# Special & General Relativity Questions and Answers

## How is it that, according to General Relativity, that space can be curved?

Between 1905 and 1912 Einstein had worked on many aspects of how to go beyond special relativity before he finally published his theory of general relativity in 1915. Shortly after August 10, 1912 by his own recollections of this period, he began to realize that an entirely new geometry for space-time was needed. The space-time used by special relativity did not fit the bill at all. Its flat geometry insured that any pair of reference frames would be tied together by a single, constant relative velocity which would preclude the accelerating affects of gravity. So, what then should the true geometry of space-time look like?

Einstein proposed in his general theory of relativity no less a magical thing than that there was an indivisible relationship between the curvature or 'warpage' in the geometry of space-time, and the force of gravity. Perhaps no other concept in all of physics is as impossible to comprehend as curvature of space, let alone the curvature of space-time. The equivalency between curvature and gravitational forces also appears profoundly counter-intuitive, but in fact geometry and force are linked together in our everyday lives more than we might at first suspect. The closest we come to experiencing what curved space must be like are the distorted and multiplied reflections of ourselves we see in fun house mirrors at the amusement park. But these are very passive distortions in geometry which have no other consequence than simple visual disorientation.

A very simple thought experiment devised by physicist Alfred Schild in 1960 shows exactly why we have to abandon flat space-time. Ultra precise measurements have demonstrated that light suffers a loss of energy, a so-called gravitational red shift, as it attempts to leave a gravitational field like the one provided by the earth. In 1959, R. Pound and G. Rebka at Harvard University measured the energy of a gamma ray produced by the decay of a radioactive isotope of iron, and discovered that after climbing 74 feet that its energy had been lowered by nearly 1 part in 25 thousand trillion.

To see why this is such a pivotal experiment for general relativity, consider the following simplified experiment. Imagine two experimenters, Thelma and Louise, where Thelma is standing on the ground and Louise is on top of a tall step ladder. Thelma sends exactly 10 cycles of yellow light to Louise, who then waits until her equipment has registered all 10 beats of the light. Because the light signal Louise received has lost some energy climbing up the earth's gravity well to the top of the ladder, it has been red shifted to a lower frequency. Louise has to wait longer than it took Thelma to send all ten beats of the light signal in the first place. The reason that this experiment contradicts special relativity is because, first of all, both Thelma and Louise are at rest relative to each other. They are moving with exactly zero velocity with respect to each other. This means that they share exactly the same reference frame in special relativity. But, this necessarily means that a single, flat space-time patch should be exactly sufficient upon which to describe both their world lines, and those of the wavelets in the light ray they exchanged. Their world lines should form the opposite sides of a rectangle in which the light ray forms the diagonal. By a simple geometric construction, however, we see that this cannot occur because the light wavelets take longer to receive than to send. This can only be accommodated if the space-time patch is not flat, thereby showing that the gravitational redshift demands space-time curvature. This is a direct deduction from observation, and is the essential underpinning of the entire rubric of curved space-time.

The conclusion is inescapable, though intuitively troubling. In the presence of gravity, space-time must be curved.

Copyright 1997 Dr. Sten Odenwald Return to Special & General Relativity Questions and Answers page.