Two predictions, a factor of two apart
At its defaults it draws two predictions, a factor of two apart. The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit.
light-bending 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.
The deflection of light passing a mass, against impact parameter, on logarithmic axes. The lower line is what a Newtonian photon does — it falls while it crosses, and comes out bent by 2GM/bc². The upper line is what a geodesic does in curved spacetime, which is exactly twice that. At the surface of a body of 1.99·10³⁰ kg the two are 0.88″ and 1.75″. Both are straight lines of slope minus one, so the ratio is two everywhere and the measurement is a choice between two theories rather than a fit.
The bend that depends on when the planet was there
The options are the ones The bend a moving planet gives 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 difference, in microarcseconds, between the quasar's deflection computed from Jupiter's position when the light passed it — its position as seen from the Earth, 53 minutes earlier — and from its position at the moment of observation, over the day of the 2002 pass. In those 53 minutes Jupiter moves 40998 km, 0.57 of its own radius, and the deflection, a vector of 1.14 mas pointing away from the planet, turns and shrinks by up to 46 µas — about one twenty-fifth of itself. That small term was what the experiment was designed to see. The VLBA measured 51 ± 9 µas, the value general relativity predicts for a deflection pointing away from the position light reveals.
How far a planet bends starlight
The options are the ones The bend a moving planet gives 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 deflection of a ray passing a planet, 4GM/c²b, in microarcseconds, against the impact parameter in units of the planet's own radius, for Jupiter (16.3 mas at the limb), Saturn (5.8 mas) and the Earth (0.57 mas); every line falls as 1/b. The Sun bends a grazing ray by 1.75 arcseconds, a hundred times more than Jupiter. On 8 September 2002 a quasar passed 3.7 arcminutes from Jupiter's centre, 14.2 Jupiter radii, where the deflection was 1.14 mas — marked — and radio interferometry measured it.
A quasar pushed aside by a passing planet
The options are the ones The bend a moving planet gives 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 path of the quasar J0842+1835 relative to Jupiter over a day around closest approach on 8 September 2002, in Jupiter radii on the sky, with Jupiter's disc drawn to scale: the true direction of the quasar as a straight line, and its apparent direction, displaced radially away from Jupiter by the deflection, exaggerated 1600 times so that it can be seen. At closest approach, 14.2 radii from Jupiter's centre, the displacement is 1.14 mas; twelve hours either side, 27 radii away, it has fallen to 0.60 mas. Dots mark six-hour intervals, Jupiter moving about 30 arcseconds an hour against the distant quasars. The measurement compared the quasar's position with two reference quasars nearby throughout the pass, so that the atmosphere's and the instruments' errors, which affected all three alike, cancelled.
What the measured term was said to measure
The options are the ones The bend a moving planet gives 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 predicted velocity-dependent term, 46 µas × c/c′, if the deflection pointed away from Jupiter's position retarded by its distance divided by some speed c′, against c′/c; the measured 51 ± 9 µas as a horizontal band. Where they cross, in this simplified geometry with Jupiter's motion taken square to the line of sight, c′ = 0.90 c, the band spanning 0.77 to 1.10 c; the published analysis of the full geometry gave 1.06 ± 0.21, and was announced in 2003 as the first measurement of the speed of gravity. The algebra is not in dispute. The dispute was over what c′ is: in general relativity the delay that enters is the time for light to travel from Jupiter to the Earth, so the experiment confirmed that the deflection points away from where Jupiter is seen, which a critic called a measurement of the speed of light; a theory in which gravity's speed differed from light's would change the deflection only at order (v/c)², far below this.
The terms a moving planet's bend is built from
The options are the ones The bend a moving planet gives 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 parts of a ray's deflection by Jupiter, in microarcseconds on a logarithmic scale, against how far from the planet's centre it passes, in Jupiter radii: the ordinary bend 4GM/c²b; the velocity-dependent term, of order (v/c)(D/b) of it, from the planet's motion during the light time to the Earth; the extra bend from Jupiter's flattening, of order its quadrupole moment times the square of the ratio of its radius to the impact parameter; the second-order bend in GM/c²b; and a term of order (v/c)² of the bend, the first place a speed of gravity different from light's could enter. At the 2002 pass, 14.2 radii out: monopole, 4GM/c²b 1143 µas; velocity term 46 µas; flattening 0.083 µas; second order in GM 4.7·10⁻⁶ µas; order (v/c)² 2.1·10⁻⁶ µas. The measurement reached about ten microarcseconds, enough to see the first two and none of the others: the term that was argued over is the second, and the one that would have measured gravity's own speed is 7 orders of magnitude below it.
What checks it
physicscheck asserts something about light-bending 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 bend a moving planet gives
Jupiter bends starlight by sixteen thousandths of an arcsecond at its edge, a hundredth of what the Sun manages, and in 2002 radio telescopes spread across a continent measured the bend on a quasar three and a half arcminutes from the planet. The experiment was designed to see something smaller still: the fact that Jupiter moves, by half its own width in the time light takes to reach the Earth, so that the bend depends on where the planet was when the light went past. The measured term was announced as the speed of gravity, and the argument over whether it was anything of the kind is the clearest lesson there is in what an experiment's number is a number of.
AstrophysicsThe bend Newton got half right
A photon treated as a falling body passes a mass and comes out deflected. So does a photon treated as a straight line in curved spacetime — by exactly twice as much. The factor of two is not a refinement; it is the sharpest available statement that gravity is geometry.
AstrophysicsThe circle light cannot leave
A black hole has three radii and they are all sometimes called its size. The horizon is where nothing can come back from; the photon sphere, half again as far out, is where light can circle and cannot keep circling; and the shadow a distant observer sees is larger than either, because the ray that just escapes was bent on its way out.
AstrophysicsThe delay that is not a bend
The same metric that bends a ray also slows it, and the two are different tests. A radar echo from Venus arrives 233 microseconds late when its path grazes the Sun — and half of that delay is accumulated more than twenty solar radii away, in a field thousands of times weaker.
AstrophysicsThe fifth image a galaxy hides
A single star bends light into two images of whatever lies behind it. A galaxy, whose mass is spread out, has to make an odd number — one, three or five — because the map from the sky to the source is smooth and has nowhere to put an even count. The famous cross of four quasar images is therefore five images, with the fifth thousands of times too faint to see, sitting at the centre of the galaxy that made it. Images are born and destroyed only in pairs, and only on the curves where the map folds.
AstrophysicsThe half of the bend a slow body never feels
Light passing the Sun is bent by twice Newton's angle, and the factor of two is usually read as a fact about light. It is the end point of a curve every body lies on. A body at speed v is bent by (2GM/bv²)(1 + v²/c²) — the 1 from the curvature of time, which a slow body feels in full, and the v²/c² from the curvature of space, which is the same absolute angle for a comet as for a photon and is simply swamped when the body is slow.
AstrophysicsThe lens a long wave walks round
Geometric optics says that a star passing exactly in front of a source magnifies it infinitely. No real wave agrees, because a wave cannot be focused to a point, and for the gravitational waves the detectors hear the disagreement is the whole story. A star, or a black hole of a few tens of suns, is smaller than one of those waves in the only measure that matters — the delay between paths round it — and the wave flows past it almost untouched. What decides whether a mass lenses a wave is not the mass's size, which is millions of kilometres, but its Schwarzschild radius set against the wavelength.
AstrophysicsThe lens with no focal length
A glass lens bends a ray by an angle proportional to how far off the axis it passes, which is precisely the condition for every ray to arrive at one point. Gravity bends by an angle that falls with distance off the axis, so every ray has its own focus and there is no image plane anywhere — only a half-line of foci, beginning 548 astronomical units from the Sun and running outward for ever.
AstrophysicsThe star whose far side can be seen
Look at a ball and half of it faces you; the other half is hidden behind it. That half is a fact about straight lines, and near a neutron star the lines are not straight. Light leaving the surface is bent round the star on its way out, so a distant observer sees three-quarters of a typical neutron star at once — and a star squeezed to 1.76 times its own Schwarzschild radius would show every point of its surface, the point directly behind it included.