Illustrated Physics
physics, drawn
Series

Mass-energy — the series

13 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · relativity
  2. 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.

    part 2 · relativity
  3. The curve that makes both fusion and fission release energy. Binding energy per nucleon against mass number: how much energy would have to be supplied, per particle, to take a nucleus apart into free protons and neutrons. The curve is the semi-empirical mass formula, evaluated at whichever proton number binds most tightly for each mass number rather than at a guessed one; the points are measured values. It rises steeply at the light end, peaks at mass number 58, and falls slowly thereafter. Everything about nuclear energy follows from that shape and from nothing else. Two light nuclei joined move up the curve and release the difference; one heavy nucleus split moves up it too, from the other side. Both directions are downhill in energy because the peak is in the middle, and the peak is in the middle because two effects fight — the surface term, which penalises small nuclei for having most of their nucleons on the outside, and the Coulomb term, which penalises large ones because every proton repels every other. The energy released is the height climbed times the number of nucleons carried, and it is a million times a chemical bond for the same reason the vertical axis is in millions of electronvolts rather than in single ones.

    The mass that is missing

    A helium nucleus weighs less than the two protons and two neutrons it is made of. The shortfall is not an error in the weighing; it is the binding energy, converted at the going rate. One curve of that shortfall against size explains why both fusion and fission release energy.

    part 3 · relativity
  4. 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.

    part 4 · relativity
  5. 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.

    part 5 · relativity
  6. 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.

    part 6 · relativity
  7. 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.

    part 7 · relativity
  8. An atom's line and a nucleus's, on one scale. Three energies, in electronvolts on a logarithmic scale, for an atomic transition, sodium's yellow line, and a nuclear one, the 14.4 keV gamma ray of iron-57: the amount by which emission and absorption by free atoms miss each other because of recoil, E²/Mc²; the natural width of the line, ħ over the excited state's lifetime; and the Doppler width from thermal motion at room temperature. For sodium the recoil miss is 2.1·10⁻¹⁰ eV, 197 times smaller than the natural width, so an atom absorbs its neighbour's light. For iron-57 it is 0.00392 eV, 839,015 times its natural width of 4.67·10⁻⁹ eV. The recoil grows as the square of the transition energy and the width does not, so a gamma ray emitted by a free nucleus misses an identical nucleus by nearly a million line widths.

    The gamma ray its own kind cannot absorb

    An atom can absorb the light another atom of its kind emits; that is how a sodium flame casts a shadow in sodium light. A nucleus cannot. When it emits a gamma ray it loses the photon's energy as mass, and the photon's momentum kicks it backward, taking a small share of the energy with it; to absorb the same gamma ray, an identical nucleus would need that share added rather than removed. For iron-57's 14.4 keV line the shortfall is nearly a million times the line's own width. In 1958 Rudolf Mössbauer found that a nucleus held in a crystal can hand its recoil to the whole crystal and lose almost nothing — and the resulting line is so sharp that it could weigh the fall of light down a twenty-two metre tower.

    part 8 · relativity
  9. 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.

    part 9 · relativity
  10. A peak at half the mass, made by a sine. The momentum across the beam of the electron from 200,000 simulated W decays, as a probability density per GeV, for a W of mass 80.37 GeV. For a W exactly at rest across the beam, of exactly its mass (dashed), the electron carries M/2 = 40.19 GeV in the W's frame and shows a fraction sin θ of it across the beam, so the distribution climbs to a sharp peak at M/2 and stops: the Jacobian peak, piled up because sin θ hardly changes near θ = 90°. The W's natural width of 2.1 GeV rounds the peak (solid, peak at 39.9 GeV); its motion across the beam, a few GeV on average, smears it further and pushes a tail past M/2 (lighter). The position of the peak measures the mass, and is shifted by anything that moves the W.

    The mass weighed without one of its pieces

    A W boson decays into an electron and a neutrino, and the neutrino leaves every detector without a trace. Its mass cannot be found the usual way, by adding up the energy and momentum of what it decayed into, because half of what it decayed into is missing. What survives is the part of the motion across the beam, where the neutrino's momentum can be inferred from the imbalance it leaves. There the electron's momentum piles up at half the W's mass, and a quantity built from both particles ends at a sharp edge at the full mass. The world's best measurements of that edge disagree by seventy-five parts in eighty thousand.

    part 10 · relativity
  11. 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.

    part 11 · relativity
  12. 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.

    part 12 · relativity
  13. The price of a pair depends on who pays it. The lowest energy of a photon that can make an electron–positron pair when struck on a stationary partner, against the partner's mass: 2mc²(1 + m/M), from requiring that the total four-momentum have the invariant mass of what is left at rest in the centre of momentum. For a lead nucleus it is 2mc² to better than one part in a hundred thousand, the figure every textbook quotes. For a proton it is 0.054 per cent above. For a free electron it is exactly twice, 4mc², or 2.04 MeV, even though the partner is the very same mass as the pair's members. The rest energy created is unchanged; what rises is what has to be carried off by the partner and the pair together.

    The partner that sets the price of a pair

    Every textbook says it takes 1.02 MeV to make an electron and a positron from light, twice the electron's rest energy. Done in empty space, it cannot be done at any energy at all. Next to a lead nucleus it costs 1.02 MeV. Next to a lone electron, a particle exactly as heavy as the ones being made, it costs 2.04. Nothing about the pair has changed. What changes the price is who has to carry the momentum away.

    part 13 · relativity

All series