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

## Main Question or Discussion Point

This discussion started as a side clarification of something in this thread https://www.physicsforums.com/showthread.php?p=3971013#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?

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A.T.
Work is frame dependent.
I maintain that the table is doing work on the book against the gravitational fictitious force
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.

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
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.

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

In the inertial free falling frame, there is no gravitational fictitious force on the book. Fictitious forces exist only in non-inertial frames.
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.
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.
Right. This work is felt by the non-inertial observer.
In the non-inertial rest frame of the table & book there is no work done on the book, because there no displacement.
This seems to contradict what you just wrote. The non-inertial observer is not at rest wrt to the inertial frame.

This true in the non-inertial frame rest frame of the table.
I'm not sure what you mean by this.
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.
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
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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.
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?

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?
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.

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
Looks like a good summary.

PeterDonis
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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.
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".

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".
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)

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

PeterDonis
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I meant a rest frame in a static spacetime

...

The latter. The geodesic path in a exterior Schwarzschlld geometry.
Ok, got it.

I agree with A.T.'s summary.

Dale
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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
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.

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

Book rests on the table:
- Free falling frame: one force (upwards real), book accelerates upward, table force is doing work on the book, KE increases, PE does not exist.
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.

Dale
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even less I uderstand how could you not admit you were wrong, I can only in your case attribute it to something personal.
Please point out exactly where you think that anything I said there conflicts in any way with anything I said here.

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

This is an incorrect understanding of energy. Once the small deformation is done there is no motion, so no work is being done.
None of that energy is going into the book whose KE and PE are remaining constant.
No, I am not considering the table an inertial object, if the table were inertial then the force on the book would be 0.

So, the book, sitting on the table at rest, has no change in KE. It also has no change in PE. No work is being done on it.
However here:
Book rests on the table:
- Free falling frame: one force (upwards real), book accelerates upward, table force is doing work on the book, KE increases, PE does not exist.
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.
Just a few:
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:
Book rests on the table:
- Free falling frame: one force (upwards real), book accelerates upward, table force is doing work on the book, KE increases, PE does not exist.

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:
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.

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A.T.
So then why was he saying I was wrong when I claimed that in the free-falling frame work was done on the book.
Where?

Where?
Where what? where was he saying I was wrong? In every post he answered mine.
where was I saying I was considering the inertial frame, implicitly in most of my posts. Explicitly in 317# and the posterior discussion with stevendaryl made it even clearer. Besides we were only considering two frames the free-falling and the non-inertial, what else could I mean if I kept saying that in the earth's frame it was clear work was not done.

In any case if he actually knew it and kept saying I was wrong that can only mean he was not acting on good faith.

A.T.
where was I saying I was considering the inertial frame, implicitly in most of my posts. Explicitly in 317#
Where in post 317# do you explicitly state that you consider the free falling frame?

Where in post 317# do you explicitly state that you consider the free falling frame?
"In which the table would be preventing the book from following its geodesic path. And in order to do that Work in the form of a quantity proportional to the EM force in the table material times distance from the theorical geodesic path of the book, must be done." My bold.

In this case geodesic path and free-falling path are accepted synonims. If here it is not obvious I'm considering the geodesic frame, in the context of my insistence on not considering the earth's non-inertial....

This is a not very productive exchange, you may think what you like. I can see there is certain subjective bias on your part to favor what is obvious DS thinks.

A.T.
Where in post 317# do you explicitly state that you consider the free falling frame?
"In which the table would be preventing the book from following its geodesic path. And in order to do that Work in the form of a quantity proportional to the EM force in the table material times distance from the theorical geodesic path of the book, must be done." My bold.

In this case geodesic path and free-falling path are accepted synonims.
That says nothing about the reference frame you are considering. Not even implicitly. The table is preventing the book from flowing a geodesic path in any frame.

You seem to confuse:
- objects being inertial / non-inertial
- reference frame being inertial / non-inertial

The non-zero proper acceleration of the book on the table is frame invariant. So the book on the table is non-inertial in any frame. So stating that the book is non-inertial (or doesn't follow a geodesic path) doesn't imply anything about the reference frame you are considering.

This seems to be the root of the misunderstanding between you and DS about which frame is being considered.

That says nothing about the reference frame you are considering. Not even implicitly. The table is preventing the book from flowing a geodesic path in any frame.
Are you serious? I was referring to the next sentence(we have different notions of what is obvious it seems): when considering the usual formula W=F.d you need to reference the distance d to some reference, in the case of the book if the frame is the table distance is zero and no work is done, if the reference is the free falling frame the distance is not zero and that is why there is work.

A.T.
we have different notions of what is obvious it seems
We definitely have different notions of what "explicitly" means. Anyway, the whole misunderstanding between you two could have been cleared up quickly by stating the frames explicitly, like I did in post #4.

Dale
Mentor
Just a few:

However here:
As A.T. mentioned, all of those quotes in the other thread refered to the table frame since that was always the topic of that thread. (Q-reeus refuses to consider any coordinates other than Schwarzschild and is interested in coordinate effects, so all of his threads deal only with the non-inertial frames). So they cannot possibly be in conflict with the quote above regarding physics in the free-falling frame.