"Greg Egan - Foundations 2 - From Special To General" - читать интересную книгу автора (Egan Greg)

Foundations
by Greg Egan

2: From Special to General
Copyright © Greg Egan, 1998. All rights reserved.



The 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

The 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.


Manifolds

Before exploring curved geometry, it will be useful to take a look at a kind of geometry
that's neither flat nor curved: geometry without any metric at all. Essentially, this is like
asking what you can say about lines drawn on a sheet of rubber that remains true
 Egan: "Foundations 2"/p.2


however much you stretch or squeeze the sheet: distances and angles lose all meaning,
but you can still talk about such things as whether or not two lines intersect. Why is this
relevant to general relativity, which does assign a metric to every part of spacetime?