Read · Special relativity: from clock operations to electrodynamics

§1 · operational simultaneity

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Detail

Letters

Show modern qualifications

Einstein's letters change 12 of this paper's 40 formulas. 28 stay in today's letters, with a note saying so.

A definition

Time at a distant clock is an operation

How can clocks at different places share a time coordinate?

The 10 printed paragraphs this passage explains

§1, paragraph 1: Einstein starts from a coordinate system in which Newton's equations of mechanics hold, and calls it the stationary system.
§1, paragraph 2: The position of a point at rest in that system can be found with rigid measuring rods by the methods of Euclidean geometry.
§1, paragraph 3: Describing a motion means giving positions as functions of time, which has a meaning only once it is said what time is. Every statement about time is a statement about simultaneous events, such as a train arriving when the small hand of a watch points to 7.
§1, paragraph 4: Reading time from the hand of one clock works only at the place of that clock. It does not connect events that happen at other places, far from the clock.
§1, paragraph 5: An observer at the clock could assign each distant event the time at which its light signal arrives, but that assignment depends on where the observer stands. Einstein looks for a more practical rule.
§1, paragraph 6: Clocks at A and at B each give a time only for events near them. Einstein defines a common time by declaring that light takes as long to go from A to B as to come back, so the two clocks are synchronous when the outward and the return times agree.
§1, paragraph 7: He assumes this definition can be applied without contradiction to any number of clocks: if the clock at B is synchronous with the clock at A, the one at A is synchronous with the one at B, and two clocks synchronous with a third are synchronous with each other.
§1, paragraph 8: This fixes what synchronous clocks at rest in different places are, and with it what simultaneous and time mean: the time of an event is the reading, at that event, of a resting clock at its place that is synchronous with one chosen clock.
§1, paragraph 9: In agreement with experience, twice the distance between two clocks divided by the time light takes to go there and back is taken to be a universal constant V, the speed of light in empty space.
§1, paragraph 10: Because this time is defined by clocks at rest in the stationary system, Einstein calls it the time of the stationary system.

t_B - t_A = t_{A,\mathrm{return}} - t_B

Shown in modern letters; Einstein's are in §1 of the German source face.

The outward assigned travel time equals the return assigned travel time.

t_B = \frac{t_A + t_{A,\mathrm{return}}}{2}

Shown in modern letters; Einstein's are in §1 of the German source face.

The assigned reflection time is the midpoint of departure and return times.

If clock A sends a flash at time 0 and the reflection comes back at time 10, the rule says the flash reached B at time 5. Seeing B later, when its light arrives, is a separate event and does not change that assignment.

The rule assumes light takes as long to go as to come back, and Einstein states that as a definition rather than a measurement. It sets clocks within one frame; §2 shows that clocks in another frame, set by the same rule, disagree about which distant events happen at the same time. The clock-synchronization laboratory keeps departure, reflection, return and each clock reading as separate records.

Open the foundation: Events and distant clocks

Show every step here: Time at a distant clock is an operation

Letters: a light ray leaves A at the A-time $t_A$ , is reflected at B at the B-time $t_B$ , and arrives back at A at the A-time $t'_A$ . This explanation writes that return reading $t_{A,\mathrm{return}}$ . V is the speed of light, c here.

Start from one coordinate system in which Newton's equations of mechanics hold; the paper calls it the stationary system. A point at rest in it has a position measured with rigid rods by the methods of Euclidean geometry.
Describing a motion means giving the coordinates as functions of time, and that has a physical meaning only once it is said what time means. For an event at the place of a clock, the time is the position of the clock's hand at that event.
A clock at A times the events at A, and a clock at B times the events at B. So far there is an A-time and a B-time, but no time common to A and B. Comparing an event at A with an event at B needs a further rule.
The rule, set by definition: the time light takes to go from A to B equals the time it takes to come back from B to A. With the three readings above, that is the first formula below.

t_B - t_A = t_{A,\mathrm{return}} - t_B

Shown in modern letters; Einstein's are in §1 of the German source face.

The outward assigned travel time equals the return assigned travel time.

Solve for $t_B$ . Add $t_B$ to both sides and $t_A$ to both sides: $2t_B=t_A+t_{A,\mathrm{return}}$ . Divide by 2.

t_B = \frac{t_A + t_{A,\mathrm{return}}}{2}

Shown in modern letters; Einstein's are in §1 of the German source face.

The assigned reflection time is the midpoint of departure and return times.

Worked case: A sends the flash at time 0 and the reflection comes back at time 10, so the reflection is given the time 5. If B's clock read 5.3 at the reflection, it is 0.3 ahead and is set back.
The paper assumes that this definition can be applied without contradiction to any number of points: if the clock at B runs in step with the clock at A, then A's runs in step with B's; and if A's runs in step with both B's and C's, then B's and C's run in step with each other.
The speed of light comes out of the same readings: twice the distance AB between the stations, divided by $t'_A-t_A$ , is V, a universal constant, in agreement with experience. With A and B $3\times10^{8}$ metres apart and a round trip of 2 seconds, V is $3\times10^{8}$ metres per second.
The time of an event is then the reading, at the event, of a clock at rest at its place and in step with the chosen clock. Since it is set with clocks at rest in one system, the paper calls it the time of the stationary system.
Keep two things apart: the time this rule gives to a distant event, and the moment its light reaches you. Seeing the reflection from A at time 10 does not move it from time 5.
The status: equal travel times out and back are a definition, not a measurement, and a round trip read on one clock cannot measure the two one-way times separately. §2 shows that clocks set this way in a moving system disagree with the stationary system about which distant events are simultaneous.

A tool for this step

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.

One flash, out and back

A flash leaves clock A, turns round at clock B and comes back. By the rule the two trips take equal times, so the turn-round happened half way between leaving and returning: at 5 on A's clock.

B reads 5 at the turn-round, which is what A's clock says the moment was. The two are synchronous.

Now the same flash, against a clock that is running ahead.

B reads 7 where the rule says 5, so it is 2 seconds fast. Set it back 2 and the two agree. Nothing about the flash changed between the drawings; only what B's hands were showing.

What it shows, in words

Two clocks stand apart, drawn as vertical lines with time running down the page. A flash leaves A when A reads 0, reaches B, is reflected, and returns to A when A reads 10. The rule says the trip out took as long as the trip back, so the turn-round was half way between: 5 by A's clock. In the first drawing B reads 5 at that moment and the clocks are synchronous. In the second B reads 7, which is 2 seconds more, so B is 2 seconds fast and setting it back 2 makes them agree.

The same round trip also measures light, if the distance is known separately: with B at 1.5 × 10⁹ metres, a there-and-back of 10 seconds gives 3 × 10⁸ metres per second.

What it does not show, and this is the lesson's own point rather than a limitation of the drawing: the rule is a definition, and no picture can show it being tested. Checking that the trip out really took as long as the trip back would need two clocks already synchronised, which is the thing the rule exists to supply. Nor is there a second observer anywhere here. Only one frame is drawn, so nothing in it bears on how someone moving past would time the same three events; that question needs the paper's §2, and the geometric aid usually reached for is Minkowski's of 1908, which is later than the argument and not its presentation.

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.

Explore the equations in this step

Read each operation, check its units and assumptions, or open the mathematical step behind it. These are modern teaching equations, not the equations as printed.

Explore the equation · Modern model notation

The rule: out and back take equal times

t_B - t_A = t_{A,\mathrm{return}} - t_B

Shown in modern letters; Einstein's are in §1 of the German source face.

minus equals minus .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

The outward assigned travel time equals the return assigned travel time.

This is a definition of what B's clock should read, not a law found by measurement: it sets the time light takes from A to B equal to the time it takes from B back to A.

Model assumptions and every term’s meaning

Clocks A and B are at rest in the same system, and a light signal goes from A to B, is reflected, and returns to A.
Equal travel times out and back is the definition of B's reading, not a measurement: nothing timed at one place can test it.
Modern teaching notation. The paper marks A's return reading with a prime; here it carries the word return instead, so that a prime keeps meaning the moving system.

Time assigned to the return trip: A's return reading minus B's reading: how long the light is said to take from B back to A. Read the prerequisite
Time assigned to the outward trip: B's reading minus A's departure reading: how long the light is said to take from A to B. Read the prerequisite
Set the two equal: Requiring the two trips to take equal times is the synchronization convention. It could be chosen otherwise without contradicting any single measurement at A or B. Read the prerequisite
Reading at A as the signal leaves: Clock A's reading at the moment the light signal leaves A. It is read at A, where the event happens, so no convention is needed to know it. Read the prerequisite
The reading B is to show: Clock B's reading at the reflection. This is the unknown the rule is for: B is far from A, so A's clock cannot be read at that event. Read the prerequisite
Reading at A when the signal is back: Clock A's reading when the reflected signal arrives back at A. Also read locally, at the event. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Explore the equation · Modern model notation

So B reads the midpoint

t_B = \frac{t_A + t_{A,\mathrm{return}}}{2}

Shown in modern letters; Einstein's are in §1 of the German source face.

is halfway between and .

Select a term or operation. In the formula, Down enters an operation, Up returns to its parent, and Left/Right move between siblings. Escape clears selection. Tab leaves the formula.

Explain this equation

Each term and operation, in words

No live values

This equation states a relation. No instrument on this page computes its terms.

Read the equation aloud in words

The assigned reflection time is the midpoint of departure and return times.

Solving the equal-times rule for B's reading: B's clock is set to read halfway between the moments the signal leaves A and returns to A.

Model assumptions and every term’s meaning

Clocks A and B are at rest in the same system, and a light signal goes from A to B, is reflected, and returns to A.
Equal travel times out and back is the definition of B's reading, not a measurement: nothing timed at one place can test it.
Modern teaching notation. The paper marks A's return reading with a prime; here it carries the word return instead, so that a prime keeps meaning the moving system.

Halve the sum: Half the sum of two readings is their midpoint. If A sends at 0 s and the signal is back at 10 s, B is to read 5 s at the reflection. Read the prerequisite
B's reading, assigned: The value B's clock must show for the two clocks to count as synchronous under the rule. Read the prerequisite
Add the two readings at A: The departure and return readings, both taken at A. Read the prerequisite
Reading at A as the signal leaves: Clock A's reading at the moment the light signal leaves A. It is read at A, where the event happens, so no convention is needed to know it. Read the prerequisite
The reading B is to show: Clock B's reading at the reflection. This is the unknown the rule is for: B is far from A, so A's clock cannot be read at that event. Read the prerequisite
Reading at A when the signal is back: Clock A's reading when the reflected signal arrives back at A. Also read locally, at the event. Read the prerequisite

Every term’s units were checked when this page was built. That checks the units, not the model. This equation is written for this edition in modern notation, not transcribed from the paper.

Modern qualifications

Philosophers since Reichenbach (1928) have debated how much of the equal-travel-time rule is a convention rather than a fact; Einstein calls it a definition. Either way, simultaneity set this way belongs to one frame, which is the point of §2.

Assumptions and limits: Time at a distant clock is an operation

Assumed here

The clocks and the endpoints are at rest in one inertial frame.
The rule assumes light takes as long to go as to come back.

What this does not establish

A round trip read on one clock cannot measure the two one-way times separately.
Setting clocks does not move them or change their rates.

Source context: German source · English · Interlinear gloss · Facsimile

References

A. Einstein, Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt. Annalen der Physik (4), 17, 132–148 (1905).

A. Einstein, On the motion of particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Annalen der Physik (4), 17, 549–560 (1905), §§1–5. Bibliographic pointer; this preview is not a source transcription or translation.

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

A. Einstein, Does the inertia of a body depend upon its energy content? Annalen der Physik (4), 18, 639–641 (1905). External 1923 Perrett–Jeffery translation, electronically transcribed by John Walker; its notation was modernized. A reference for this explanatory preview, not this edition’s translation or its pinned facsimile.

28 foundation readings sit behind this argument.

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§1 · operational simultaneity

Original journal scan

Time at a distant clock is an operation

The rule: out and back take equal times

So B reads the midpoint

Modern qualifications

What this relies on

Assumed here

What this does not establish

References

Reading a graph

Fractions and ratios

A letter stands for a quantity

Marks that look alike and do different jobs

A sign records direction

Powers of ten and physical units

Squares and square roots

Adding and averaging

Rates of change and derivatives

Density is not probability

Dot and cross products

Charges, fields and light

Exponential change

Fields and waves

Counting what crosses a boundary

Events and distant clocks

Functions and graphs

Hyperbolic functions and rapidity

Adding continuously

Logarithms: turning products into sums

Matrices and linear maps

The push of light

Partial derivatives and held-fixed quantities

Quantities and units

Approximating a curve near a point

The units of 1905

Vectors and their components

Energy of motion and inertia

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