Illustrated Physics
physics, drawn
Concept

Tidal force — where it appears

The difference between the gravitational field at two nearby places, which is what no choice of frame can transform away. It falls as the inverse cube of distance rather than the square, so a nearby small mass raises a larger tide than a distant large one.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size.

The floor that cannot be told from gravity

Seal a laboratory, take away the windows, and no experiment inside it can distinguish standing in a gravitational field from accelerating through empty space. That is not a philosophical remark — it forces light to bend, forces clocks to disagree, and has a size at which it stops being true.

astrophysics · Equivalence principle How small each mass would have to be. The Schwarzschild radius of 4 masses, on a logarithmic scale spanning 35 orders of magnitude. A horizon is not something a mass has; it is a size a mass would have to be squeezed inside. For the Sun it is 2.95 km against a real radius of 696,000 km, a factor of 2.36·10⁵. One row has no real size to set beside the number, which is the one case where the horizon is not hypothetical.

The surface that only lets things in

An eighteenth-century calculation asking where the escape speed reaches the speed of light gives exactly the right radius, by reasoning that is wrong in every step. What is actually there is not a surface in space at all, and nothing local happens when it is crossed.

astrophysics · Horizons A collision with restitution 0.4. Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.

The point that keeps moving as if nothing had happened

Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.

mechanics · Momentum How large a box may be before gravity shows in it. The largest freely falling laboratory in which nothing can be detected, against how long the experiment runs, for an instrument resolving 10⁻⁹ m. Free fall removes the field and leaves the gradient: two masses released a distance apart converge across the fall and separate along it, by (GM/r³)Lt²/2, so the box that stays undetectably flat shrinks as the square of the time. At one second the limits are 1.30 mm at the Earth's surface, 5.07 mm at the Sun's surface, 8.61 nm at a white dwarf, 1.86·10⁻¹⁷ m at a neutron star, 3.88·10⁻¹⁷ m at a stellar black hole, 388 mm at a giant black hole. Two things are worth reading off it. The equivalence principle is local in time as much as in space, on the same footing and with a worse exponent — patience is more expensive than room. And the tidal parameter GM/r³ falls with mass at a horizon, so a large enough black hole is one an experimenter could fall through with a metre-sized laboratory and a very good interferometer and detect nothing at all: the curve for the giant lies above the Earth's, not below it.

The term free fall cannot remove

Fall freely and gravity disappears. It disappears only to the extent that the falling laboratory is small — what survives is the gradient, which pulls two released masses together across the fall and apart along it. Given an instrument, the size of the box in which nothing is detectable is computable, and that number is the whole content of the word "locally".

astrophysics · Equivalence principle The bigger the hole, the gentler the horizon. The difference in gravitational pull between the two ends of a 1.8 metre body, at the moment it crosses the horizon, against the mass of the hole. Both axes are logarithmic and the line is straight, of slope -2.00: the tidal acceleration at a horizon falls as the square of the mass, because the tide goes as the mass over the cube of the radius and the radius itself is proportional to the mass. At the low end — a hole of a few solar masses — the stretch is 1.9e+7 times Earth's gravity across a person, which pulls them apart long before they arrive. At the high end it is 1.9e-11, which is nothing at all: crossing the horizon of a large enough hole is locally unremarkable, and an observer would notice no boundary being passed. The two meet at 1.0e+4 solar masses, above which the horizon can be crossed intact. That is the sharpest available statement of what a horizon is and is not. It is not a surface, nothing is there, and nothing local marks it — it is the place from which no future path leads out, which is a statement about the whole of the future rather than about anything present.

The horizon that nothing marks

A falling body is torn apart by the difference in gravity between its ends. At the horizon of a black hole that difference goes as the inverse square of the mass, so a large enough hole can be entered intact — and nothing local happens at the crossing to say it has occurred.

astrophysics · Horizons The distance that does not know how big the moon is. How close a satellite held together by its own gravity can orbit before the tide pulls it apart, in units of the primary's radius, against how much denser the primary is than the satellite. The curve is the distance at which the tidal stretch across the satellite's own body equals the satellite's surface gravity. Setting those two equal cancels the satellite's radius on both sides, so a boulder and a thousand-kilometre moon of the same material break up at exactly the same distance — the limit is a ratio of densities and nothing else, and it goes as the cube root of that ratio, measured here as 0.3333. For ice around a planet of density 687 kg/m³ the rigid limit is 1.15 radii and the limit for a body that can deform under the tide is 2.23, because a satellite pulled into an egg presents a longer body to the tide and gives way sooner. Saturn's rings end at 2.27 radii, just outside that second number, and its innermost round moon orbits at 3.08 — so the boundary between a ring and a moon falls where this calculation puts it. What this calculation leaves out is strength: a body small enough for its material strength to beat its own gravity ignores the limit entirely, which is why Phobos is well inside Mars's and still in one piece, and why the Shoemaker–Levy fragments were held together by nothing at all.

The distance that forgets the moon

A satellite held together by its own gravity comes apart if it orbits too close, and the distance at which it does contains no reference to its size. Both the tide pulling it apart and the gravity holding it together are proportional to its radius, so the radius cancels twice over and what is left is a ratio of two densities.

astrophysics · Self-gravity Six ways a wave of gravity could move a ring of free masses. Sixteen free masses on a ring (dashed) and where a passing gravitational wave of each of the six possible polarisations moves them at its crest, drawn at a strain of 0.32, some 10²⁰ times larger than a real wave's. Top row: plus and cross, the two general relativity allows, which stretch one direction across the wave while squeezing the other and change no area; and breathing, which swells the ring evenly. Bottom row, drawn in planes containing the direction of travel (arrow): longitudinal, which stretches along the travel; and the two vector modes, which shear those planes. Any metric theory of gravity permits these six and no others; which of them a theory keeps is decided by the field that carries gravity.

The four shapes a wave of gravity leaves out

A passing gravitational wave squeezes a ring of free masses in one of two patterns, and that is usually presented as the whole menu. It is general relativity's choice from a longer one. The tide a wave carries is a symmetric table of six numbers, so any theory in which gravity is geometry allows six shapes; a theory keeps two only if its graviton has no mass and nothing else rides along with it. Give the graviton the smallest mass imaginable and it gains three more states, and one of them changes how much the Sun bends light — by a quarter, however small the mass.

astrophysics · Gravitational waves How much of a wave of gravity a bar can catch. A resonant bar's cross-section for gravitational waves, integrated over its resonance, (8/π)(GM/c)(v/c)², against its mass, on logarithmic axes, for sound speeds of 3.0 km/s, 5.4 km/s, 10.0 km/s (aluminium is about 5.4). It is proportional to the mass, which is why the bars weighed tonnes, and to the square of the speed of sound over the speed of light — the bar answers a strain only by being stretched against its own stiffness, and (v/c)² is about 3 × 10⁻¹⁰. Weber's first bar caught 2.3·10⁻²⁵ m² Hz: to absorb a single joule, a burst must carry 4.4·10²⁴ J per square metre per hertz at the bar's frequency. Dots: Weber's bar, and the two cryogenic aluminium bars that ran longest, ALLEGRO in Louisiana and NAUTILUS near Rome.

The bar a wave of gravity barely touches

The first gravitational-wave detectors were tonnes of aluminium, hung in a vacuum and listened to for ringing. A passing wave stretches and squeezes the bar, and the bar can answer only by vibrating against its own stiffness, so the fraction of the wave it catches is set by the speed of sound over the speed of light, squared: about three parts in ten billion. That one ratio explains why the bars had to be enormous and cold, why Joseph Weber's reported detections of 1969 would have required the Galaxy to be converting dozens of suns a day into gravitational waves, and why the waves were finally found by light instead.

astrophysics · Gravitational waves

Named alongside it

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

Spacetime curvatureEquivalence principleFree fallGeodesic deviationSchwarzschild radiusEvent horizonGravitational massInertial massLocal inertial frameProper timeScalingBlack hole

All concepts