## Curved spacetime

A central tenet of general relativity is that the presence of a gravitational field alters the rules of geometry in space-time. The effect is to make it seem as if space-time is "curved". To see what we mean by geometry in a curved space, we look at geometry on the surface of a sphere, as illustrated in Fig. 8.2. The surface is two-dimensional. We need only two coordinates (say latitude and longitude) to locate any point on the surface. However, it is curved into a three-dimensional world, and that curvature can be detected.

To discuss the geometry of a sphere, we must first extend our concept of a straight line. In a plane, the shortest distance between two points is a straight line. On the surface of the sphere it is a great circle. Examples of great circles on the Earth are the equator and the meridians. (A great circle is the intersection of the surface of the sphere with a plane passing through the center of the sphere.) In general, on any surface, the shortest distance between two points is called a geodesic.

People on the surface of the Earth can tell that it is curved, and can even measure the radius, without leaving the surface. For example, two observers can measure the different position of the Sun as viewed from two different places at the same time. (Thus, even the ancient Greeks knew the Earth was round. When Columbus sailed the only issue being seriously debated was how big the Earth is, since there was some confusion in interpreting the Greek results, which had been given in "stadia". Columbus believed the "small Earth" camp, explaining why he thought he had reached India.)

Surveying the surface will also tell you that the rules of geometry are different. For example, consider the triangle in Fig. 8.2. In a plane, a triangle has three sides, each made up of a straight line. The sum of the angles is 180°. On the surface of a sphere, we replace straight lines by great circles. A triangle should therefore be made up of parts of three great circles. In the figure we use sections of two meridians and a section of the equator. Each meridian intersects the equator at a right angle, so the sum of those two angles is 180°. When we add the third angle, between the two meridians, that makes the sum of the angles greater than 180°. The results of Euclidean (flat space) geometry no longer apply. The greater the curvature of the sphere, the more non-Euclidean the geometry appears. On the other hand, if we stick to regions on the surface that are much smaller than the radius of the sphere, the geometry will be very close to Euclidean.

H In this image of a cluster of galaxies, the light of an even more distant galaxy is bent into an arc (Einstein ring) by the severe curving of the geometry of space-time.This curvature is caused by the large mass of an intervening galaxy. [ESO]

We now look at what we mean when we say that gravity curves the geometry of space-time. This is illustrated in the space-time diagram in

Fig. 8.3. In the absence of gravity, objects move in straight lines at constant speeds. If we throw a ball straight up with no gravity, the world line for the ball is a straight line. If we turn on gravity, the world line looks like a parabola. We can

Fig. 8.3. In the absence of gravity, objects move in straight lines at constant speeds. If we throw a ball straight up with no gravity, the world line for the ball is a straight line. If we turn on gravity, the world line looks like a parabola. We can

H Space-time diagram for a ball thrown up from the ground. (a) With no gravity, the space-time trajectory is a straight line. (b) With constant gravity, the trajectory is a parabola.

say that it follows this path because the spacetime surface on which it must stay is curved. Ultimately, to represent fully the trajectory of the ball we would have to consider all of the four space-time dimensions. The effect of gravity is then to curve that four-dimensional world into a fifth dimension. It is hard to represent that dimension in pictures, but we can still measure the curvature by doing careful geometric measurements.

In this geometric interpretation of gravitation, we need two parts to a theory. The first is to calculate the curvature of space-time caused by the presence of a particular arrangement of masses. The second is to calculate the trajectories of particles through a given curved space-time. Einstein's theory of general relativity provides both. However, the mathematical complexity goes well beyond the level of this book. (Supposedly, even Einstein was upset when he realized the area of formal mathematics into which the theory had taken him.) However, we can still appreciate the underlying physical ideas, and we can even carry out some simple calculations that bring us close to the right answers.

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