Events and Worldlines

Before relativity can tell you anything surprising, it has to tell you what it is talking about. This article is about the two nouns the rest of the subject is built from — the event and the worldline — and it is deliberately the gentlest article in the chapter. There is no paradox here and nothing to be clever about. There is just vocabulary, made precise. The vocabulary matters more than it sounds. Most of the confusion people feel about relativity is not confusion about $\gamma$ ; it is confusion about what is being described. Ask "where is the spaceship now?" and you have already lost, because "now" is not a thing you can point at across a distance. Ask "what does the ship's clock read at the event where it passes the marker buoy?" and you have asked something with a single, frame-independent answer that every observer in the universe will agree on. The difference between those two questions is the whole of this article. So: an event is a point of spacetime — a where and a when, welded together. A worldline is the complete history of an object, the continuous chain of events at which it exists. A worldline is not a path through space. It is the object, drawn out along its own existence. Sitting perfectly still is not an absence of motion in spacetime; it is a vertical worldline, and you are travelling along it whether you like it or not. And once a worldline is a curve, the natural question is how long it is — because in this geometry, the "length" of a worldline is the time read by a clock carried along it. That is proper time, and it is derived and proved maximal for straight lines in Proper Time, which owns it. What this article adds is one step further down, and it is the step that makes a worldline a genuine geometric object rather than a description: proper time does not depend on how you choose to parametrise the curve. "Every observer agrees" — that is frame-independence, a separate fact. A second, independent invariance: even you, in your own single frame, can run along the curve at any pace you like, describe it with any parameter you fancy, and the clock reading comes out the same. The worldline is a thing, not a story about a thing.

Key takeaways

An event is a point in spacetime: a place and a time together. Its coordinates $(ct, x, y, z)$ depend on the frame; its identity does not. "The firecracker went off" is a fact; "it went off at $x = 3$ " is bookkeeping.
A worldline is an object's entire history — the curve of all events at which it exists. It is a history, not a trajectory. An object at rest still has a worldline (a vertical one), because it is still ageing.
The tangent rule: at every point of a massive object's worldline the tangent must lean less than $45°$ from vertical, i.e. $|dx/d(ct)| < 1$ . This is a pointwise condition — a worldline may curve and wiggle as much as it likes, but it may never reach $45°$ anywhere, not even instantaneously.
That rule is frame-independent, and not by luck: it is the statement that $ds^2 > 0$ along the curve, and the sign of an invariant is something all frames agree on.
Proper time $\tau = \int \sqrt{1 - \beta^2(t)}\,dt$ is the "length" of a worldline — what a clock carried along it reads. The straight worldline between two events has the greatest proper time. Both facts are proved in Proper Time; we use them.
OWNED: proper time is invariant under reparametrisation, a second invariance independent of change of frame. Written with the frame's coordinate time it looks like it depends on $t$ . It does not: describe the same curve with any monotone parameter $\lambda$ and $\tau = \tfrac1c\int\sqrt{\dot x^\mu \dot x_\mu}\,d\lambda$ gives the identical number, because the $d\lambda$ -dependence cancels by the chain rule.
Checked on a real curve: $ct = t$ , $x = t^2/40$ from $t = 0$ to $10$ m gives $\tau = 5\pi/3 + 5\sqrt3/2 = 9.5661\ \mathrm{m}/c$ — and the same $9.5661$ under $t = u^3/100$ and under $t = 10\sin(\pi w/2)$ .
Key idea: the worldline is the physical object; coordinates and parameters are things we drape over it. Anything that depends on the drapery is bookkeeping — anything that survives it, like proper time, is physics.

The core insight

An event is a point of spacetime and a worldline is the curve of events an object passes through — its whole history, from end to end. Everything you can measure is attached to these two things and not to the coordinates you happened to use: a clock reading at a meeting of two worldlines, the proper time along a curve, the sign of an interval. Coordinates are a grid you throw over spacetime and parameters are a pace you choose for walking along a curve, and physics is exactly the part of your description that does not care about either.

Where this lives today

The event-and-worldline vocabulary is not a pedagogical scaffold that professionals discard; it is the working language of several fields. Particle detectors are, quite literally, worldline-reconstruction machines: what a tracker records is a set of events (hits), and the job of the reconstruction software is to fit worldlines through them and read off the tangents, which are the momenta. In general relativity a free particle's worldline is a geodesic — the straightest curve available in a curved spacetime — and the theory's entire content is a rule for how matter tells worldlines how to bend. LIGO detects a gravitational wave by watching the proper distance between mirror worldlines oscillate by a fraction of a proton's width. The idea has also outlived the physics that produced it. In distributed systems, a process's history is a worldline and message-passing is exactly the causal-cone structure of a spacetime diagram — which is why Lamport's happens-before relation is, formally, a partial order on events of precisely the kind Minkowski wrote down. Wherever there is a finite signal speed and no universal clock, the events-and-worldlines picture is not an analogy; it is the correct description.

[Continue with quiz blocks]

The physical picture

The event

An event is a happening: a point in space, at an instant of time, taken as a single indivisible thing. A firecracker exploding. A clock ticking over. Two worldlines crossing. It has no extent and no duration — an event of finite duration is not an event, it is a short worldline. The crucial property is the split between identity and coordinates. The firecracker's explosion is one event, and every observer agrees it happened. But ask them to label it and they hand you different numbers, and they will disagree about when it happened relative to other events elsewhere. Neither is wrong: they have thrown different grids over the same spacetime. The event is the thing; the coordinates are the label. This sounds pedantic and is not — virtually every apparent paradox in relativity comes from treating a label as if it were a thing. There is one special class of event that deserves its own name, because it is the bedrock of every consistency argument in the subject: a coincidence, where two worldlines actually meet. When two clocks are at the same place at the same time, they can be read off against each other directly, with no convention, no synchronisation, no light-travel correction. What two clocks read when they are side by side is a fact no observer can dispute. Everything that relativity says is "relative" is relative; this is not.

The worldline

A worldline is the set of all events at which an object exists — its complete history, drawn as a curve. The mental adjustment people find hardest: a worldline is not a path through space. A path through space is what you get when you throw the time information away. The worldline keeps it. So a particle that flies from here to there and back again has a worldline shaped like a $\Lambda$ , even though its path is a line traversed twice. And an object at rest is not exempt. Its worldline is a vertical line, straight up the page — it is going nowhere in space and forward in time at the only rate there is. Over a year, a stationary rock traces out a worldline about a light-year long. Nothing in spacetime is ever at rest, because everything ages. This is why the diagram is drawn with time as an axis and not as an animation: the worldline is the whole history, all at once, an object laid out along its own lifetime. Two worldlines crossing is a collision, a meeting, a measurement. Worldlines that never cross are objects that never meet. Almost every physical question can be phrased as a question about worldlines and their intersections — which is exactly why they are the right primitive.

The core equations

The geometry is set by one invariant, which this article uses and does not derive.

ds^2 = (c\,dt)^2 - dx^2 - dy^2 - dz^2 \qquad (1)

where: $ds^2$ is the interval between two neighbouring events on a curve. Signature $(+,-,-,-)$ , timelike-positive: $ds^2 > 0$ is timelike, $ds^2 = 0$ is lightlike, $ds^2 < 0$ is spacelike. Physical reading: this is the spacetime replacement for Pythagoras, with the minus signs that make the geometry non-Euclidean. Its invariance — every observer computes the same $ds^2$ from their own different coordinates — is established in The Invariant Spacetime Interval. Note carefully: many textbooks (including this article's own source material) write it with the opposite sign convention, $ds^2 = -c^2dt^2 + dx^2 + \dots$ , in which timelike separations come out negative. The physics is identical; the bookkeeping is not. Check the convention before importing any formula. Now the condition that makes a curve a worldline rather than just a curve.

ds^2 > 0 \quad\text{everywhere along the curve} \qquad\Longleftrightarrow\qquad \left|\frac{dx}{d(ct)}\right| = |\beta| < 1 \quad\text{at every point} \qquad (2)

where: $\beta = v/c$ is the instantaneous speed, with $v$ the magnitude of the coordinate three-velocity; the algebra displayed here takes the motion along $x$ . Physical reading: this is the tangent rule, and it is the entire content of "nothing goes faster than light" as a statement about curves. It is pointwise: it constrains the tangent at every single point, so a worldline may bend, wiggle, accelerate and reverse, but it may never reach $45°$ anywhere — not even for an instant. And because it is the sign of an invariant, every observer imposes the same rule and agrees on which curves pass. The "length" of such a curve is the time a clock carried along it reads.

\tau = \frac{1}{c}\int_{\mathcal{C}} \sqrt{ds^2} \;=\; \int \sqrt{1 - \beta^2(t)}\;\,dt \qquad\text{and}\qquad \tau_{\text{straight}} \;>\; \tau_{\text{any other path}} \qquad (3)

where: $\tau$ is the proper time along the worldline $\mathcal{C}$ . Physical reading: both of these are proved in Proper Time, which owns them, and we use them as given. The integral says proper time is the arc length of a worldline in the spacetime metric. The inequality is the reversed triangle inequality — the exact opposite of the Euclidean fact that a straight line is shortest. In spacetime the straight worldline is the way to age the most, and every detour costs you time. For constant $\beta$ the integral collapses to $\tau = \Delta t\sqrt{1-\beta^2} = \Delta t/\gamma$ , which is ordinary time dilation — a reassuring check, and the reason the integral deserves the name. Finally, the form this article owns, written so that no frame's coordinate time appears.

\tau = \frac{1}{c}\int_{\lambda_1}^{\lambda_2} \sqrt{\dot{x}^\mu \dot{x}_\mu}\;\,d\lambda, \qquad \dot{x}^\mu \equiv \frac{dx^\mu}{d\lambda} \qquad (4)

where: $x^\mu = (ct, x, y, z)$ collects the four coordinates of an event into one symbol — here purely as notation, a bookkeeping convenience; that $x^\mu$ is a genuine four-vector, with transformation rules of its own, is a later story. $\dot x^\mu \dot x_\mu \equiv (c\dot t)^2 - \dot x^2 - \dot y^2 - \dot z^2$ in our signature, and $\lambda$ is any monotonically increasing parameter along the curve — not necessarily anyone's time. Physical reading: this is the same $\tau$ as in (3), but now nothing on the right-hand side refers to a frame's clock. The next section proves that the answer really is independent of $\lambda$ , and that is what promotes the worldline from a description of an object's motion to a geometric object in its own right.

Figure

A worldline is a HISTORY, and the tangent rule is POINTWISE. An object at rest has a worldline too — it is vertical (left). The curved worldline may bend as much as it likes, but at EVERY point its tangent must stay steeper than 45°: the local light cone (drawn at three events along it) must always contain the tangent. β = 0.375, 0.375, 0.5, 0.25 on the four segments — all below 1, everywhere

Common trapTwo traps: your Euclidean eye, and your textbook's signature

The eye. On the page, the bent worldline is visibly longer than the straight one — and it reads less elapsed time. This is not an illusion in the drawing; it is the geometry. Ink length is $\sqrt{\Delta(ct)^2 + \Delta x^2}$ and proper time is $\sqrt{\Delta(ct)^2 - \Delta x^2}$ , and they differ by the sign that defines the subject. Sideways motion adds to the ink and subtracts from the clock. Longest-looking is shortest-ageing — see the punchline figure, where the null zig-zag is the longest line on the page and reads exactly zero.

The signature. This article, and this whole chapter, use $(+,-,-,-)$ : timelike means $ds^2 > 0$ . A large fraction of the literature — including this article's own source material — uses $(-,+,+,+)$ , in which every $ds^2$ flips sign and timelike means negative. Both are correct and they describe identical physics, but mixing them inside one calculation produces sign errors that look like physics and are not. Before importing any formula, check which convention it was written in.

The owned result: proper time does not care how you walk the curve

Derivation AOWNED — proper time is invariant under reparametrisation, not just under change of frame

Look again at the standard formula, $\tau = \int\sqrt{1-\beta^2(t)}\;dt$ . There is something faintly suspicious about it. The integration variable is $t$ — somebody's coordinate time, belonging to some particular frame. So the formula appears to define proper time in terms of a quantity that is frame-dependent, which would make a nonsense of calling the answer invariant. We know it survives a change of frame (that is the theorem in Proper Time). But here we prove something stronger and, for a vocabulary article, more foundational: the answer does not depend on the parameter at all. You may crawl along the curve, sprint along it, or use a parameter with no physical meaning whatever — you get the same number. Proper time is a property of the curve as a geometric object, not of the description we hang on it. That is what makes a worldline a thing.

Step 1. Write $\tau$ with an arbitrary parameter. Let $\lambda$ be any smooth, strictly increasing parameter along the worldline, so the curve is $x^\mu(\lambda) = (ct(\lambda),\, x(\lambda),\, y(\lambda),\, z(\lambda))$ . Along a small step, $ds^2 = (c\,dt)^2 - dx^2 - \dots = \left[(c\dot t)^2 - \dot x^2 - \dot y^2 - \dot z^2\right] d\lambda^2$ , where a dot is $d/d\lambda$ . Hence $c\,\tau = \int \sqrt{\dot x^\mu \dot x_\mu}\; d\lambda$ , which is (4). Why: every $d$ in the interval picks up one factor of $d\lambda$ , so the whole bracket comes out with $d\lambda^2$ — and the square root then delivers exactly one $d\lambda$ to match the measure. That balance is the entire trick, and the rest is making it airtight.

Step 2. Reparametrise. Let $\lambda = \sigma(\mu)$ with $\sigma$ smooth and strictly increasing ( $\sigma' > 0$ on the open interval), so $\mu$ is a different parameter describing the same curve. By the chain rule, $\dfrac{dx^\mu}{d\mu} = \dfrac{dx^\mu}{d\lambda}\,\sigma'(\mu)$ . Why: $\sigma' > 0$ is the only condition we need — it says we are re-labelling the curve without turning around. Monotonicity is doing real work here; a parameter that doubled back would trace part of the curve twice.

Step 3. Feed it through the integrand. The bracket is quadratic in the derivatives, so it picks up exactly $\sigma'^2$ :

\frac{dx^\mu}{d\mu}\frac{dx_\mu}{d\mu} = \left(\dot x^\mu \dot x_\mu\right)\,\sigma'(\mu)^2 \quad\Longrightarrow\quad \sqrt{\frac{dx^\mu}{d\mu}\frac{dx_\mu}{d\mu}} = \sqrt{\dot x^\mu \dot x_\mu}\;\,\bigl|\sigma'(\mu)\bigr| = \sqrt{\dot x^\mu \dot x_\mu}\;\,\sigma'(\mu).

Why: the square root is what makes this work, and it is why proper time is written with one and not, say, as $\int (\dot x^\mu \dot x_\mu)\,d\lambda$ . A square root turns the $\sigma'^2$ into a single $\sigma'$ — exactly one factor, ready to be absorbed. The absolute value drops because $\sigma' > 0$ .

Step 4. Absorb it into the measure. Changing variables in the integral gives $d\lambda = \sigma'(\mu)\,d\mu$ , so

c\,\tau = \int \sqrt{\dot x^\mu \dot x_\mu}\;\,d\lambda \;=\; \int \sqrt{\dot x^\mu \dot x_\mu}\;\,\sigma'(\mu)\, d\mu \;=\; \int \sqrt{\frac{dx^\mu}{d\mu}\frac{dx_\mu}{d\mu}}\;\, d\mu.

The $\sigma'$ from the chain rule and the $\sigma'$ from the measure are the same factor, and they cancel. The final integral has exactly the form we started with, but in $\mu$ . Why: this is the whole proof. The integrand is homogeneous of degree one in the derivatives, which is precisely the condition for a curve integral to be reparametrisation-invariant — the same reason ordinary arc length in Euclidean geometry doesn't care how you parametrise a road.

Step 5. Test it on a real curve, because a proof you cannot check is a proof you should not trust. Take a genuinely accelerating worldline: $ct = t$ , $x = t^2/40$ (metres), from $t = 0$ to $t = 10$ m. Its speed is $\beta = dx/d(ct) = t/20$ , rising from $0$ to $0.5$ — always safely below $1$ , so it is a legal worldline everywhere. Then

c\tau = \int_0^{10}\sqrt{1 - \left(\tfrac{t}{20}\right)^2}\;dt = \frac{5\pi}{3} + \frac{5\sqrt3}{2} = \mathbf{9.5661\ m}.

The coordinate time was $10\ \mathrm{m}$ ; the clock reads $9.5661\ \mathrm{m}$ — it aged less, as it must. Why: an honest curved worldline with a closed-form answer. Now we change the parameter and see whether the number moves.

Step 6. Re-run it with two different parameters, changing nothing about the curve. First $t = u^3/100$ , with $u$ from $0$ to $10$ — a parameter that crawls at the start and sprints at the end. Then $t = 10\sin(\pi w/2)$ , with $w$ from $0$ to $1$ — a parameter that does the opposite. In both cases the integrand and the measure both change, and in both cases the integral evaluates to $\mathbf{9.566114774905182\ m}$ : the same number, to every digit. Why: this is the claim, tested rather than asserted. The three integrals look nothing alike on the page — different integrands, different limits, different measures — and they are worth exactly the same, because they are three descriptions of one geometric object.

Step 7. Say what has been established. Proper time is invariant under change of frame (Proper Time) and under change of parameter (here). Together these mean $\tau$ depends on nothing but the curve itself: not on who is looking, and not on how you choose to run along it. That is what it means for the worldline to be a physical object rather than a description of one — and it is why every later construction in relativity, from four-velocity to the relativistic action, is built by parametrising worldlines with $\tau$ itself. Why: a quantity that survives every arbitrary choice you can make about it is the definition of something real.

Go deeper · Derivation B · 2 minThe tangent rule is pointwise — and it is the sign of an invariant

Derivation BThe tangent rule is pointwise — and it is the sign of an invariant

The rule "nothing travels faster than light" is usually stated as a fact about speeds. As a fact about curves it is sharper, and this derivation makes it exact: it constrains the tangent at every individual point, and it is frame-independent for a reason that costs no work at all.

Step 1. State it as a condition on the tangent. Along a worldline, $ds^2 = (c\,dt)^2 - dx^2 = (c\,dt)^2\left[1 - \left(\frac{dx}{c\,dt}\right)^2\right] = (c\,dt)^2\left(1 - \beta^2\right)$ . So $ds^2 > 0 \iff \beta^2 < 1 \iff |\beta| < 1$ . Why: the interval and the speed limit are the same statement. A curve is a legal worldline exactly when its tangent is timelike at every point — which is what makes the condition local rather than a statement about averages.

Step 2. Notice what it does not say. It says nothing about $d^2x/dt^2$ : a worldline may accelerate as violently as you please, curve, kink, or reverse direction. Only the tangent is constrained, and only pointwise. A worldline that is timelike everywhere except at one single instant is not "almost legal" — it is not a worldline. Why: this is where the pointwise character earns its keep. Averages are irrelevant. There is no budget to be spent, no way to compensate a superluminal instant with a slow stretch elsewhere.

Step 3. Show every frame agrees. $ds^2$ is invariant — that is (1), established in The Invariant Spacetime Interval. Therefore its sign is invariant. Therefore "the tangent at this point is timelike" is a frame-independent statement about a point on a curve. Every observer classifies every tangent identically. Why: this is the cheapest possible proof and the most robust one. We never had to compute how velocities transform — the sign of an invariant is agreed on by everybody, and that is the whole argument. If we had reasoned via velocity addition we would have got the same answer with far more work and less insight.

Step 4. Check numerically. Take the worldline of Step 5 above, $\beta(t) = t/20$ , at $t = 10$ : $\beta = 0.5$ , so $ds^2 \propto 1 - 0.25 = 0.75 > 0$ , timelike. Boost to a frame moving at $\beta_0 = 0.6$ and the same point of the same curve has a new speed, but $ds^2$ is unchanged and still positive — it remains a legal worldline for every observer. Why: the point of the check is that the curve does not become illegal in somebody else's frame. It could not, and now we have seen it not do so.

Figure

PUNCHLINE — four worldlines, the SAME two events, four different clock readings. From A = (0,0) to B = (ct = 10, x = 0) m. The straight worldline reads τ = 10.000 m/c — the MAXIMUM (theorem proved in Proper Time). Every detour reads LESS, and the null zig-zag out to the corner reads exactly ZERO. Note the trap your eye is setting: the longest-LOOKING path is the shortest-AGEING one

Limiting cases

Case	Result	Meaning
$\beta = 0$	worldline vertical, $\tau = \Delta t$	at rest: the clock reads the coordinate time. Even here the worldline exists
$\beta$ constant	$\tau = \Delta t\sqrt{1-\beta^2} = \Delta t/\gamma$	the integral collapses to ordinary time dilation — the check that (3) is the right generalisation
$\beta \to 1$	$\tau \to 0$	the null limit: a light ray accumulates no proper time. The worldline is not attainable by anything massive
straight line	$\tau$ maximal	the reversed triangle inequality (proved in Proper Time) — the opposite of Euclid
reparametrise	$\tau$ unchanged	the owned result: $\tau$ belongs to the curve, not to the description
$c \to \infty$	$\tau \to \Delta t$ always	the Newtonian limit: every worldline between two events reads the same universal time

Worked examples

1 · Two twins, one pair of events, two clock readings

Twin A stays home; twin B flies out at $\beta = 0.6$ , turns around, and comes back. They part at event $P$ and reunite at event $Q$ , $10$ years of A's time later. What does each clock read?

Step 1. Identify the events. $P$ and $Q$ are the two events where the twins' worldlines cross — coincidences, where both clocks are side by side and can be compared with no convention at all. This is what makes the question answerable at all.

Step 2. Twin A's worldline is straight (she is inertial), from $P$ to $Q$ , with $\beta = 0$ . So $\tau_A = \int\sqrt{1-0}\;dt = \mathbf{10.000\ years}$ .

Step 3. Twin B's worldline has two straight legs at $|\beta| = 0.6$ , each lasting $5$ years of A's coordinate time. On each leg, $\sqrt{1 - 0.36} = 0.8$ , so each contributes $5 \times 0.8 = 4$ years: $\tau_B = \mathbf{8.000\ years}$ .

Step 4. Read the result. B is two years younger, and there is no paradox to resolve: the two worldlines are different curves between the same two events, and proper time is a property of the curve. B's worldline is bent; A's is straight; the straight one has the greater proper time, by the theorem in Proper Time. The situation is not symmetric — B turned around and A did not, and you can see the kink in the diagram. Notice, too, that this is the $\tau = 2\sqrt{25 - x^2}$ family of the punchline figure in disguise, with $x = 3$ : $2\sqrt{16} = 8$ . Same numbers, same geometry.

2 · Is this curve a legal worldline?

Test each of these curves, given in metres with $ct$ as the time coordinate.

Step 1. $x(t) = t^2/40$ from $t = 0$ to $10$ . Tangent: $\beta = dx/d(ct) = t/20$ , which runs from $0$ to $0.5$ . Since $|\beta| < 1$ everywhere, this is a legal worldline. Its proper time is $5\pi/3 + 5\sqrt3/2 = \mathbf{9.5661\ m}/c$ versus $10\ \mathrm{m}$ of coordinate time.

Step 2. $x(t) = t^2/10$ from $t = 0$ to $10$ . Tangent: $\beta = t/5$ , which reaches $1$ at $t = 5$ and exceeds it after. Illegal — and note where it fails. It is a perfectly good curve up to $t = 5$ ; from there on it is not a worldline at all. The condition is pointwise, so failing at one point is failing.

Step 3. $x(t) = 3\sin(t/2)$ , any range. Tangent: $\beta = 1.5\cos(t/2)$ , whose maximum magnitude is $1.5 > 1$ . Illegal, even though the curve is bounded, smooth, and looks entirely reasonable. The test is on the tangent, never on the shape.

Step 4. The general recipe, and the boundary number. Differentiate, form $|\beta| = |dx/d(ct)|$ , and find its maximum. If $\max|\beta| < 1$ it is a worldline; if it reaches $1$ anywhere it is not — and $\beta = 1$ is the boundary in every case, because that is $ds^2 = 0$ , the light cone.

Physics of the portrait

Origins · 2 minWorldline and how this was discovered

Worldline

1908–present · Minkowski's "Weltlinie" · the primitive object of relativistic physics

The word is Minkowski's. In "Raum und Zeit" (Cologne, 21 September 1908) he introduced Weltpunkt — world point, an event — and Weltlinie, the worldline: "the everlasting career of the substantial point may be regarded as a curve in the world, a world-line". It sounds like terminology. It was a change of ontology. In Newtonian physics the fundamental object is a particle at a time, and its history is assembled by watching it move; in Minkowski's picture the fundamental object is the whole history, laid out at once, and "motion" is just the shape of that curve. That reframing is why the twin paradox stops being a paradox — the twins' worldlines are simply two different curves between the same two events, and curves have lengths. It is also why the worldline survived every subsequent revolution in physics: in general relativity a free-falling body's worldline is a geodesic, the straightest available curve in a curved spacetime, and Einstein's equations are the rules for how matter bends the curves. In quantum field theory the worldline reappears as the trajectory summed over in a path integral, and the "worldline formalism" computes loop diagrams by integrating over the histories of a single particle. The line of the young mathematician's chalk turned out to be the most durable object in the subject.

History

1632

— Galileo argues in the Dialogue that no experiment inside a smoothly moving cabin can reveal its motion — the first statement that "at rest" is not an absolute property of an object.

1905

— Einstein derives special relativity from two postulates, stating it entirely in terms of clocks, rods and coordinates, and without any geometry.

1907

— Minkowski recasts electrodynamics in four-dimensional form and coins "Eigenzeit" — proper time, the reading of a clock carried along a worldline.

1908

— In "Raum und Zeit", delivered at Cologne on 21 September, Minkowski introduces "Weltpunkt" (event) and "Weltlinie" (worldline) as the primitive objects of physics.

1909

— Minkowski dies on 12 January, four months after the lecture; "Raum und Zeit" appears in Physikalische Zeitschrift 10, 104–111.

1966

— Taylor and Wheeler's Spacetime Physics makes events and worldlines the starting point of the subject rather than a late refinement.

Going further

Texts. Taylor & Wheeler, Spacetime Physics, 2nd ed., Ch. 1 — the definitive gentle treatment of events and worldlines · Minkowski, "Raum und Zeit", Physikalische Zeitschrift 10 (1909) 104–111; translated in The Principle of Relativity (Dover) · Rindler, Introduction to Special Relativity, 2nd ed., §§7–8 · Misner, Thorne & Wheeler, Gravitation, §1.2 for worldlines as the primitive. The event/worldline vocabulary and the tangent condition are standard, and the proper-time integral $\tau = \int\sqrt{1-\beta^2(t)}\,dt$ together with the theorem that the straight worldline has the greatest proper time are owned and proved in Proper Time — this article uses, illustrates and attributes them, and does not re-derive them. What this article owns is the reparametrisation invariance of proper time: written in the form $c\tau = \int\sqrt{\dot x^\mu \dot x_\mu}\,d\lambda$ , the integrand is homogeneous of degree one in the derivatives, so a change of parameter $\lambda = \sigma(\mu)$ produces a factor $\sigma'$ from the chain rule that cancels exactly against the $\sigma'$ from the measure — and hence $\tau$ is a property of the curve as a geometric object, independent not only of the frame but of the description. This is confirmed on the accelerating worldline $ct = t$ , $x = t^2/40$ , $t \in [0, 10]$ m, whose proper time is $5\pi/3 + 5\sqrt3/2 = 9.566114774905182$ m — reproduced to $10^{-9}$ under the reparametrisations $t = u^3/100$ and $t = 10\sin(\pi w/2)$ . The punchline values $\tau = 10.000$ (straight), $8.000$ (one kink at $x=3$ ), $8.667$ (two kinks), $0.000$ (null zig-zag) are likewise confirmed, and every SVG tangent satisfies $|\beta| < 1$ . The symbol $x^\mu$ appears here as notation only; that it is a genuine four-vector is a later story. Signature: $(+,-,-,-)$ , timelike-positive — this article's source material uses the opposite convention throughout, and every sign here has been re-derived rather than transcribed.

Builds on. Inertial Frames of Reference. Leads to. Spacetime Diagrams · The Invariant Spacetime Interval · The Light Cone · Proper Time · Causal Structure · Four-Vectors.

If an object is at rest, why does it have a worldline at all? Nothing is happening.

Something is happening: it is ageing. A worldline is a history, not a route, and the history of a stationary object is a perfectly good history — it exists at noon, and at one o'clock, and at every instant in between, and those are all distinct events. Draw them and you get a vertical line. The deeper point is that "at rest" is not a property an object has on its own; it is a relation to a frame, and a rocket flying past sees that same object moving and draws its worldline as a slanted line. Same worldline, same events, different grid. What both agree on is that the worldline exists and has a definite proper time along it — the reading of the clock the object carries.

What is the difference between a worldline and a trajectory?

A trajectory is a curve in space: the set of places an object visits, with the timing thrown away. A worldline is a curve in spacetime: the set of events at which it exists, timing included. A ball thrown straight up and caught has a trajectory that is a vertical segment traversed twice — and a worldline shaped like an arch, which visits each event exactly once. The worldline carries strictly more information, and it is the object relativity actually cares about: you cannot compute a proper time from a trajectory, because you do not know how long anything took.

Why insist that the tangent condition is "pointwise"? Isn't an average speed below c enough?

It is not enough, and the distinction is not a technicality. If a curve were superluminal even for an instant, there would exist a frame in which that portion of the "history" runs backwards in time — the object would exist at the later event before the earlier one, and cause and effect would be inverted. Averages cannot repair that, because the pathology is local: it lives at that point, in that frame, regardless of what the curve does elsewhere. This is why relativity's speed limit is best stated as a condition on tangents rather than on speeds: it applies at every point independently, and there is no budget to be spent.

Is τ = ∫√(1 − β²) dt not obviously frame-dependent? There's a coordinate time t sitting right in it.

That is exactly the right suspicion, and answering it is this article's owned result. The formula is written with one frame's coordinate time — but the value it returns does not depend on that choice. Rewrite it as $c\tau = \int\sqrt{\dot x^\mu \dot x_\mu}\,d\lambda$ with an arbitrary parameter $\lambda$ , and the chain-rule factor from reparametrising cancels exactly against the factor from the measure, because the integrand is homogeneous of degree one in the derivatives. So any parameter gives the same number — a frame's time, a different frame's time, or a parameter with no physical meaning at all. We check this on a genuinely accelerating worldline with three different parameters and get $9.566114774905182$ m every time. Proper time belongs to the curve.

Can two different worldlines connect the same two events? Isn't that a contradiction?

Not only can they, they are the whole point — and it is no more contradictory than two roads joining the same two cities. Between any two timelike-separated events there are infinitely many worldlines, and they have different lengths, meaning clocks carried along them read differently on arrival. That is the twin paradox, and once you see it as a statement about curves it stops being paradoxical: nobody is troubled that a scenic route is longer than a direct one. The only genuinely surprising part is the sign — in spacetime, the straight worldline is the longest in proper time, so the detour makes you age less, not more. That theorem belongs to Proper Time.

The article writes x^μ. Do I need to know what a four-vector is to read this?

No — here $x^\mu$ is nothing more than shorthand, a single symbol for the four numbers $(ct, x, y, z)$ that label an event, with $\mu$ running over $0,1,2,3$ . That is all it means in this article. The genuinely interesting fact — that these four numbers transform together as a unit under a boost, that this is what makes them a four-vector, and that whole families of physical quantities (momentum, current, potential) share the same transformation law — is a later development, and it is built on top of exactly the event-and-worldline vocabulary established here. You can read every line of this article treating $x^\mu$ as a convenient abbreviation, and lose nothing.