A photon climbing a tower
At its defaults it draws a photon climbing a tower. A photon emitted at the foot of a tower 22.5 m high and received at the top. It arrives with its frequency lower by gh/c² = 2.455·10⁻¹⁵ — two and a half parts in a thousand million million. Nothing was done to the photon on the way up; the two ends of the tower disagree about how fast time passes, and the frequency is the evidence. The same fraction says a clock at the foot loses 0.21 nanoseconds a day against one at the top.
redshift-tower is one function in lib/figures/gravity.js —
geometry, horizons, and the waves in it. 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.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
A photon emitted at the foot of a tower 22.5 m high and received at the top. It arrives with its frequency lower by gh/c² = 2.455·10⁻¹⁵ — two and a half parts in a thousand million million. Nothing was done to the photon on the way up; the two ends of the tower disagree about how fast time passes, and the frequency is the evidence. The same fraction says a clock at the foot loses 0.21 nanoseconds a day against one at the top.
Two corrections, opposite in sign and different in size
The options are the ones The clock that is wrong in two directions passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
How fast a clock in a circular orbit runs compared with one on the ground, in microseconds a day, against the height of the orbit — with the two effects drawn apart rather than added. Being high speeds a clock up, by an amount that saturates: the potential term is bounded because there is only so much potential to climb out of. Moving slows it down, and a higher orbit is a slower one, so that term shrinks toward zero. They cancel at 3186 km — a radius of exactly 1.5 Earth radii, which follows from setting the sum to zero and contains neither G, nor the Earth's mass, nor the speed of light. At 20200 km the gravitational term is 45.7 µs a day and the speed term −7.2, leaving 38.5. Left uncorrected, that is 11.5 km of position error a day, growing without limit, from a clock that is working perfectly.
A photon climbing a tower
The options are the ones The clock that is wrong in two directions passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
A photon emitted at the foot of a tower 22.5 m high and received at the top. It arrives with its frequency lower by gh/c² = 2.455·10⁻¹⁵ — two and a half parts in a thousand million million. Nothing was done to the photon on the way up; the two ends of the tower disagree about how fast time passes, and the frequency is the evidence. The same fraction says a clock at the foot loses 0.21 nanoseconds a day against one at the top.
Where a clock gains, and where it loses
The options are the ones The clock that is wrong in two directions passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The rate of a clock in a circular orbit against one on the ground, in microseconds per day, plotted against altitude. Height makes it gain and speed makes it lose, and the two cancel exactly at 3186 km — where a satellite keeps the same time as the ground for two reasons that have nothing to do with each other. At 20200 km the total is 38.5 µs a day, which is about ten kilometres of position error if it is ignored.
Two corrections, opposite in sign and different in size
The options are the ones The clock that is wrong in two directions passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
How fast a clock in a circular orbit runs compared with one on the ground, in microseconds a day, against the height of the orbit — with the two effects drawn apart rather than added. Being high speeds a clock up, by an amount that saturates: the potential term is bounded because there is only so much potential to climb out of. Moving slows it down, and a higher orbit is a slower one, so that term shrinks toward zero. They cancel at 3186 km — a radius of exactly 1.5 Earth radii, which follows from setting the sum to zero and contains neither G, nor the Earth's mass, nor the speed of light. At 400 km the gravitational term is 3.6 µs a day and the speed term −28.3, leaving −24.7. Left uncorrected, that is 7.4 km of position error a day, growing without limit, from a clock that is working perfectly.
Where a clock gains, and where it loses
The options are the ones The clock that is wrong in two directions passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The rate of a clock in a circular orbit against one on the ground, in microseconds per day, plotted against altitude. Height makes it gain and speed makes it lose, and the two cancel exactly at 3186 km — where a satellite keeps the same time as the ground for two reasons that have nothing to do with each other. At 20200 km the total is 38.5 µs a day, which is about ten kilometres of position error if it is ignored.
What checks it
physicscheck asserts something about redshift-tower 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.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
The clock that is wrong in two directions
A satellite clock loses 7.2 microseconds a day to its speed and gains 45.9 to its height. The two effects have opposite signs, different sizes and different dependence on the orbit, so there is exactly one altitude where they cancel — and 38.6 microseconds a day, left alone, is eleven and a half kilometres of position error.
AstrophysicsThe clock that measures a height
A clock a metre higher runs faster by a part in ten thousand million million million. That was once so small it took a tower and a Mössbauer source to see; the best clocks now resolve a centimetre of height, at any distance, without a line of sight. What began as a test of general relativity has become a surveying instrument that measures the quantity surveying actually wants.
AstrophysicsThe clock that runs slow lower down
Two identical clocks, one on the floor and one on a shelf, do not keep the same time — and the difference is large enough that a satellite navigation system which ignored it would be useless within a morning. The derivation needs nothing but a photon and a conservation law.
AstrophysicsThe clocks that must all slow together
Every clock on the Earth runs slower in January than in July, by three parts in ten thousand million, because the orbit carries the planet deeper into the Sun's potential at perihelion. No clock on the Earth can see this, and that invisibility is the claim worth testing. If the redshift is a property of time rather than of clocks, two clocks built on different physics must slow by exactly the same fraction, and their ratio must not move with the seasons. A ratio that did move would mean the constants of nature depend on where they are measured.
RelativityThe column that is hotter at the bottom
Two bodies in equilibrium have the same temperature — that is what equilibrium was supposed to mean. In a gravitational field it is false. A column left alone until nothing in it changes is warmer at the bottom by exactly the factor by which clocks there run slow, a part in ten million billion per metre on the Earth and more than a per cent across the outer kilometre of a neutron star, and near a black hole's horizon the equilibrium temperature grows without limit.
AstrophysicsThe 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.
AstrophysicsThe parallelogram that will not close
Two clocks twenty-two metres apart in a lift shaft run at different rates, by two parts in a thousand million million. That measurement, on its own, is enough to prove that spacetime cannot be flat — and the proof needs no field equation, no curvature tensor and no astronomy. It needs one drawing and the fact that opposite sides of a parallelogram are the same length.
AstrophysicsThe 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".
RelativityThe two clocks that flew in opposite directions
Two caesium clocks were flown round the world in 1971, one each way, and came back disagreeing with the clock left behind — one having lost 59 nanoseconds and the other gained 273. Height alone would have made both gain. The sign flip comes from the ground already moving eastward at 400 metres a second before the aircraft took off.