3, 1, 1 and 3 marbles. Pour them together and there are 8. Share the 8 equally between the four jars and each holds 2. Two is the average, although no jar held 2 to begin with.

The average keeps the total and the count and nothing else, so 3, 1, 1, 3 and 2, 2, 2, 2 have the same average. Listing every value twice doubles both the total and the count, and the average stays at 2.

Worked example: Three mornings with the same average

One morning the four jars hold 3, 1, 1 and 3 marbles. Add them: 8. Divide by the four jars: an average of 2.
Another morning they hold 8, 0, 0 and 0. Add them: 8 again. Divide by four: 2 again, and now three jars are empty.
A third morning every jar holds 2, which is the average itself. Add them: 8. Divide by four: 2.
Write any of those three lists out twice over. The total doubles to 16 and the count doubles to 8 jars, and 16 divided by 8 is still 2.
Four descriptions and one average. It keeps the total and the count, and the differences between the mornings are what it leaves behind.

Where this lesson stops

An average is the total shared equally among the observations, and two very different lists can have the same one.

Return to the chapter

Events and distant clocks

What does it mean to say that two distant events happen at the same time?

An event is something that happens at one place at one instant: a spark, a collision, a flash leaving a lamp. To give it a time, you read a clock at that place. Reading a distant clock by looking at it would mix in the time the light takes to reach you.

A reference frame is a set of rigid measuring rods and clocks at rest, with a clock at every place, so that each event gets three coordinates of position and one of time from the rods and the clock beside it. Before that works, the clocks must agree, and for clocks far apart agreeing needs a rule.

§1 of the relativity paper gives the rule. Send a flash of light from clock A to clock B and back. Note A's reading when it leaves, B's reading when it arrives, and A's reading when it returns. The clocks are called synchronous if the trip out takes as long, by the clocks, as the trip back:

t_B - t_A = t'_A - t_B

t B minus t A equals t A prime minus t B: the trip out takes as long as the trip back.

The paper calls this a definition. It cannot be checked by measurement, because checking it would need clocks that are already synchronous. §1 adds two assumptions: that the rule is consistent, so that if B agrees with A then A agrees with B, and two clocks that agree with a third agree with each other; and that light's speed measured out and back is a universal constant, which the paper writes V.

Worked example: Setting a distant clock with one light signal

Clock A reads 0 s as a flash leaves it for clock B.
The flash is reflected at B and arrives back at A when A reads 10 s.
By the rule the two trips take equal times, so the reflection happened at 5 s on A's time.
If B's clock read 5 s at the reflection, it is synchronous with A. If it read 7 s, it is 2 s fast, and setting it back 2 s makes the two agree.

t_B = \frac{1}{2}\left(t_A + t'_A\right)

The time at B equals one half of the sum of A's reading when the flash left and A's reading when it came back.

The rule needs no distance. If B is known to be 1.5 × 10⁹ metres away, the same round trip also gives light's speed: 2 × 1.5 × 10⁹ m divided by 10 s is 3 × 10⁸ metres per second.

Where this lesson stops

An event is something that happens at one place and one time. Whether two distant events are simultaneous depends on a rule for setting clocks, which §1 of the relativity paper supplies.

This lesson builds on

Functions and graphs

Return to the chapter

Functions and graphs

How does a graph record the way one quantity depends on another?

A bath fills from a tap. After one minute it holds 10 litres, after two 20, after three 30. For each time there is exactly one amount of water, and that pairing, one output for each input, is a function. Plot the pairs with time along the bottom and litres up the side and they fall on a straight line that climbs 10 litres for every minute. The slope, in litres per minute, is how fast the bath fills.

In the papers the pairs are physical: the mean square displacement of a Brownian particle for each elapsed time, or the concentration for each position along a tube. A function says how one thing responds when another changes.

A graph puts the input along the horizontal axis and the output up the vertical one, so each point is one pair. Both axes carry units, and so does everything read off the graph: the slope of mean square displacement against time is in square micrometres per second, and the area under a probability density is a plain probability.

A formula such as ⟨x²⟩ = 2Dt describes an average over many particles. One particle's squared displacement scatters widely around the line; the average over many lies close to it. The Brownian paper predicts the average, and ends by hoping that a researcher will soon decide the question by observation.

Worked example: Mean square displacement plotted against time

\langle x^2\rangle = 2Dt

The mean square displacement equals two times the diffusion coefficient times the time.

Take D = 0.5 square micrometres per second, so 2D = 1 square micrometre per second.
At t = 1 s the mean square displacement is 1 µm²; at t = 4 s it is 4 µm²; at t = 9 s, 9 µm².
Plotted against time, these points lie on a straight line through the origin whose slope is 2D, 1 µm² per second.
Their square roots, the root-mean-square displacements, are 1, 2 and 3 µm: four times the time gives twice the distance.

Where this lesson stops

A graph is a record of paired quantities, each with its units. A slope read off it has units too, and a slope stated without them explains nothing.

This lesson builds on

Reading a graph

Return to the chapter

Static laboratory examples

sr-03: a built-in worked result

Static host-calculated example, not a running experiment or an observation. Canonical units and owner-supplied statuses are retained. Open sr-03 online

Build-time scalar readouts
Quantity	Value	Unit	Status
spatialSeparationK	10	ls	value
temporalSeparationK	0	s	value
spatialSeparationKPrime	12.5	ls	value
temporalSeparationKPrime	-7.5	s	value
simultaneityK	0	1	value
simultaneityKPrime	-1	1	value
measuredLength	10	ls	value
rodLengthK	8	ls	value
rodLengthKPrime	10	ls	value
readingsOnRodEnds	0	1	value
spacetimeIntervalSquared	100	ls^2	value
causalOrder	1	1	value
gammaFactor	1.25	1	value
ellipsoidAxisLongitudinal	0.8	ls	value
ellipsoidAxisTransverseY	1	ls	value
ellipsoidAxisTransverseZ	1	ls	value

0 array outputs are not reproduced. This table does not claim to reproduce the interactive plot.

Settings (canonical units)

L0: 10
R: 1
endpointPairId: platform-simultaneous
measuringFrame: K
rodRestFrame: k
v: 0.6

Evaluator: source:sha256:5aca8eaa733b21262f272915553a148b6395ceb2c24be06a26d30c2487a9acd4

Source references

A. Einstein, Zur Elektrodynamik bewegter Körper. Annalen der Physik (4), 17, 891–921 (1905).

Special relativity: from clock operations to electrodynamics: §2 · lengths and measurement events