Does gravity as a fictitious force do work? (GR's free-falling frame POV)?

TrickyDicky

This discussion started as a side clarification of something in this thread http://www.physicsforums.com/showthr...13#post3971013 and although tangentially related it probably deserves a thread of its own so anyone can participate without reservations.
Please keep in mind I am not referring to the question in classical mechanics with the earth's frame of reference and gravity acting as a real force in wich it is obvious there is no work done on a book placed on a table.

Also to simplify things we are using the Schwarzschild exterior solution of GR, so that we can consider the geodesic free-falling frame of reference as static.

The question is, given this setting, with gravity considered as fictitious force in GR does the table do work on a book sitting on it?
And if the book falls off the table and while is falling, does gravity fictitious force do work on the book?
I would like to take a more neutral stance in this thread,and just explain why I would consider good arguments to say that work is done in the particular situation I described.

First by analogy to other fictitious forces like centrifugal force. In my GR scenario we are considering the free-falling frame the "inertial" frame (remember the terms inertial and non-inertial are used differently in newtonian mechanics versus GR), and the book is kept in the non-inertial frame of the surface of the table on the earth.
In the centrifugal force example this force is the apparent outward force that draws a rotating body away from the center of rotation and is caused by the inertia of the body.


By analogy I consider the gravitational (fictitious) force as the apparent downward force that draws the book to the centre of the earth and is caused by the mass of the body.

Now what allows to make the analogy between this two fictitious forces is the equivalence principle that equals the inertial mass to the gravitational mass.


In the above linked thread stevendaryl argued that the equivalence failed because he could find coordinates in the schwarzschild spacetime that made the metric appear time varying.
Being obvious that coordinates can introduce artifacts that don't affect the physics of the situation I still don't know what his point was wrt the physics of the problem at hand.
I believe he was trying to make a point about being able to choose coordinate systems making the metric tensor appear time varying but that is such an obvious thing and so disconnected with the physics of the problem I chose not to follow that distracting path.

Following the analogy with the centrifugal force, I maintain that the table is doing work on the book against the gravitational fictitious force so that as soon as the book falls and as long as no object is in its way the gravitational fictitious force does work on the book as it tries catching up the free-falling frame.
Any objections?

A.T.

Work is frame dependent.
In the inertial free falling frame, there is no gravitational fictitious force on the book. Fictitious forces exist only in non-inertial frames. Here the only force on the book is the table force. And yes, it is doing work on the upwards accelerating book, as soon as it moves in that frame. Note that inertial frames exist only locally in curved space time.

In the non-inertial rest frame of the table & book there is no work done on the book, because there no displacement.

This true in the non-inertial frame rest frame of the table.

In the inertial rest frame frame of the book there are no forces acting on the book, so there is no work done on the book.

TrickyDicky

Exactly and I clearly referred to what is observed in the non-inertial frame of the book.

In the free-falling (geodesic) frame there are no forces. That's correct. We must remember we are always referring to apparent forces in the given scenario.
Right. This work is felt by the non-inertial observer.
This seems to contradict what you just wrote. The non-inertial observer is not at rest wrt to the inertial frame.

I'm not sure what you mean by this.
Exactly.

A.T.

Let's deconfuse the two scenarios and two frames.

Book rests on the table:

- Rest frame of table: no work is done on the book
- Free falling frame: table force is doing work on the book

Book falls from the table:

- Rest frame of table: inertial force of gravity is doing work on the book
- Free falling rest frame of the book: no work is done on the book

PeterDonis

I'm not sure I understand what this means. I understand what it means for a spacetime to be static, and the Schwarzschild spacetime outside the horizon is; but what does it mean for a free-falling frame of reference to be static?

TrickyDicky

I meant the inertial rest frame, and to give a well defined rest notion it is simpler to choose a static solution, instead of say an expanding one where it is trickier.

TrickyDicky

Looks like a good summary.

PeterDonis

But an inertial frame can only be at rest relative to a static observer (one who stays at a constant radius, or, to put it in invariant terms, one who follows an orbit of the timelike Killing vector field) for an instant. So it seems strange to call the frame itself "static".

Also, there is a difference between "a local inertial frame at some particular event on a free-falling observer's worldline" and "a frame in which the free-falling observer is at rest for his entire fall". The former acts just like an inertial frame in SR (within its local range); the latter does not. I'm still not sure which of the two you mean by "the inertial rest frame".

TrickyDicky

It seems strange because as I explained it was just a slip, I meant a rest frame in a static spacetime rather than a static frame (I don't think that is even a conventional form to refer to a frame of reference)

The latter. The geodesic path in a exterior Schwarzschlld geometry.


What is your answer to the OP question?

PeterDonis

Ok, got it.

I agree with A.T.'s summary.

DaleSpam

I also agree with this summary, but would add a little detail as follows:

Book rests on the table:

- Rest frame of table: two forces on book (upwards real, downwards fictitious), no motion, no work is done on the book, KE constant, PE constant.
- Free falling frame: one force (upwards real), book accelerates upward, table force is doing work on the book, KE increases, PE does not exist.

Book falls from the table:

- Rest frame of table: one force (downwards fictitious), acceleration downwards, inertial force of gravity is doing work on the book, KE increases, PE decreases
- Free falling rest frame of the book: no forces on book, no motion, no work is done on the book, KE constant, PE does not exist

Hope I got those right.

TrickyDicky

This was the only point come to think of it that you (and stevendaryl) were disagreeing with me in Q-reeus thread, I can't really understand why if you admitted that in the falling book case gravity force was doing work on the book, even less I uderstand how could you not admit you were wrong, I can only in your case attribute it to something personal.
By the way what do you mean potential energy doesn't exist? It does if you want to conserve energy.

DaleSpam

Please point out exactly where you think that anything I said there conflicts in any way with anything I said here.

TrickyDicky

Just a few:


However here:
But none of this quoting was really necessary, I started this thread because you were saying in the other thread that my claim that-in a different frame than the usually considered- work was done on the book was nonsense. And now you are saying that "you hope you got those right".

A.T.

I checked the whole posts by DS you quoted, and it is quite obvious that he talks about the non-inertial rest frame of the table. So there is no contrdiction to this:

TrickyDicky

So then why was he saying I was wrong when I claimed that in the free-falling frame work was done on the book. That's pretty absurd. I kept saying I was not referring to the earth's non-inertial frame. And even made a heuristic formula for work in the geodesic (free-falling) frame that he said was wrong.

A.T.

Where?