Total energy against speed, in units of the rest energy
At its defaults it draws 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.
relativistic-energy is one function in lib/figures/spacetime.js —
worldlines, slicing and the factor that governs both. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
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.
Total energy against speed, in units of the rest energy
The options are the ones Mass is a form of energy, which is not the same as a source of it passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What fraction of the mass each process actually converts
The options are the ones Mass is a form of energy, which is not the same as a source of it passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The share of a kilogram's rest energy released by five processes, on a logarithmic axis spanning ten decades. Burning coal converts 3.6e-10 of it; fission 9.1e-4; deuterium–tritium fusion 3.8e-3; annihilation all of it. The equation applies to the chemistry too — the mass change is simply far too small to weigh.
The beam energy a discovery costs, two ways
The options are the ones Mass is a form of energy, which is not the same as a source of it passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The energy per beam required to reach a given invariant mass, on logarithmic axes, for a stationary proton target and for two beams meeting head on. antiproton, p̄p pair needs 3.8 GeV of invariant mass, which is 1.9 GeV per beam head-on and 7 GeV against a target; the W boson needs 80.4 GeV of invariant mass, which is 40.2 GeV per beam head-on and 3442 GeV against a target; the Z boson needs 91.2 GeV of invariant mass, which is 45.6 GeV per beam head-on and 4430 GeV against a target; the Higgs boson needs 125.3 GeV of invariant mass, which is 62.6 GeV per beam head-on and 8359 GeV against a target; the top quark, in pairs needs 345.5 GeV of invariant mass, which is 172.8 GeV per beam head-on and 63618 GeV against a target. The names on the plot are the products far enough apart in mass to be labelled without collision; the rest are in the list on the right. The antiproton row is the one that was actually built: the Bevatron was designed at 6.2 GeV precisely because the threshold is a kinetic energy of 5.63, and the machine's energy was chosen from this arithmetic before there was anything to find. Above about ten GeV of invariant mass the fixed-target column becomes absurd, and every discovery on the list after the antiproton was made at a collider — not for want of engineering but because the requirement grows as the square of what is wanted.
The mass of two things that have none
The options are the ones Mass is a form of energy, which is not the same as a source of it passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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 curve that makes both fusion and fission release energy
The options are the ones The box of light that weighs something passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What checks it
physicscheck asserts something about relativistic-energy that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
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.
RelativityThe 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.
RelativityThe centre that is not a place
The centre of mass is replaced in relativity by the centre of energy, which moves uniformly and does everything the old point did — except be the same point for everybody. Boost a spinning body and its centre moves, so a spinning object has no centre at all.
RelativityThe collision that wastes most of the energy
The LHC's two beams carry 6,500 GeV each and 13,000 GeV are available. Fire one of those beams at a stationary block of copper instead and 110 GeV are available — the other 12,890 have gone into the motion of the wreckage and cannot be used for anything. The difference is a square root, and every accelerator built since 1970 is a consequence of it.
RelativityThe cone a decay cannot leave
A particle at rest breaks into two and they go opposite ways. Set the parent moving and the whole pattern folds forward — into a cone with a hard edge, beyond which nothing is emitted at any rest-frame angle at all. The energy spectrum that comes out is exactly rectangular, and the one number the boost leaves alone is how the parent is identified at all.
RelativityThe energy above which faster arrives later
In a racing circuit a faster car laps sooner. In a proton synchrotron it does too, at first — and then, past a definite energy, the faster proton laps later. Extra momentum sends a proton round a slightly longer orbit and makes it slightly faster, and relativity decides which wins: at low energy the speed does, but near the speed of light more momentum buys almost no more speed, and the longer road takes over. At the crossing, called transition, a spread of momenta stops producing a spread of arrival times, the bunch stops oscillating in time, and the accelerating wave has to jump from one slope to the other in a few turns. The first machine to try it did so in 1959.
RelativityThe flat spectrum a fast pion must paint
A neutral pion falls apart into two photons, each with half its rest energy, in a hundred-million-billionth of a second. Seen in flight, those photons could have any energy between two edges, and every energy in that range is equally likely, whatever the pion's speed. Add up a whole population of pions and something unexpected appears: a peak at exactly the 67.5 MeV that no individual decay shows. Astronomers took it as evidence of where cosmic rays are made.
RelativityThe 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.
RelativityThe 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.
RelativityThe 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.
RelativityThe kick a fast particle can give an electron
A slow proton crossing matter can give any one electron at most a fifth of a per cent of its energy, because an electron struck head-on by something two thousand times heavier rebounds at no more than twice the heavier one's speed. That rule of thumb fails at relativistic speeds, and fails upward: the electron's speed is capped below light, but its energy is not, and the largest kick grows as the square of the projectile's Lorentz factor until, above a teraelectronvolt, one collision can take more than half of everything a proton carries.
RelativityThe 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.
RelativityThe light that moves an electron and pays it nothing
A pulse of light strong enough to throw an electron forward at a large fraction of the speed of light passes over it, and the electron is left exactly as it was found — at rest, with none of the light's energy — except that it is somewhere else. The reason is two quantities the electron cannot change while a plane wave is passing, and the same two quantities fix the figure eight it traces inside the wave and the angle at which it leaves a real, focused beam carrying the energy it managed to keep.
RelativityThe 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.
RelativityThe light that takes back an electron's energy
An electron at rest in the most intense laser light ever made feels its own radiation as a force a millionth of the light's push — the textbook reason radiation reaction never matters. Send the electron into the light at nearly the speed of light and the arithmetic turns over. In its own frame the light is stronger by twice its Lorentz factor, the power it radiates grows as the square of that, and a pulse a few tens of femtoseconds long can strip it of most of its energy as gamma rays. That is where the classical theory of a radiating charge, argued over for a century, finally has to answer to experiment — and where it starts to fail for quantum reasons.
RelativityThe 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.
RelativityThe 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.
RelativityThe neutrino energy an angle fixes
A pion breaks into a muon and a neutrino, and the neutrino carries off a fixed 29.8 MeV in the pion's own frame. In the laboratory, straight along the pion's path, it carries a fixed share of whatever energy the pion had, so a beam of pions of every energy makes neutrinos of every energy. Two and a half degrees to one side, something else happens. The neutrino's energy, as a function of the pion's, rises to a maximum and falls again, and around the maximum pions from two to ten GeV all give neutrinos within a tenth of a GeV of each other. An experiment in Japan builds its detector deliberately off the axis of its own beam for that reason — trading most of its neutrinos for the ones at the energy it needs.
RelativityThe 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.
RelativityThe 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.
RelativityThe plane in which three bodies are flat
A particle breaking into two gives each product a fixed energy; one breaking into three gives none of them one. What it gives instead is a plane of two invariant masses in which a decay with no forces spreads perfectly evenly inside a curved boundary — so every band, dark stripe and bright crossing a real decay draws there is a force, its spin, or a phase between two routes to the same three particles.
RelativityThe right angle a fast collision closes
When one billiard ball strikes an identical one at rest, the two always leave at right angles — a fact every snooker player uses without knowing its proof. The proof is Newton's, and it fails for fast particles. Two protons leaving a collision at 1.9 GeV open at 70 degrees, not 90, and the narrowing follows one law, tan θ₁ tan θ₂ = 2/(γ + 1), in which the right angle is the slow limit. The circle that forced the right angle has been squashed into an ellipse by the energy the incoming particle carries as inertia.
RelativityThe slope a spin leaves in a spectrum
An unpolarised parent decaying in flight gives its products a rectangle of energies. Give the parent a spin along its line of flight and the rectangle tilts, while its two edges stay exactly where they were. The tilt is the polarisation, it can be read without ever seeing which way the parent was going — and which variable it is read from decides how many decays the reading costs.
RelativityThe sound speed a neutron star needs
Heat any gas without limit and the speed of sound in it climbs towards a ceiling: not the speed of light, but light's speed divided by the square root of three, the sound speed of a gas of massless particles. For decades it was a natural guess that no matter could beat that ceiling, since the densest matter ought to behave like a gas of free quarks. Neutron stars say otherwise. A star of twice the Sun's mass, packed into a ball twenty-five kilometres across, can only be held up if its core is stiffer than the ceiling allows — unless nuclear physics fails at densities where it is believed to hold. Somewhere inside the heaviest neutron stars, sound almost certainly travels faster than it can in any free gas.
RelativityThe speed that stops while the energy keeps coming
In 1964 William Bertozzi gave electrons energies from half a megaelectronvolt to fifteen, timed them over eight and a half metres, and caught them in an aluminium disc to measure the heat they brought. By Newton the fastest should have crossed in under four nanoseconds, seven times faster than light. They took twenty-eight, and so did the slower ones, more or less. The heat, meanwhile, rose in proportion to the energy put in. The experiment is famous for showing that nothing outruns light, but its sharper result is the second half: the energy did arrive, all of it, carried at very nearly the same speed.
RelativityThe spring that becomes a light-clock
A mass on an ideal spring keeps the same period however far it swings — that is what makes it a clock. Give the mass the inertia relativity demands, pull it back far enough, and the clock breaks. The speed can never pass c, so a large swing spends almost all its time moving at nearly the speed of light, turning round abruptly at each end: the sine becomes a triangle, the period grows until it is simply the time light takes to cross the swing four times, and a clock riding on the mass ages a small fraction of what one beside it does.
RelativityThe 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.
RelativityThe throw a uniform field bends into a chain
Throw a stone sideways across the Earth's gravity and it draws a parabola. Throw a charge sideways across a uniform electric field and, at low speed, it does the same. At high speed it draws a different curve, a hyperbolic cosine — the shape of a chain hanging between two posts, stretched — and the reason is that nothing pushes it sideways yet its sideways speed falls. Its sideways momentum is fixed, the field keeps adding energy, and energy is inertia. The charge gets only logarithmically far across the field however long it flies, and its sideways distance turns out to be a reading of its own clock.
RelativityThe wave an electron rides to a gigaelectronvolt
A laser pulse fired into a thin plasma pushes its electrons aside and leaves them oscillating behind it, a wave of charge travelling at nearly the speed of light. The field in that wave is a thousand times stronger than any metal accelerator can hold, and an electron caught in it at the right moment is carried along like a surfer, gaining a gigaelectronvolt in a few centimetres. Then it outruns the wave. The distance it can ride, and the energy it can gain, are both set by how nearly the wave keeps pace with it — which is to say, by relativity.