Re: Letter to Dirk Van de moortel
From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 03/04/05
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Date: 4 Mar 2005 13:27:22 -0800
RP says...
>I've engaged Dirk in the same argument. He doesn't seem to grasp that
>he's deriving a GR relationship from SR and the premise of
>nonreciprocity (broken symmetry) in an accelerating field. I could be
>wrong, but then so necessarily was Einstein, since this was the very
>argument that he used to support the necessity of GR, an argument that
>I quoted for Dirk, to no avail. I suppose one could state that SR
>requires GR, and thus *everything* is resolvable with SR alone, in
>effect. But then the extension of SR to accelerating frames *is* GR,
>so why not call it properly GR in which the solution to the twin
>paradox lies.
There's really a number of steps from Special Relativity to General
Relativity. Some of the steps involve extra physics, while other steps
only require more complex mathematics.
1. Use inertial coordinates to describe inertial motion in flat
spacetime.
2. Use inertial coordinates to describe noninertial motion
in flat spacetime.
3. Use noninertial coordinates in flat spacetime.
4. Use noninertial coordinates in curved spacetime.
5. Use the equivalence principle and noninertial coordinates
to describe the effect of gravity on matter.
6. Use Einstein's GR to describe the effect of matter on gravity.
To get from Step 1 to Step 2, the only additional physical assumption
that you need is that under gentle acceleration, the proper rate of
clocks and the proper lengths of rigid objects remain unchanged. But
the mathematics becomes a little more complex, because you do a little
calculus.
To get from Step 2 to Step 3, no additional physical assumptions are
needed, but the mathematics becomes yet more complicated (because
non-inertial coordinates are messier to work with).
To get from Step 3 to Step 4, you need to assume that in curved spacetime,
Special Relativity still holds approximately in small enough regions of
spacetime. The rest is just the mathematics of non-Euclidean space.
To get from Step 4 to Step 5, you need to assume that gravity is a
manifestation of spacetime curvature.
To get from Step 5 to Step 6, which is full General Relativity, you
need to use Einstein's equation relating spacetime curvature to the
density of mass and energy.
The big intuitive leap towards GR, in my opinion, was the equivalence
principle, Step 5. This is what allowed him to connect gravity to
Special Relativity, and led him to curved spacetime. The next step,
to Step 6, took a long time for Einstein, but that's just because
the mathematics is so hard. The physics wasn't that hard once you
had Step 5---Einstein's equation is the simplest generalization of
Newton's law of gravitation that incorporates gravity as spacetime
curvature.
-- Daryl McCullough Ithaca, NY
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