## Overview of general relativity

by Wm. Robert Johnston
last updated 3 November 2008

Einstein published his work on the general theory of relativity in 1915. This theory, like much of modern physics, is not intuitive. D'Inverno says in Introducing Einstein's Relativity:
Indeed, the intellectual leap required by Einstein to move from the special theory to the general theory is, there can be little doubt, one of the greatest in the history of human thought.
Additionally, general relativity is far more mathematically complex than special relativity. Einstein wrote to a colleague in 1912 and said:
But one thing is certain, that in all my life I have never struggled so hard... Compared to this problem the original relativity theory is child's play.
Recall that special relativity declared that space and time were not separate concepts, but intertwined:
If you are moving relative to me, then your time is a mixture of my space and time, and vice versa.
This means, among other things, that events seen as simultaneous for one observer may not be simultaneous as seen by another observer moving relative to the first.

Now additionally, general relativity says that what we call gravity is actually a curvature in this space-time caused by matter, like the Earth:

Matter tells space-time how to curve, and space-time tells matter how to move.
Einstein's equation of general relativity elaborates on this statement. This theory has been confirmed by a variety of experiments. It also calls for the existence of such things as black holes and gravitational waves.

Now for a little more detail:

Given that space and time are related, measurements in space can be related to measurements in time by the speed of light (which was the limiting speed in special relativity). Multiply the speed of light, 299,792,458 meters/second, by a length of time and you get a corresponding length of distance. One second translates to a distance of about 300,000 kilometers.

General relativity describes gravity not as a force, but as a consequence of curvature of this combined space-time. In our experience this curvature can be ignored (and indeed was until Einstein).

Throw a ball at an upward angle: it will follow a curved path, reaching a highest point, and then curving back down to the ground. Newton explained this curved motion as a result of the force of gravity. He also pointed out that in the absence of gravity and any other influence, it would continue in a straight line, same speed, forever.

In Einstein's general relativity, however, the original path is actually a "straight" path in curved space-time. The path of the ball we see is a curve reaching, say, one meter high before coming back to the ground one second after you threw it. The ball traveled one meter vertically and one second time-wise-which translates to 300,000 kilometers in relativity. That one meter up-and-down motion over a distance of 300,000 kilometers is actually very nearly a straight line. The difference between that and a truly straight line is the degree to which spacetime is curved because of the Earth's presence.

North-south lines on a globe are not truly straight since they curve around the globe. They are "straight" in the sense that if you move confined to the surface of the globe without changing direction you will follow such a path. Such a path--on the globe or in curved space-time--is a geodesic. The motion of the ball thrown above is such a path.

At the Earth's surface, this spacetime curvature (otherwise known as gravity) is comparable to the curvature of the surface of a sphere with a radius of one light year. Since this curvature is so slight, it is no wonder that it was never noticed.

The curvature near the Sun is greater--sufficient to measurably bend light coming from stars behind the Sun.

In flat (uncurved) space, we are familiar with its mathematical nature. For example, the length of a circle's circumference divided by its diameter is equal to the constant pi, 3.141592653589793... However, the curvature of space-time due to a massive object means that such rules of flat space may not work. Were one to measure the diameter through the Earth, the result would be 2.2 millimeters greater than the expected value obtained from dividing the circumference by pi. In the case of the Sun, the difference is 4 kilometers. (For a derivation of these values, see this .pdf document.)

General relativity has been observationally confirmed in a variety of ways:

• Measurements of the precession of Mercury's perihelion (1915).
• Observed deflection of starlight from stars near the edge of the Sun during a solar eclipse (1919).
• Measured red shifting of light (stretching of light waves) from white dwarf stars (1924).
• Measured red shifting of gamma rays in the Earth's gravitational field (1960).
• Measured time delay of radar signals bounced off Venus and Mercury (1968).
• Measured time delay of radio signals from space probes passing other planets (1970).
• Measurements of the precession of asteroid Icarus's perihelion (1971).
• Measured time dilation due to the Earth's gravity, by comparing atomic clocks on Earth and in space (1976).
• Observed gravitational lensing or focusing of distant star light by massive objects between us and the star (1980).
• Observed inspiral and orbital precession of binary neutron stars, in accordance with predicted loss of energy due to radiation of gravitational waves (1982).

(Note: some of these listed observations are not unique confirmations of Einstein's theory of general relativity as opposed to other relativistic theories of gravitation. However, they all represent confirmations of predictions made by Einstein or others on the basis of his theory.)

As surprising as it may be, general relativity already has practical applications which affect many people in their work and play. The U.S. Global Positioning System, or GPS, used by military and civilians for navigation and other purposes, includes corrections for the effects of both special and general relativity. The portable GPS receiver determines position by passively receiving signals from several GPS satellites at one time, then determining the receiver's location, altitude, and velocity with precision. This determination depends on the time signals transmitted from the GPS satellites, which carry atomic clocks. Since the satellites are moving, corrections for special relativity are required; since the satellites are at a distance from the Earth, corrections for general relativity are required. The system is precise enough for relativistic effects to make a difference.

When the first prototype GPS satellite with an atomic clock was launched in 1977, reportedly some involved still doubted that the relativistic effects were real. The satellite carried a component that could be switched on to correct the signal if the satellite's clock proved to be affected by relativity. It did, measuring time at a rate almost one part in two billion faster than clocks on the ground. Currently, the GPS satellite atomic clocks are routinely calibrated with ground clocks which correct for relativistic effects and other effects. One way or another, the relativistic effects must be accounted for to maintain the positional accuracy which GPS provides.

One of the implications of general relativity is gravitational waves.

© 2001-2005, 2008 by Wm. Robert Johnston.