Illustrated Physics
physics, drawn
Concept

Proper acceleration — where it appears

The acceleration measured by an accelerometer carried along with a body, which is what its occupants feel as weight. Holding it constant produces a hyperbolic worldline rather than an ever-increasing speed, and it fixes a horizon a distance c²/a behind — 0.97 light years at one gravity.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

Pushing one way and going another. The angle between an applied force and the acceleration it produces, against the angle between the force and the body's velocity, at four speeds. At 0° and 90° the two are parallel, because those are the two directions the γ³ and γ divisors do not mix. Everywhere between, they are not: at 0.99c the worst case is 73.9°, reached with the force at 8.0°. A body under a steady sideways-ish push does not travel along it.

The push that does not point where the body goes

Newton's second law survives relativity in the form F = dp/dt and in no other. Written as F = ma it fails outright, and not merely by a factor — at high speed a body pushed at forty-five degrees accelerates at eighty, because the same force is divided by γ³ along the motion and by γ across it.

relativity · Relativistic dynamics A horizon 0.97 light years behind, made by nothing but the motion. Position across and time up, in units where light travels at 45°, for a rocket holding a constant proper acceleration of 1 gravity. The worldline is the hyperbola x² − c²t² = (c²/a)², asymptotic to the light line it never crosses. Three light signals are drawn: one released at x = 0.55 catches up at t = 0.63, one released at x = 0 never arrives, one released at x = -0.6 never arrives. The dividing line is the asymptote itself. Everything at or behind it is permanently out of reach, and for one gravity that boundary sits 0.97 light years behind the rocket's starting point. Nothing is there — no mass, no field, no surface. The horizon is a consequence of never stopping.

The wall of silence behind a rocket that never stops

Hold a constant acceleration and the worldline is a hyperbola asymptotic to a light ray — so there is a light ray that never catches it. An observer who never stops accelerating has a horizon a distance c²/a behind, made by nothing but the motion, and at one gravity it sits 0.97 light years back. Nothing is there. No mass, no surface, no field.

relativity · Accelerated frames How far a 1-gravity ship gets, against its own clock. The distance covered by a ship accelerating steadily at 1 gravity for half the trip and braking for the other half, against the time on its own clock, on a logarithmic vertical axis. The curve is a cosine hyperbolic and therefore an exponential once the ship is relativistic, which it is after about a year: Proxima Centauri in 3.5 shipboard years, Sirius in 4.6 shipboard years, the Pleiades in 11.9 shipboard years, the galactic centre in 19.8 shipboard years, Andromeda in 28.6 shipboard years. Nothing about this violates anything. The speed never reaches c — it is the hyperbolic tangent of the rapidity and after 3.5 years it is 0.9495 at turnover — and every one of those journeys takes slightly more than the distance in years as measured from home. What is growing exponentially is not the speed but the length contraction, and the traveller's honest description of the trip is that the distance shrank. The rapidity is what accumulates steadily: it grows by one unit every 0.969 years of shipboard time, for ever, with no ceiling anywhere in the arithmetic. That is the whole reason the numbers come out survivable, and the reason the fuel does not.

The ship that never arrives at c

Accelerate at one gravity and never stop. The speed creeps toward light and never reaches it, and meanwhile the galactic centre is twenty shipboard years away and Andromeda twenty-nine. What makes the journey survivable is that rapidity has no ceiling; what makes it impossible is that the fuel goes as the exponential of the same quantity.

relativity · Accelerated frames The temperature of an acceleration. The Unruh temperature against proper acceleration, both axes logarithmic. An observer accelerating through empty space finds it is not empty: the state that an inertial observer calls the vacuum, an accelerated one finds populated, with a thermal spectrum at T = ħa/2πck — which is 4.06e-21 kelvin for every metre per second squared. The line is straight because the relation is exactly proportional, and the numbers on it are what make the effect so hard to see: one gravity gives 3.98·10⁻²⁰ K, a centrifuge at 10⁵ g gives 3.97·10⁻¹⁵ K, an electron in a strong laser gives 40.6 K, the surface of a solar-mass hole gives 6.17·10⁻⁸ K. Reaching one kelvin requires 2.5·10²⁰ m/s², which is 2.5·10¹⁹ gravities and beyond anything that can be sustained. What makes the effect worth taking seriously despite that is not its size but its structure: it says the number of particles present is not a property of the field alone but of the observer as well, and the same expression with a black hole's surface gravity in place of the acceleration is the Hawking temperature exactly, checked here to a part in 10¹².

The temperature of an acceleration

Empty space is empty for an observer who is not accelerating. For one who is, the same state of the same field is a thermal bath at a temperature proportional to the acceleration — and the constant of proportionality is 4 × 10⁻²¹ kelvin for every metre per second squared, which is why nobody has felt it. What the effect changes is not what can be measured but what a particle is.

relativity · Accelerated frames Radiated while the push holds, paid for when it stops. The power a charge radiates, the power the radiation reaction force takes from its motion, and the rate of change of the Schott term mτ a·v, through a push that rises over the first 20 per cent of its duration, holds steady, and falls away over the last 20, in units of mτa₀² where a₀ is the steady acceleration. Radiated power is a², the reaction force's take is −ȧv, and the Schott rate is found by differencing a·v along the trajectory; at every instant the first equals the sum of the other two. While the push is steady the reaction force is exactly zero and the charge still radiates at the full rate, all of it drawn from the Schott term. When the push stops, ȧ is large and negative while the charge is moving fast, and the reaction force takes 0.775 units, more than the 0.750 radiated over the whole push. The difference is what it handed back while the push was starting: then the charge is still slow, the reaction force points along the rising acceleration, and it does 0.025 units of work on the charge instead of taking any. The totals agree to a part in a hundred thousand.

The bill that arrives when the pushing stops

A charge accelerating steadily radiates at the full Larmor rate while the radiation reaction force on it is exactly zero, so for as long as the push holds, nothing about the charge's motion pays a single watt. The energy is lent by the field that travels with the charge, the loan is called the Schott term, and it is repaid the moment the acceleration changes.

astrophysics · Radiating charge

Named alongside it

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

RapidityEquivalence principleHawking temperatureHorizonsHyperbolic motionProper timeAccelerated framesCausalityEnergy conservationEvent horizonForceHorizon

All concepts