Re: Does gravity do work on the freely falling body?



Tom Roberts wrote:

Phil wrote:

Let's say one takes the
position that freely-falling motion is unaccelerated inertial motion.
How does that position stack up to "all the evidence" that gravity
accelerates the object which is inertial?


That is a reasonable question to ask (unlike most of this thread).

In GR, indeed we do consider freely-falling motion to be inertial, and therefore unaccelerated, in the sense that the 4-acceleration of such an object is identically zero. Note that I qualified what I meant by "acceleration", because in GR that word has several meanings; some apply to what I said, some don't.

In GR one must be careful of coordinate-dependent quantities, because artifacts of the coordinate system can confuse the discussion. It's usually MUCH better to discuss invariants (quantities independent of coordinate choice).

4-acceleration is a 4-vector, and is completely independent
of coordinates. That's why it is so useful, and why I used
it above.

In coordinates fixed to my office, a falling object has nonzero acceleration. Note that this statement is explicitly coordinate dependent. When I consider the 4-acceleration of the object, it is zero (neglecting air resistance and other minuscule effects). The un-qualified word "acceleration" could refer to either of these, and the resulting confusion is the basis of your question.

When you say "gravity accelerates the object" you mean a coordinate acceleration using coordinates that are fixed to the earth (or similar). This COORDINATE acceleration of the object is really an artifact of the coordinates, as any object at rest in them has nonzero 4-acceleration. Those coordinates are not inertial coordinates, so naturally any object moving inertially is accelerated relative to them. In GR, we say that an object "moves inertially" when its 4-acceleration is zero [%].

[%] Technically this holds only for "test particles" --
objects that have no internal structure and are so small
that their effect on the geometry can be completely neglected.
For example, this marble (1 cm glass sphere) can be considered
to be a test particle here on earth.

In summary, in GR the idea that freely-falling motion is unaccelerated inertial motion is completely consistent with "all the evidence that gravity accelerates the object". One must apply the appropriate meanings to those words, that's all. This is why such rather loose descriptions are inappropriate in discussions of physics -- they are insufficiently precise, and discussions get bogged down because people interpret the loose wording in different ways. Worse, such sloppy statements encourage one to think in sloppy ways that often lead to errors....


It wasn't as much about
Work as it is about what work is.


In GR we do not normally attempt to discuss work, because it is exceedingly complicated and not very useful [#]. That's primarily because work is a type of energy, and energy is ALWAYS coordinate dependent; re-read my caveat about coordinate-dependent quantities above. In short, once one studies GR in any detail, it becomes QUITE CLEAR that whenever possible, one wants to discuss invariant quantities, not coordinate-dependent ones.

This is true even in Newtonian mechanics. Compare solving
F=ma to solving the equations of Lagrangian mechanics....
The Lagrangian equations are independent of coordinates,
while F=ma is not.

[#] One aspect of the complexity is related to the fact
that work is inherently an integral, and integration in a
curved manifold has thorns....


You were correct IMHO in your
reference to evidence supporting the notion that Gravity does work on
the freely falling body. There are many other ways to cite evidence
in this regard also.


Sure. But it's all COORDINATE-DEPENDENT evidence, using NON-inertial coordinates. See above for my caveat about coordinate-dependent quantities.

Use co-moving locally-inertial coordinates, and it's obvious that no work at all is being done on the object (it of course remains at rest in such coordinates).

So simply asking whether or not "work is done" is not very useful....

Remember that selecting coordinates is an ARBITRARY human
choice, and such choices cannot possibly have any effect on
physical phenomena. Since I can make "work" appear or disappear
merely by thinking differently about the situation, it should
be clear that this "work" is unrelated to any physical
phenomena -- it's just human bookkeeping. Understand the
implications of this and you're on the path to understanding
how truly enormous is the change in outlook that GR brings.

[pmb has not made this leap of understanding, with his repeated
insistence on notions like "relativistic mass".]

Note that for a true force, such as from a spring or E&M, discussing work is VASTLY easier, and can make sense in GR as long as one is careful about the coordinate system used. Still, it's usually better to discuss an appropriate invariant.


In that [pmb] and I find agreement that gravity does work, what does
that mean about free-fall motion?


You're being overly naive. See above.

Careful, phil has a tantrum when someone comments on his ideas.


Tom Roberts
.

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