Illustrated Physics
physics, drawn
Concept

E = mc² — where it appears

The identity of a body's rest mass with an energy it already has, whether or not any process ever releases it. The mass of a bound system is less than the sum of its parts by the binding energy over c², so a nucleus weighs less than its nucleons and the difference is measurable.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

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 The wavelength shift against scattering angle. How much longer a scattered photon's wavelength is, against the angle it scattered through. The shift runs from nothing at 0 degrees to 4.853 picometres straight back, passing through the electron's Compton wavelength of 2.4263 picometres at 90 degrees. Nothing about the incident light or the target material appears anywhere on this axis.

A photon with a momentum, and a collision that proves it

X-rays bouncing off electrons come back with a longer wavelength. How much longer depends on the angle they turned through, and on nothing else — not the incident wavelength, not the target material, not the intensity.

quantum · Photon The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 200 square centimetres — 0.0200 square metres — against the separation of its plates, drawn twice. Held at 15 nC the energy rises in proportion to the separation; held at 169 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 2.00 mm, where they cross at 1.27 µJ, and there their slopes are equal and opposite: the attraction between the plates is 635 µN, or 6.353·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Where the energy of a field actually is

A charged capacitor holds 1.27 µJ, and two entirely different accounts agree on the number: one built from charges and potentials, one built from joules per cubic metre of empty space. They part company at a resistor, where the power arrives sideways through the surface at 1.67 W.

electromagnetism · Field 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 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 The mass of two things that have none. The invariant mass of a pair of photons of equal energy, in units of E/c², against the angle between them. It is computed from the total energy and the vector sum of the two momenta, and agrees with 2E·sin(θ/2) to 1.0e-14. at 0° the pair weighs 0.000 E/c²; at 30° the pair weighs 0.518 E/c²; at 60° the pair weighs 1.000 E/c²; at 90° the pair weighs 1.414 E/c²; at 120° the pair weighs 1.732 E/c²; at 180° the pair weighs 2.000 E/c². Two photons flying in the same direction have no mass between them at all, because their momenta add to exactly the energy over c; anything else and they do. Nothing has been added: the constituents are massless at every angle, and the mass of the system is a property of the arrangement. At 180° the pair weighs 2E/c², which is every joule it contains — the case of a sealed box of light, where the two beams cancel in momentum and the whole energy shows up on the scales.

The box of light that weighs something

Two photons flying apart have a mass between them, though neither has one. Seal them in a mirrored box and the box is heavier than it was empty, by exactly the energy inside divided by c². Mass is not a property of stuff and it does not add up — it is a property of a system, and 99 per cent of the mass of everything anybody has ever weighed is of this kind.

relativity · Mass-energy The fuel a starship needs, which is most of the universe. The mass of fuel a rocket must start with, divided by the mass it ends with, against the final speed as a fraction of light's — the vertical axis being the number of decades in that ratio, because the numbers do not fit on any other scale. Each solid curve is one exhaust speed, and each dashed one beside it is what Tsiolkovsky's Newtonian formula would have said. Below about a tenth of light speed the two are indistinguishable; above it they part, and the relativistic curve turns upward without limit as the final speed approaches light's, because what adds linearly is the rapidity rather than the velocity. The photon rocket — exhaust at exactly the speed of light, which is the best any engine can do — needs a mass ratio of 1.7321 to reach half light speed, which is √3 exactly, and 4.4 to reach nine tenths. Those are modest numbers and they are the whole of the good news. A chemical exhaust at 4 km/s needs 10^2968 even to reach a tenth of light speed, which is not a difficult engineering problem but an arithmetic impossibility — there are about 10⁵⁰ atoms in the Earth. What the chart cannot show is the other half of the trip: stopping at the far end squares the ratio, and coming home squares it again.

The fuel a starship needs

Tsiolkovsky's logarithm survives relativity with one substitution: what adds is the rapidity rather than the velocity. The result is that a photon rocket reaches half light speed on a mass ratio of the square root of three, and a chemical one reaches a tenth of it on a mass ratio with three thousand digits.

relativity · Mass-energy What a beam's energy buys, three ways. The speed reached against the energy intercepted, measured in the body's own rest energy, for a perfect mirror pushed by a beam, a perfect absorber pushed by the same beam, and a photon rocket that carries the same energy as fuel and throws it out behind. The mirror's curve is γ(1 + β) = 1 + 2E/mc², a rapidity of ln(1 + 2E/mc²); the dots integrate the reflected beam's force, (2P/c)(1 − β)/(1 + β), and agree with it to 10⁻¹⁵. To reach 0.2c the mirror needs 0.1124 of its rest energy, the absorber 0.2500 and the photon rocket 0.2247 — for a 1 g sail, 10.1 terajoules against 20.2. The rocket's rapidity is ln(1 + E/mc²), so the mirror is the rocket with its fuel left at home and each joule used twice, once arriving and once leaving. The absorber does worst, because the energy it keeps becomes rest mass it then has to carry.

The rocket that leaves its fuel at home

A mirror pushed by a beam from the ground carries no propellant, and relativity gives its speed in closed form: its rapidity is ln(1 + 2E/mc²), the photon rocket's equation with the fuel left behind and every joule used twice. What stops it is not the energy, which can be stored for days, but diffraction, which fixes the distance over which the energy can be handed over — and so demands an acceleration of tens of thousands of g.

relativity · Mass-energy Equilibrium is where the entropy peaks, and there the temperatures differ. Two cavities of radiation, one high in a gravitational field and one low, free to exchange energy, with the clock at the bottom running at 0.8 of the rate of the one at the top. What is conserved is the energy either would deliver to a distant observer, so energy held at the bottom counts for 0.8 of its local value. Across: the share of that conserved energy held at the top. Above: the total entropy of the two gases. Below: the temperature a thermometer in the top cavity reads, as a fraction of one in the bottom cavity. The entropy peaks at a share of 0.339, found by search, and there the top cavity is at 0.800 of the bottom's temperature — the clock-rate ratio exactly. Where the two local temperatures are equal, at a share of 0.556, the entropy is 1.82 per cent below its peak and energy still flows downward, into the deeper cavity.

The column that is hotter at the bottom

Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.

relativity · Relativistic thermodynamics A beam of light pulls hardest on what comes towards it. The sideways pull of a long, thin beam of light on a body moving parallel to it at velocity βc, as a multiple of the pull the same body would feel from a static mass with the same energy per unit length, against β: 2(1 − β)²/(1 + β²). A body at rest beside the beam is pulled twice as hard as by a static mass — the beam's pressure along its length counts. A body moving with the beam is pulled less, and a ray of light running alongside it is not pulled at all (β = 1). A ray running against it is pulled 4 times as hard as a static mass of the same energy would pull it. Tolman, Ehrenfest and Podolsky found this in 1931. Two parallel laser beams do not attract each other; two opposed ones do.

The light that pulls twice as hard as its mass

Energy has mass, E/c², and a box of light weighs that much more than an empty one. But the gravity something produces is not set by its energy alone. Pressure counts too, three times over, and light's pressure is a third of its energy — so a beam of light pulls on things beside it twice as hard as its mass would suggest. The box weighs only E/c² because its walls, straining to hold the light in, pull with a negative amount that exactly cancels the difference. Two parallel beams of light do not attract each other at all; two opposed ones attract four times as hard. And a substance whose pressure is negative enough does not pull at all — it pushes.

relativity · Mass-energy One lever, two frames, two different torques. A right-angled lever pivoted at its corner, with arms of length L, held still by a force F at the end of each arm, perpendicular to it. Left, in the lever's own frame: torque FL anticlockwise from the horizontal arm's force, FL clockwise from the vertical arm's, and they cancel. Right, the same lever seen moving to the right at 0.8 of the speed of light: the horizontal arm, along the motion, is contracted to L/γ = 0.60L, and the force on it, which is across the motion, is reduced to F/γ = 0.60F; the vertical arm and its force, along the motion, are unchanged. The torques are now 0.36FL and FL, and the lever is twisted clockwise by 0.64FL — the square of its speed as a fraction of light's. It does not turn.

The lever that is twisted and does not turn

A right-angled lever held still by two equal forces has no net torque on it. Watch the same lever go by at high speed and one arm is shorter, one force is weaker, and the torques no longer cancel — yet the lever goes by without turning. Nothing is wrong with either frame's arithmetic. The forces are doing work on the moving lever, the energy they put in is flowing through its arms, and a flow of energy carries momentum that points across the lever's motion: a body that moves straight with a momentum that does not.

relativity · Mass-energy The parabola a family of isobars lies on. The atomic masses of the nuclei with 91 nucleons, as energy above the lightest, against the number that are protons, from the semi-empirical mass formula: a parabola with zirconium-91 at the bottom, which is where the stable isobar is. Left of the floor the nuclei have too many neutrons and β-decay towards it, each step turning a neutron into a proton and releasing the mass difference — 8.2 MeV from Kr, 5.8 MeV from Rb, 3.4 MeV from Sr, 1.0 MeV from Y — the steps shrinking as the floor nears, because the slope of a parabola does. Right of it the nuclei have too many protons and decay the other way, by capturing an electron or, where the drop exceeds 1.022 MeV, by emitting a positron: 6.2 MeV from Tc, 3.8 MeV from Mo, 1.4 MeV from Nb. Every arrow points downhill in mass, and nothing else decides its direction.

The parabola that decides which way a nucleus decays

A weak decay turns a neutron into a proton or a proton into a neutron and leaves the number of nucleons alone, so it moves a nucleus sideways among its isobars. Whether it can happen, and in which direction, is settled by weighing: a decay goes wherever the atom ends up lighter, by any amount at all, except that emitting a positron needs two electron masses more. Laid out against proton number, the masses of a family of isobars fall on a parabola — or, for an even number of nucleons, on two — and the shape of that parabola is why potassium-40 decays both ways and germanium-76 can only take two steps at once.

relativity · Mass-energy A mass that goes missing as light. For six captures of a slow neutron — ¹H to ²H, ¹²C to ¹³C, ⁵⁶Fe to ⁵⁷Fe, ²⁸Si to ²⁹Si, ³⁵Cl to ³⁶Cl and ³²S to ³³S — the mass that disappears, the atom plus the neutron less the atom formed, in thousandths of an atomic mass unit, against the energy released as gamma rays, in MeV, both computed from the tabulated atomic masses. All six lie on the line Δm = E/c², whose slope is one atomic mass unit per 931.494 MeV: ¹H → ²H loses 2.388 mu and releases 2.225 MeV; ¹²C → ¹³C loses 5.310 mu and releases 4.946 MeV; ⁵⁶Fe → ⁵⁷Fe loses 8.208 mu and releases 7.646 MeV; ²⁸Si → ²⁹Si loses 9.097 mu and releases 8.474 MeV; ³⁵Cl → ³⁶Cl loses 9.211 mu and releases 8.580 MeV; ³²S → ³³S loses 9.277 mu and releases 8.642 MeV. Drawn this way the line is a definition — the energies were computed from the masses. The experiment of 2005 measured the two quantities for silicon and sulphur independently, the masses as frequencies and the energies as angles, and asked whether they still fell on it.

The test of mass and energy that needs a ruler

The most direct test of E = mc² weighed two silicon atoms and measured a gamma ray. A silicon-28 nucleus that captures a slow neutron becomes silicon-29 and loses three ten-thousandths of its mass as gamma rays; weigh both atoms in a trap where they circle a magnetic field, measure the gamma rays by the angle at which a silicon crystal diffracts them, and the two sides agree to four parts in ten million. What the comparison needed is the interesting part: the recoil of the atom, a correction four hundred times larger than the precision, computed from the momentum relativity gives to light; angles of a few hundred picoradians; and a chain of constants tying a frequency and a ruler to the same number.

relativity · Mass-energy

Named alongside it

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

Energy conservationMomentum conservationBinding energyPhotonConservation lawsExhaust velocityInterstellar travelInvariant intervalThe Lorentz factorMomentumPhoton rocketRadiation pressure

All concepts