Re: "acceleration due to gravity"

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding Michael Edmiston's (possibly rhetorical?) question:

> ...                     Isn't a main conclusion of general relativity
> that there is an equivalence of the concepts of gravity and acceleration,
> similar to a main conclusion of special relativity that there is an
> equivalence of the concepts of energy and mass?

To be honest, I think the answer to the question is 'sort of, in a very
weak sense, but not really'.  GR does *not* consider all gravitational
effects as equivalent to merely acceleration effects.  OTOH, it *does*
consider the the passive effect *locally* of a given gravitational field
acting *on* a test particle as equivalent to the effect on the test
particle in an appropriately accelerating coordinate system *in a
sufficiently small local region of spacetime containing a small segment
of the particle's world line and its neighboring events.  But GR
considers the active real gravitational effects produced *by* sources of
gravitation (i.e. nonzero concentrations of energy, momentum, and stress)
as geometric effects induced by the fact that these sources actually
curve the spacetime manifold, and the global effects of such curvature is
*not* eliminatable by *any* judicially chosen reference frame.

Gravitational effects are real physical effects that go well beyond those
induced by describing the world in an accelerated coordiante system.
It's just that these intrinsic effects do not show up at the level of a
global gravitational *force* field, e.g. g, itself.  Rather, they show up
at the level of the nonuniformity or gradients (in spacetime) of such
force fields, and at the level of any coordinate system in spacetime that
is locally inertial near its origin being *necessarily* non-inertial in
more distant regions of spacetime due to the geometric effects of the
curvature of the spacetime manifold itself (induced by the real presence
of the gravitational sources that act to curve it).  Real gravitational
effects show up globally, not locally, *because* they really are curvature
effects.

The geometric effects of curvature never show up on a sufficiently small
local scale. (For instance, we could probably not detect the spheroidal
geometry of the Earth's surface by surveying just one city block.)  Any
gravitational *force* (not tidal field) can only *locally* be eliminated
by a judicious choice of coordinate system.

*If* all gravitational forces *could* be eliminated globally by a
particular choice of coordinate system, then in *that* world's spacetime
there really is *no* "real" gravitation at all, and any gravitational
force field effects that might happen to appear in any given coordinate
system are merely those inertial effects caused by the fact that that
coordinate system as a whole does not describe an inertial frame, but is
accelerating and/or rotating somehow.  *That* hypothetical world was just
described using an accelerated frame, and that's the only reason, *that*
world had any gravitational forces showing up.  But *that* pretend world
does *not* describe our world which really *does* manifest measurable
curvature effects, i.e. gravitation, caused by the gravitational sources
that *are* present.

> So, if the equivalence of energy and mass allow us to mix energy and mass
> units and terminology, shouldn't the equivalence of gravity and acceleration
> allow us to mix these units and terminology?

Not in general, but *maybe* under certain well-controlled special
circumstances.

David Bowman
David_Bowman@georgetowncollege.edu