In article <
99013d69-da41-4847...@googlegroups.com>,
"
kqui...@yahoo.com" <
kqui...@yahoo.com> wrote:
[...]
> Consider that objects also move in curved paths in response to EM forces. Why
> is that not equivalent to another curvature of space - that caused by EM
> forces? And so on. All motion is the result of forces applied to objects.
> Does every force 'curve space'?
This is a good question, and the answer should
help explain why general relativity describes
gravity as geometry.
Two different objects in an electric field will,
in general, react very differently. Positively
and negatively charged particles will accelerate
in opposite directions. Two positively charged
particles with different charges or masses will
undergo different accelerations. So while an
electric field determines "curves" -- paths of
particles -- they're different curves for each
type of particle. That is, the paths are really
characteristics of the particles, not the space.
The same is true of almost all other interactions.
But gravity is different. An enormous collection
of observations and measurements (together summarized
as the "principle of equivalence") show that any
two objects in a gravitational field will follow
the same path, regardless of their mass, composition,
shape, or any other feature. So gravity really
determines a set of paths, independent of what is
moving on those paths.
Mathematically, such a set of preferred paths is,
by definition, a geometry. Ordinary Euclidean
geometry is completely determined by giving the
straight lines and their mutual relationships. The
spherical geometry of the surface of the Earth is
determined by giving the great circles. Whether
or not you want to think of gravity as "really"
being geometry, the principle of equivalence
guarantees that it can be given a geometrical
description.
(There's one subtlety here -- you have to think
about paths in spacetime rather than space. The
path of an object in three-dimensional space depends
on its initial velocity, so it's not really unique.
But different initial velocities also affect the
time it takes for the object to move from one
point to another.
In Euclidean geometry, you get a unique straight
line by giving an initial and final endpoint.
In a gravitational field, if you only give an
initial and final *spatial* point, you don't
get a unique path. You could, for example, drop
a coin to the floor, or throw it straight up in
the air, and it would end at the same place. But
if you give an initial and final position *and
time*, the path is unique.)
There have been lots of efforts to give similar
geometric descriptions to other interactions.
But these require extra structure. For example,
you can "geometrize" electromagnetism by going
to more than four spacetime dimensions -- the
"momentum" in an extra dimension can mimic the
effect of different charges. But these models
involve extra assumptions, and are still pretty
speculative.
> Why isn't the 'curvature of space', this brilliant
> insight, merely taking an easy way out? It doesn't
> really explain anything, it merely puts Newton's Law
> of Gravitation into other words. It's appealing - as
> poetry. I'm not sure what additional information it
> conveys about the real nature of gravity.
Well, for one thing, if you work out the details of
the math of curved spacetime, you get modifications
of Newtonian gravity, which can be tested. So far,
these tests have all confirmed general relativity.
(More concretely, if you assume curved spacetime,
an "action principle," and that gravity is *only*
due to spacetime curvature with no extra structures,
you get an almost unique set of equations for gravity,
the Einstein field equations. It's already a bit of
a miracle that these give back Newton's inverse square
law as a very good approximation. The fact that they
also predict small modifications which then match
observation is a pretty good argument that the picture
is probably correct.)
Steve Carlip