Illustrated Physics
physics, drawn
Concept

Signal velocity — where it appears

The speed at which information actually travels, which is bounded by light even where a pulse's peak moves faster or backwards. What limits it is the arrival of a new front rather than the motion of an existing shape, and no reshaping of a pulse can produce a front that was not already there.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

A peak that leaves before it should have arrived. Two pulses at the far face of a cell, both normalised to the peak the vacuum one reaches. One has crossed empty space; the other has crossed a medium with two gain lines either side of its carrier, whose group index there is -3.85 — negative, so the envelope's peak should emerge early, and it does. Measured off the two curves the advance is 616 in units where the carrier period is 2π, against 582 predicted from the group index alone; the difference is the higher-order dispersion the group index leaves out. The advance is 0.21 of the pulse's own duration, and the peak leaves the far face before the input peak has entered the near one. Nothing has outrun anything. The emergent pulse is a reshaped version of the input's leading edge, which arrived in plenty of time and already contained — for a smooth pulse — everything needed to reconstruct the rest; the medium amplifies it by 1.14× and delivers it early. Give the pulse a genuine front, a moment before which it is exactly zero, and that front travels at the speed of light in every medium there is.

The speed that carries no signal

In the right medium a pulse's peak emerges from the far side before it entered the near one. The measurement is real, it has been made, and nothing has outrun light — because the peak of a smooth pulse was never carrying any information in the first place.

waves · Wave packets Which happened first, asked of several observers. Two events on a spacetime diagram: one at the origin and one 3 light-seconds away and 1 second later, so that light leaving the first cannot reach the second. Through the second event runs a family of lines, each one the set of events some observer calls simultaneous with it; an observer moving at a fraction β of the speed of light has such a line of slope β on these axes. Where a line meets the vertical axis is the time that observer assigns to the second event. For a slow observer that meeting point is above the origin and the second event happens later; for a fast one it is below and the second event happens EARLIER. The changeover is at β = 0.3333, which is the time separation divided by the space separation, and it is a legal speed only because the separation is spacelike. So the order of these two events is not a property of the events. What every observer does agree on is that neither could have caused the other, because the two lie outside each other's light cones — drawn here as the diagonals — and that agreement is what causality rests on rather than on any shared notion of before.

Which came first, and who decides

Two events far apart can happen in either order, depending on who is asked, and both answers are correct. That is not a loophole in causality but the reason causality survives at all — because the pairs whose order is negotiable are exactly the pairs neither of which could have caused the other.

relativity · Simultaneity A source moving faster than its own waves. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

The cone the source leaves behind

Take the Doppler construction past the speed of the wave and the wavefronts acquire an envelope. Its half-angle obeys sin θ = 1/M, an expression with no pressure, no density and no shape of the object in it — so a photograph of the cone is a speedometer. And the bang is not an event at the moment of crossing — it is a signature dragged along the ground for the whole of the flight.

waves · Doppler The delay that stops caring how thick the wall is. The Wigner phase time — how late the transmitted packet's peak arrives, compared with a free particle covering the same distance — for an electron under a 3 eV barrier, at 0.2 nm, 0.5 nm, 1 nm thick. 0.2 nm: 0.3728 fs at half the barrier height; 0.5 nm: 0.4372 fs at half the barrier height; 1 nm: 0.4388 fs at half the barrier height. The widths span a factor of 5.0 and the times span 1.18. A thicker barrier is exponentially harder to cross and the delay in crossing it barely moves — which is either a remarkable fact about tunnelling or a sign that the phase time is not a traversal time, and the rest of the essay is about which.

How long the crossing takes

Tunnelling has a probability, and asking how long it takes turns out to be a different kind of question. The delay a transmitted packet shows stops growing once the barrier is opaque, so a thicker wall is crossed in the same time — and several defensible clocks give several different answers.

quantum · Tunnelling

Named alongside it

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

CausalityDispersionGroup velocityPhase velocitySuperpositionWave packetAnalytic continuationAnomalous dispersionEnvelopeEvanescent waveFourier analysisFront velocity

All concepts