Illustrated Physics
physics, drawn
Concept

Systematic error — where it appears

An error that biases every measurement in the same direction, so that repeating the experiment does not reduce it. It is what a mature precision measurement is limited by, and separating one from the signal usually means finding a parameter the two depend on differently and sweeping it.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Two pivots with one period, and the length between them. The period of a uniform bar 1 m long swung about a pivot, against the distance of that pivot from its centre of mass. Close to the centre the period runs away, because there is almost no restoring torque; far from it the bar behaves more and more like a simple pendulum. In between is a minimum at h = k = 0.2887 m, the radius of gyration, and because there is a minimum every period above it belongs to two pivots at once — here 0.4 m and 0.2083 m, whose product is k² and whose periods agree to 2.2e-16 seconds. Suspend the bar from those two knife edges in turn, adjust until the periods match, and the equivalent simple pendulum is the distance between them: 0.6083 m. Then g = 4π²(h+h′)/T² gives 9.8100 m/s², recovered from the drawing to 1.8e-16. The mass of the bar, the distribution of that mass, and the position of the centre of mass appear nowhere in the answer.

The length nobody has to measure

A pendulum's period depends on its length, so a pendulum will measure gravity — except that a real swinging bar has no single length, and finding where its mass effectively sits is far harder than timing it. Kater's answer was to build the instrument so that the awkward quantity cancels, leaving a distance between two knife edges that a rule can read to five figures.

mechanics · Pendulum Three centuries of finding nothing. The upper limit on η against the year it was set, on a logarithmic scale. Newton (1687) reached 1e-3; Bessel (1832) reached 2e-5; Eötvös (1922) reached 5e-9; Dicke (1964) reached 1e-11; Braginsky (1972) reached 1e-12; Eöt-Wash (2008) reached 2e-13; MICROSCOPE (2022) reached 1e-15. That is 12 orders of magnitude in 335 years, and every one of those measurements returned zero. A sequence of null results is not a sequence of failures. Each one is a statement that a principle assumed by every theory of gravity holds to a new level, and each new level excludes a class of theories that would have shown a departure there — a long-range force coupling to something other than mass-energy, a scalar partner to the graviton, a violation arising at some energy scale. The measurement is worth making again precisely because it has always come out the same way, which is what makes any departure decisive.

The fall that does not depend on what is falling

Everything falls at the same rate, and the statement has been tested for three hundred years by people looking for the exception. Twelve orders of magnitude have been added to the limit and every measurement has returned zero. The last three orders came not from a better instrument but from finding something bigger to fall towards.

astrophysics · Equivalence principle The shift a member galaxy shows against the centre. The predicted gravitational shift of a cluster member's light relative to the central galaxy's, in km/s, against the member's projected distance from the centre, averaged along the line of sight with the members assumed to follow the mass out to twice R200, for clusters of 1 × 10¹⁴ M☉, 3 × 10¹⁴ M☉, 1 × 10¹⁵ M☉. Every member is blueshifted relative to the centre, because its light climbs out of a shallower part of the well. For the heaviest the shift is −13.2 km/s at half a megaparsec and −21.6 at two; for the lightest, whose members end at 1.9 Mpc, −4.4 and −5.7 at one and a half. Projection blunts the profile: members seen near the centre include many that lie far in front or behind it.

The light that climbs out of a cluster of galaxies

A cluster of galaxies is a well a few megaparsecs wide, and its galaxies sit at different depths in it. Light from the one at the bottom climbs furthest, so every other member should look slightly blue beside it — by ten kilometres a second, against galaxies that rush about at a thousand. No single cluster can show it. Thousands stacked together can, but only once a second effect of exactly the same size, the slowing of the galaxies' own moving clocks, has been taken out — and it cannot be separated by choosing heavier clusters, because both are set by the same depth.

astrophysics · Gravitational redshift The deeper the line, the closer to gravity. The shift of a solar line in sunlight seen from the Earth, against the line's depth, in the two-stream model with the convective velocities dying away above the photosphere over 150 km and a deeper line taken to form higher (600 km × depth²). Weak lines, formed low among the fastest flows, show 411 m/s — about two-thirds of the gravitational value; the deepest lines, whose cores form in the upper photosphere where the granules' motion has nearly stopped, show 627 m/s and are still climbing towards the 633 m/s that gravity alone gives, drawn as the dashed line. That is the measurement made with integrated sunlight reflected from the Moon and calibrated with a laser frequency comb: plot the shift against line depth, and read the gravitational redshift off where the strongest lines are heading. Its published value, 638 ± 6 m/s, agrees with general relativity.

The redshift the Sun's boiling hides

Light leaving the Sun climbs out of a gravitational well and should arrive redder by 633 metres a second in velocity terms — a prediction Einstein made in 1907, before he had general relativity. It took half a century to confirm, and the reason was not the size of the effect but the Sun's surface, which boils. Bright hot gas rising in granules and fainter cool gas sinking between them blueshift every line by a few hundred metres a second, by an amount that depends on the line and on where on the disc it is seen. The gravitational shift was found by treating that nuisance as a second measurement: deeper lines and lines near the limb both converge on the same number.

astrophysics · Gravitational redshift

Named alongside it

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

Gravitational redshiftMeasurementPendulumCentre of massCentrifugal forceConvective blueshiftDoppler effectEquivalence principleEquivalent lengthFree fallGalaxy clusterGranulation

All concepts