Although we cannot go into details here, Einstein's argument shows that the same effect is expected in a normal gravitational field such as that on the earth. Here the field is directed downward, so that a clock at sea level runs slower than one on top of a mountain. The slowing down applies equally to all other physical, chemical and biological phenomena, and a typist working on the first floor of the Empire State Building will age slower than her twin sister working on the top floor. Stronger fields produce greater retardation. A clock on the surface of the sun would run .0001 per cent slower than a terrestrial clock.
Obviously we cannot put a clock on the sun, but we can watch the rate of atomic vibrations that produce the various lines in the solar spectrum. If these natural clocks are slowed down, the light they emit should be shifted toward the low-frequency, or red, end of the spectrum. This "gravitational red shift" was predicted by Einstein. Such a shift is indeed found in the lines of the solar spectrum, but it is so small as to be almost at the limit of observational precision. Spectra of the much denser white-dwarf stars, where the red shift is expected to be 40 times larger than on the sun, agree quite well with the theory.
Astronomical evidence is not so satisfying as experiments that can be performed in a terrestrial laboratory. Until a couple of years ago, however, there seemed to be no hope of measuring the minute difference predicted between clocks at different heights in the earth's gravitational field. Then R. L. Mössbauer, working at the University of Munich, found a way to produce nuclear gamma rays of very pure frequency and to measure extremely small changes in their frequency [see "The Mössbauer Effect," by Sergio De Benedetti; SCIENTlFIC AMERICAN, April, 1960]. Seizing on the new opportunity, several workers proceeded to show that two nuclear "clocks" separated by only a few tens of feet in the earth's field run at measurably different rates, and the difference is exactly that predicted by Einstein, within the limits of experimental error. Still another verification, if any more are needed, will almost certainly be obtained when an atomic clock in an artificial satellite is compared with one on the ground.
So we see that in a gravitational field clocks run slow, light rays bend in the direction of the field and a straight line is not the shortest distance between two points. Yet how can one define "straight line" other than as the path of light in a vacuum, or the shortest distance between two points? Einstein's idea was to retain this definition. Instead of saying that light rays and shortest distances are curved, he suggested that space itself (more accurately space-time) is curved. It is difficult to conceive of a curved three-dimensional space, let alone a curved four-dimensional space-time, but some idea of what it means can be gained from an analogy with two-dimensional surfaces. The Euclidean geometry we all learned at school pertains to figures that can be drawn on a plane. If geometrical figures are drawn on curved surfaces, for example a sphere or a surface shaped like a saddle, many of the Euclidean theorems do not hold.
In particular, the sum of the angles of a plane triangle is equal to 180 degrees. In a spherical triangle the sum of the angles is greater than 180 degrees, and in a triangle drawn on a saddle surface it is less. True, the lines forming triangles on spherical and saddle surfaces are not straight from the three-dimensional point of view, but they are the "straightest" (i.e., shortest) lines between the points if one is confined to the surface in question. Mathematicians call such lines geodesic lines, or simply geodesics.
In three-dimensional space a geodesic line is by definition the path along which a light ray would propagate. Consider a triangle formed by three such geodesics. If the sum of the angles is equal to 180 degrees, the space is said to be flat. If the sum is more than 180 degrees, we say that the space is spherelike, or positively curved; if it is less than 180 degrees, we say that it is saddle-like, or negatively curved. Because of the bending of light toward the sun, astronomers located on earth, Mars and Venus would measure more than 180 degrees in the angles of the triangle formed by light rays traveling between the planets. Hence we can say that the space around the sun is positively curved. On the other hand, in the merry-go-round type of gravitational field, the sum of angles of a triangle is less than 180 degrees, and this space is curved in the negative sense.