Spacetime diagrams

Units and the diagram

The lab works in one space dimension and in units with c = 1. You measure time by how far light gets in that time, so a year goes with a light year and a metre with about 3.3 nanoseconds. Velocities are fractions of the speed of light, and every light signal runs at 45 degrees on the diagram. Time points up, space points right. This is the convention of Taylor and Wheeler's Spacetime Physics, which uses the same units throughout [4, Ch. 1].

A point on the diagram is an event, a place at a time. A worldline is a body's whole history, every event it occupies. Minkowski gave the reason for drawing them this way in one sentence: "No one has observed a place except at a particular time, or has observed a time except at a particular place" [2]. He opened his 1908 lecture in Cologne with the claim that made the diagram famous: "Henceforth, the old conception of space for itself, and time for itself shall reduce to a mere shadow, and some sort of union of the two will be found consistent with facts" [2].

Events, the interval and light cones

Two observers moving steadily past each other will argue about the time \Delta t and the distance \Delta x between two events. But they agree on one combination, the interval

s^2 = \Delta t^2 - \Delta x^2 .

Its sign sorts every pair of events into three classes [4, Ch. 6]. If s^2 > 0 the pair is timelike: a clock can travel from one to the other, and \sqrt{s^2} is the time it reads if it moves uniformly. If s^2 < 0 the pair is spacelike: no signal can join them, and \sqrt{-s^2} is their distance in the frame where they happen at once. If s^2 = 0 the pair is lightlike: one flash of light connects them.

The lightlike events through a point form its light cone, two lines at 45 degrees in this one-dimensional world. The upper wedge is the event's future and the lower wedge its past. The regions to either side are elsewhere, events it can't reach and that can't reach it.

Two frames on one diagram

The Lorentz transformation from frame S to a frame S' moving at velocity v is

t' = \gamma (t - v x), \qquad x' = \gamma (x - v t), \qquad \gamma = \frac{1}{\sqrt{1 - v^2}} .

Einstein derived it in 1905 from two postulates: the laws of physics take the same form in every inertial frame, and light moves at the same speed in all of them [1, §3]. Substituting shows that t'^2 - x'^2 = t^2 - x^2, which is why the interval is invariant.

On the diagram, the t' axis is the worldline x' = 0, the line x = v t, tilted towards the light line. The x' axis is the line t' = 0, that is t = v x, tilted by the same angle the other way. Both axes squeeze towards the light line, like closing scissors, as v grows. The units along them aren't the paper's units. The event one unit up the t' axis is where that axis crosses the invariant hyperbola t^2 - x^2 = 1, because every observer's unit of proper time lies on it [4, Ch. 5; 2]. Minkowski drew exactly this construction, a hyperbola c^2 t^2 - x^2 = 1 with its asymptotes and a tilted radius vector marking the new unit of time [2]. The lab draws the hyperbolae t^2 - x^2 = \pm n^2 and puts the tick marks of both frames on them.

Because the transformation preserves the interval, a boost slides every event along its own hyperbola. Switch the view between S and S' and you'll see it happen: the lab eases the rapidity of the displayed frame from one value to the other and redraws, so the Lorentz transformation plays out as a motion along the hyperbolae.

Relativity of simultaneity

In section 1 of his paper Einstein defined simultaneity operationally: clocks at A and B are synchronous if light takes as long from A to B as from B to A [1, §1]. In section 2 he showed what follows. In the 1920 translation: "two events which are synchronous when viewed from one system, will not be synchronous when viewed from a system moving relatively to this system" [1, §2].

On the diagram, events simultaneous in S lie on a horizontal line. Events simultaneous in S' lie on lines parallel to the x' axis, which the lab draws as its simultaneity grid. Two lightning strikes at t = 0 and x = \pm 2 happen together in S, but t' = -\gamma v x gives them different times in S'. For the moving frame, the strike ahead comes first.

Time dilation and length contraction

A clock at rest in S' ticks at x' = 0, t' = 0, 1, 2, \dots. In S those ticks land at t = \gamma t', so the moving clock runs slow by \gamma. A light clock makes this concrete. Light bounces between two mirrors, and each round trip is one tick. Each clock in the lab ticks every 2 units of its own time, and the moving one's ticks are 2\gamma apart in S.

A rod at rest in S' with length L_0 has worldlines x' = 0 and x' = L_0. Measured in S, at one instant t, its ends are L_0/\gamma apart [4, Ch. 3; 5, Ch. 3]. That sets up the barn and pole puzzle. A pole of length 2 at v = 0.8 (\gamma = 5/3) is 1.2 long in the barn frame and fits inside a barn of length 1.5. In the pole frame the barn is 1.5 \times 0.6 = 0.9 long. Both are right. "The pole is inside" means "the rear has entered before the front reaches the exit", and those two events are spacelike apart, so their order depends on the frame.

Proper time and the twin paradox

The time a clock reads along its own worldline is its proper time,

\tau = \int \sqrt{dt^2 - dx^2} = \int \frac{dt}{\gamma(t)} .

Minkowski introduced the name: the integral along the world-line "may be called the 'Proper-time'" [2]. Taylor and Wheeler call it wristwatch time [4, Ch. 5]. The lab sums \sqrt{\Delta t^2 - \Delta x^2} over the points of each worldline.

Einstein already drew the consequence in 1905: "if we set in motion one of them with a constant velocity along a closed curve till it comes back to A, the journey being completed in t-seconds, then after arrival, the last mentioned clock will be behind the stationary one", by \tfrac12 t v^2/c^2 to lowest order [1, §4]. That's the twin paradox. A twin who travels out at v = 0.8 for 5 years and back for 5 comes home having aged 10/\gamma = 6 years, while the twin who stayed aged 10.

Draw the worldlines and the paradox goes away. The twins aren't symmetric. The one at home stays on a straight worldline, and the traveller's bends. In Minkowski geometry a straight timelike worldline between two events has the longest proper time of any path between them. That's the reverse of Euclid, where the straight line is shortest, and it happens because the space part enters the interval with a minus sign. Taylor and Wheeler: "In spacetime the greatest aging between two events is experienced by the traveler who does not change direction" [4, Ch. 4; see also Ch. 5]. Every bend, sharp or smooth, costs proper time. The traveller's lines of simultaneity also swing at the turnaround, pointing one way before it and the other way after. That's why the traveller can't just apply the time dilation formula to the twin at home.

Hyperbolic motion

A body with constant proper acceleration a, the acceleration it feels, moves on a hyperbola. Starting from rest at the origin,

x(t) = \frac{\sqrt{1 + (a t)^2} - 1}{a}, \qquad v(t) = \frac{a t}{\sqrt{1 + (a t)^2}}, \qquad \tau(t) = \frac{\operatorname{asinh}(a t)}{a} .

Its velocity creeps towards 1 and never gets there. Minkowski noted that such a hyperbolic branch, read as a world-line, describes a motion that approaches the speed of light in the distant past and future [2]. The light line through the hyperbola's centre is a horizon. Send a signal from beyond it and it never catches the accelerating body.

The red twin uses four such legs: accelerate away for a quarter of the trip, decelerate to the turnaround, accelerate home, decelerate to a stop. Each leg of coordinate time T/4 takes proper time \operatorname{asinh}(a T/4)/a, so the red twin ages 4 \operatorname{asinh}(aT/4)/a. For a = 1 and T = 10 that is 6.59.

Rapidity and velocity addition

Write v = \tanh\varphi. Then \gamma = \cosh\varphi, \gamma v = \sinh\varphi and the boost becomes a hyperbolic rotation:

t' = t \cosh\varphi - x \sinh\varphi, \qquad x' = x \cosh\varphi - t \sinh\varphi .

The parameter \varphi is the rapidity. Robb named it in 1911 and saw its point at once: for small values velocity and rapidity are practically equal, "but we shall see later that, for large values, it is the rapidity and not the velocity which follows the additive law" [3, p. 9]. Two boosts compose into one boost with the summed rapidity, so the velocities combine by the tangent addition formula,

w = \tanh(\varphi_1 + \varphi_2) = \frac{u + v}{1 + u v},

which is Einstein's addition theorem [1, §5; 5, Ch. 3]. Add up as many finite rapidities as you like and you never reach infinity, so no combination of speeds below 1 ever reaches 1. The lab's ladder stacks six boosts of the same rapidity. Their rapidities add linearly, and the unit ticks of every rung fall on the same hyperbolae.

Causality

For a timelike or lightlike pair, |\Delta x| \le |\Delta t|, so \Delta t' = \gamma(\Delta t - v \Delta x) has the sign of \Delta t for every |v| < 1: all observers agree which came first. For a spacelike pair |\Delta t| < |\Delta x|, and any frame with v \Delta x > \Delta t reverses the order [4, Ch. 6]. So if anything could travel faster than light, some observer would see the effect before the cause. In special relativity the light cone is the boundary of cause and effect.

The Doppler factor

A source moving towards an observer at speed v emits pulses every \Delta\tau of its own time. In the observer's frame they leave every \gamma\Delta\tau, and each starts v\gamma\Delta\tau closer, so they arrive every \gamma(1 - v)\Delta\tau. The received frequency is raised by

k = \frac{1}{\gamma (1 - v)} = \sqrt{\frac{1 + v}{1 - v}} = e^{\varphi} .

For a receding source v < 0 and k < 1. Einstein derived the general form with the angle of observation in his section on Doppler's principle and aberration [1, §7]; Rindler treats it in his chapter on relativistic optics [5, Ch. 4]. Notice k = e^{\varphi}: Doppler factors multiply when rapidities add. The lab measures k from the arrival times of the drawn pulses and plots it against the exact curve.

Experimental tests

None of this is only on paper. People have measured it. Rossi and Hall measured how the decay rate of cosmic ray mesotrons, now called muons, depends on their momentum [6]. Ives and Stilwell measured the rate of a moving atomic clock through the Doppler-shifted light of fast hydrogen ions [7].

In 1971 Hafele and Keating carried four caesium clocks around the world on ordinary commercial flights, once eastward and once westward. Their prediction, which includes the gravitational shift from altitude as well as the motion, was a loss of 40 \pm 23 ns eastward and a gain of 275 \pm 21 ns westward [8]. Relative to the US Naval Observatory the clocks lost 59 \pm 10 ns eastward and gained 273 \pm 7 ns westward. In their words: "These results provide an unambiguous empirical resolution of the famous clock "paradox" with macroscopic clocks" [9].

At CERN, Bailey and collaborators stored muons with \gamma = 29.33 in a ring and measured lifetimes of 64.419(58)\ \mu\text{s} for positive muons, against about 2.2\ \mu\text{s} at rest. Their abstract reports that "the Einstein time dilation factor agrees with experiment with a fractional error of 2×10⁻³ at 95% confidence" [10].

Things to try in the lab

• Relativity of simultaneity. The strikes sit on one horizontal line; the passenger's flashes arrive apart. Turn on the grid and drag the strikes onto one blue simultaneity line: now the passenger sees them together and the embankment does not.
• Time dilation with light clocks. Count ticks: the blue clock ticks every 2.5 in S at v = 0.6. Switch "Drawn from" to S' and watch the diagram boost; now the ink clock's ticks are spread out.
• Barn and pole. Select the two events and read t and t' for each. Their order flips between the frames, and s^2 < 0 says it is allowed to.
• Twin paradox. Compare the three proper times and the proper time plot. Raise a: the red path approaches the sharp-cornered one. Lower v: the blue twin's advantage disappears as 1 - \gamma^{-1}.
• Velocity addition. Push both sliders to 0.95. The combined velocity reads 0.9987 and the red worldline hugs, but never crosses, the light line.
• Causality. Slide v slowly through 0.3. The sign of t'_B - t'_A changes; drag C around inside A's cone and its order never does. Drag B onto the cone and the verdict turns lightlike.
• Relativistic Doppler. Read the measured factor and compare the dot with the curve. Set v = -0.6: the factor is exactly 0.5.
• Ladder of rapidities. Change v and watch the rungs crowd towards the light line while their ticks stay on the hyperbolae.
• Free events. Click to add events, drag P outside O's cone and back and watch the blue worldline stop being a possible clock. Change a and compare the red clock with \operatorname{asinh}(a t)/a.

Reading the lab's instruments

• Diagram. Ink axes t and x belong to S, blue axes t' and x' to S'. Small dots on the axes and on worldlines mark whole units of proper time; on the axes they sit on the faint invariant hyperbolae. Dashed lines are light. Shaded wedges, when on, are the light cones of A and B.
• Lorentz factor and rapidity of S′. \gamma and \varphi = \operatorname{atanh} v for the frame slider.
• Interval s², A to B. \Delta t^2 - \Delta x^2 for the two selected events; the line under the diagram classifies it and states their time order in both frames.
• Coordinates table. t, x, t' and x' of A and B.
• Proper time, ink / blue / red. Proper time summed along that colour's worldlines; n/a where a scene has none.
• Combined velocity w. The red worldline's slope in the addition scene, the top rung's velocity in the ladder, (u + v)/(1 + uv) elsewhere.
• Doppler factor, measured. Ratio of emission to reception intervals for the pulse train at the current v.
• γ against v. The exact curve, with a dot at the current v computed by transforming the event one unit up the t' axis into S.
• Proper time of the twins. \tau against t for the three twins at the current v and a; the dots are their ages at the reunion.
• Doppler factor against v. The exact curve \sqrt{(1+v)/(1-v)} with the measured value as a dot.

References

[1] A. Einstein, "Zur Elektrodynamik bewegter Körper," Annalen der Physik 322, 891-921 (1905). DOI: 10.1002/andp.19053221004. Quotations from the English translation by M. N. Saha, "On the Electrodynamics of Moving Bodies," in The Principle of Relativity (University of Calcutta, 1920), as transcribed on Wikisource

[2] H. Minkowski, "Raum und Zeit," lecture at the Versammlung Deutscher Naturforscher und Ärzte, Cologne, 21 September 1908, Jahresbericht der Deutschen Mathematiker-Vereinigung 18, 75-88 (1909); German text on Wikisource. Quotations from the English translation by M. N. Saha, "Space and Time," in The Principle of Relativity (University of Calcutta, 1920), pp. 70-88, as transcribed on Wikisource

[3] A. A. Robb, Optical Geometry of Motion: A New View of the Theory of Relativity (W. Heffer and Sons, Cambridge, 1911), p. 9. Internet Archive

[4] E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (W. H. Freeman, New York, 1992). Ch. 1 "Spacetime: Overview"; Ch. 3 "Same Laws for All"; Special Topic "Lorentz Transformation"; Ch. 4 "Trip to Canopus"; Ch. 5 "Trekking through Spacetime"; Ch. 6 "Regions of Spacetime". Free under CC BY 4.0 at eftaylor.com

[5] W. Rindler, Relativity: Special, General, and Cosmological, 2nd ed. (Oxford University Press, 2006). Ch. 2 "Foundations of special relativity; The Lorentz transformation"; Ch. 3 "Relativistic kinematics"; Ch. 4 "Relativistic optics"; Ch. 5 "Spacetime and four-vectors". DOI: 10.1093/oso/9780198567318.001.0001

[6] B. Rossi and D. B. Hall, "Variation of the Rate of Decay of Mesotrons with Momentum," Physical Review 59, 223-228 (1941). DOI: 10.1103/PhysRev.59.223

[7] H. E. Ives and G. R. Stilwell, "An Experimental Study of the Rate of a Moving Atomic Clock," Journal of the Optical Society of America 28, 215 (1938). DOI: 10.1364/JOSA.28.000215

[8] J. C. Hafele and R. E. Keating, "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains," Science 177, 166-168 (1972). DOI: 10.1126/science.177.4044.166

[9] J. C. Hafele and R. E. Keating, "Around-the-World Atomic Clocks: Observed Relativistic Time Gains," Science 177, 168-170 (1972). DOI: 10.1126/science.177.4044.168

[10] J. Bailey et al., "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit," Nature 268, 301-305 (1977). DOI: 10.1038/268301a0