General relativity

General relativity

Einstein, 1915

The plate from the eclipse of 29 May 1919. The corona is not the point and the Moon is not the point: the point is the short horizontal marks, ruled by hand beside stars that were photographed a little further from the Sun than they belong. The displacement being argued over is a good deal smaller than the thickness of the marks drawn to point at it.

Field equations	25 November 1915, Prussian Academy[1]
Full exposition	March 1916, Annalen der Physik[1]
Gravity, in the theory	Not a force[2]
What it is instead	The shape of the region
Founding observation	That falling feels like nothing
Adjustable constants	One (Λ)
Times removed	Once
Times restored	Once
First exact solution	Schwarzschild, 1915, from a trench[3]
Parameters fitted to Mercury	None available
Rotation still detectable?	Yes, locally

General relativity is the theory of gravitation published by Albert Einstein in November 1915 and set out in full the following March. It extends special relativity to gravitation, and in doing so disposes of gravity as a force. Nothing in the theory pulls on anything. Bodies in free fall travel the straightest available lines through a spacetime whose shape is determined by whatever mass and energy is in it, and what had been a force becomes a geometry.^[2]

The theory was not built from an anomaly, and this is unusual. It was built from a sensation: that a man falling freely does not feel his own weight. Einstein noticed this in 1907 and called it afterwards the happiest thought of his life, which for a man of his output is a considerable claim.^[4]

The happiest thought

Stand in a lift. If the cable is cut, you and the lift and everything loose in it fall together, and for the duration you float. Nothing in the lift with you can tell you whether you are falling in a gravitational field or drifting in deep space with no field at all, because gravity acts on every part of you equally and there is therefore nothing left over to feel.

Einstein's move was to take this seriously as a principle rather than as a curiosity: that a freely falling frame is, locally, exactly as good as an inertial frame, and that no experiment performed inside a small enough box over a short enough time can distinguish gravitation from acceleration. This is the equivalence principle, and everything else follows from insisting on it.

It is worth dwelling on the shape of the reasoning, because this encyclopedia is largely occupied with arguments of the opposite shape. The argument from sensation is the oldest one there is and is usually the weakest: I feel no motion, therefore there is none. Einstein began from a sensation too, and did the reverse thing with it. He did not conclude that the falling man was at rest. He asked what the laws of physics would have to look like if the falling man's account were exactly as good as everybody else's, and then found out. The difference between the two procedures is eight years and a field equation.^[5]

The theory also states the limit of its own founding observation, which is the other half of the difference. The equivalence holds only locally. Make the box large enough and the floor gives it away: two dropped apples on opposite sides of a falling lift converge slightly, because they are each falling towards the same centre and the centre is not infinitely far off. Those are tidal effects, they cannot be transformed away by any choice of frame, and they are the curvature itself.^[6]

What replaced the force

Matter and energy tell spacetime what shape to take; the shape tells matter how to move. The relation is one tensor equation, which reads in full as ten coupled non-linear partial differential equations and on the page as this:

The left is geometry, the right is contents, and the constant in front of the contents is fixed by requiring that the whole thing reduce to Newton where Newton works. There is no freedom in it. Given a distribution of mass and energy, and the conditions at the edge of the region, the equations return one geometry and will not be persuaded to return another. Solving them is a separate misery: exact solutions exist for almost no realistic arrangement of matter at all, which is why the few that do exist are known by the names of the people who found them, in the manner of comets.

Spacetime tells matter how to move; matter tells spacetime how to curve.
– John Archibald Wheeler

An orbit, in this account, is not a body being pulled off a straight line. It is a body going as straight as it can through a region that is not flat – a geodesic, in the theory's own word – in the same sense that an aircraft flying the shortest route from London to Tokyo goes nowhere near the straight line on the wall map and does not thereby require a force.^[7]

Λ, the cosmological constant, is the theory's one genuinely adjustable number. Einstein added it in 1917 so that the equations would permit a universe that stayed the same size, that being what everyone then believed the universe did. When it turned out not to, the term was dropped. It was reinstated in 1998, by two teams measuring distant supernovae independently of each other, and it is now the largest single term in the inventory of the universe. Its value is not predicted by anything.^[8]

What an accelerometer reads

An accelerometer is a weight on a spring in a box. Push the box and the weight lags behind, the spring gives, and the instrument reports how hard the spring is having to work. That is the whole of the mechanism, and it is worth following through two cases, because they come out the opposite way round from the way nearly everyone expects.

Push the box along a level road and the reading grows. The push arrives at one surface and has to be handed on from part to part – bolts to case, case to spring, spring to weight – and a spring that is handing something on is a spring under strain, which is the quantity being reported. It was reading the road before the push began and goes on reading it throughout, so what it reports is the two together, the upward and the forward, and never the forward by itself.

Now drop the box. It reads zero the whole way down, and goes on reading zero until it lands. Nothing has been switched off. Gravity is acting on the case, the spring and the weight in precisely the same measure, so the weight does not lag, the spring is neither stretched nor squeezed, and there is nothing left over for the instrument to notice. This is the lift again, in hardware: gravity cannot be felt, because it does not take hold of one part of a thing at a time. It is alone in that, and the reason is that it acts in proportion to mass, which is exactly the quantity that resists being moved. Every part of the instrument is therefore given precisely the acceleration it was going to have anyway, and nothing is left over to strain a spring. A push has to be handed on. This does not.

The corollary is the part that unsettles people. An accelerometer standing still on a bench reads about 9.8 m/s², and the reading points upward. It is not measuring gravity. It is measuring the bench, which is the only thing in the arrangement that is pushing anything at all, and an instrument anywhere on the Earth's surface cannot be brought to read zero except by dropping it.

Physics keeps two words for this and it is worth keeping them apart. Coordinate acceleration is how a velocity changes with respect to some chosen set of coordinates: a bookkeeping quantity, which comes out differently if the coordinates are chosen differently. Proper acceleration is what the instrument reads: the departure from free fall, on which every observer agrees and which no choice of coordinates can adjust. In the two cases above they are not merely different figures. They are each other's opposites.

The falling instrument is the one going straight. Free fall is travel along a geodesic, and proper acceleration is the measure of being held off one by something solid: the bench does that continuously, and what the instrument reports is the bench. The theory's inversion of the ordinary account is complete here and can be put in a sentence. The body that is accelerating is the one sitting still.

The three classical tests

Mercury. The perihelion of Mercury advances by about 43 seconds of arc per century more than the pull of the other planets accounts for. The discrepancy had been known since Le Verrier reported it in 1859 and had been attacked for fifty years, most seriously by supposing an unseen planet inside Mercury's orbit; it was named Vulcan and it was searched for and it was not there. Einstein computed the advance from the new equations in November 1915 and got 43 seconds of arc per century, with nothing to adjust and nothing adjusted. He is reported to have had palpitations.^[9]

Starlight. A ray grazing the Sun should be deflected by 1.75 seconds of arc. This is the test that made him famous, and it is also the one where he was publicly and recoverably wrong. In 1911, working from the equivalence principle alone and before he had the field equations, he published a value of about half that, and campaigned to have it measured. An expedition went to the Crimea for the eclipse of August 1914 to do so, and was interned as enemy aliens when the war began three weeks before totality. Had it not been, it might well have measured a deflection twice the size of his published one, five years before he could have accounted for it.^[10]

The measurement was eventually made on 29 May 1919 at Sobral in Brazil and on Príncipe off West Africa, and announced in London that November. Sobral returned 1.98 seconds of arc and Príncipe 1.61, against a prediction of 1.75 and a Newtonian half-value of about 0.87. The error bars were large, the argument about which plates to keep has run ever since, and the answer has been confirmed many times over by people with better equipment and no stake in it.^[11]

Redshift. A clock deeper in a gravitational field runs slow, and light climbing out of one is reddened. This is the hardest of the three to measure and the last to be done properly: Pound and Rebka managed it in 1959 over a height of twenty-two and a half metres, using the recoil-free emission of gamma rays from iron to get the precision. The fractional shift they were looking for was about two and a half parts in a thousand of a million million.^[12] They made the measurement up the lift shaft of the Jefferson Laboratory at Harvard, so that a theory which had begun in an imaginary lift was settled, on its third point, in a real one.

Since then

The theory is now tested continuously rather than occasionally, and the tests are mostly no longer experiments but appliances.

The last line is the one worth dwelling on. It is not a confirmation performed by physicists in a laboratory; it is a correction applied by engineers in a factory, to hardware sold commercially, because the equipment does not work without it.^[13]

Challenges arrive steadily and mostly do not survive contact with the apparatus. The Pioneer anomaly was the most durable of the recent ones: two spacecraft drifting sunward of their predicted tracks for as long as they transmitted, offered for years as a possible failure of gravity at large distances, and traced in 2012 to the heat coming off their own generators.

The redshift has since been turned into an instrument twice over. It governs the definition of the world's time scale, which has to name the surface it is realised on, a clock one metre higher running measurably faster; and it has been run backwards, so that two optical clocks compared against each other now measure the difference in height between them to a few centimetres.

General covariance is not what it sounds like

The theory is written so that its equations hold their form in any coordinate system whatever: rotating, accelerating, arbitrarily contorted. This is called general covariance, and it is where nearly all of the trouble comes from, because it sounds exactly like a licence to declare any state of motion as good as any other.

The objection to reading it that way was made almost immediately, in 1917, by Erich Kretschmann, and Einstein conceded it the following year. Kretschmann's point was that general covariance is not a physical claim at all. Any theory can be written in generally covariant form, Newton's included, if one is willing to carry the extra apparatus; what covariance expresses is a freedom in the description, not a symmetry of the world.^[14]

The distinction is the whole of the matter and is worth putting plainly. You are entirely free to adopt a coordinate system rotating with the Earth, in which the Earth does not turn and the sky does. General relativity permits this, and will hand you correct answers in it, and no physicist objects. What arrives with the coordinates is a spacetime laced with enormous inertial terms, precisely arranged to sweep the entire sky round once a day, every one of them a consequence of the rotation you have declined to name. The arrangement also has an edge. At about twenty-seven and a half astronomical units the co-rotating coordinate stops describing anything that could be at rest in it, since holding station there would mean outrunning light; that boundary falls a little inside the orbit of Neptune, which is a peculiar place for a worldview to end. The description has changed. The dynamics have not, and neither has the Earth.^[15]

The use made of it

General relativity is cited in this encyclopedia's usual quarters for three propositions. It declines the first two outright, and grants the third about half of what it asks.

The first is that it makes all motion relative and therefore permits a stationary Earth at the centre of a revolving cosmos. It does not. It makes all coordinates admissible, which is a different and much cheaper claim, and it leaves rotation plainly detectable: by a pendulum, by any gyroscope, by the Sagnac effect in a fibre coil costing rather less than a telescope. A theory in which you cannot tell whether you are turning would be a remarkable theory. This is not it.^[16]

The second is that the theory is unfalsifiable, being patched with dark matter and dark energy whenever it fails. This is the reverse of its history. Mercury was a retrodiction with nothing to tune, the light deflection was a prediction made in advance and made wrong once in public before it was made right, and the waveform of a black hole merger was computed before any instrument existed that could have heard one, which is how the detectors knew what to listen for. What is genuinely open is whether the theory holds at galactic scales without additional matter – that is a live question, and it is being pursued by people who would very much like the answer to be no.^[17]

A third is newer, and is made less about the theory than with its vocabulary: that the world is a plane accelerating upward at 9.8 m/s², and that the accelerometer's reading is the proof of it. The reading is real, and the first half of the reading is granted here without argument, because it is the same half this article has just spent a section on: you are being pushed, and the push comes from below.

What the doctrine cannot supply is the variation. A rigid floor under uniform acceleration reads the same at every point across itself, and the Earth does not: about 9.78 m/s² at the equator against 9.83 at the poles, which is half a per cent, in the pattern a spinning oblate body gives. Nor can it supply the fall with height. An accelerating floor does permit one, of about one part in 10¹⁶ for every metre climbed; the measured figure is three parts in 10⁷, which is some three thousand million times larger, and a good gravimeter sees it between one stair tread and the next.^[18] The instrument was granted first, and it is now the difficulty.

It remains, a century and more on, incompatible with quantum mechanics, and every physicist alive knows this and says so in print. That is the actual scandal in the subject, and it is not the one usually alleged.

See also

• Special relativity – the theory of uniform motion that this one generalises
• Albert Einstein – who spent eight years on the extension and ten minutes on the idea
• Kinematic and dynamic equivalence – where the argument from covariance is set out and answered
• Newton's bucket – the question of what rotation is measured against
• Ernst Mach – whose answer to that question Einstein took up, named, and could not make work
• Foucault's pendulum – rotation, detected in a closed room
• GPS – where the theory is applied to hardware before it leaves the ground
• Time – where the redshift stopped being a test and became the definition of the second's surface
• Vulcan – the planet invented to explain Mercury, and abolished by explaining it
• Orbits – the Newtonian account of the same motion, which this one does not so much correct as reinterpret
• Gravity – the force this theory does without, and the alternatives that do without it less successfully
• The speed of light – which gravitational waves travel at, to a few parts in a thousand million million
• The Pioneer anomaly – the best-known candidate for a failure of it, which turned out to be a warm generator

References

1. ^ "Die Feldgleichungen der Gravitation", presented 25 November 1915. Four communications were given to the Academy that month, in successive weeks, the theory changing between them. The full exposition followed in the Annalen the next March, running to some fifty pages against the thirty of the special theory, and citing rather more than nobody.
2. ^ It is worth being exact about how thorough this is. There is no gravitational force in general relativity, no gravitational potential energy in the Newtonian sense, and nothing that gravity does to a freely falling body. What remains is the shape of the region and the requirement that bodies go straight through it. Note also what the theory is not: it is often described as special relativity extended to accelerated motion, and that is not right. Special relativity handles acceleration without difficulty, and always could. What it cannot accommodate is curvature, which is to say gravity, and that is the whole of the extension. The other description survives because it was Einstein's own motivation, and the section on general covariance is the story of his giving it up.
3. ^ Karl Schwarzschild produced the exact exterior solution while serving with the German artillery on the Russian front, and sent it to Einstein within weeks of the equations appearing. Einstein replied that he had not expected the exact solution could be stated so simply. Schwarzschild died the following May of an autoimmune disease contracted at the front, aged forty-two.
4. ^ Recalled in a manuscript of about 1920: der glücklichste Gedanke meines Lebens. He was in the patent office at Bern at the time and, by his own account, sitting in a chair.
5. ^ 1907 to 1915. Much of that time went on learning the differential geometry, which he did not know and which his friend Marcel Grossmann did. See Albert Einstein.
6. ^ Which is why the principle is always stated with locally in it, and why quoting it without that word produces most of the paradoxes attributed to the theory.
7. ^ The analogy is imperfect in the usual way: the aircraft's shortest path is through curved space only, and an orbit is a straight line through a curved spacetime, most of the curvature being in the time direction. This is a genuine difference and not a quibble, since it is why things fall at all rather than merely being deflected.
8. ^ George Gamow reported Einstein calling the constant his greatest blunder; the remark is not recorded anywhere in Einstein's own hand, and is repeated here with that noted. Two independent supernova programmes restored the term in 1998, and it has been the largest single component of the universe ever since.
9. ^ Vulcan was announced, named, observed by several people, and assigned an orbit, none of which is the same as existing. The 43 seconds of arc had been the outstanding failure of Newtonian gravitation for half a century and was cleared in a fortnight by a theory built for other reasons entirely.
10. ^ Erwin Finlay-Freundlich's party reached the Crimea and was arrested when war broke out, the eclipse arriving on 21 August 1914 with the instruments impounded. Einstein remarked afterwards that he had been fortunate. It is the only recorded case in this encyclopedia of a war improving a theory.
11. ^ The Sobral astrographic plates were set aside as defocused and the argument over whether that was justified has outlived everybody concerned. The modern radio measurements settle the physics and do not settle the argument, which was never really about the physics.
12. ^ R. V. Pound and G. A. Rebka, "Apparent Weight of Photons", 1960. The Mössbauer effect had been discovered two years earlier and made the measurement possible; the shift over twenty-two and a half metres is about $2.5 \times 10^{-15}$. It should be said that this test is the weakest of the three as evidence for these particular equations: the redshift follows from the equivalence principle alone, and any theory built on a curved metric predicts it. Mercury and the deflection discriminate; this one only qualifies.
13. ^ See GPS, where the gravitational and the special-relativistic corrections run in opposite directions and do not cancel.
14. ^ E. Kretschmann, Annalen der Physik, 1917. Einstein's acknowledgement appeared in 1918, and he replaced the appeal to covariance with an appeal to simplicity, which is a weaker and more honest thing to appeal to.
15. ^ The inertial terms are not a rhetorical flourish. They are the metric components you obtain by transforming to the rotating frame, they grow with distance from the axis, and they are the reason the same manoeuvre performed on a merry-go-round does not persuade anybody that the fairground is spinning. The twenty-seven-and-a-half-unit figure is $c/\omega$ for one rotation of the Earth in a sidereal day, and marks where the co-rotating frame ceases to contain observers at rest rather than where the coordinates cease to be usable; the coordinates remain perfectly legal, and empty. See kinematic and dynamic equivalence.
16. ^ A fibre-optic gyroscope is a consumer article. It reports the rotation of the Earth from a sealed box on a bench, and has to be told about it in software so that it can subtract it and get on with reporting the rotation of the vehicle.
17. ^ Modified-gravity programmes are a going concern in professional astrophysics and are pursued by people who publish, predict and lose arguments in the ordinary way. They are not usually the people citing the theory's alleged unfalsifiability.
18. ^ The floor's own gradient is $a^{2}/c^{2}$ per metre, which for $a = 9.8\ \text{m/s}^{2}$ is 1.07 × 10⁻¹⁵ m/s² per metre, or 1.1 × 10⁻¹⁶ of the reading. The free-air gradient actually measured is 3.086 × 10⁻⁶ m/s² per metre, or 3.1 × 10⁻⁷ of it. One tread of a staircase is about 55 microgal, and a modern gravimeter resolves rather better than that. The departures from the smooth figure are surveyed commercially, to find what is buried under them.

External links

• Einstein, "Die Grundlage der allgemeinen Relativitätstheorie" – Annalen der Physik 49 (1916)
• Dyson, Eddington & Davidson, "A determination of the deflection of light by the Sun's gravitational field" – Phil. Trans. R. Soc. A (1920)
• The 1919 eclipse negative, as reproduced from that paper – Wikimedia Commons
• Pound & Rebka, "Apparent Weight of Photons" – Phys. Rev. Lett. 4 (1960)
• Everitt et al., "Gravity Probe B: Final Results of a Space Experiment to Test General Relativity" – Phys. Rev. Lett. 106 (2011)
• LIGO, "Observation of Gravitational Waves from a Binary Black Hole Merger" – Phys. Rev. Lett. 116 (2016)
• Event Horizon Telescope, "First M87 Results I: The Shadow of the Supermassive Black Hole" – ApJL 875 (2019)
• General relativity – general reference