Pushing one way and going another

At its defaults it draws 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.

relativistic-force is one function in lib/figures/spacetime.js — worldlines, slicing and the factor that governs both. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

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.

A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

Pressure against energy density for three descriptions of dense matter, with density measured against nuclear saturation and both axes in units where the speed of light is one. The straight line at forty-five degrees is not a model: it is the ceiling. The speed of sound in a material is the square root of the slope of this curve, so a curve steeper than the diagonal anywhere describes a substance in which a push travels faster than light — and no substance does. The steepest slope each model reaches is 0.231c for the ideal gas of nucleons, 0.577c for the relativistic degenerate, 0.962c for the a stiff nuclear model, measured off the drawn curves. That ceiling is why the argument in this essay is not a curiosity. A neutron star is held up by the stiffness of nuclear matter, so a stiffer equation of state supports a heavier star — and the causal limit therefore puts a hard cap on the mass of any neutron star whatever, at a little under three solar masses, regardless of what nuclear physics turns out to be. Anything heavier and more compact than that is a black hole because no material may be stiff enough to stop it. What the picture cannot show is the direction of the inference: the ceiling is a statement about signals, and it constrains matter only because matter is made of things that have to be told.

A rod 3 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 10.0 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 8.8 ms at 3.4e+2 m/s, 600.0 μs at 5.0e+3 m/s, 250.0 μs at 1.2e+4 m/s, 100.0 ns at 3.0e+7 m/s, 20.0 ns at 1.5e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

A rod 1 metres long, pushed at one end at time zero. On the left, position across and time upwards, for the two fastest disturbance speeds drawn here, with the light cone beside them: nothing may lean further to the right than that line, which crosses the rod in 3.3 nanoseconds. A rigid rod would be the vertical dashed line — the far end moving at the same instant as the near one — and it is not a limit that a hard material approaches. It is a signal at infinite speed. On the right, how long the far end actually waits, against how fast the disturbance travels, both logarithmic, with every material on it: 666.7 μs at 1.5e+3 m/s, 200.0 μs at 5.0e+3 m/s, 83.3 μs at 1.2e+4 m/s, 10.0 ns at 1.0e+8 m/s. The line has slope −1 and the light cone is a hard floor beneath it. Ordinary materials sit four to five decades above that floor, which is why rigidity is such a good approximation and why it is still not a limit: steel's delay is not small compared with light's, it is 6e+4 times larger. Everything usually derived from rigid bodies survives, because the delay is beneath notice in ordinary circumstances. What does not survive is the use of rigidity in an argument about simultaneity, which is where it does real damage: a rod pushed at one end is compressed for as long as the wave takes to cross it, and there is a frame in which its far end is still at rest while its near end is moving.

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.

physicscheck asserts something about relativistic-force that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.