Re: [Phys-L] Nonequivalence of equivalence principles

[John Denker <jsd@av8n.com>]

On 12/29/2014 06:45 AM, Savinainen Antti wrote:
> It turns out that the standard formulations are not equivalent, and
> they can be used in analyzing competing theories of gravity.

It's surprisingly tricky to come up with a good statement
of "the" equivalence principle.  Most of the typical informal
versions are provably wrong.

Here's how I think about it:

Executive summary:  The equivalence principle says that
/one particular term/ in the Taylor series is indistinguishable
from an accelerated reference frame.  Almost anything else
you might like to say turns out to be not true.


1) In a wide range of classical and "nearly classical"
 situations, the predominant part of the gravitational 
 interaction can be described in terms of a potential.

2) If we stay away from singularities, we can expand
 this potential in a Taylor series as a function of
 position.

3) The leading term in this series is the background
 potential.  This has absolutely no physical significance.
 We can make it go away by means of a gauge transformation.

4) The next term is the one we think of as "the"
 gravitational acceleration.  The equivalence principle
 states that this term is indistinguishable from an
 acceleration of the reference frame.  Therefore it
 has "almost" zero physical significance.  If everybody
 is accelerating the same, nobody notices.

 This term gives a potential that varies linearly with
 position.

5) At this level of approximation, the equivalence of
 inertial mass and gravitational mass is a direct
 corollary of item (4).

6) The next term is the tidal stress.  It gives us a
 lowest-order description of how the previous term (the 
 acceleration) varies from place to place.  This term
 has unmistakable physical significance.  It is gauge
 independent and frame independent.

 This term give a potential that varies quadratically
 with position.  The tidal stress on the earth has
 roughly the symmetry of x^2 + y^2 - 2 z^2.

7) The "predominant" part mentioned in item (1) is not
 actually the whole story.  Just as in electromagnetism
 the electrostatic potential is not the whole story,
 in gravitation the gravitational potential is not the
 whole story.  A moving source produces gravimagnetic
 fields that are not the gradient of any potential.
 This has been observed experimentally.  This is
 approximately the hardest experiment ever done,
 because the effect is so small.
   http://einstein.stanford.edu/highlights/status1.html

 Therefore any claim that gravitation and inertia are 
 equivalent cannot possibly be true.  You have to make
 a much more careful statement about certain things
 being asymptotically equivalent in certain limits.

8) It's even worse than that.  There are three things:
  a) Inertial mass.
  b) Mass that /responds/ to a given gravitational field
  c) Mass that serves as a /source/ for the gravitational
   field.

 Asserting that (a) is equivalent to (b) does not mean
 that either of them is equivalent to (c).  The third 
 law of motion imposes some symmetries between the source
 point and the field point, but there is still a lot of
 wiggle room.  For example, rather than the usual bilinear
 form

         φ = G m1 m2 / r                      [1]

 you could have perhaps a trilinear form

         φ = G m1 m2 m3 / r                   [2]

 This would not violate the third law of motion, and would 
 not violate the equivalence of gravitational mass and
 inertial mass for a test particle.  Weirder things happen
 all the time.  Quantum mechanics is highly nonlinear.
 Just look at the periodic table:  there is not a smooth
 linear variation of properties as we go from H to U.

 I'm not suggesting that [2] is a viable theory, but I am
 saying that there are a lot of ways to have an equivalence
 principle that is true if interpreted one way but not another.
 There is a vigorous cottage industry devoted to cooking up
 nonlinear theories of gravity.  Indeed GR is already nonlinear.

==========

Bottom line:  There is one piece of the the gravitational
interaction that is indistinguishable from an acceleration 
of the reference frame.  This has tremendous historical and
heuristic importance, but it should not be taken as gospel.
If you extend it in almost any direction, you get nonsense.