5.8.1 The Basic Idea (General Relativity)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
5.8.1 The Basic Idea (General Relativity)
Let's get started with the basic ideas which combine the concepts we
have discussed to produce GR. Here I will simply state the main ideas
without an explanation of their application. You will get some feel for
their application in our two examples to follow.
So, here are the main claims of GR which involve the concepts we have
discussed. First, the space-time in which we live is a four dimensional
manifold. On that manifold there is a metric tensor (or just "a metric")
which describes the geometry of space-time. The metric can be used to find
geodesics on the space-time manifold, and when an object (only being acted
on by gravity) goes from one point in space-time to another point in
space-time (note: these are not just two points in space, but two
points--i.e. events--in space-time), it moves between the points by
following a space-time geodesic. Therefore, all the information necessary
for us to determine how such objects move through space-time is held within
the form of the metric. How, then, do we determine the metric? Well, the
metric of space-time in a region is itself determined (in a not-too-trivial
way) from the stress-energy tensor (T) which is affecting the region. This
then is the new theory of gravity which relativity has produced. The
stresses and pressures and momenta in a nearby region produces a
stress-energy tensor which, in turn, changes the metric of the nearby
space-time (making its geometry "curved"). This forces objects in the region
to follow specific paths (geodesics) through the "curved" space-time, and we
attribute this motion to gravitational effects.
As a conceptual example, consider a football being thrown from the
surface of the earth. Because of the mass of the earth, the space-time the
football is traveling through is a curved manifold, and the football follows
a "straight line" geodesic in the four dimensional curved space-time. To us,
the football's path is curved through three-space, but if we could somehow
experience the time dimension as a spacial dimension (i.e. if we were four
dimensional beings) and if we followed the path of the football in the
four-space, we would seem to be following a straight line on our four
dimensional curved manifold. However, in reality, the fourth dimension of
time does not act like the other dimensions in our perception of the
space-time manifold. Thus we do not see the actual four dimensional path of
the football, we only see the path in three dimensions while the fourth
component of the path is revealed to us as a dynamic component of the ball's
motion through time. That's why we can't see that its path is a "straight
line" in curved space-time. The stright-line is revealed to us as curved
motion, and we attribute that motion to gravitational effects.
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