Mass
A 50 kilogram object weighs 491 newtons on Earth and only 81.5 newtons on the Moon. The object never changed. The number on the scale did. That gap holds a puzzle about mass, an intrinsic property of a body, that took physicists centuries to untangle.
For a long time mass was thought to be simply the quantity of matter in a body. Then came the atom and particle physics, and that picture broke. Different atoms and different elementary particles, in theory carrying the same matter, turned out to have different masses.
How do you measure something that resists every simple definition? Why do a hammer and a feather, dropped together, land at the same instant in a vacuum? And how does a particle come to have mass at all?
Inertial mass measures an object's resistance to being accelerated by a force. Push two objects equally, and the one with more mass speeds up less. Newton's second law states it cleanly. A body of fixed mass m under a single force F has acceleration F divided by m.
Active gravitational mass sets the strength of the field an object generates. Place a body of mass mA a distance r from a body of mass mB, center to center. Each feels an attractive pull governed by the universal gravitational constant. Passive gravitational mass measures the force on an object sitting in a known field.
These three phenomena are conceptually distinct. Some theorists speculated they could be independent. Current experiments find no difference, however mass is measured. Trials since the 17th century show inertial and gravitational mass identical. Since 1915 the equivalence principle of general relativity has assumed it.
Galileo, by scientific folklore, dropped objects from the Leaning Tower of Pisa to prove their descent time was independent of mass. The story is most likely apocryphal. He more plausibly used balls rolling down nearly frictionless inclined planes, slowing the motion to sharpen his timing.
Galileo offered a theoretical argument that needs no tower. Tie a heavy body to a light one. Does the system fall faster because it is now more massive? Or does the slow light body hold back the heavy one? The only consistent answer is that all bodies fall at the same rate. This universality of free fall holds only where gravity is the sole force. Drop a hammer and a feather through air and the feather lags, because air resistance against it rivals gravity. Remove the air and they land together.
David Scott demonstrated exactly this on the Moon during Apollo 15. Lóránd Eötvös pushed the precision further on Earth, using the torsion balance pendulum in 1889. No deviation from Galilean equivalence has been found, at least to one part in a million. Inertial and gravitational mass match to one part in a trillion.
The Romans weighed gold against carob seeds. An object equal to 1728 seeds weighed one Roman pound; one equal to 144 seeds weighed one Roman ounce. Weight standards were defined as amounts, since people noticed the weight of similar objects rose in direct proportion to their number. A balance scale exploited this across a gap small enough that both sides felt similar gravity.
Johannes Kepler sought work with Tycho Brahe in 1600. Using Brahe's precise observations of Mars, he spent five years characterizing planetary motion. In 1609 he published three laws, describing orbits as ellipses with the Sun at a focus. The square of each planet's orbital period, he found, is proportional to the cube of its orbit's semi-major axis.
Isaac Newton bridged Kepler's orbits and Galileo's falling stones. Robert Hooke had published a concept of gravitational forces in 1674. In letters of 1679 and 1680 he conjectured gravity might fall off with the square of distance. Hooke urged Newton to work the mathematics. Newton verified Hooke was right but, owing to personal differences, kept silent. In 1684 he told Edmond Halley he had solved gravitational orbits but misplaced the solution. Halley pressed him, and Newton wrote his findings into a three-book set, Philosophiæ Naturalis Principia Mathematica. His cannonball thought experiment, from his 1728 book A Treatise of the System of the World, completed the picture. A stone thrown hard enough from a high mountain would clear the whole Earth and return to where it began.
Mass and weight blur in everyday speech. A person's weight gets stated as 75 kilograms. In a constant gravitational field weight is proportional to mass, so the shared unit causes no trouble. The distinction sharpens only when precision beats a few percent, or far from Earth, in space or on other planets. Mass, in kilograms, is intrinsic. Weight, in newtons, is the force keeping an object from free fall. Objects in free fall are weightless yet still have mass.
Newton's theory introduced universal gravitational mass. Every object has it, so every object generates a field that weakens with the square of distance. Gather an immense number of carob seeds into an enormous sphere, and its field would be proportional to the seed count. Any traditional mass unit can, in principle, measure gravitational mass.
In principle is not in practice. The fields of small objects are extremely weak. Newton's work appeared in the 1680s. The first successful measurement of Earth's mass in traditional units, the Cavendish experiment, came only in 1797. Henry Cavendish found Earth's density to be 5.448 plus or minus 0.033 times that of water. As of 2025 Earth's mass is known to only about five significant figures. The product of Earth's mass and the gravitational constant is known to over nine. Earth's mass is roughly three-millionths of the Sun's.
The kilogram, the SI base unit of mass, equals 1000 grams. It was first defined in 1795 as the mass of one cubic decimetre of water at the melting point of ice. Measuring that volume precisely proved difficult. So in 1889 the kilogram became the mass of a metal object, freed from water and the metre. The lineage ran from a copper prototype of the grave in 1793 to the platinum Kilogramme des Archives in 1799. It ended with the platinum-iridium International Prototype of the Kilogram in 1889.
The International Prototype and its copies drifted in mass over time. A redefinition took effect on the 20th of May 2019, after a final vote by the CGPM in November 2018. It rests only on invariant quantities of nature: the speed of light, the caesium hyperfine frequency, the Planck constant, and the elementary charge.
Smaller masses got their own scale. The dalton is one-twelfth of the mass of a free carbon-12 atom, so a carbon-12 atom has a mass of exactly 12 daltons. Other units sit outside SI entirely. The slug runs about 14.6 kilograms, the pound about 0.45 kilograms. The Planck mass derives from fundamental constants. The solar mass weighs stars and galaxies against the Sun.
Ernst Mach championed inertial mass as resistance to acceleration. Percy W. Bridgman developed the idea into operationalism. The method is elegant. Isolate two objects so the only forces are mutual, let them collide, and Newton's third law ties their accelerations together. Call one the reference, set its mass to one kilogram, and any other mass follows from the measured accelerations.
The trouble is binding energy. Mach's definition ignores the potential energy needed to bring two masses close enough to measure. Compare a proton inside a deuterium nucleus with a free proton. The free one is heavier by about 0.239 percent, owing to deuterium's binding energy. Mach's formula yields only ratios of masses, never an absolute. Henri Poincaré found a further flaw he called insurmountable. Instantaneous acceleration cannot be measured at all. It demands several readings of position and time, then a calculation.
Einstein's relativity of 1905 unsettled the older picture. The measurable mass of an object increases when energy is added, by heating it or forcing it near something that repels it. Barré de Saint-Venant had argued in 1851 that mass was proportional to the number of interchangeable points an object contained. That amount-of-matter view, still taught in basic education, no longer held at the frontier.
Special relativity splits mass in two. Rest mass, or invariant mass, is the Newtonian mass an observer moving with the object would measure. Relativistic mass is the total energy in a body divided by c squared, and it grows with velocity through the Lorentz factor. Invariant mass is the same in every inertial frame, so physicists favor it. Because relativistic mass merely tracks energy, it has gradually fallen out of use.
Mass and energy are two names for one thing, joined by E equals mc squared. Melt exactly one kilogram of ice and the meltwater outweighs a kilogram. The excess is about 3.7 nanograms, the mass of the latent heat that did the melting. That figure is the latent heat of melting ice, 334 kilojoules per kilogram, divided by the speed of light squared. In bound systems binding energy escapes as gamma rays, so an atomic nucleus weighs less than its free nucleons.
In the Standard Model, developed in the 1960s, particle mass comes from coupling to the Higgs field through the Brout-Englert-Higgs mechanism. A stranger corner remains. A tachyonic field carries imaginary mass, signaling not faster-than-light travel but instability. It resolves through tachyon condensation, a phase transition driving the system to a stable state. Gerald Feinberg coined the word tachyon in a 1967 paper, only for it to emerge that his model allowed no superluminal speeds at all.