INTRODUCTION.
The Special Theory of Relativity (SR) dealt with the laws of physics regardless of the relative uniform motion of the frame of reference. The General Theory of Relativity (GR) attempts to formulate the laws of physics regardless to the relative accelerated motion of the frame of reference. This idea occurred to Einstein as perhaps the key to understanding the equivalence of gravitational mass and inertial mass. It had long been known that the mass as measured by gravitational attraction (that is, its weight) was equivalent to the mass of the same body measured as determined by its inertia (that is, its resistance to acceleration). A careful series of experiments carried out first by Eotvos in 1889 and finally in 1922 had established this equivalence with great precision of five parts in 109. From this equivalence of gravitational mass and intertial mass, Einstein concluded that there could be no difference between physical systems in an uniform gravitational field and in an accelerating frames of reference with no gravitational field. This suggested to Einstein a general law of nature, which he called the "Principle of Equivalence." This principle says that all the laws of nature are the same in an uniform gravitational field as they are in an uniformly accelerating observer in the absence of a gravitational field. According to this Principle of Equivalence all phenomena in an accelerating vehicle (in no gravitational field) are the same as if the vehicle were at rest in an uniform gravitational field. That is, in an uniform gravitational field, as at the surface of the earth, light, like everything else, falls at the same acceleration, g. The distance that the light falls in one second is far too small to be measurable, because of the very large speed of light, 186,000 miles per second; that is, in one second light travels 186,000 miles. Using Galileo's law, s = ½gt2, the distance that light would fall in one second is 6 × 10-10 or 0.0000000006 ft. Thus the distance that light falls as the light travels horizontally across an ordinary size room would be very, very small.
In 1911, Einstein published his first tenative conclusions in
a paper, "On the Influence of Gravitation on the Velocity
of Light." In this paper he proposed an experimental test.
In his special theory, Einstein had found that mass and energy
were related, E = mc2, and were fundamentally
the same. If this was so, then, he concluded, light energy should
be acted on by gravity and it should be accelerated; that is,
change its direction along a curved path, falling toward the source
of the gravitational field. This should occur when light from
a star passed near to the sun. He calculated the angular deflection
of the ray of light glazing close to the sun by the formula he
derived,
θ = 2GM/Rc2,
where θ is the angle of deflection measured in radians (2π radians
= 360 degrees),
G is the universal constant of gravitation
(G = 6.67 × 10-11
newton-m2 per kgm2),
M is the mass of the sun, R is the radius of the sun, and
c is the speed of light.
The first calculation gave a deflection of 0.87 sec of arc;
this was later modified.
In 1911, Einstein had not fully developed his General Theory of Relativity in which the laws of physics are invariant between coordinate systems that are accelerating relative to each other. He relates later in an autobiographical note his thinking at that time.
"The fact of the equality of inert and heavy mass leads quite naturally to the recognition that the basic demand of the special theory (invariance under Lorentz transformation) is too narrow, i.e. that an invariance of the laws must be postulated also relative to non-linear transformations of the coordinates in the four-dimensional continum. This happened in 1908. Why were another seven year required for the construction of the general theory? The main reason lies in the fact that it is not so easy to free oneself from the idea that coordinates must have an immediate metrical meaning." [1]Gradually the idea was taking shape in Einstein's mind that ordinary Euclidean geometry with its fixed metric must be abandoned as the only universal way of defining events in space and time. In the special theory of relativity, which used an Euclidean geometry of four dimensions, the space was isotropic, that is, it had the same properties in all directions, regardless of the presence of masses. The transformation equations between events in different inertial systems were linear and easily expressed in a Euclidean metric. But, to include accelerating frames of reference as well as those in unform motion into a more general theory, Einstein found he had to use a non-Euclidean geometry in which the coordinates defining space and time are "curved" by the presence of the masses. In this new geometry, space is non-isotropic, that is, space no longer has the same properties in all directions.
After nine years of searching for a theory of gravitation, Einstein proposed his Theory of General Relativity. In 1916 Einstein published another paper, titled "The Foundations of the General Theory of Relativity". [Annalen der Physik, Vol. 49, 1916.] In this theory he sought to extend the Special Theory to explain the experimental fact that Galileo had observed that all free falling bodies regardless of their masses followed the same trajectory in a gravitational field. Einstein reasoned that since the trajectory of a freely falling body does not depend upon its mass or its internal composition, its motion under gravity must be related to the properties of space-time itself. He then showed how to interpret the gravitational force as a manifestation of a property of space-time called its curvature.
The force of gravity is described classically by Newton's Law of Gravitation. This inverse square law states that the magnitude of the force of gravity F12 between two bodies with masses m1 and m2 is directly proportional to the product of their masses and inversely proportional to the square of the distance r between them:
F12 = Gm1m2/r2,
where G is the constant of proportionality that is called the
gravitational constant.
The region of space about a body, like the earth or the sun, may be considered
as a field of gravitational force, with that body considered to be the
source of the gravitational field; thus the force of gravity exerted upon
another body in the field may be considered as exerted by the gravitational
field, rather than directly by the source acting at a distance. This
gravitational field extends from the mass-source to infinity. Since the
force exerted by the gravitational field is not constant, but varies from
point to point in the field, physicist have defined a quantity to describe
this variation, called gravitational field intensity, which is equal to
the ratio of the gravitational force F exerted on a test body at a point
in the gravitational field to the mass m of the test body, that is,
F/m is the gravitational field intensity.
Since the weight w of the test body is equal to the force of gravity
F exerted on it by the mass M and by Newton's second law of
motion w = mg, where g is the acceleration due to
force of gravity, then
the gravitational field intensity is equal to
w/m = mg/m = g.
Thus g, the acceleration due to gravity, is a measure of
gravitational field intensity. The acceleration due to gravity is not
constant, but varies with the distance from the center of the source of the
gravitational field. That is, since w = mg, then
g = w/m = (1/m)w = (1/m)(GmM/r2) = GM/r2,
where M is the mass of the body that is the source of the gravitational field and r is the radial distance between the test body of mass m and the center of the body that is source of the field. From this point of view the test body interacts not directly with the source mass M, but with the gravitational field. This means that at each point in space there is a knowledge of source mass and of what force to exert on the test mass. But according to the Newtonian theory of gravity, when the source mass is moved, the gravitational field changes instantaneously to adjust to the new position of the source mass. But this instantaneous change is fundamentally incompatible with the theory of special relativity, which holds that distrubances cannot propagate faster than the speed of light. This problem motivated Einstein to extend his special theory of relativity and to formulate his General Theory of Relativity in 1915.
Another feature of Newtonian physics that puzzled Einstein was the equivalence of inertial mass and gravitational mass. Gravitational mass is the mass in Newton's Law of Universal Gravitation and inertial mass is the mass in Newton's Second Law of Motion: F = ma. This equivalence had been known by many generations of physicists but no explanation why they were equivalent. This lack of explanation led Einstein to the Principle of Equivalence of gravity and acceleration, that is, the acceleration of a body not in gravitational field is identical to the effect of an uniform gravitational field. Einstein also noted in this connection that all bodies in the same gravitational field fall with the same acceleration, regardless of their mass. Einstein reasoned tha since the trajectory of a freely falling body does not depend upon its mass or its internal composition, its motion under gravity must be related to the properties of spacetime itself. He then showed how to interpret the gravitational force as a manifestation of a property of spacetime called it curvature.
Consider the surface of a sphere. The surface of a sphere is two-dimensional because one must give two coordinates, such as latitude and longitude, in order to specifiy a point on the surface. Now the shortest distance between two points on the sphere that lies entirely on the surface is the shorter arc of a great circle that passes through the points. This basic geometric fact is constantly applied in determining the most efficient air route over the earth. This concept can be applied to any curved surface that is more complicated than a sphere; there is a path of shortest distance on the surface that connects any two points. This path of shortest distance is called a geodesic, from the Greek words meaning the "division of the earth".
In general relativity, spacetime is the four-dimensional analogue of the curved surface. It is four-dimensional because it takes four coordinates to specify a point. A point in spacetime is a physical event, such as the collision of two particles. The event is specified by where and when it happens: by three spatial coordinates and its time. A geodesic in spacetime is the analogue of the geodisic on a surface. It is the shortest path in spacetime between two events that is specified by the geometry of spacetime. According to general relativity, any particle acted upon solely by the force of gravity follows a geodisic in spacetime. In this way, general relativity explains Galileo's observation that all freely falling bodies follow a common trajectory, irrespective of the mass of the falling bodies.
Thus the General Theory of Relativity explains the reality of gravitational
fields as a warping of spacetime around a mass-source of the field. Thus
a mass will distort spacetime like a bowling ball laid on a rubber sheet.
The effect of gravity on the trajectory of a passing particle will be
analogous to rolling a marble across the curved rubber sheet.
Thus, general relativity suggests that instead of thinking of bodies as
moving under the influence of gravitational forces, we should think of
them as moving freely through warped, or curved, spacetime. Hence,
gravity is reduced to the curved geometry of spacetime. Geometry
has different rules on a curved surface than for a flat or Euclidean
surface. For example, on the curved surface of the earth, two north
pointing lines which are parallel at the equator of the earth actually
meet at the north pole; on a flat surface, two parallel lines never meet.
As we saw above, in the curved spacetime, straight lines must be replaced
by geodesics as the shortest path between two points; free particles move
along the geodesics. Einstein expressed this interpretation of gravity as
geometry in his
Field Equations of General Relativity:
Rmn - ½gmnR = 8πGTmn,
where Rmn is the Ricci-tensor or contracted curvature tensor,
gmn is the components of symmetric curvature tensor
called the "Fundamental Tensor",
R is the scalar = gmnRmn, and
Tmn is the Minkowski's mass-energy tensor.
These equations may be loosely translated,
(geometry of spacetime) = 8πG × (mass and energy),
where G is the Newtonian constant of gravitation.
So mass and energy determines the curvature and geometry of space-time;
and the curvature and geometry of spacetime determines the motion of a
particle of matter. In other words, "matter tells spacetime how to bend",
and "spacetime tells matter how to move".
The theory also predicts the existence of gravitational waves propagating through space as the result of changes in a mass-source, such as the collapse of a star into a neutron star or black hole. In this event, distortions of space-time will spread out spherically in space at the speed of light, somewhat like the ripples spread out circularly across the surface of a still pond into which a stone is dropped. Although much experimental effort has been devoted to the attempt to detect such waves resulting from cosmic events out in space, so far none has succeeded, probably because the disturbances are too small to be detected even by the most sensitive detectors.
CONFIRMATION OF THE GENERAL THEORY.
Developing the Theory of General Relativity, using a mathematical method
called tensor analysis, in his 1916 paper Einstein returned to the problem
of the deflection of a light ray passing close to the sun. He showed
that under the new formulation, a ray of light will not only be
deflected as predicted on the basis of the special relativity,
but, due to the non-Euclidean curvature of space-time close to
the sun, there is an additional deflection. The calculated value
on this basis comes out as a first approximation to be
θ = 4GM/Rc2 = 1.75 sec of arc
(which is twice as great as his previous calculation).
A test of the theory had to wait for a solar eclipse to occur because
only then could a star be seen near the sun. On May 29, 1919, the
shadow of a total eclipse of the sun swept across the Atlantic Ocean
from western Africa to northern Brazil. Expeditions mounted by
the British government at the instigation of Sir Arthur Stanley
Eddington were ready to observe the stars near the darkened disk
of the Sun. One of Eddington's objective of his expedition was
to test the theory put forward by Einstein four years earlier.
The observation of the eclipse made Einstein world famous. The
stars were shifted by just the predicted amount, and the success
of the Einstein's geometric approach to gravity was dramatically
confirmed.
The General Theory of Relativity, in addition to showing that space is curved and distorted in the presence of a mass, it predicted that time would be changed in a gravitational field. A clock will slow down a very small amount in the presence of a gravitational field. But this amount is so small that it cannot be detected by ordinary clocks. However, by 1960 an instrument that kept extraordinarily precise time measurement based on atomic phenomena was invented. And in 1960 this carefully designed "atomic clock" was used in a 70 foot tower built at Harvard University to test whether an atomic clock at the foot of the tower (70 feet closer to the center of the earth) would keep slower time than one at the top of the 70 foot tower. The test agreed with the predictions of Einstein's General Theory of Relativity.
Another confirmation of the General Theory of Relativity was the precession of the orbit of Mercury. This is not the only "perturbation" of the orbits of the planets that had been observed, but most of them can be explained by the presence of other planets. About seventy years earlier, in 1845, two mathematicians, Urban J. J. Leverrier (1811-1877) and John Couch Adams (1819-1892), had studied the motions of the planet Uranus and after the effects of the other known planets, especially Jupiter and Saturn, had been taken in account, Uranus' orbital motion could not be explained by Newton's Law of Universal Gravitation. But they did not conclude that the Law of Gravitation was wrong. But rather, they guessed that another undiscovered planet was responsible for the descrepancy. They calculated where the unknown planet should be and told the astronomers where to point their telescopes. And when they did, the planet Neptune was discovered. An astronomer in Berlin found the planet with his telescope on the very evening of the day that he received a letter from the Paris mathematician Leverrier telling him were to look. Leverrier had made extensive calculations of the probable path of a planet near Uranus, capable of causing the perturbations that had been observed in the planet's orbit. (These calculations were made independently by the English mathematician, Adams.) Thus Neptune was added to the solar system in 1846. It was considered the verification of Newton's Law of Universal Gravitation.
In 1859, Leverrier found a similar problem with the planet Mercury. Mercury's orbit is an ellipse with the sun at one of its foci, like the other planets, but its orbit was more elongated, more cigar-shaped, than the others. And centuries of observation had shown that Mercury did not quite repeat its orbit about the sun on each revolution. Rather, while the shape of its orbit did not change, the orientation of its ellipical orbit gradually rotated. The point on its orbit closest to the sun, called its "perihelion", was gradually changing its position, rotating about the sun. A line drawn between the sun and Mercury's perihelion was rotating almost ten minutes of arc every century. This was known as the precession of the perihelion of Mercury. Leverrier tried to explain this precession of the perihelion of Mercury by the gravitational attraction of other planets, and was able to account for most of it. But a tiny descrepancy remained between the calculated value and the observed value. It amounted to 43 seconds of arc per century. This amount could be accounted for as observational error. But on the basis of the success in explaining the orbit of Uranus, Leverrier thought that Mercury's orbital motion could be explained by another undiscovered planet closer to the sun than Mercury. Lost in the sun's glare, it was not unreasonable that the planet was not detected. He calculated its mass and orbit, and even gave it a name, Vulcan. For decades astronomers searched for the planet Vulcan, occasionally thinking they found it, but always losing it again.
Using his General Theory of Relativity, Einstein calculated the orbit of a planet about a massive object like the sun. The orbits were ellipses, closed on themselves, to a very close approximation. But not quite. Very close to the sun, objects do not move quite as predicted by Newton's inverse square law of gravitation. At the end of his 1916 paper introducting the General Theory, Einstein wrote:
"According to this, a ray of light going past the sun undergoes a deflection of 1.7", and a ray going past the planet Jupiter a deflection of about 0.02".And so Einstein solved a century old puzzel. And again, very accurate measurements proved decisive in the verification of a theory. And again, the motion of the planets had revealed the laws of physics.If we calculate the gravitation field to a higher degree of approximation, and likewise with corresponding accuracy the orbital motion of a material point of relatively infinitely small mass, we find a deviation of the following kind from the Kepler-Newton laws of planetary motion....
Calculation gives for the planet Mercury a rotation of the orbit of 43" per century, corresponding exactly to astronomical observation (Leverrier); for the astronomers have discovered in the motion of the perihelion of this planet, after allowing for disturbances by other planets, an inexplicable remainder of this magnitude." [2]
In 1917, the accepted cosmological view was that our Milky Way Galaxy was the entire Universe, a stable collection of stars. Individual stars might be wandering within the cloud of the Galaxy, but as a whole the preeminent characteristic of the Universe was it stability. The Universe was not getting any larger or smaller, and as far as any scientist knew it had been there forever, throughout eternity. Some stars might be born and others die, but the overall appearance of the Milky Way stayed much the same, in a steady state. Isaac Newton's theory of gravity had predicted a collasping universe, and Newton had been troubled by the implication of his theory. Every object in the universe was being pulled towards every other object on a cosmic scale. This would cause every object to move closer to every other object. The attraction might start a steady creep, but it would gradually turn into an avalanche which would end in an almighty crutch; the universe was appartently destined to destroy itself. One of Newton's solutions was to envisage an infinite, symmetric universe, in which every object would therefore be pulled equally in all directions, and there would be no overall movement and no collapse of the universe. Unfortunately, he soon realized that the carefully balanced universe would be unstable. An infinite universe could theoretically exist in a state of equilibrium, but in practice the tiniest distributance in this gravitational equilibrium would upset this balance and the universe would end in catastrophe. For example, a comet passing through the Solar System would momentarily increase the mass density of each part of space through which it passed, and thus initiating a process of total collapse. Even turning the page in a book would alter the balance of the universe and given enough time, this also would trigger a cataclysmic collapse. To solve this problem, Newton suggested that God intervened from time to time to keep the stars and other celestial objects apart and prevent such a collapse.
So when Einstein took the equations of his new theory of General Relativity and applied them to this description of the Universe, he expected his theory to describe this steady state universe. He was surprised and disappointed by his theory's prediction of how the universe operates. What he found implied that the universe was ominously unstable. Every object in the universe was being pulled towards every other object on a cosmic scale. This would cause every object to move closer to every other object. The attraction might start a steady creep, but it would gradually turn into an avalanche which would end in an almighty crunch; the universe was appartently destined to destroy itself. This was a preposterous result. At the beginning of the twentieth century the scientific establishment was confident that the universe was static and eternal, not contracting and temporary. So not surprisingly, Einstein disliked the concept of collapsing universe. "To admit such a possibility seems senseless." Einstein was not prepared to acknowledge a role for God in holding the universe apart, but at the same time he was anxious to find a way in maintain an eternal and static universe in keeping with the scientific consensus.
In 1917, Einstein presented a paper called "Cosmological Considerations on the General Theory of Relativity" to the Berlin Academy of Science, in which he describes his surprise at what the equations told him and how he managed to force the equations to conform to that steady state universe. Einstein says in the paper at the beginning of the paragraph in which he presents the boundary conditions of the universe according to the General Theory of Relativity,
"In the present paragraph I shall conduct the reader over the road that I have myself traveled, rather a rough and winding road, because otherwise I cannot hope that he will take much interest in the result at the end of the journey. The conditions I shall arrive at is that the field equations of gravitation which I have championed hitherto still need slight modification, so that on the basis of the general theory of relativity those fundamental difficulties may be avoided which have been set forth in § 1 as confronting the Newtonian theory." [3]Einstein tried to describe the simplest model of the universe that bore any similarity to reality, one that contain matter spread out uniformly through space. That universe was closed, like the surface of a sphere, wrapped around upon itself so that there could be no edge. But it would not stay still. When Einstein put the equations describing such a static universe through the manipulations of the General Relativity, the results said that the Universe must either be expanding or contracting, but it could not be standing still. After re-examining his theory of General Relativity, he discovered a mathematical trick that would rescue the universe from collapse. The only way he could hold the model universe still, to mimic the Milky Way Universe, was to add a extra term to the equations of General Relativity, a term he called the "cosmological constant" and often represented by the Greek letter lambda (Λ).
Rmn - ½gmnR + gmnΛ = 8πGTmn,
Then and only then would the equations provide a description of a universe that was neither expanding nor contracting. The last sentence of the 1917 paper said,
"That term is necessary only for the purpose of making possible a qusai-static distribution of matter, as required by the fact of the small velocities of the stars." [4]A dozen years later, the American astronomer, Edwin P. Hubble (1889-1953), in 1925 showed that the Milky Way was not the whole Universe and with Milton L. Humason (1891-1972) he discovered from the observation of the redshift of the radiation from distant galaxies that these distant galaxies were moving with very large velocities, all of them in recession. In 1929, Hubble announced his famous law: all galaxies recede from us with a velocities proportional to their distance from us. The whole universe was expanding. The necessity for the lambda term in the General Relativity equations was then removed, and these equations had predicted the expansion of the universe. The existence of an expanding Universe was the greatest prediction of General Theory of Relativity. What the General Relativity equations were trying to say to Einstein and to all astronomers was that the Universe in which we live is a dynamic, expanding Universe, not a static universe.
Later Einstein said concerning the introduction of the lambda term that it was the "biggest blunder of my life". But it is hard to see, given the view of Universe in 1917, that anyone could have hardly avoided making that blunder. The Theory of Relativity had run ahead of the observations, and it was only when Hubble and Humason showed that we do live in a expanding Universe, that the General Theory of Relativity was recognized as the prediction of the origin of that dynamic Universe.
[1] Albert Einstein, Philosopher-Scientist,
edited by Paul A. Schilpp,
Library of Living Philosophers, Inc.,
(LaSalle, Ill.: Open Court Publishing Co., 1949), p. 67.
Quoted by Ripley, J. A.,
The Elements and Structure of the Physical Sciences
(New York, London, Sydney: John Wiley & Sons, Inc., 1964), p. 463.
[2] Albert Einstein, "The Foundations of
the General Theory of Relativity" in
The Principle of Relativity,
(New York: Dover Publications, Inc., 1952), pp. 163-164.
[3] Albert Einstein, "Cosmological Considerations
on the General Theory of Relativity" in
The Principle of Relativity,
(New York: Dover Publications, Inc., 1952), pp. 179-180.