Illustrated Physics
physics, drawn
Concept

Invariant interval — where it appears

The combination of a time and a distance between two events that every observer agrees about, and whose sign classifies the pair. A time-like interval means one event can influence the other and every observer agrees which came first; a space-like one means neither can, and the order is a matter of frame.

Named by 10 essays across one field — each of them below, with the objects they name alongside it.

A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

Two axes, one speed, and a diagram that does the arguing

Put position across and time up, insist that light travels at forty-five degrees for everyone, and nearly every result in special relativity becomes something to read off rather than derive.

relativity · Spacetime diagram A light clock at β = 0.6. The same clock at rest and moving. Light covers the hypotenuse rather than the height, and since its speed is the same for both observers, the moving clock must take longer to tick.

The clock that has to slow, and why no clock can refuse

One constant speed and one right-angled triangle force a moving clock to tick slower. The argument is Pythagoras, which is what makes it inescapable rather than merely surprising.

relativity · Time dilation A spacetime diagram at β = 0.5. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.5 of the speed of light.

The length that depends on when, and is not really about length

A moving object is measured shorter. The contraction is real, it is not an illusion of light travel time, and it turns out to be a disagreement about simultaneity wearing a different costume.

relativity · Length contraction Total energy against speed, in units of the rest energy. The total energy of a moving body divided by its rest energy, against speed as a fraction of the speed of light. The Newtonian answer, one plus half v squared over c squared, is drawn beside it: the two agree to 0.004 per cent at a tenth of light speed and disagree by 39 per cent at nine-tenths. The relativistic curve has a vertical asymptote at c, which is why nothing with mass reaches it.

Mass is a form of energy, which is not the same as a source of it

The famous equation is usually read as a promise that mass can be turned into energy. It says something stricter and stranger — that a mass is an energy already, sitting there, whether or not anything ever releases it.

relativity · Mass-energy What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The quantity nobody argues about

Relativity takes away the length of a rod and the duration of an event and hands back exactly one thing in their place. Its hyperbolae are what put a scale on the tilted axes of a spacetime diagram — without which the diagram is a picture with no units on it.

relativity · Spacetime diagram What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The invariant that survives a boost

Energy and momentum are both answers to the question "how fast is it going, and according to whom". One combination of them is not, and that combination is the mass — which is why two photons of 511 keV can be a thing of mass 1.022 MeV or a thing of no mass at all, depending only on the angle between them.

relativity · Mass-energy Simultaneity at β = 0.866. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

The pole that fits and does not fit

A twenty-metre ladder is carried through a ten-metre barn at 0.866 of light speed, and both doors shut behind it. In the ladder's own frame the barn is five metres long and there is plainly no room. Both accounts are correct, and the doors' closings are 57.8 nanoseconds apart in one of them.

relativity · Length contraction A horizon 0.97 light years behind, made by nothing but the motion. Position across and time up, in units where light travels at 45°, for a rocket holding a constant proper acceleration of 1 gravity. The worldline is the hyperbola x² − c²t² = (c²/a)², asymptotic to the light line it never crosses. Three light signals are drawn: one released at x = 0.55 catches up at t = 0.63, one released at x = 0 never arrives, one released at x = -0.6 never arrives. The dividing line is the asymptote itself. Everything at or behind it is permanently out of reach, and for one gravity that boundary sits 0.97 light years behind the rocket's starting point. Nothing is there — no mass, no field, no surface. The horizon is a consequence of never stopping.

The wall of silence behind a rocket that never stops

Hold a constant acceleration and the worldline is a hyperbola asymptotic to a light ray — so there is a light ray that never catches it. An observer who never stops accelerating has a horizon a distance c²/a behind, made by nothing but the motion, and at one gravity it sits 0.97 light years back. Nothing is there. No mass, no surface, no field.

relativity · Accelerated frames Where the second tick actually goes. A spacetime diagram with a second observer's axes at β = 0.6. The hyperbolae are the sets of events one second and one metre from the origin — invariantly, by the interval — and every observer's unit tick is where their own axis crosses them. That is checked here rather than drawn by eye. The moving observer's one-second mark sits 1.458 times further from the origin on the page than the stationary one's, so a ruler laid on this picture reads the two frames on different scales. The picture is not distorted; the page is Euclidean and spacetime is not. The Lorentz factor here is 1.2500.

The diagram a ruler cannot read

A spacetime diagram is drawn on flat paper, and the geometry it depicts is not flat. The tick marking one second on a moving observer's axis sits further from the origin than the stationary observer's, by an amount that is not the Lorentz factor and means nothing at all.

relativity · Spacetime diagram Every detour costs time. 4 routes between the same two events, 10 seconds apart in the frame drawn, each swinging out and back 1 time on the way. The proper time each carries is the integral of the square root of one minus the speed squared, computed by Simpson's rule along each curve: the straight route, 10.0000 s; wandering 1 light-seconds, 9.7485 s; wandering 2 light-seconds, 8.9245 s; wandering 3 light-seconds, 7.0935 s. The straight one carries the most, and every other one carries less — checked, on each drawn route. That is the opposite of what a length behaves like on paper, where the straight line is the shortest, and the whole difference is the minus sign in front of the space term.

The longest way round is the shortest clock

Of all the routes between two events, the one with no acceleration in it carries the most time on its own clock. That is the opposite of the Euclidean statement about straight lines, it comes entirely from one minus sign, and in a gravitational field it is why a thrown ball follows the path it does.

relativity · Time dilation

Named alongside it

The objects these essays reach for when they reach for this one.

The Lorentz factorProper timeThe Lorentz transformationCausalityLight coneLength contractionSpacetime diagramWorldlineEnergy conservationEquivalence principleE = mc²Momentum conservation

All concepts