# general theory of relativity

Space-time is represented in this diagram geometrically, as it is in the general theory of relativity. Space-time is shown as a flexible sheet that is distorted by the presence of masses. The large mass creating a space-time "crater" in the center is the Sun, around which the Earth rotates.

The original telegram to Einstein concerning Eddington's successful observation of the bending of starlight near the Sun in the eclipse of May 29, 1919. The telegram was sent by the the Dutch physicist H. A. Lorentz and states that "Eddington has found a stellar deflection at the solar limb provisionally between 0.9 seconds of arc and twice that." Image copyright: Museum Boerhaave, Leiden.

The general theory of relativity is Albert Einstein's theory of gravity, which describes gravitational forces in terms of the curvature of spacetime caused by the presence of mass. As the American physicist John Wheeler put it: "Space tells matter how to move; matter tells space how to curve".

The starting principle of the general theory, known as the equivalence principle, is that frames of reference undergoing acceleration and frames of reference in gravitational fields are equivalent. Among its predictions, which have been borne out by observation, are the advance of the perihelion of Mercury, the bending of light in a gravitational field (including gravitational lenses), and the spin-down of pulsars (due to the emission of gravitational waves, which have yet to be detected directly). Also predicted by general relativity is that time runs more slowly in strong gravitational fields.

General relativity treats special relativity as a restricted sub-theory that applies locally to any region of space sufficiently small that its curvature can be neglected.

## How the general theory of relativity came about

By the early years of the twentieth century, physicists had begun to appreciate,
through relativity theory, the intimate connections between energy and mass, and space and time. Yet gravity
stubbornly remained outside this picture. Henri Poincare,
in a paper submitted in July 1905, just days before Einstein's special relativity
paper, suggested that all forces ought to transform according to the Lorentz
transformation. But if this were the case, he pointed out, then Newton's
law of gravitation couldn't be valid because Newton's law allows *instantaneous* action at a distance. Drawing
an analogy with electromagnetic theory, Poincaré proposed that gravitational
interactions take place at the speed of light and involve waves that propagate at this fixed rate. Earlier, in 1900, Lorentz
had also hinted that gravitation could be put down to actions that travel
at light-speed.

It was in 1907 that Einstein began seriously to look into the problem of gravity. Two years after putting forward the special theory of relativity, he was sitting in his patent office in Berne wondering what would have to be done to Newtonian gravitation to make it fit in with his newly-hatched theory. Suddenly, he recalled, he had "the happiest thought" of his life:

[F]or an observer falling freely from the roof of a house there exists – at least in his immediate surroundings – no gravitational field. Indeed if the observer drops some bodies then these remain to him in a state of rest or uniform motion... The observer therefore has the right to interpret his state as 'at rest' [at least until he hits the ground!].

To drive home this point, imagine a slightly different situation. You're in a windowless room and are told that one of two circumstances is true: either the room is floating in space far away from any source of gravity or it's an elevator whose cable has been cut. Your task is to decide which, without leaving the room or otherwise obtaining information from outside. According to Einstein the task is impossible because there's no experiment you can carry out that will help you decide between the two scenarios. Nor, for the same reason, could you tell whether you were in a room that was sitting on the Earth or being smoothly accelerated at 9.8 meters per second per second – the rate at which things fall freely in Earth's gravity – by a rocket.

There's simply no observable difference, Einstein realized, between acceleration and gravity. On some deep level, they're one and the same. Consequently, he said:

[W]e shall ... assume the complete physical equivalence of a gravitational field and the corresponding acceleration of the reference frame. This assumption extends the principle of relativity to the case of uniformly accelerated motion of the reference frame.

It's an assumption that broadens the equivalence principle introduced by Galileo, which asserts that all objects
fall at the same rate, with the result that mass measured gravitationally
is indistinguishable from mass measured by its inertia.
What's now called the Einstein or strong equivalence goes beyond this older,
weaker version by stating that *all* the laws of physics, not just
the law of gravity, are the same in all small regions of space, regardless
of their relative motion or acceleration.

In the same year, 1907, that Einstein announced this broader principle of
equivalence, he also began linking his mass-energy
relationship, *E* = *mc*^{2}, with gravity. It
had long been known that gravity acts on everything with mass. Now that
mass and energy turned out to be two sides of the same coin, it seemed reasonable
to Einstein that gravity could act on energy too. In particular, it ought
to be able to influence the movement of light rays.

Einstein's very first scientific paper, published in March 1905, had been
on the nature of light. In it, he argued that
a well known phenomenon in physics called the photoelectric
effect could be explained if light behaved as if it consisted of tiny
discrete particles. Later, these particles came to be known as photons.
Because photons contained energy, and therefore, from the *E* = *mc*^{2} relation, an equivalent mass, their paths ought to be bent by gravity, just
as the path of a bullet is curved by gravity as it travels from gun to target.
But in 1907, when Einstein realized this, he was thinking only in terms
of how light might be influenced by gravity here on Earth and there seemed
little chance of experimentally verifying an effect that would be so small.

For four years, Einstein published nothing else on gravity. Then, in 1911, it dawned on him that the bending of light by gravity could be checked by astronomical means. Light from a background star ought to follow not a straight line but a gentle arc as it passed close to the sun as seen from Earth. Einstein came up with a figure for this bending, completely unaware that the same answer had been obtained back in 1803 by a little-known Bavarian astronomer Johann van Soldner, who used Newtonian gravitational theory and treated light as a stream of little projectiles. In 1913, Einstein wrote to the American astronomer George Hale to ask if it were possible to look for the minuscule deflection of starlight by the sun without waiting for a total eclipse. Hale replied that it wasn't: the sun's blindingly bright disk needed to be completely blotted out before any deflection of starlight would show as an apparent displacement of stars from their normal positions. The German astronomer Erwin Finley-Freundlich planned an expedition to Russia to observe an eclipse due to occur there in 1914 and thus to test Einstein's prediction. But World War I intervened and the expedition was canceled. For Einstein it proved to be a lucky break – his prediction would have turned out wrong.

By 1912 Einstein was hot on the heels of a new theory of gravity that would incorporate his strong equivalence principle. By calling on this principle, he realized, he could avoid dealing with gravity as a force altogether. Move in the right way, by free-falling, and you don't feel gravity: in an inertial frame,you're weightless and gravity drops out of the picture. But Einstein also realized that the Lorentz transformation of special relativity wouldn't carry over to a more general setting because the way you have to move to cancel out gravity is different in different locations. What he needed was some mathematical way to stitch together local inertial frames in different places so that gravity canceled out everywhere. Although he wasn't yet sure what form his new theory of gravity would take, he did know this: If all accelerated systems are equivalent [with respect to the laws of physics], then Euclidean geometry cannot hold in all of them.

Euclidean geometry is the geometry we learn in high school, with its familiar
straight lines, circles and triangles. It's the geometry of the plane or
"flat space" and was fully described around 300 B.C. by Euclid in his monumental book *Elements*.
Euclid starts out by listing five axioms, or self-evident truths, together
with five postulates, or additional assumptions. The last of these postulates,
which has come to be called the parallel
postulate, has always been a bit of an oddball. One way to state it
is that given any straight line and any point not on it, we can draw through
that point one, and only one, straight line parallel to the given line.
On the face of it, this seems commonsensical and obvious (try it with pencil
and paper). But there'd always been a lingering doubt about whether the
properties of parallel lines as presupposed in Euclidean geometry could
be derived from the other postulates and axioms, or whether the parallel
postulate had to be assumed as an extra fact. In the early 1800s, three
mathematicians, working independently, found good reason for this doubt.
Remarkably, they discovered geometric systems that satisfy all the axioms
and postulates of Euclidean geometry except the parallel postulate. These
geometries showed not only that the parallel postulate must be assumed in
order to obtain Euclidean geometry but, more importantly, that other geometries
– non-Euclidean geometries – can and do exist.

## Beyond Euclid

The first to hint that there were geometric realms undreamt of by Euclid was the mighty Carl Gauss, German mathematician, astronomer and physicist, who in 1817 wrote:

I became more and more convinced that the necessity of our [Euclidean] geometry cannot be demonstrated... [W]e must consider geometry as of equal rank, not with arithmetic, which is purely logical, but with mechanics, which is empirical.

In other words, argued Gauss, the geometry of the space we live in can't simply be assumed to be Euclidean; its nature must be determined by measurement and experiment. And this is exactly what he did. Commissioned by the government in 1827 to make a survey map of the region for miles around Göttingen, Gauss found that the sum of the angles in his largest survey triangle was different from the expected, Euclidean 180 degrees. The observed deviation – almost 15 arc-seconds – was both inescapable evidence for, and a measure of, the curvature of the surface of Earth. It was also the first concrete proof of a world that lay beyond Euclid's ken.

Gauss had many brilliant ideas that he didn't publish, and his pioneering
thoughts on non-Euclidean geometry were among them. His motto, *pauca
sed matura* ("few but ripe") and his fear of "the clamor of the Boetians"
– a reference to the people from a region of ancient Greece famous
for their obtuseness – conspired to keep him silent on this topic.
Only many years later, after his death in 1855, did the diary come to light
in which Gauss had written down his manifesto for a non-Euclidean revolution.

The first mathematician actually to go to press with his views on the subject was the Russian Nikolai Lobachevsky in 1826. He describes a geometry in which Euclid's parallel postulate isn't obeyed and in which the sum of the angles of a triangle add up to less than 180 degrees: a kind of geometry said to be hyperbolic and the sort found on the surface of a saddle. Unbeknownst to him, a young Hungarian mathematician, János Bólyai, had made the same startling breakthrough a few years earlier. Bólyai could hardly believe what he'd found: "Out of nothing I have created a strange new universe." His father, Wolfgang, a friend of Gauss, had spent much of his life trying to prove Euclid's fifth postulate and reacted with alarm to János's revelation:

For God's sake, I beseech you, give it up. Fear it no less than sensual passions because it, too, may take all your time, and deprive you of your health, peace of mind and happiness in life.

Gauss, however, reassured the elder Bólyai that the concept of geometries beyond that of Euclid wasn't as insane as it sounded and that, in fact, he'd held similar beliefs for several years. Finally and reluctantly, in a book published by Wolfgang in 1832, he included his son's revolutionary work on geometry as an appendix.

None of these contributions to exploring the non-Euclidean landscape had much impact on mathematics in the first half of the 19th century: the ideas were too arcane and bizarre, too heretical. Yet their time was coming. In 1853, when Gauss was seventy-six, his star pupil Bernhard Riemann had to give a lecture at the University of Göttingen to confirm his position as a faculty member. It was the tradition in such circumstances to offer three possible topics, but that the choice would be made between only the first two. Not surprisingly, given this normal course of events, Riemann hadn't fully prepared for his third choice: the foundations of geometry. Gauss, however, couldn't resist the prospect of hearing his wunderkind speak on a subject that he (Gauss) had grappled with for much of his life and so he asked Riemann to deliver his third topic. After several postponements, Riemann gave his lecture "On the Hypotheses Which Lie at the Foundation of Geometry" in June 1854. It proved to be a triumph and marked a turning point in our understanding of non-Euclidean math.

Gauss, earlier in his career, had published results in which he hugely advanced
the theory of surfaces in two dimensions. He'd shown that it isn't necessary
to consider a two-dimensional surface, such as a sphere,
to be embedded in a three-dimensional space in order to define its geometry.
It's enough to consider measurements made entirely within that two-dimensional
geometry, such as an intelligent ant might make that was forever restricted
to live on its surface. The ant would know that the surface was curved by
measuring that the sum of the internal angles of a large triangle differs
from 180 degrees (as Gauss had done during his geodetic survey), or by measuring
that the ratio between a large circumference and its radius differs from
2π. As a result of his study of surfaces, Gauss gave a precise mathematical
meaning to the idea of curvature and a
way of evaluating it. So-called *Gaussian curvature* is positive
on the surface of a sphere, negative at every point on a saddle-shaped surface
such as a hyperboloid, and zero for a plane. It thus determines whether
a surface has elliptic (Riemannian)
or hyperbolic geometry.

But Gauss didn't confine his thinking to a curved two-dimensional surface floating in a flat three-dimensional universe. In a letter to Ferdinand Schweikart in 1824, he dared to conceive that space itself is curved: "Indeed I have therefore from time to time in jest expressed the desire that Euclidean geometry would not be correct." This brilliant inspiration was to take root in the mind of Gauss's most talented apprentice.

Riemann extended Gauss's work to spaces of any number of dimensions and
put on a firm footing the type of non-Euclidean geometry that Gauss had
hinted at: the kind known as elliptic geometry, in which there are no parallel
lines and in which the angles of a triangle always add up to more than 180
degrees. He also generalized the notion of the shortest distance between
two points. In Euclidean geometry this is simply a straight line. But step
out of Euclid's domain and the quickest way to get from *A* to *B* involves a change of tack. The easiest way to grasp this idea is to think
about traveling on the Earth's surface, which isn't flat but (roughly) spherical:
a special case of Riemann's elliptic geometry. To take a ship on the shortest
route between two ports you sail, wherever possible, along an arc of a great
circle – the circle that goes all the way around the Earth and
on which both ports lie. Any such minimum-length path on a surface, the
special case of which on a plane is a straight line, is called a geodesic,
meaning "Earth divider".

In Euclidean geometry, the shortest distance between two points can be found using Pythagoras's theorem. What Riemann discovered was a more powerful, general form of Pythagoras's theorem that works on curved surfaces, even when the curvature is in more than two dimensions and varies from one place to another. In this looking-glass world of curved space, the familiar idea of distance is replaced by the broader concept of something called a metric, from the Greek for "measure," while curvature is similarly described by a more elaborate mathematical object. Gauss had found that the curvature in the neighborhood of a point of a specified two-dimensional geometry is given by a single number: the Gaussian curvature. Riemann showed that six numbers are needed to describe the curvature of a three-dimensional space at a given point, and that 20 numbers at each point are required for a four-dimensional geometry: the 20 independent components of the so-called Riemann curvature tensor.

In his famous lecture of 1854, Riemann emphasized, as Gauss had done, that the truth about the space we live in can't be found by poring over 2,000-year-old books of Greek geometry. It has to come from physical experience. He pointed out that space could be highly irregular at very small distances and yet appear smooth on an everyday level. At very great distances, he also noted, a large-scale curvature of space might show up, perhaps even bending the universe into a closed system like a gigantic ball:

Space [in the large] if one ascribes to it a constant curvature, is necessarily finite, provided only that this curvature has a positive value, however small... It is quite conceivable that the geometry of space in the very small does not satisfy the axioms of [Euclidean] geometry... The properties which distinguish space from other conceivable triply-extended magnitudes are only to be deduced from experience.

So far ahead of his time was Riemann that, having arrived at his great mathematical description of space curvature, he began working on a unified theory of electromagnetism and gravitation in terms of it. Riemann grasped that forces might be nothing more nor less than a manifestation of the geometry of space. Flat beings on a wrinkly two-dimensional landscape, like that of a crumpled sheet of paper, would, when they tried to move around, experience what felt to them like gravitational effects. By analogy, he reasoned, forces in our world might best be explained in terms of warps in a higher dimension. And the effect would work both ways. If space told mass how to move, then space must itself – by the principle of action and reaction – be affected by mass.

With these extraordinary possibilities, the 39-year-old Riemann wrestled in the summer of 1866, even as he lay dying of tuberculosis at Selasca on Lake Maggiore. He came so close – astonishingly close – to a geometric theory of gravity, half a century before Einstein, who later remarked of Riemann's contribution:

Physicists were still far removed from such a way of thinking: space was still, for them, a rigid, homogeneous something, susceptible of no change or conditions. Only the genius of Riemann, solitary and uncomprehended, had already won its way by the middle of the last century to a new conception of space, in which space was deprived of its rigidity, and in which its power to take part in physical events was recognized as possible.

One major obstacle had blocked Riemann's further progress. He thought only of space and its topography. Einstein's great epiphany was that, in building a new theory of gravity, he also had to deal with time – with spacetime and spacetime curvature. But, to begin with, he didn't have the mathematical tools to do this. They existed: Einstein simply didn't know about them.

## The mathematics of curved spacetime

How to stitch together countless tiny inertial patches to make a large, smoothly undulating quilt of curved spacetime? As Einstein began thinking about this, he remembered that he'd studied Gauss's theory of surfaces in college and suddenly realized that the foundations of geometry had physical significance. To pursue the problem further he contacted his old friend and talented mathematician Marcel Grossman. Einstein and Grossman had been students together at the ETH in Zurich; when Einstein skipped classes he'd often borrow Grossman's lecture notes. Einstein's overall mark at graduation was a marginal 4.91 out of 6, which left him the only member of his class not to be offered a place in the ETH's physics department. He'd been written off, he said later, as "a pariah, discounted and little loved," virtually unemployable. Toward the end of 1901, still having found no permanent position, he wrote to Grossman explaining his plight. Fortunately, Grossman's father happened to be chums with Friedrich Haller, the chief of the Swiss Patent Office, and so it was that Einstein got a desk job there despite Haller's opinion that he was "lacking in technical training."

Now, with gravity and curved spacetime on his mind, Einstein once again turned for help to his trusty ally. Grossman had been appointed professor of descriptive geometry at the ETH in 1907 and had gone on to build a reputation as an outstanding teacher. He told Einstein of Riemann's work and of a subject called tensor calculus, especially to contributions made in the 1860s by Elwin Christoffel and more recently by Gregono Ricci-Curbastro and Tullio Levi-Civita at the University of Padova. Abruptly thrown into a new and difficult field of math of which he'd previously been unaware, Einstein wrote:

[I]n all my life I have not labored nearly so hard, and I have become imbued with great respect for mathematics, the subtler part of which I had in my simple-mindedness regarded as pure luxury until now.

He had to learn about tensors – mathematical
objects that behave in certain well-defined ways when you switch coordinate
systems. (Vectors, for example, are a simple
type of tensor). And soon it became clear to him that tensor calculus gave
the perfect language for describing four-dimensional spacetime. In 1913,
Einstein and Grossman jointly published a paper in which they used the tensor
calculus of Ricci-Curbastro and Levi-Civita to portray gravity in terms
of a metric tensor (a tensor that gives a generalized way of measuring distance).^{1} But their theory was still far from complete. When Max Planck,
the father of quantum mechanics visited Einstein in 1913 and Einstein told
him how things stood with his new scheme of gravity, Planck said: "As an
older friend I must advise you against it for in the first place you will
not succeed, and even if you succeed no one will believe you."

For a while, it looked as if Planck might be proved right: not many scientists at the time thought Einstein was on the right track. Then, in October 1914, Einstein wrote a paper, nearly half of which was a treatise on tensors and differential geometry (the mathematics of surfaces). It proved to be a turning point because it led to a correspondence between Einstein and Levi-Civita in which the Italian pointed out technical errors in Einstein's analysis of tensors. Einstein was delighted by the exchange. Yet he continued to struggle with the equations that linked gravity with the geometry of spacetime.

At the end of June 1915 Einstein spent a week at the University Göttingen
where he lectured for six two-hour sessions on his (still incorrect) October
1914 version of what would become general relativity. Two of those present
were colossi in the world of mathematics, David Hilbert and Felix Klein. "To my great joy,"
Einstein later recalled, "I succeeded in convincing Hilbert and Klein completely."
Shortly after, Einstein and Hilbert began an intense exchange of letters
on the outstanding problems in Einstein's theory. And now matters quickly
came to a head. After chopping and changing the equations in his theory
several times in the autumn of 1915 – totally confusing his scientific
colleagues in the process – Einstein made a monumental breakthrough.
On November 18, 1915, he applied his new theory of gravitation to the old
problem of Mercury's orbit and, lo and behold, found that it predicted,
for the extra advance of the perihelion, exactly the 43 arc-seconds per
century that astronomers had measured and that had foiled every other attempt
at explanation. "For a few days," he remembered, "I was beside myself with
joyous excitement." To Hilbert, he wrote: "Today I am presenting to the
[Prussian] Academy a paper in which I derive quantitatively out of general
relativity, without any guiding hypothesis, the perihelion motion of Mercury
discovered by Leverrier. No gravitation theory had achieved this until now."
The Mercury figure was correct, but not yet the precise formulation. On
November 25, Einstein submitted yet another paper, called "The Field Equations
of Gravitation," which at last contained the correct mathematical scaffolding
of general relativity.^{2}

There's a postscript. Five days earlier, Hilbert had submitted a paper to
a journal in Göttingen containing exactly the same field equations.^{3} Suggestions have been made that Hilbert plagiarized Einstein, or perhaps
vice versa; certainly over those final frantic weeks before publication,
each man came to know the other's thoughts well. But if the relationship
between Einstein and Hilbert was strained for a while over the question
of priority, it ended amicably enough and Hilbert was able to write: "Every
boy in the streets of Göttingen understands more about four-dimensional
geometry than Einstein. Yet, in spite of that, Einstein did the work and
not the mathematician."

## Newton eclipsed

Aristotle saw gravity as a property of matter, Newton considered it a somewhat mysterious force. But in general relativity it's neither of these things. Gravity, in the brave new world of Einstein, is a manifestation of curvature in the geometry of spacetime. As John Wheeler put it: "Matter tells space how to curve. Space tells matter how to move." (Here Wheeler is using "space" as shorthand for "spacetime".) The Newtonian equivalent of this neat aphorism would be: "Matter tells matter how to move."

In many ways, general relativity turns our everyday notion of gravity on its head. Throw a ball straight up in the air and a graph of its height versus time, seen through Newton's eyes, traces out a parabola. Einstein, however, recognizes that a massive body – in this case, the Earth – curves the coordinate system itself. Rather than following a curved path in a flat (Cartesian) coordinate system, the ball actually follows a minimum-distance path, or geodesic, in a curved coordinate system, returning to the thrower's hand at a later time because the geodesic leads it there.

This remarkable new view of things immediately removes two of the unanswered questions in Newtonian theory: How does gravity work? And, why is the inertial mass of an object exactly equal to its gravitational mass? Einstein dismisses the first of these by showing that gravity isn't a force but simply a consequence of geometry. The second mystery also evaporates because, in general relativity, gravitational motion is seen as being nothing other than inertial motion in curved spacetime. In other words, the equivalence of inertial and gravitational mass, which, under Newton, appears to be a curious and accidental fact, is seen in general relativity to be a necessary and unavoidable feature of the theory. In Einstein's scheme, inertial mass and gravitational mass aren't just accidentally numerically equal, they're ontologically identical.

Though seemingly counterintuitive when first encountered, general relativity is a beautiful piece of work – mathematically and conceptually. But beauty alone isn't enough to ensure survival. The acid test of any good scientific theory is whether the predictions it makes are borne out by experiment and observation. Chalk one up for general relativity for getting right the advance of the perihelion of Mercury. Then add credits for two other classic I-told-you-so's: one concerning the deflection of light rays from faraway stars that graze the sun, the other the phenomenon of gravitational redshift.

We saw earlier that Einstein was lucky to escape having his (erroneous)
1911 divination of how much light is bent by the sun's gravity put to the
test. The new value that followed from the field equations of general relativity
in 1915 was a factor of two larger at 1.74 arc-seconds. In 1919, the two
British expeditions, one led by Eddington,^{4} triumphantly confirmed
this value to within the limits of experimental error, recording 1.98" ±0.30"
and 1.61" ±0.30". As for a check on Einstein's prediction of a gravitational
redshift, this had to wait much longer, until 1960, after Einstein's death.
Very accurate (atomic) clocks were needed to test that time really does
slow down to the extent foretold. But when these clocks became available,
general relativity was again completely vindicated.

Einstein's new vision of gravity superceded that of Newton. It explained what the older theory could not, in the most elegant way imaginable, and it survived the classic tests of its accuracy.

## References

1. Einstein, A., and M. Grossman. Entwurf einer Verallgemeinerten Relativitätstheorie
und eine Theorie der Gravitation (Draft of a Generalized Theory of Relativity
and a Theory of Gravitation). I. Physikalishcher Teil von (Physics Part
by) A. Einstein. II. Mathematischer Teil von (Mathematics Part by) M.
Grossman. *Zeitschrift für Physik* 62 (1913).

2. Einstein, A. Feldgleichungen der Gravitation (The Field Equations
of Gravitation). *Sitzungsberichte der Preussischen Akadamie der
Wissenschaften* (1915).

3. Hilbert, D. Die Grundlagen der Physik (Erste Mitteilung) (The Foundations
of Physics, First Communication). *Nachrichten von der Gesellschaft
der Wissenschaften zu Gottingen, Mathematisch-physikalische Klasse* (1915): 395-407.

4. Dyson, F. W., A. S. Eddington, and C. Davidson. *Philosophical
Transactions of the Royal Society of London* 220A (1920): 291-333.