The cables break and the elevator goes into free fall.
Newton told us that the elevator accelerates and, therefore, there is a
force that makes it accelerate.
Then Einstein came along and told us that this is not true and that
there is no force that accelerates the elevator in free fall.
But if there is no force that accelerates the elevator, it means that
the elevator does not accelerate.
And if it does not accelerate, then it moves with uniform speed.
But speed is not absolute: it is relative.
And so I ask: is there any reference system with respect to which its
speed is uniform?
This is for Newton's second law: force that accelerates mass.
Instead, for the first law, Einstein says that a body in the elevator
in free fall is at rest with respect to the elevator itself.
So, why does a body placed below the center of gravity of a
free-falling elevator accelerate downwards, and if it is above the
center of gravity, it accelerates upwards?
Luigi Fortunati

In article <vk0k8g$2p4uk$1@dont-email.me>, Luigi Fortunati discussed,
in Newtonian mechanics and in general relativity (GR), the behavior of
an elevator which is (a) initially suspended stationary from an elevator
cable or cables, and then (b) free-falling downwards after the cable(s)
break. Luigi then went on to (c) ask some questions about the motion
of bodies placed above/below the center of gravity of a free-falling
elevator.

In this article I'll try to analyze the same system Luigi described,
and clarify a few of the tricky parts of how this system is modelled
in GR. In this article I'll focus on (a) and (b) above; I'll discuss
(c) in a following article.

Let's start with the Newtonian perspective, where gravity is a force,
and where we measure acceleration with respect to (i.e., relative to)
an inertial reference frame (IRF). To a very good approximation, we can
treat the surrounding building housing the elevator shaft as an IRF.

BEFORE the cable-break, i.e., when the elevator is suspended from
the cables, stationary with respect to the surrounding-building IRF,
there are 2 (vertical) forces acting on the elevator:
* elevator weight (F=mg, pointing down), where m is the elevator's
mass and g is the local gravitational acceleration (about 9.8 m/s^2
near the Earth's surface)
* cable tension (F=T, pointing up, i.e., the cable(s) are pulling up
on the elevator)
The net (vertical) force acting on the elevator is thus T - mg in the
upward direction.

Since we observe that the elevator is stationary with respect to the
surrounding-building IRF, i.e., the elevator's acceleration with respect
to that IRF is a=0, we use Newton's 2nd law to infer that the net force
F_net acting on the elevator must also be zero:
F_net = ma = 0
and hence
T - mg = 0
and hence the cable tension must be
T = mg

Luigi wrote
> The cables break and the elevator goes into free fall.
> Newton told us that the elevator accelerates and, therefore, there is a
> force that makes it accelerate.

That's correct.

AFTER the cable-break, when the elevator is in free-fall downwards
(let's neglect air resistance for simplicity), the only force acting
on the elevator is the elevator's weight (F=mg, pointing down), so the
net force acting on the elevator is F_net = mg (pointing down) and the
elevator's acceleration with respect to the surrounding-building IRF
is a = F_net/m = g (again pointing down).

Now let's look at the same system from a GR perspective, i.e., from a
perspective that gravity isn't a force, but rather a manifestation of
spacetime curvature. In this perspective it's most natural to measure
accelerations relative to *free-fall*, or more precisely with respect
to a *freely-falling local inertial reference frame* (FFLIRF). An
FFLIRF is just a Newtonian IRF in which a fixed coordinate position
(e.g., x=y=z=0) is freely falling.

Like Newtonian IRFs, there are infinitely many FFLIRFs at a given
position, with differing relative positions, velocities, and
orientations, but all these FFLIRFs have zero acceleration and
rotational velocity with respect to each other. If all we care about
is acceleration, we often ignore the freedom to choose different
relative positions, velocities, and orientations, and refer to "the"
FFLIRF.

Any FFLIRF is (by definition) freely-falling, so it's accelerating
*downwards* at an acceleration of g relative to the surrounding-building
Newtonian IRF.

Since the g vector points (approximately) towards the center of the
Earth, we see that the FFLIRF *changes* if you go to a different place
near the Earth's surface. For example, my FFLIRF differs from (i.e.,
has a nonzero relative acceleration with respect to) the FFLIRF of
someone 1000 km away on the surface of the Earth, or even of someone
at my latitude/longitude but 1000 km above me. This why we have the
word "local" in the phrase "freely-falling *local* inertial reference
frame". This is related to Luigi's questions (c); I'll elaborate on
this in a following article.

In GR it's easiest to first consider the situation AFTER the cable-break,
when the elevator is freely falling (we're neglecting air resistance).

Luigi wrote:
> Then Einstein came along and told us that this is not true and that
> there is no force that accelerates the elevator in free fall.

That's correct. There are no forces acting on the elevator (remember
we're not considering gravity to be a "force"), so the net force acting
on the elevator is zero, so Newton's 2nd law
a = F_net/m
says that a=0, i.e., the elevator has zero acceleration
*with respect to (i.e., relative to) a FFLIRF*.

Since we've already established that a FFLIRF is accelerating *downwards*
at an acceleration of g relative to the surrounding-building IRF, we
conclude that the elevator is accelerating *downwards* at an acceleration
of g relative to the surrounding-building IRF.

Luigi wrote:
> But if there is no force that accelerates the elevator, it means that
> the elevator does not accelerate.
> And if it does not accelerate, then it moves with uniform speed.

This two sentences both leave out a key qualification, namely "with
respect to a FFLIRF". That is, a more accurate statement is that if
there is no force that accelerates the elevator, it means that the
elevator does not accelerate *with respect to a FFLIRF*, and hence it
moves with uniform speed *with respect to a FFLIRF*. The qualification
"with respect to a FFLIRF" is essential here -- without it the statement
is ambiguous (acceleration with respect to what?).

Luigi wrote:
> But speed is not absolute: it is relative.
> And so I ask: is there any reference system with respect to which its
> speed is uniform?

Yes, the elevator's speed is uniform with respect to any FFLIRF. Since
a FFLIRF is accelerating (downwards) with respect to the surrounding
buildint's IRF, the elevator's speed is NOT uniform with respect to the
surrounding building's IRF.

Now let's consider the situation BEFORE the cable-break from a GR
perspective. Now there *is* an external force acting on the elevator,
namely the cable tension (F=T pulling up on the elevator). In the
Newtonian perspective we found that T = mg, and this turns out to still
be true in GR.
[Aside: What I just wrote is true for weak gravitational
fields like the Earth's. If we were in a very strong
gravitational field (e.g., close to a neutron star or
black hole) then we might have to be more careful with
many of the statements I'm making.]
So, the net force acting on the elevator is F_net = mg, pointing up.

Newton's 2nd law then says
a = F_net/m
= mg / m
= g
i.e., the elevator (which is stationary relative to the surrounding
building) must be accelerating *up* at an acceleration of g with respect
to (i.e., relative to) any FFLIRF.

This seems a bit counterintuitive, but in fact it's correct: Since a
FFLIRF is accelerating *down* at an acceleration of g with respect to the
surrounding-building IRF, the (stationary) surrounding building (and the
elevator, which is stationary with respect to the building) must be
accelerating *up* at an acceleration of g with respect to the FFLIRF.

[Aside: It's instructive to compare the previous paragraph
with what we'd think about a different physical system:
suppose that the building and elevator were in space far from
any other masses, and the building's foundation were replaced
by a huge rocket that's accelerating the whole building (and
the elevator suspended inside the building from cables which
haven't yet broken) upwards at an acceleration of g relative
to a Newtonian IRF.

Given our assumption of "in space far from any other masses",
a Newtonian IRF is a FFLIRF, and vice versa. So, this
"rocket-accelerated elevator in space" would have the same
upward acceleration with respect to a FFLIRF as our ordinary
elevator here on Earth (again, BEFORE the cable-break) in the
GR perspective.

This is an example of the "equivalence principle" (EP) which,
in its simplest form, says (roughly) that a uniform gravitational
field has the same local effects as a steady acceleration.
In Newtonian mechanics it's not apparent why the EP should
be true; GR sort of assumes the EP as a postulate. In fact,
assuming the EP can take you most of the way to deriving GR,
and this was roughly the route that Einstein took in originally
obtaining GR. (I'm glossing over lots of technical details here.)]

To summarize, then, in GR *free-fall* plays a similar role to that which
*uniform motion* plays in Newtonian mechanics. Newton's 2nd law
a = F_net/m
is formally the same in GR and in Newtonian mechanics, but a and F_net
are interpreted somewhat differently:
* In Newtonian mechanics, "a" is interpreted as acceleration with respect
to (relative to) an IRF, and gravity is viewed as a force contributing
to F_net.
* In GR, "a" is interpreted as acceleration with respect to (relative to)
a FFLIRF, and gravity is *not* viewed as a force and does *not* contribute
to F_net. As I'll explain in a following article, gravity actually shows
up as spacetime curvature, evidenced by the relative acceleration of
FFLIRFs at different places (e.g., the relative acceleration of my
FFLIRF and the FFLIRF of someone 1000 km away).

Jonathan Thornburg [remove -color to reply] il 21/12/2024 09:27:44 ha
scritto:
> ...
> Now let's look at the same system from a GR perspective, i.e., from a
> perspective that gravity isn't a force, but rather a manifestation of
> spacetime curvature. In this perspective it's most natural to measure
> accelerations relative to *free-fall*, or more precisely with respect
> to a *freely-falling local inertial reference frame* (FFLIRF). An
> FFLIRF is just a Newtonian IRF in which a fixed coordinate position
> (e.g., x=y=z=0) is freely falling.

Can we define the interior space of the elevator as "local" or is it
too big?

If it is too big, how big must it be to be considered "local"?

If it is shown that there are real forces inside the free-falling
elevator, can we still consider this reference system inertial?

Are tidal forces real?

Do we mean by "freely falling bodies" only those that fall in the very
weak gravitational field of the Earth or also those that fall in any
other gravitational field, such as that of Jupiter or a black hole?

Luigi Fortunati.

On 22/12/2024 09:57, Luigi Fortunati wrote:
> Jonathan Thornburg [remove -color to reply] il 21/12/2024 09:27:44 ha
> scritto:
>> ...
>> Now let's look at the same system from a GR perspective, i.e., from a
>> perspective that gravity isn't a force, but rather a manifestation of
>> spacetime curvature. In this perspective it's most natural to measure
>> accelerations relative to *free-fall*, or more precisely with respect
>> to a *freely-falling local inertial reference frame* (FFLIRF). An
>> FFLIRF is just a Newtonian IRF in which a fixed coordinate position
>> (e.g., x=y=z=0) is freely falling.
>
> Can we define the interior space of the elevator as "local" or is it
> too big?
>
> If it is too big, how big must it be to be considered "local"?
>
> If it is shown that there are real forces inside the free-falling
> elevator, can we still consider this reference system inertial?
>
> Are tidal forces real?
>
> Do we mean by "freely falling bodies" only those that fall in the very
> weak gravitational field of the Earth or also those that fall in any
> other gravitational field, such as that of Jupiter or a black hole?
>
> Luigi Fortunati.

This problems in understanding GR is, in my opinion, due to too much
emphasis on the geometrical point of view. Of course, geometry is the
theoretical foundation of all of modern physics, i.e., a full
theoretical understanding of physics is most elegantly achieved by
taking the geometric point of view of the underlying mathematical
models. However, there's also a need for a more physical, i.e.,
instrumental formulation of its contents.

Now indeed, from an instrumental point of view, the gravitational
interaction is distinguished from the other interactions by the validity
of the equivalence principle, i.e., "locally" you cannot distinguish
between a gravitational force on a test body due to the presence of a
gravitational field due to some body. In our example we can take as a
test body a "point mass" inside the elevator, with the elevator walls
defining a local spatial reference frame. The corresponding time is
defined by a clock at rest relative to this frame at the origin of the
frame (say, one of the edges of the elevator). Now, the equivalence
principle says that it is impossible for you to distinguish by any
physics experiment or measurement inside the elevator, whether you are
in a gavitational field (in our case due to the Earth), which can be
considered homogeneous (!!!), for all relevant (small!) distances and
times around the origin of our elevator reference frame or whether the
elevator is accelerating in empty space. A consequence is also that if
you let the elevator freely fall in the gravitational field of the
Earth, you don't find any homogeneous gravitational field, i.e., free
bodies move like free particles locally, and thus the free-falling
elevator defines a local inertial frame of reference.

Translated to the "geometrical point of view" that means that you
describe space and time in general relativity as a differentiable
spacetime manifold. The equivalence principle means that at any
space-time point you can define a local inertial frame, where the
pseudometric of Minkowski space (special relativity) defines a
Lorentzian spacetime geometry.

If you now look at larger-scale physics around the origin of the
freely-falling-elevator restframe, where the inhomogeneity of the
Earth's gravitational field become important, there are "true forces"
due to gravity. In the local inertial frame these are pure tidal forces,
named because they are responsible for the tides on the Earth-moon
system freely falling in the gravitational field of the Sun.

So it's important to keep in mind that the equivalence between
gravitational fields and accelerated reference frames in Minkowski space
holds only locally, i.e., in small space-time regions around the origin
of your coordinate system, in which external gravitational fields can be
considered as homogeneous (and static). T

he physically interpretible geometrical quantities are tensor (fields),
and the general-relativistic spacetime at the presence of relevant true
gravitational fields due to the presence of bodies (e.g., the Sun in the
solar system) is distinguished from Minkowski space by the non-vanishing
curvature tensor, and this is a property independent of the choice of
reference frames and (local) coordinates, i.e., you can distinguish from
being in an accelerated reference frame in Minkowski space (no
gravitational field present) and being under the influence of a true
gravitational field due to some "heavy bodies" around you, by measuring
whether there are tidal forces, i.e., whether the curvature tensor of
the spacetime vanishes (no gravitational interaction at work, i.e.,
spacetime is described as a Minkowski spacetime) or not (gravitational
interaction with other bodies present, and you have to describe the
spacetime by some other pseudo-Riemannian spacetime manifold, which you
can figure out by solving Einstein's field equations, given the
energy-momentum-stress tensor of the matter causing this gravitational
field, e.g., the Schwarzschild solution for a spherically symmetric mass
distribution).

--
Hendrik van Hees
Goethe University (Institute for Theoretical Physics)
D-60438 Frankfurt am Main
http://itp.uni-frankfurt.de/~hees/

On 12/20/24 12:51 AM, Luigi Fortunati wrote:
> The cables break and the elevator goes into free fall.
> Newton told us that the elevator accelerates and, therefore, there is a
> force that makes it accelerate.
> Then Einstein came along and told us that this is not true

This is one of your problems. Physics is not about "true or false"; we
make MODELS of the world we inhabit -- Newtonian mechanics and GR are
DIFFERENT MODELS. The problem is that you intermix nomenclature
willy-nilly between them.

> and that
> there is no force that accelerates the elevator in free fall.
> But if there is no force that accelerates the elevator, it means that
> the elevator does not accelerate.
> And if it does not accelerate, then it moves with uniform speed.

This is another of your problems. You did not specify the coordinates
relative to which "speed" is measured. Relative to an elevator-fixed
LOCALLY inertial frame, it moves with uniform speed.

> But speed is not absolute: it is relative.

This is another of your problems. Your words are HIGHLY ambiguous.

> And so I ask: is there any reference system with respect to which its
> speed is uniform?

Yes. Any elevator-fixed locally inertial frame.

> This is for Newton's second law: force that accelerates mass.
> Instead, for the first law, Einstein says that a body in the elevator
> in free fall is at rest with respect to the elevator itself.
> So, why does a body placed below the center of gravity of a
> free-falling elevator accelerate downwards, and if it is above the
> center of gravity, it accelerates upwards?

For this last to happen the elevator does not meet the criteria of a
locally inertial frame (see my recent post in this newsgroup). It is of
course due to tidal forces inside the elevator, the acceleration of
which exceeds one's measurement accuracy. Objects separated horizontally
will also accelerate towards each other (due to tidal forces that exceed
measurement accuracy).

I repeat: you need more precision in thought and words.

Tom Roberts

On 12/22/24 2:57 AM, Luigi Fortunati wrote:
> Can we define the interior space of the elevator as "local" or is it
> too big?

This depends on: a) the curvature of spacetime where the elevator is
located, b) the size of the elevator (including duration), and c) one's
measurement accuracy. For an ordinary-sized elevator near earth falling
for ten seconds, and a measurement accuracy of microns and microseconds
(or larger), its interior can be considered a LOCALLY inertial frame.

> If it is shown that there are real forces inside the free-falling
> elevator, can we still consider this reference system inertial?

Depends on the details (and the meanings of words). For internal forces
that are small enough to not significantly distort the steel elevator,
it can be considered LOCALLY inertial, as long as it meets the criteria
above.

> Are tidal forces real?

This depends on the meanings of words, and is therefore ambiguous and
uninteresting to me.

> Do we mean by "freely falling bodies" only those that fall in the very
> weak gravitational field of the Earth or also those that fall in any
> other gravitational field, such as that of Jupiter or a black hole?

"freely falling" means not subject to any external forces. This is
independent of the size of nearby bodies. Note that an object with size
comparable to curvature cannot be considered freely-falling. (Here
gravity is not a force.)

The size of a locally-inertial frame depends on the criteria of my first
paragraph above. The inside of the elevator above but near Jupiter can
be considered a locally-inertial frame. For a one-solar-mass black hole
just outside its horizon, such an elevator is too big. For a
billion-solar-mass black hole just outside its horizon, it can be
considered a LOCALLY inertial frame. (More massive black holes have
smaller spacetime curvatures at their horizon.)

Tom Roberts

Tom Roberts il 24/12/2024 07:32:18 ha scritto:
> On 12/20/24 12:51 AM, Luigi Fortunati wrote:
>> The cables break and the elevator goes into free fall.
>> Newton told us that the elevator accelerates and, therefore, there is a
>> force that makes it accelerate.
>> Then Einstein came along and told us that this is not true
>
> This is one of your problems. Physics is not about "true or false"; we
> make MODELS of the world we inhabit -- Newtonian mechanics and GR are
> DIFFERENT MODELS. The problem is that you intermix nomenclature
> willy-nilly between them.

Physics talks about models? Ok.

Newton's MODEL states that there is a *real* gravitational force
(F=GmM/d^2) that accelerates the mass <m> of the elevator towards the
mass <M> of the Earth (and vice versa), while Einstein's MODEL denies
that such *real* gravitational forces are present in the free-falling
elevator.

So, it is the presence or absence of *real* gravitational forces in the
"local" frame of the elevator that establishes which of the two models
is correct and which is not.

If in the free-falling elevator the *real* gravitational forces are NOT
there, Newton's model is wrong, if they are there, Einstein's model is
wrong.

Since the real gravitational tidal forces in the "local" space of the
elevator are there, Einstein's model is (conceptually) wrong.

Someone will ask: but then there are no elevators at rest on Earth,
effectively devoid of any real gravitational force?

I answer that such an elevator, where real *terrestrial* gravitational
forces are totally absent, exists but it is not the one in free fall.

In a thousand years they will have built an elevator that descends
downwards without limits.

I enter this elevator and press the down button.

The elevator descends downwards at a constant speed without falling.

At first, I do not notice anything but after a certain time I feel
lighter and then, as I descend more and more, my weight decreases until
it disappears completely when the elevator stops at the center of the
Earth.

That is the only point where the elevator has no trace of real
(terrestrial) gravitational forces inside it and, therefore, it is the
only inertial terrestrial reference system!

With respect to this truly inertial reference, all the other
terrestrial elevators in free fall are accelerated.

Luigi Fortunati

A contestation that I expected and that no one has made is this: if
General Relativity is wrong, why are its results more consistent with
observed phenomena than those of Newton's gravitation?

The answer is that the errors of RG are only conceptual, that is, they
only concern the presumed causes of gravity (space-time curvature
instead of forces) and not the excellent innovation that exposes
Newton's error.

For Newton, the mass M acts directly on the mass m (and this is wrong
but I will talk about it in the next specific discussion), while for
Einstein the mass M acts on the space around itself and the space
transmits this action (force) on the mass m (and this is correct).

I deliberately wrote space and not space-time.

Luigi Fortunati

On 12/29/24 2:34 PM, Luigi Fortunati wrote:
> [...] *real* gravitational force

You are basing your entire argument on the meaning of one word. That is
NOT physics; at best it is linguistics.

> So, it is the presence or absence of *real* gravitational forces in
> the "local" frame of the elevator that establishes which of the two
> models is correct and which is not.

Complete nonsense.

First, you give no method to determine whether a given force is "real"
or not. You seem to assume that this distinction is obvious, but it most
definitely is NOT, and depends on the meaning one gives to that word.
The validity of physical models doe NOT depend on the meanings of words
-- the models themselves are mathematical, not linguistic. IOW: the word
"real" does not appear in any physical theory.

Second, the ONLY way to determine whether a given model (theory) is
valid is to perform experiments and make measurements on them, then
compare to the predictions of the model for the same measurements of the
same experiments.

Bottom line: if you perform experiments near the surface of the earth,
with measurement accuracies of microns and microseconds over distances
<~ 100 meters and durations <~ 100 seconds, you'll find that both
Newtonian gravity and General relativity are valid. But for certain
astronomical measurements (such as the advance of mercury's perihelion
with arc-second/century accuracy) only GR is valid.

> A contestation that I expected and that no one has made is this: if
> General Relativity is wrong, why are its results more consistent
> with observed phenomena than those of Newton's gravitation?

This is more nonsense. See above. GR is valid over a LARGER DOMAIN than
Newtonian gravity. As I keep saying, "right and wrong" (also "true and
false") are not applicable to physical theories; theories are valid over
a given domain with a given measurement accuracy, and in this case those
domains are different for NG and GR.

I repeat: you REALLY need to learn what physics actually is -- you have
wildly incorrect notions about it. Until you do, you will keep confusing
and mystifying yourself.

[This is getting overly repetitive and boring. Don't expect
me to continue.]

Tom Roberts