![]() ![]() These are just those trajectories in spacetime along which the greatest proper time elapses. We saw in an earlier chapter that free bodies in the Minkowski spacetime of special relativity move along timelike This very slight difference is enough to make a noticeable difference to bodies that are free to move above the earth's surface. This very slight difference is big enough to require correction if the GPS satellite clocks are to be the basis of reliable position data. Throught the curvature of spacetime, their altitude above the surface of the earth is enough to produce a very slight difference in the rates of their clocks compared to the those on the surface of the earth. Their proper functioning depends on the satellites having very accurate clocks. GPS depends on signals sent from satellites orbiting above the surface of the earth. As it happens, it makes a difference to the "GPS" Global Positioning System that many of us use everyday through our cellphones. You might imagine that this is a delay that is so tiny that it could not make a different to anything that matters to us. The numbers count off the ticks marked by the clocks. It shows the world lines of clocks suspended at different altitudes above the earth. The spacetime diagram greatly exaggerates the magnitude of the slowing effect. It would be behind by just 6.977 x 10 -10 second that is roughly seventy billionth of a second. When the clock in distant space ticks one second, the one on earth would tick by almost one second. To see it, compare a clock on the surface of the earth with one in distant space. If we had clocks suspended at different altitudes above the surface of the earth, those closer to the earth would run more slowly. Merely saying that "time runs more slowly" is dangerously vague. More briefly, time runs more slowly closer to the earth's surface. Very loosely speaking, it appears as a very slight slowing down of time the closer we get to the earth's surface. If we consider the spacetime region above the surface of the earth, we can be more specific about the curvature of the space-time sheets. (There is more to say here and it will be said in the next section.) Curvature in the space-space sheets produces no easily observed effect in the vicinity of the earth. ![]() The only parts of the spacetime curvature that make any sensible difference are those in the space-time sheets. We can be a little more specific for the very weak gravitational effects in the vicinity of the earth. Spacetime is curved and the spacetime trajectories of free bodies follow the straightest lines of this curved spacetime geometry. Why does it do this? Why doesn't a stone like this just hover in space above the earth? Or, if it has some initial upward velocity, why does it not just fly off into space? The answer given by general relativity was already described in the last chapter. It will rise and, if not hurled too quickly, will slow to a halt and then fall back down again. Consider, for example, a stone hurled vertically. Why, according to the theory, do things fall down? We can take the simple and familiar case of bodies above the surface of a big mass like the earth. ![]() Let us return to the most basic question of gravity in general relativity. These were the firstĪpplications of Einstein's new theory. The first place to start is the most familiar, the gravitational effects arising near a massive object like our earth or sun. We now need to understand what those elements entail for gravity. In the last chapter, we learned the barest elements of Einstein's general theory of relativity. Linked document: Geodesics of Space Near the Sun ![]()
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