Re: photon in uniform gravity (??)

Cyberkatru wrote:
I don't have Callahan's book, so I can't tell whether he's not
presenting the material clearly or whether you are misunderstanding
what he says. But the description you offer above should not have the
word "curvature" in it anywhere.

There are several places in Callahan's book where he brings up the word
curvature in discussing accelerating frames relative to an inertial
frame --such as his section on rotating frames. But I think we can all
agree that an accelerating frame doesn't imply curvature.

Agreed. In that case, I would (again, based solely on your description)
say that Callahan misspeaks when identifying the presence of curvature
and non-trivial Christoffel symbols.

In fact, I often say that in Einsteins theory, gravity is the curvature
of spacetime but apparently this is not always what is meant by gravity
since an idealized constant gravitational field is still a flat
situation. As a differential geometer, I think I am clear on the math
but the way words are used in some physics books is what I want to be
clear on. For example, Callahan thinks of the g_i_j as being the
potential while the Christoffel symbols are the field. This is foreign
to me since the Christoffel symbols can be nonzero even when the space
is flat (choose polar coordinates for example). Thus, to me, that
definition of the field is too coordinate dependent. I suppose that it
is consistent with the idea that noninertial frames are associated with
gravity but to me it is an illusion rather than something
physical--what doesn't depend on the chart is whether the curvature
tensor is zero or not. I would prefer lauguage that respected the idea
that whatever gravity is, it is an invariant notion. Thats why I would
consider curvature to "be" gravity while nonzero Christoffel symbols
indicate "graviational effects" that could possibly be chart dependent
and therefore in some sense illusory. Such effects occur for certain
frames (accelerating) even in a prefectly globally flat Minkowski
space.

The terminology used in physics books is heavily historically
influenced. The existence of so-called pseudo-forces, which arize as
corrections to the free particle equations of motion when expressed in
an accelerating frame predate General Relativity by a quite a margin. A
discussion of such forces (the most famous of which is the Coriolis
force) takes place in any introductory as well as more advanced course
on classical mechanics. As such, they are very familiar to the
physicist.

One way to state the Equivalence Principle, which takes advantage of
this familiarity, is to say that (locally) gravity is a pseudo-force.
But, at the same time, (locally) gravity can also be described as a
conservative force (i.e. coming from a potential). Matching up these
two descriptions gives the correspondence between Christoffel symbols
and the gravitational force field. By the same token, we are forced to
identify the components of the metric tensor and the gravitational
potential.

All of the above identifications are of course coordinate dependent.
And, even though physics does not care about coordinates, physicists
do. That's why it's nice to be able to give names different terms that
enter a particle's equations of motion in any given coordinate system.
That's where the notion of a pseudo-force comes from. Another well
known example of the same nomenclature tendency is the effective radial
potential in the Keppler problem. Of course, the shape of this
potential strongly depends on the fact that we expressed the equations
of motion in spherical coordinates.

Of course you may say that gravity as curvature is itself a
pseudo-force since objects follow geodesics and experience no
acceleration in the covariant sense. Fine, thats great. But at least
curvature is not fframe dependent and it is curvature that is affected
by matter and visa versa.
Am I off base?

Not off base at all. If you encounter a physics book claiming that
"curvature" is present in some non-inertial coordinate system, even
though the underlying space is flat, you can safely conclude that this
claim is either wrong or misguided.

Hope this helps.

Igor

.

Relevant Pages

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