# Mass–energy equivalence **Mass–energy equivalence** is the statement that the [[Energy|energy]] of a body at rest and its mass are the same quantity in different units, related by E = mc². The claim is not that mass can be turned into energy as though the two were substances, but that they are one property measured on two scales: a hot object weighs more than a cold one, a compressed spring more than a slack one, and a bound [[Atomic_nucleus|nucleus]] less than the particles it is made of.[^idema-partii][^murphy-be] Because c² is 8.99 × 10¹⁶ J/kg, the conversion is lopsided in one direction — a single gram of mass is worth 8.99 × 10¹³ J — and negligible in the other, which is why chemistry got away for two centuries with a law of conservation of mass.[^ball-md][^derived-me] In the microsim below the reader pushes β = v/c toward 1 and watches two curves diverge. The equation that answers is `E = gamma·m·c^2` with `gamma = 1/sqrt(1 − beta^2)`, so that the kinetic part `(gamma − 1)·m·c^2` sits on Newton's `m·v^2/2` at low speed, rises 7.3 % above it by β = 0.3, and diverges as β → 1 — the reason nothing with mass reaches c.[^idema-partii][^derived-me] The companion relation `E^2 = (p·c)^2 + (m·c^2)^2` covers massless particles as well, for which the first term is all there is. A mass-defect preset switches the same slider to a Δm axis and converts mass loss to energy on a logarithmic scale, with ticks at the H–H bond (4.5 [[Electronvolt|eV]]), [[Deuterium|deuterium]] plus [[Tritium|tritium]] (17.6 MeV), [[Uranium|uranium]]-235 fission (about 200 MeV) and the [[Proton|proton]]'s own rest energy, 938.27 MeV.[^raven-proton][^spec-p41] This sim is a sibling of the [[Special_relativity|special relativity]] root and is built from the same spec family: the same β slider and the same γ, read for energy instead of for geometry, so a reader who has tilted the Minkowski axes on that page is looking at the same dial here.[^spec-p41] On the [[Physics]] flagship this article is the *Mass and energy* section of Part II — Core theories, and the mass-defect ladder it carries is the rule that the [[Nuclear_fission|fission]] row P51 and the fusion row P53 of the spine spend. ## Description E = mc² is a statement about rest energy. A body of mass m that is not moving, not rotating and not interacting still has energy mc², and that energy is not potential energy relative to some reference: it is a property of the body itself.[^idema-partii] The equation is read most usefully backwards. Whenever a system gives up energy ΔE by any mechanism at all — radiating light, forming a chemical bond, binding a nucleus — its mass falls by ΔE/c².[^murphy-be] A flashlight is lighter after it has shone; a battery is lighter after it is flat; a kilogram of water at 100 °C is heavier than the same water at 0 °C. The Portal Book puts numbers on the everyday end: 4 MJ, a day's food intake, corresponds to 4.5 × 10⁻¹¹ kg, and winding a spring adds about 1.1 × 10⁻¹⁷ kg per [[Work_(physics)|joule]] stored.[^murphy-be][^derived-me] Nothing about the equation restricts it to nuclear processes; what nuclear processes have is a large enough ΔE for the mass change to be measurable. Breaking a mole of H₂ takes 436 kJ, which is a mass change of 4.85 × 10⁻⁹ g per mole — far below any balance ever built.[^likharev-h2][^derived-me] Splitting a mole of uranium-235 releases 1.65 × 10¹³ J, a mass change of 0.183 g per mole, which is easy to weigh.[^ball-md] The physics is identical; only the size differs. ## Mass in special relativity Relativity forces two decisions about the word *mass*, and most of the confusion in this subject comes from mixing the answers. The convention used here and in the Portal Books is that m means the [[Invariant_mass|invariant mass]] — the same number in every frame — and that the frame-dependent growth of inertia at high speed is carried by γ, written explicitly.[^idema-partii] ### Relativistic mass The older convention absorbs γ into the mass, defining a "relativistic mass" γm that increases without limit as β → 1, so that `p = m_rel·v` keeps its Newtonian form. It is not wrong, but it is a poor bargain: `F = m_rel·a` still fails, because force and acceleration are not parallel in general, and the single symbol then means different numbers in different directions. Modern practice keeps m invariant and writes `p = gamma·m·v` and `E = gamma·m·c^2`, which is the form the sim uses.[^idema-partii] Read that way, the divergence at β → 1 is a property of γ, not of the body. ### Conservation of mass and energy Before relativity there were two conservation laws, of mass and of [[Energy|energy]]. Afterwards there is one. The conserved quantity is total energy, of which rest energy is a part, so mass is conserved only to the extent that energy fails to leave the system.[^murphy-be] A sealed bomb weighs exactly the same before and after it explodes; an unsealed one weighs less afterwards, by the energy that escaped divided by c². The chemical law of conservation of mass survives as an excellent approximation because chemical energies are of order an eV per bond while rest energies are of order a GeV per nucleon, a ratio of 10⁻⁹.[^derived-me] ### Massless particles For a particle with m = 0 the relation `E^2 = (p·c)^2 + (m·c^2)^2` collapses to E = pc. A [[Photon|photon]] therefore carries momentum in proportion to its energy and travels at exactly c, having no rest frame in which to be weighed.[^idema-partii] This is not an exception to the equivalence but the clearest case of it: a box of photons has more inertia than an empty box, because energy trapped inside contributes to the mass of the system even though each constituent has none. ### Composite systems The mass of a composite object is not the sum of the masses of its parts. It is the total energy of the system in the frame in which its total momentum is zero, divided by c², which includes the kinetic energy of the parts and the negative potential energy that binds them. A [[Helium|helium]]-4 nucleus is 28.29 MeV lighter than two free protons and two free neutrons; that deficit, 0.76 % of the total, is the [[Nuclear_binding_energy|binding energy]].[^murphy-be] The effect runs the other way for the [[Proton|proton]] itself, whose three valence quarks account for about 9 grams in every 938 — roughly 1 % — with the remainder supplied by the energy of the field that confines them.[^raven-proton] ### Relation to gravity Because gravity couples to energy rather than to a separate gravitational charge, the mass that appears in the equivalence is also the mass that weighs. A hot brick is very slightly heavier than a cold one in the same gravitational field, and the binding-energy deficit of a nucleus lowers its weight by exactly the same fraction as its inertia. [[General_relativity|General relativity]] makes this a structural statement rather than a coincidence: the source of the gravitational field is the full energy–momentum content of matter, of which rest mass is one component. ## Efficiency The useful measure of a process is what fraction of the rest mass it converts, and the range is enormous. Burning hydrogen releases about 4.5 eV per molecule out of a rest energy of 1,878 MeV, an efficiency of 2.4 × 10⁻⁹.[^likharev-h2][^derived-me] Fusing [[Deuterium|deuterium]] with [[Tritium|tritium]] releases 17.6 MeV out of 4,685 MeV, an efficiency of 3.8 × 10⁻³ — 1.6 million times better.[^murphy-fusion][^derived-me] Fission of uranium-235 releases about 200 MeV out of 219,800 MeV, an efficiency of 9.1 × 10⁻⁴, or 7.8 × 10⁻⁴ on the exact figure of 172 MeV.[^murphy-be][^derived-me] Fusing four hydrogen nuclei into helium in the [[Sun|Sun]] converts 0.7 % of the mass, the largest fraction any common process reaches.[^murphy-fusion] | process | energy released | fraction of rest mass | |---|---|---| | H–H bond, per molecule | 4.5 eV | 2.4 × 10⁻⁹ | | U-235 fission, per nucleus | ≈ 200 MeV | 9.1 × 10⁻⁴ | | D + T → He + n | 17.6 MeV | 3.8 × 10⁻³ | | 4 ¹H → ⁴He (stellar) | 26.7 MeV | 7.0 × 10⁻³ | | complete annihilation | mc² | 1 | The gap between the first row and the rest is the whole practical content of nuclear energy: per gram of fuel the Portal Book gives roughly 10 kcal for chemistry, 16.8 million for fission and 81 million for D–T fusion.[^murphy-fusion][^derived-me] Complete conversion, the last row, is not available to any engineering process; it happens only in matter–antimatter annihilation. ## Extension for systems in motion For a system that is moving, the rest energy is replaced by the full relativistic energy, and the quantity that behaves properly under a change of frame is the [[Four-momentum|four-momentum]] (E/c, p). Its invariant length is the mass: `m^2·c^4 = E^2 − (p·c)^2`, the same number in every frame even though E and p separately are not.[^idema-partii] This is the working form of the equivalence in [[Elementary_particle|particle physics]], where the invariant mass of a set of decay products is computed from their measured energies and [[Momentum|momenta]] and identifies the particle that produced them. Two consequences are worth stating plainly. A system's mass includes the kinetic energy of its parts but not its own bulk motion, so heating a gas raises its mass while carrying the cylinder across the room does not. And mass is not additive across a system with internal motion: two photons flying apart have zero mass each but a non-zero invariant mass together, which is why the equivalence is a statement about systems rather than about particles. ## Low-speed approximation Expanding γ for small β gives γ ≈ 1 + β²/2 + 3β⁴/8, so that `E = gamma·m·c^2 ≈ m·c^2 + m·v^2/2`, and Newton's [[Kinetic_energy|kinetic energy]] emerges as the first correction to the rest energy.[^idema-partii] The constant term mc² is exactly what Newtonian mechanics discards as an unobservable additive constant, which is why the equivalence went unnoticed for so long: as long as no process changes m, the rest energy never appears in an energy balance. The sim's first panel is this expansion made visible. The two curves — (γ − 1)mc² and mv²/2 — are indistinguishable below about β = 0.1, where they differ by 0.4 %. By β = 0.3 the exact value is 0.0483 mc² against Newton's 0.0450, an excess of 7.3 %; by β = 0.6 it is 0.25 against 0.18, an excess of 39 %; and by β = 0.9 it is 1.294 against 0.405.[^derived-me] The Newtonian curve is a parabola that keeps climbing at a fixed rate, while the relativistic one has a vertical asymptote at β = 1, and the reader who drags the slider to its stop sees the energy readout run away rather than reach a value. That is the whole argument for the speed limit: not that something forbids c, but that the bill for the last increment of speed is infinite. ## Applications The equivalence is used in two distinct ways. Where ΔE is known it predicts a weight change, and where a weight change can be measured it gives the energy — which is how nuclear energies were first obtained, from mass spectrometry rather than from calorimetry. ### Application to nuclear physics The mass defect of a nuclide is the sum of its constituent masses minus its measured mass, and the [[Nuclear_binding_energy|binding energy]] is that defect times c².[^murphy-be] With the conversion 1 [[Atomic_mass|atomic mass unit]] = 931.494 MeV/c² the arithmetic is direct: iron-56 is assembled from parts totalling 56.46340 u but measures 55.934942 u, a defect of 0.528447 u, which is 492.25 MeV, or 8.79 MeV per nucleon.[^murphy-be][^derived-me] Binding energy per nucleon against mass number is the ladder the sim's preset climbs.[^manual-ladder] It rises from 1.11 MeV for deuterium to 7.07 for helium-4 and 7.68 for carbon-12, peaks near iron-56 at 8.79, and falls slowly to 7.59 for uranium-235.[^murphy-be] Every nuclear energy source is a move toward that peak. Fusion pays on the light side: D + T → ⁴He + n releases 17.6 MeV, and the stellar chain 4 ¹H → ⁴He releases 26.7 MeV from a mass loss of 0.0287 u, which is 4.28 × 10⁻¹² J.[^murphy-fusion][^derived-me] [[Nuclear_fission|Fission]] pays on the heavy side, and the graphical estimate from the curve — 8.7 × 95 plus 8.4 × 140 against 7.6 × 235 — gives about 210 MeV against an exact 172 MeV, the difference being the spare neutrons and later [[Beta_decay|beta decays]] the sketch omits.[^murphy-be][^manual-ladder] The ladder also explains why nothing lies beyond iron: past the peak neither joining nor splitting releases energy, and a star with an iron core has run out of fuel. ### Practical examples Reactor arithmetic is done in grams. Fission of a mole of uranium-235 loses 0.1834 g and releases 1.65 × 10¹³ J, against roughly 650 kJ per mole of CH₂ in hydrocarbon [[Combustion|combustion]].[^ball-md] Fusing a mole of D with T loses 0.01888 g for 1.70 × 10¹² J.[^ball-md] Lead-208, were it assembled from free nucleons, would lose 0.1002 g per mole for 9.02 × 10¹² J, which is 4.34 × 10¹⁰ J per gram.[^ball-md] Natural uranium is 0.7 % U-235, reactor fuel about 3 %, and weapons-grade material 70 % or more, a separation problem that exists because the [[Nuclear_chain_reaction|chain reaction]] needs a critical density of fissile nuclei rather than a critical mass of uranium.[^ball-md] At the far end of the scale, 12.0 u of matter — a mole-scale sample seen one atom at a time — is 1.99 × 10⁻²⁶ kg and 11,178 MeV of rest energy, which no process short of annihilation can release.[^murphy-be][^derived-me] ## History The equation has a longer prehistory than its date suggests, and its first quantitative use came from radioactivity rather than from mechanics. ### Developments prior to Einstein Several nineteenth-century results pointed the same way without arriving. Electromagnetic theory gave a charged body extra inertia from the energy of its own field, and calculations of that "electromagnetic mass" produced expressions of the form E/c² with factors that depended on the model of the charge distribution. Attempts to attribute all of an electron's inertia to its field were the closest approach, and they failed for a characteristic reason: without relativity there was no way to make the result independent of the shape assumed. ### Einstein: mass–energy equivalence Einstein's September 1905 paper, a short sequel to the paper that introduced [[Special_relativity|special relativity]], asked what happens to a body that emits two equal pulses of light in opposite directions. Its momentum is unchanged, so it stays at rest, but its energy has fallen by L; transforming the same process into a moving frame and comparing kinetic energies shows that the body's mass has fallen by L/c².[^einstein1905b] The argument uses no model of matter, which is precisely its strength: it applies to any body that emits energy by any means. ### Radioactivity and nuclear energy The equation became quantitative when nuclear masses could be measured well enough to see the defect. Radioactive decay supplied energies millions of times larger than chemistry's, and mass spectrometry supplied the corresponding mass differences, so the two could be compared directly. The binding-energy curve that resulted is the source of every figure in the sections above, and it converted E = mc² from a deduction about light pulses into the design equation of [[Nuclear_reactor|reactors]] and [[Fusion_power|fusion]] devices.[^murphy-be][^openstax-v3-ch10] ## See also - [[Invariant_mass]] — the m in the formula, defined frame-independently - [[Four-momentum]] — the object whose invariant length is that mass - [[Kinetic_energy]] — the C17 Energy page, whose low-speed form this one corrects - [[Special_relativity]] — the root of this sim's spec family - [[Nuclear_binding_energy]] — the mass defect read as an energy - [[Nuclear_fission]] and [[Nuclear_fusion]] — the two ways down the ladder - [[Conservation_of_energy]] — the law that absorbs conservation of mass ## Notes Explanatory notes are folded into the body of this page; the footnotes under References carry both the sources and the derivations. ## References [^idema-partii]: Idema, Timon (2018). *Mechanics and Relativity*. Part II, "Special relativity" (pp. 120–169): relativistic momentum `p = gamma*m*v`, energy `E = gamma*m*c^2`, the energy–momentum relation `E^2 = (p*c)^2 + (m*c^2)^2`, the invariant mass, and the low-speed expansion returning `m*v^2/2` (page to pin). https://open.umn.edu/opentextbooks/textbooks/mechanics-and-relativity [^murphy-be]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15: `dE = dm*c^2` with c = 2.99792458 × 10⁸ m/s (p. 266); the mass defect and binding energy per nucleon, with 1 u = 1.66054 × 10⁻²⁷ kg = 931.49432 MeV/c² and the proton, neutron and electron masses (pp. 262, 266–269); Table 15.5, giving Δm·c² and binding energy per nucleon for ²H (2.22 MeV, 1.11), ⁴He (28.29, 7.07), ¹²C (92.16, 7.68), ⁵⁶Fe (492.25, 8.79) and ²³⁵U (1,783.85, 7.59), the ⁵⁶Fe worked example (parts 56.46340 u, measured 55.934942 u) and the ⁶²Ni value of 8.795 (pp. 267–269); the everyday figures of 4.6 × 10⁻¹¹ kg for 4 MJ and ~10⁻¹⁷ kg per joule in a wound spring (p. 266); and the graphical fission estimate of ≈210 MeV against an exact 172 MeV (pp. 274–275). https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^murphy-fusion]: Murphy, Thomas (2021). *Energy and Human Ambitions on a Finite Planet*. Chapter 15, the fusion reactions 4 ¹H → ⁴He + 26.7 MeV, ²H + ²H → ⁴He + 23.8 MeV and ²H + ³H → ⁴He + n + 17.6 MeV, the p–p mass loss of 0.0287 u (0.7 %) = 4.2 × 10⁻¹² J, and the energy densities of 153, 137 and 81 million kcal/g against 16.8 million for fission and ~10 kcal/g for chemistry (pp. 273–274, 285). https://open.umn.edu/opentextbooks/textbooks/energy-and-human-ambitions-on-a-finite-planet [^ball-md]: Ball, David (2011). *Introductory Chemistry*. Chapter 15 (nuclear chemistry): `E = mc^2` with Δm taken as products minus reactants (pp. 752–753); U-235 + n with Δm = −0.1834 g/mol → −1.65 × 10¹³ J, against ~650 kJ/mol per CH₂ in hydrocarbon combustion (pp. 752–753); U-238 at −1.35 × 10¹³ J (pp. 755–756); 1.00 g fully converted → 9.00 × 10¹³ J, Pb-208 losing 0.1002 g/mol → 9.02 × 10¹² J (4.34 × 10¹⁰ J/g), and D + T losing 0.01888 g → 1.70 × 10¹² J (pp. 758–760); natural uranium 0.7 % U-235, reactor fuel ~3 %, weapons ≥70 % (p. 757). https://open.umn.edu/opentextbooks/textbooks/introductory-chemistry [^raven-proton]: Raven, Will (2025). *Atomic Physics for Everyone*. Chapter 11: the proton's rest energy, 938.27 MeV for m = 1.6726219 × 10⁻²⁷ kg (p. 231), and the quark-mass comparison in which 2.2 + 2.2 + 4.7 units of quark mass stand against the proton's 938, about 1 % (p. 229). https://open.umn.edu/opentextbooks/textbooks/atomic-physics-for-everyone-an-introduction-to-atomic-physics-quantum-mechanics-and-precision-spectroscopy-with-no-college-level-prerequisites [^likharev-h2]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the two-coupled-wells model of a covalent bond, with H₂ ≈ 435 kJ/mol ≈ 4.5 eV per molecule (p. 69; the sub-manual records this page label as approximate). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^manual-ladder]: Wikitube MICROSIM_GUIDE sub-manual 10, *Earth and Energy*, §4.2 "Binding energy per nucleon: why fusion and fission both pay", and sub-manual 05, *Chemistry*, §9.3 "Mass defect to energy: fission, fusion and chemistry". §4.2 supplies the binding-curve microsim (control A from 2 to 240, with `Q_split` and `Q_fuse` readouts) and the pitfall that the graphical fission estimate overshoots because it omits the spare neutrons and later β⁻ decays; §9.3 supplies the Δm-to-energy log axis this page's preset uses, its tick list, and the derived observation that breaking H₂ changes the mass by about 4.8 × 10⁻⁹ g/mol. [^derived-me]: Computed for this article from the constants of [^murphy-be]. 1 g fully converted: 8.988 × 10¹³ J; 1 J of stored energy: 1.11 × 10⁻¹⁷ kg; 4 MJ: 4.45 × 10⁻¹¹ kg. H₂ at 436 kJ/mol: Δm = 4.851 × 10⁻⁹ g/mol. Fractions of rest mass released, using 1 u = 931.49432 MeV/c²: H–H bond 4.5 eV of H₂'s 1,877.9 MeV = 2.40 × 10⁻⁹; D + T 17.6 MeV of 4,685.4 MeV = 3.76 × 10⁻³ (1.57 × 10⁶ times the chemical figure); U-235 + n 200 MeV of 219,833 MeV = 9.10 × 10⁻⁴ (3.8 × 10⁵ times), or 7.82 × 10⁻⁴ on the exact 172 MeV; the stellar 0.7 % is 2.9 × 10⁶ times. Iron-56: 0.528447 u = 492.25 MeV = 8.790 MeV per nucleon, matching Table 15.5. 4 ¹H → ⁴He: 0.0287 u = 26.73 MeV = 4.28 × 10⁻¹² J. 12.0 u = 1.993 × 10⁻²⁶ kg = 1.79 × 10⁻⁹ J = 11,178 MeV. The proton's 938.27 MeV reproduces from 1.6726219 × 10⁻²⁷ kg to 938.26 MeV. Kinetic energy (γ − 1)mc² against Newton's β²/2·mc², in units of mc²: 0.00504 against 0.00500 at β = 0.1 (+0.8 %), 0.04828 against 0.04500 at β = 0.3 (+7.3 %), 0.25 against 0.18 at β = 0.6 (+39 %), 1.294 against 0.405 at β = 0.9. Energy-density ratios from [^murphy-fusion]: fission 1.68 × 10⁶ times chemistry per gram, D–T 8.1 × 10⁶ times. [^spec-p41]: Matter & Energy Cluster contract, `_registry/plans/PHYSICS_SECTIONS.md` row P41: new sibling of the `Special_relativity` root (row P40) in the same spec family, driven by the same β slider. `E = gamma*m*c^2` and `E^2 = (p*c)^2 + (m*c^2)^2`; as β is pushed toward 1 the kinetic energy `(gamma - 1)*m*c^2` climbs off Newton's `m*v^2/2` past β ≈ 0.3 and diverges at c, so nothing with mass gets there. A mass-defect preset converts Δm to energy on a log axis with ticks at the H–H bond (4.5 eV), D + T (17.6 MeV), U-235 fission (~200 MeV) and the proton's rest energy 938.27 MeV — the rule that the fission row P51 and the fusion row P53 of the spine spend. The sim runs live; every quantity is closed form. [^openstax-v3-ch10]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 10, "Nuclear Physics" (pp. 441–492), binding energy, the binding-energy curve, fission and fusion at first-course level (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^einstein1905b]: Einstein, Albert (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" *Annalen der Physik* 18. Pages and DOI to pin. The two-light-pulse argument giving Δm = L/c². ## External links - The Wikipedia pair's *External links* section is the place to look for the original papers and for explanatory material; this page lists only the open texts cited above, which it has read. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Mass–energy_equivalence.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Mass–energy equivalence* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Mass–energy_equivalence.html" data-title="Mass–energy equivalence"></div> *Built from `MICROSIM_GUIDE/specs/sims/Mass–energy_equivalence.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Mass–energy_equivalence) : [Wikitube](https://en.wikitube.io/wiki/Mass–energy_equivalence) · pinned revision [1367907771](https://en.wikipedia.org/w/index.php?oldid=1367907771) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Physics row P41 · sim pending (matter/Mass–energy_equivalence).*