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VBHT13: General Relativistic Spacetime
A description of the curved spacetime of general relativity.
The spacetime of general relativity is a manifold in which the metric is free to vary from event to event, but in such a way that any infinitesimally small region looks just like the Minkowski space of special relativity. Small, freely falling test particles follow timelike geodesics of the spacetime. The way in which the metric varies from place to place is determined by the distribution of mass and energy by way of the Einstein field equations. There: that's just about the entire general theory of relativity in one paragraph, and if you've been following this series carefully then I can dash off a quick description of the field equations and you'll know all about the theory. Don't just give up, though, because I'm going to explain it all again anyway.
The first important thing about the general theory of relativity is that it places all coordinate systems on the same footing. Remember that in Newtonian spacetime there is only really one coordinate system that everybody pretty much agrees on. In special relativity the coordinates are slightly diminished in importance because people in different inertial frames (that is, moving with different velocities) carve up spacetime in different ways so they can't even agree on which way the coordinate axes ought to point. You can change your viewpoint between one inertial frame and another using a Lorentz transformation. There is a gadget called the metric that turns the steps across anybody's coordinate grid into a proper distance or proper time that anybody agrees on. In general relativity, as described in the last part, the inertial frames only work in really tiny regions of spacetime and so even given the worldline of an observer there isn't any best way to carve up spacetime. Coordinate systems are now just convenient ways to label the various events in the spacetime, and one set of labels is just as good as another. The metric now assumes an absolutely central position in the theory because regardless of how Roxanne and I choose to label the points of a worldline between two events we both agree on how much proper time elapses on a clock carried along that worldine between the events (you can see why this must be so by reading my careful description of proper time in Part 2). A further consequence is that "simultaneous" has lost all meaning. Remember that in special relativity two inertial observers that were moving relative to each other disagreed about whether two events were simultaneous. In general relativity things are even worse: an observer can carve up spacetime into stacks of spacelike "simultaneous" surfaces in a myriad different ways and there's no reason to choose one over another. Even a single observer cannot decide if two events are simultaneous, for if a moment of time doesn't fix a slice of space, then what can "simultaneous" mean? On the other hand, the metric still determines the light cones, so that each event still has an absolute future and an absolute past (locally at least, as in some spacetimes the past and future of an event can overlap and closed timelike curves [time machines] become possible).
Now, as I also described in the last part, time runs differently in different parts of spacetime and the paths of lightrays are curved. This means that the metric varies from place to place. For the moment assume that the metric at each event is known (I'll describe later how the mass and energy distribution determines the metric). In a spacetime with a varying metric, the nearest thing to a straight line is a geodesic, a curve of extremal elapsed proper time (in flat spacetime the geodesics are straight worldlines, and the elapsed proper time between two events is maximal for the straight worldline between them). If you know what the metric at each event then you know the proper distances or times between any event and all events near it, and so you can calculate all the geodesics (I spoke about how to do this a bit in Part 11, but it should be obvious that if you know all the proper distances or times then you can find the curves with the longest length or greatest elapsed time).
Remember that Newton's first law of motion says that particles with no forces acting on them carry on moving in a straight line at the same speed. A generalisation of this idea determines the paths of small test particles (that is, objects small enough that their effects on spacetime can be ignored). If you take a test particle and release it at some event with some speed then it just goes shooting off along the relevant geodesic (there's only one geodesic through each event for each possible speed). We say that the worldline of the freely falling particle is a timelike geodesic of the spacetime manifold (the geodesic is timelike because in the local inertial frame at the event in which you release it it is moving in a timelike direction in spacetime; and if a geodesic is timelike at one event along it then it is timelike at all the events it passes through). If it looks like the particle is being pulled about by some force then all that's really happening is that the straightest possible route that it can follow is twisting and turning in your coordinate system. The fact that all objects fall along the same paths in a gravitational field (a fact that seems mysterious in Newtonian physics) becomes pretty obvious - what else are they going to do except fall along the straightest possible path in spacetime?
A simple example is the orbit of the Moon about the Earth (well, perhaps the Moon is a bit big to be a test particle, but never mind). It certainly looks like the path of the moon is pretty curved, but that's only because we aren't thinking about its path in spacetime. If you look at it the right way then you'll realise that it's entirely plausible that the Moon is following a geodesic. The Earth clearly isn't curving spacetime very much, so we can get away with plotting the Moon's orbit on a spacetime diagram just like those I described in the parts on special relativity (at least approximately). The Moon is moving about the Earth in an approximately circular orbit with a radius of about 384 000km and it takes 28 days to complete one orbit. In the frame of the Earth the orbit of the Moon will look like a helix. Now to get the scale right we have to use seconds to measure time and light-seconds to measure distance. In these units the radius of the helix is just over 1 light-second, whereas the time taken for one revolution is some 2.5 seconds. Unless you looked really carefully at the wordline of the Moon you wouldn't notice that it was curved at all with respect to the coordinate system. It's because the curvature produced by the mass of the Earth is so mild that we don't see any more obvious manifestations of general relativity in everyday life. The more surprising effects of spacetime curvature will have to wait until we look at more extreme objects like black holes or the universe itself.