Glossary

A

Aberration

In plain words Run through falling rain and the drops seem to come at you from in front. Moving through starlight does the same to light: the directions you'd point to see the stars crowd toward the way you're heading.

Precisely The change in a light ray's direction between two frames. A ray at angle \(\theta'\) to the motion in the source frame arrives at angle \(\theta\) in the lab, with \(\cos\theta = (\cos\theta' + v)/(1 + v\cos\theta')\). Light sent sideways (\(\theta' = 90^\circ\)) appears at \(\cos\theta = v\): 53.13° at \(v = 0.6\).

First met in Chapter 12 · See also Doppler factor, photon

B

Boost

In plain words Switching from one free-float frame to another that moves at a steady speed relative to it. On a spacetime diagram, a boost scissors the axes and slides every event along its own teal hyperbola.

Precisely A Lorentz transformation along one direction with no rotation, labeled by the relative speed \(v\) or by the rapidity \(\theta\). Two boosts along the same line combine by adding their rapidities, not their speeds.

First met in Chapter 6 · See also Lorentz transformation, rapidity, calibration hyperbola

C

Calibration hyperbola

In plain words The curve that tells you where "one meter of time" lands on a moving clock's axis. Spacetime has a minus sign in it, so the curve is a hyperbola where you might expect a circle.

Precisely The curves \(t^2 - x^2 = \pm n^2\). The rocket's \(t'\) axis crosses \(t^2 - x^2 = n^2\) at the event \(t' = n\); its \(x'\) axis crosses \(x^2 - t^2 = n^2\) at \(x' = n\). On a lab drawing, unit ticks on the rocket axes are spaced \(\sqrt{(1+v^2)/(1-v^2)}\) apart: 1.458 at \(v = 0.6\). Drawing equal ticks on both frames' axes is a classic mistake.

First met in Chapter 10 · See also interval, spacetime diagram, boost

Causality

In plain words The rule that a cause comes before its effect, for everyone. One event can affect another only if a signal going at light speed or slower could get from the first to the second.

Precisely Event A can influence event B only if B lies inside or on A's future light cone: the pair is timelike or lightlike, with \(\Delta t > 0\). Every free-float frame agrees on the time order of such pairs. Spacelike pairs can't influence each other, and their time order depends on the frame.

First met in Chapter 6 · See also light cone, timelike, spacelike, tachyon

D

Doppler factor

In plain words How much faster you receive ticks, flashes or wave crests from a source that's coming toward you, compared with how fast it sends them. If it's moving away, you receive them slower by the same factor.

Precisely \(k = \sqrt{(1+v)/(1-v)}\). An approaching source sending at frequency \(f\) is received at \(kf\); a receding one at \(f/k\). At \(v = 0.6\), \(k = 2\). Signed form: \(D = \sqrt{(1+v)/(1-v)}\) with \(v < 0\) for a receding source gives \(1/k\); that is what dopplerFactor(v) returns. It is tied to gamma by \(\gamma = (k + 1/k)/2\), and once rapidity arrives, \(k = e^{\theta}\).

First met in Chapter 10 (as the light-line factor \(k\)); counted as birthday cards in Chapter 11; explained in Chapter 12 · See also reckon versus see, rapidity, aberration

E

Elsewhere see light cone and spacelike

Energy

In plain words Everything a moving object "has" in one number: part is locked up in its mass, the rest comes from its motion. With our units, energy and mass are measured in the same unit.

Precisely For a particle of mass \(m\) at speed \(v\), \(E = \gamma m\): the time part of its momenergy arrow. Total energy is conserved, but it depends on the frame, so different observers give different \(E\) for the same particle. At rest \(E = m\), which is \(E = mc^2\) written with \(c = 1\).

First met in Chapter 14 · See also momenergy, mass, kinetic energy, momentum

Ether

In plain words An invisible substance that nineteenth-century physicists imagined filling all of space, with light as waves in it, the way sound is waves in air. If it existed, light's speed would depend on how you move through it. Nobody ever found it.

Precisely A hypothetical medium that would single out one rest frame for light. It predicts that light round trips take different times along and across a moving lab's direction of travel; Michelson and Morley's 1887 experiment and many later ones found no such difference. Relativity has no ether: light's speed is 1 in every free-float frame.

First met in Chapter 2 · See also principle of relativity, frame

Event

In plain words Something that happens at one place at one instant: a firecracker going off, a clock ticking, two bumpers touching. An event is a single point in spacetime.

Precisely A point labeled by coordinates \((t, x)\), or \((t, x, y, z)\), in some frame. Different frames give the same event different coordinates, but all of them agree that it happened. We draw events as dots with a letter beside them.

First met in Chapter 0; defined in Chapter 3 · See also worldline, interval, spacetime diagram

F

Four-momentum see momenergy

Frame (reference frame)

In plain words A way of giving every event a time and a place: a whole framework of measuring rods and clocks moving together. It is not one person's point of view.

Precisely A lattice of rods with a synchronized clock at every joint, all at rest relative to one another. In this course "frame" means a free-float (inertial) frame unless we say otherwise. Two observers at rest relative to each other share one frame, however far apart they stand.

First met in Chapter 2; the lattice is built in Chapter 3 · See also lattice of clocks, observer, lab frame, rocket frame

Free-float frame

In plain words A room, with its rulers and clocks, in which anything you let go stays put or coasts in a straight line at a steady speed. A spaceship coasting far from any star is one. A freely falling elevator is very nearly one.

Precisely A frame in which free particles move in straight lines at constant speed, to a stated accuracy, over a stated region and time. Textbooks say inertial frame. Near Earth it can only be local: two balls released side by side in a falling room slowly drift together, so the room is free-float only over limited space and time.

First met in Chapter 2 · See also frame, principle of relativity, maximal aging

G

Gamma (\(\gamma\)), the stretch factor

In plain words The number that says how much relativity matters at a given speed. At everyday speeds it is 1, to many decimal places. At 60% of light speed it is 1.25.

Precisely \(\gamma = 1/\sqrt{1 - v^2}\). It is never less than 1 and grows without limit as \(v\) approaches 1. Friendly values: \(v = 3/5\) gives \(5/4\); \(v = 4/5\) gives \(5/3\); \(v = 12/13\) gives \(13/5\). In code, 1 / Math.sqrt(1 - v * v) works at every speed below 1; very close to 1, the limit is how precisely you know \(v\) itself, so physicists work with \(1 - v\) directly.

First met in Chapter 5 · See also time stretching, Lorentz contraction, speed v

I

Inertial frame see free-float frame

Interval

In plain words The one answer to "how far apart are these two events?" that every observer agrees on. It mixes the time between them with the distance between them, using a minus sign.

Precisely The squared interval \(\tau^2 = \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2\), the same in every free-float frame. Its sign sorts pairs of events: timelike (positive), lightlike (zero) or spacelike (negative). For a timelike pair, \(\tau\) is the wristwatch time of a clock that coasts straight from one event to the other. Example: \(\Delta t = 5\), \(\Delta x = 3\) gives \(\tau = 4\).

First met in Chapter 5; the hero of Chapter 6 · See also invariant, proper time, proper distance, timelike

Invariant

In plain words A quantity every observer agrees on, however they're moving. On this site invariants are drawn in teal, with a double line or the ≡ sign.

Precisely A quantity with the same value in every free-float frame: the interval between two events, the mass of a particle, the speed of light, the wristwatch time along a given worldline. Coordinates, time differences, lengths, energies and momenta are frame-dependent. Relativity is, at heart, a theory of invariants.

First met in Chapter 0; the surveyors' parable in Chapter 1 · See also interval, mass, proper time

K

Kinetic energy

In plain words The energy an object has because it's moving: whatever it has beyond the energy stored in its mass.

Precisely \(K = E - m = (\gamma - 1)m\). At small speeds this is very close to Newton's \(\tfrac12 m v^2\), because \(\gamma \approx 1 + v^2/2\). Unlike Newton's formula, it grows without limit as \(v\) approaches 1, which is why no push gets anything with mass up to light speed. A proton at \(v = 0.6\) has \(K \approx 234.6\un{MeV}\).

First met in Chapter 14 · See also energy, mass, gamma

L

Lab frame

In plain words Lin's frame: the one we choose to call "at rest" in a given story. Drawn in blue, with solid lines.

Precisely The unprimed frame, with coordinates \(t\) and \(x\). In the standard setup the rocket frame moves at \(+v\) along the lab's \(+x\) axis, and the two frames share the origin event. Calling it "the lab" is a choice, not a privilege: the laws of physics are the same in both frames.

First met in Chapter 0 · See also rocket frame, frame

Lattice of clocks

In plain words Picture a giant jungle gym of meter rods with a clock at every joint. To time an event, ask the clock right beside it. No light delay, no guessing.

Precisely The physical build behind a frame: rods at rest relative to each other, with a clock at each joint, synchronized by Einstein's method. The reference clock flashes at \(t = 0\); every other clock is preset to its distance from the reference clock and starts when the flash arrives. Lattice readings are what we mean by reckon or measure.

First met in Chapter 0; built in Chapter 3 · See also frame, reckon versus see, simultaneity

Length contraction see Lorentz contraction

Light cone

In plain words Every event a flash of light from here could reach, or could have come from. On a spacetime diagram it's the pair of 45° lines through an event. Everything you could ever affect is inside its upper half.

Precisely For an event O, the set of events lightlike-separated from it: \(\Delta t^2 = \Delta x^2 + \Delta y^2 + \Delta z^2\). It splits spacetime into O's future (inside, later), its past (inside, earlier) and elsewhere (outside). Every free-float frame agrees on where the cone is.

First met in Chapter 6; 45° light lines are drawn from Chapter 4 · See also causality, lightlike, timelike, spacelike

Lightlike

In plain words Two events are lightlike-separated when a flash of light could go straight from one to the other: the distance between them equals the time between them.

Precisely \(\tau^2 = 0\); in one dimension, \(|\Delta x| = |\Delta t|\). Every frame agrees, because light's speed is the same in every frame. Each event sits exactly on the other's light cone.

First met in Chapter 6 · See also interval, light cone, photon

Light-line factor see Doppler factor

Light-year

In plain words The distance light travels in one year, about 9.46 trillion kilometers. Despite the name, it is a distance, not a time.

Precisely 1 ly = 9.4607304725808 × 10¹⁵ m, using the Julian year of 365.25 days. Measure time in years and distance in light-years and light's speed is exactly 1, just as with meters of time and meters.

First met in Chapter 1 · See also meters of time, speed v

Line of simultaneity see simultaneity

Lorentz contraction (length contraction)

In plain words A moving rod measures shorter, along its direction of motion, than the same rod at rest. Nothing squeezes it. Measuring a length means marking both ends at the same "now", and different frames disagree about "now".

Precisely \(L = L_0/\gamma\), where \(L_0\) is the proper length. It applies only along the motion; sizes across the motion don't change. Each frame measures the other's rods as shortened. A photograph of a fast object is a separate question, because light from its far parts set out earlier.

First met in Chapter 9 · See also proper length, simultaneity, gamma

Lorentz factor see gamma

Lorentz transformation

In plain words The recipe that turns Lin's coordinates for an event into Rae's. It is the single machine behind time stretching, length contraction and the relativity of "now".

Precisely \(t' = \gamma(t - vx)\) and \(x' = \gamma(x - vt)\), with \(y' = y\) and \(z' = z\), for a rocket moving at \(+v\) along \(x\) with a shared origin. The inverse swaps \(v\) for \(-v\). It keeps the interval unchanged: \(t'^2 - x'^2 = t^2 - x^2\). Example: \((t, x) = (5, 3)\) at \(v = 0.6\) becomes \((4, 0)\).

First met in Chapter 10 · See also boost, interval, gamma, rocket frame

M

Mass

In plain words How much "stuff" something is, in the sense every observer agrees on. In this course mass never changes with speed.

Precisely The invariant length of the momenergy arrow: \(m^2 = E^2 - p^2\), the same in every frame. A system's mass is not the sum of its parts' masses: a hot kettle has more mass than a cold one, and two photons flying apart have a system mass even though each has none. Older books use "relativistic mass" for \(\gamma m\). That quantity is the energy, so we call it energy.

First met in Chapter 14 · See also momenergy, system mass, energy, invariant

Maximal aging

In plain words Of all the ways to travel between two events, coasting straight with no pushes gives you the most time on your wristwatch. Any detour makes you younger.

Precisely Among all worldlines joining two timelike-separated events, the free-float (straight) one has the greatest proper time: the triangle inequality, reversed. It settles the twin paradox. In our canonical trip the traveler's bent path ages 8 years while the stay-at-home ages 10.

First met in Chapter 11 · See also proper time, worldline, interval

Meters of time

In plain words Measuring time in meters. One meter of time is how long light takes to cross one meter, about 3.34 billionths of a second. With time and distance in the same unit, light's speed is exactly 1.

Precisely 1 m of time \(= 1/299\,792\,458\) s \(=\) 3.3356 ns, exact because 299 792 458 m/s has defined the meter since 1983. So 1 s \(=\) 299 792 458 m of time. In these units speeds are fractions of light speed, and diagrams, \(\gamma\) and intervals all come out clean.

First met in Chapter 1 · See also light-year, speed v

Momenergy

In plain words Energy and momentum joined into one arrow in spacetime, the way time and space are joined. Observers disagree about its parts but agree about its length, which is the mass.

Precisely The arrow \((E, p) = (\gamma m, \gamma m v)\), with \(E^2 - p^2 = m^2\). In any collision the total momenergy is conserved: the pieces' arrows, added head to tail, give the same total before and after. The word comes from Taylor and Wheeler; most textbooks say four-momentum.

First met in Chapter 14 · See also energy, momentum, mass, system mass

Momentum

In plain words The "oomph" of a moving object along its direction of motion: how hard it is to stop. In relativity it grows without limit as the speed approaches light speed.

Precisely \(p = \gamma m v\), the space part of the momenergy arrow. Total momentum is conserved in collisions in every free-float frame, but only with the \(\gamma\): Newton's \(mv\) could be conserved in one frame and fail in another. Light carries momentum too; for a photon \(|p| = E\).

First met in Chapter 14 · See also momenergy, energy, photon

Muon

In plain words A heavier cousin of the electron, with the same charge and about 207 times the mass. Left alone it lives on average about 2.2 millionths of a second, then decays into an electron and two neutrinos. Cosmic rays make muons high in the atmosphere, and a steady rain of them reaches the ground.

Precisely Mass 105.658 MeV; mean lifetime 2.1969811 μs, measured on the muon's own clock (its proper time). A frame in which a muon moves at \(v\) measures it living \(\gamma\) times longer on average.

First met in Chapter 0; time stretching measured in Chapter 8 · See also time stretching, proper time, gamma

N

Now-line see simultaneity

O

Observer

In plain words In relativity an observer is a whole frame: the entire lattice of clocks and rods, not one person's eyes. Two people standing still a kilometer apart are the same observer.

Precisely A free-float frame's lattice of synchronized clocks, used to give events coordinates. Each event is timed by the clock right at it, so an observer's report never includes light-travel delay. What one person sees is a different question.

First met in Chapter 3 · See also frame, lattice of clocks, reckon versus see

P

Photon

In plain words A particle of light. It carries energy and momentum but has no mass, and it always moves at light speed.

Precisely A massless particle: \(m = 0\), so \(E = |p|\), and its momenergy arrow lies along a 45° line. No frame exists in which a photon is at rest, so "how much time passes for the photon" has no answer. A pair of photons flying apart does have a system mass.

First met in Chapter 6; its momenergy in Chapter 14 · See also lightlike, momenergy, system mass

Principle of relativity

In plain words No experiment inside a sealed, smoothly moving room can tell you that you're moving. The laws of physics, the speed of light included, are the same in every free-float frame.

Precisely Every law of physics takes the same form in every free-float frame. Galileo stated it for mechanics in 1632, using a ship's cabin. Keeping it for light as well forces light's speed to be the same in all such frames, and that is what costs us a universal time.

First met in Chapter 2 · See also free-float frame, ether, frame

Proper distance

In plain words For two events too far apart for anything to travel between them: the distance between them in the one frame where they happen at the same time.

Precisely For a spacelike pair, \(\sigma = \sqrt{\Delta x^2 - \Delta t^2}\) in one dimension (in 3D, \(\Delta x^2\) becomes \(\Delta x^2 + \Delta y^2 + \Delta z^2\)), which is \(\sqrt{-\tau^2}\). It is invariant. Example: \(\Delta t = 3\) y and \(\Delta x = 5\) ly give \(\sigma = 4\) ly.

First met in Chapter 6 · See also interval, spacelike, proper length

Proper length

In plain words The length of an object measured by someone riding along with it. Every other frame measures it shorter along the motion.

Precisely \(L_0\), the distance between an object's ends in its own rest frame. It is the longest length any frame measures along the motion; a frame in which the object moves at \(v\) measures \(L_0/\gamma\).

First met in Chapter 7; explained in Chapter 9 · See also Lorentz contraction, proper distance

Proper time (wristwatch time)

In plain words The time on a clock that is present at both events, carried along from one to the other. We call it wristwatch time. It is the time you actually age.

Precisely \(\tau\) along a worldline: the sum of \(\sqrt{\Delta t^2 - \Delta x^2}\) over its straight pieces. For a straight worldline it equals the interval between its ends, the same in every frame. The habit to build: before saying a clock "runs slow", ask which clock is present at both events.

First met in Chapter 0 (the clockheads); defined in Chapter 5 · See also interval, time stretching, maximal aging

R

Rapidity

In plain words Another way to measure speed, and the one that adds up honestly: a boost of rapidity 0.5 on top of another of 0.5 gives a rapidity of exactly 1 (a speed of about 0.76). Speed creeps toward light speed and stalls; rapidity keeps growing.

Precisely \(\theta = \operatorname{artanh} v = \tfrac12 \ln\dfrac{1+v}{1-v}\), so \(v = \tanh\theta\), \(\gamma = \cosh\theta\), \(\gamma v = \sinh\theta\), and the Doppler factor is \(k = e^{\theta}\). Boosts along one line add their rapidities. At \(v = 0.6\), \(\theta = \ln 2 \approx 0.693\).

First met in Chapter 13 · See also boost, Doppler factor, gamma

Reckon versus see

In plain words To reckon (or measure) when something happened, ask the clock that was right there. To see it, wait for its light to reach your eye, which is always late news. Much of the confusion in popular accounts of relativity comes from mixing up the two.

Precisely Reckoned coordinates come from the lattice clock at the event and contain no light-travel delay. Seen times add the travel time: \(t_{\text{seen}} = t_{\text{event}} + \text{distance}\), in meters of time. A moving clock is measured slow by \(\gamma\), yet an approaching one is seen running fast, by the Doppler factor \(k\).

First met in Chapter 3 · See also lattice of clocks, observer, Doppler factor

Relativistic mass an outdated name for energy; see mass

Rest energy see energy

Rocket frame

In plain words Rae's frame, moving at a steady speed \(v\) along the lab's \(x\) direction. Drawn in vermilion, with dashed lines.

Precisely The primed frame, with coordinates \(t'\) and \(x'\). In the standard setup it moves at \(+v\) along the lab's \(+x\) axis, and both frames call the shared origin event \((0, 0)\). On a lab diagram its time axis is the line \(x = vt\) and its space axis is the line \(t = vx\): the scissored axes.

First met in Chapter 0 · See also lab frame, Lorentz transformation, boost

S

Simultaneity

In plain words Whether two events happen "at the same time". In relativity the answer depends on the frame: two firecrackers that go off together for Lin can go off one after the other for Rae.

Precisely Two events are simultaneous in a frame when its lattice clocks give them the same \(t\). For events separated along the motion, simultaneity is relative: a lab-simultaneous pair a distance \(\Delta x\) apart has \(\Delta t' = -\gamma v \Delta x\) in the rocket. On a lab diagram the rocket's lines of "now" are the tilted lines \(t = vx + t'/\gamma\).

First met in Chapter 7 · See also spacelike, lattice of clocks, Lorentz contraction

Spacelike

In plain words Two events are spacelike-separated when they are too far apart in space for even light to connect them. Neither can cause the other, and different frames can disagree about which came first.

Precisely \(\tau^2 < 0\); in one dimension, \(|\Delta x| > |\Delta t|\). Each event lies in the other's "elsewhere", outside its light cone. Some frame sees the two as simultaneous, and their separation in that frame is the proper distance.

First met in Chapter 6 · See also interval, proper distance, causality, simultaneity

Spacetime diagram

In plain words A graph with time running up and space running across. Each event is a dot and each object's history is a line. Light always runs at 45°.

Precisely A plot of events in one frame's \((x, t)\) coordinates, on equal scales. Worldlines of anything carrying a clock are steeper than 45°; light lines are exactly 45°. Another frame's axes appear scissored toward the light line, with their tick marks set by the calibration hyperbolas.

First met in Chapter 0; drawn properly in Chapter 4 · See also event, worldline, light cone, calibration hyperbola

Speed \(v\)

In plain words Speed as a fraction of light speed. \(v = 0.6\) means 60% of light speed. Because it's a fraction, it has no units.

Precisely \(v = \Delta x/\Delta t\) in some frame, with distance and time in the same unit, so light has \(v = 1\) and anything carrying a clock has \(-1 < v < 1\). Textbooks write \(\beta = v/c\); with our units, that is \(v\).

First met in Chapter 0 · See also meters of time, gamma, rapidity

Stretch factor see gamma

System mass

In plain words The mass of a whole group of things taken together. It is not the sum of the pieces' masses: their motion, and the energy stored between them, count too.

Precisely For a system with total energy \(E\) and total momentum \(p\), \(M^2 = E^2 - p^2\). Two 1 kg clay balls meeting head-on at \(\pm 0.6\) stick into a 2.5 kg lump. Two photons of energy \(E\) each, flying apart in opposite directions, make a system of mass \(2E\).

First met in Chapter 14 · See also mass, momenergy, photon

T

Tachyon

In plain words A made-up particle or signal that goes faster than light. None has ever been found, and relativity says you should hope none ever is: with them you could get a reply before you asked the question.

Precisely Anything whose worldline is flatter than 45° (\(|v| > 1\)), so it links spacelike-separated events. The time order of such events depends on the frame, so two tachyon relays moving differently could deliver a message into its sender's past: the "tachyon telephone".

First met in Chapter 6; the tachyon telephone in Chapter 7 · See also causality, spacelike, simultaneity

Time dilation see time stretching

Time stretching (time dilation)

In plain words A clock moving through a frame's lattice is measured, by that lattice, to tick slower than the lattice's own clocks. Each frame measures the other's clocks as slow, and both are right, because they compare different pairs of events.

Precisely \(\Delta t = \gamma\tau\). If one clock is present at both events and reads \(\tau\) between them, a frame in which that clock moves at \(v\) measures \(\gamma\tau\) between the same two events, using two of its own lattice clocks. It is not a trick of light delay: muons really do reach the ground.

First met in Chapter 5; all of Chapter 8 · See also proper time, gamma, simultaneity

Timelike

In plain words Two events are timelike-separated when something slower than light could be present at both. Every frame agrees on which came first.

Precisely \(\tau^2 > 0\); in one dimension, \(|\Delta t| > |\Delta x|\). Each event is inside the other's light cone. Some frame has both events at the same place, and the time between them in that frame is the proper time \(\tau\).

First met in Chapter 6 · See also interval, proper time, causality

Twin paradox

In plain words One twin stays home while the other flies to a star and back. When they meet again, the traveler is younger. It feels like a paradox, because each twin saw the other moving. But the two trips are not alike: only the traveler turns around and changes frames.

Precisely Each twin ages by the proper time along their own worldline, and between the same two events the straight worldline has the most. Out to a star 3 light-years away and back at \(v = 0.6\), the twin at home ages 10 years and the traveler 8.

First met in Chapter 11 · See also maximal aging, proper time, Doppler factor

V

Velocity addition

In plain words How speeds combine when a rocket launches a rocket. Speeds don't add: a probe fired at half light speed from a rocket moving at half light speed goes 0.8, not 1. No combination of speeds below light's ever reaches it.

Precisely If something moves at \(u\) in a frame that itself moves at \(v\), its speed in the original frame is \(w = \dfrac{u + v}{1 + uv}\). Rapidities, not speeds, are what add.

First met in Chapter 13 · See also rapidity, boost, speed v

W

Worldline

In plain words The line on a spacetime diagram that traces something's whole history: where it was at every moment. A thing sitting still has a straight, vertical worldline.

Precisely The set of all events an object is present at. Its slope gives speed, \(v = \Delta x/\Delta t\), so steeper means slower, and a clock's worldline is always steeper than 45°. Proper time is measured along it.

First met in Chapter 4 · See also event, spacetime diagram, proper time

Wristwatch time see proper time