Thursday, May 27, 2010

Foundations 2: From Special To General - Greg Egan

he first article in this series described some of the ways in which the geometry of spacetime affects travellers moving (relative to their destinations, or each other) at a substantial fraction of the speed of light. By generalising from the Euclidean metric, which captures such familiar aspects of geometry as Pythagoras's Theorem, to the Minkowskian metric suggested by the fact that the speed of light in a vacuum is the same for everyone, we analysed the “rotated” view of spacetime that two observers in relative motion have with respect to each other, and derived formulas for time dilation, Doppler shift and aberration.

This article and the next will build the framework needed to provide a similar account of the strange effects that have been predicted to take place in the vicinity of a black hole. To do this, we need to generalise yet again: from flat geometry, to curved.

Gravity as Spacetime Curvature

he basic premise of general relativity is simple: the correct way to account for the acceleration of objects due to gravity is to consider spacetime to be curved in the presence of matter and energy. How does curvature explain acceleration? If two explorers set off from different points on the Earth's equator, and both head north, their paths will grow steadily closer together, despite the fact that they started out in the same direction. In spacetime, if two nearby stars start out being motionless with respect to each other, their world lines will draw closer together, despite the fact that those world lines were initially pointing in the same direction. We could say that the force of gravity is pulling the stars together … but we don't say there's a “force” acting on the explorers, do we? Of course, the two-dimensional surface of the Earth is a visibly curved object embedded in a larger (and more or less flat) space, but we have no reason to believe that spacetime is embedded in anything larger. Rather, general relativity assumes that whatever gives rise to spacetime geometry in the first place is tied up with the presence of matter and energy in such a way that the resulting geometry is sometimes curved."

5 out of 5

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