Curved Space Help

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Introduction to Curved Space

The observable universe seems, upon casual observation, to be Euclidean. If you use lasers to “construct” polygons and then measure their interior angles with precision lab equipment, you’ll find that the angle measures add up according to the rules of Euclidean geometry. The conventional formulas for the volumes of solids such as the pyramid, cube, and sphere hold perfectly, as far as anyone can tell. Imagine a 3D space in which these rules do not hold! This is called curved 3D space, warped 3D space , or non-Euclidean 3D space . It is the 3D analog of a non-Euclidean 2D surface.

Gravity Warps Space

There is evidence that the 3D space in which we live is not perfectly Euclidean. Gravitational fields produce effects on light beams that suggest a Lobachevskian sort of warping—a negative curvature—of 3D space. Under ordinary circumstances this warping is so subtle that we don’t notice it, but it has been detected by astronomers using sensitive equipment, and in exceptional cases it can be directly observed.

The behavior of light from distant stars has been carefully observed as the rays pass close to the sun during solar eclipses. The idea is to find out whether or not the sun’s gravitational field, which is strong near the surface, bends light rays in the way that we should expect if space has negative curvature. Early in the 20th century, Albert Einstein predicted that such bending could be observed and measured, and he calculated the expected angular changes that should be seen in the positions of distant stars as the sun passes almost directly in front of them. Repeated observations have shown Einstein to be correct, not only as to the existence of the spatial curvature, but also to its extent as a function of distance from the sun. As the distance from the sun increases, the spatial warping decreases. The greatest amount of light-beam bending occurs when the photons graze the sun’s surface.

In another experiment, the light from a distant, brilliant object called a quasar is observed as it passes close to a compact, dark mass that astronomers think is an intense source of gravitation known as a black hole . The light-bending is much greater near this type of object than is the case near the sun. The rays are bent enough so that multiple images of the quasar appear, with the black hole at the center. One peculiar example, in which four images of the quasar appear, has been called a gravitational light cross .

Any source of gravitation, no matter how strong or weak, is attended by curvature of the 3D space in its vicinity, such that light rays follow geodesic paths that are not straight lines. Which causes which? It is a chicken-and-egg mystery. Does spatial curvature cause gravitation, or do gravitational fields cause warping of space? Are both effects the result of some other phenomenon that has yet to be defined and understood? Such questions are of interest to astronomers and cosmologists. For the mathematician, it is enough to know that the curvature exists and can be defined. It’s more than a product of someone’s imagination.

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