Einstein's elevator

[Luigi Fortunati]

In the lift stopped at the floor, the bodies accelerate both towards=20
the floor and towards the center of the Earth (the two directions=20
coincide).

If the cables break and the elevator goes into free fall, the two=20
gravitational accelerations no longer coincide.

The bodies stop accelerating towards the floor but continue to=20
accelerate towards the center of the Earth.

Doesn't this mean that in the free-falling elevator the force of=20
gravity has not disappeared at all but is well active?

[Richard Livingston]

It all depends on what frame of reference you are talking about. For a
"stationary" frame there is a "force" of gravity causing free objects to
accelerate downwards. Note however this this frame of reference is
not an inertial frame. Per the Equivalence Principle, this frame is
equivalent to one in space that is accelerating "upward" at 1 g.

For a reference frame that is stationary wrt the falling bodies, there
is no force acting at all, everything is in free fall and "weightless".
This IS an inertial frame because in this frame if you release a free
body, it remains stationary wrt this frame. There is no force acting
on it. With respect to this inertial frame it is the earth that is
accelerating upward. (I am ignoring tidal effects, which is a
second order effect.)

In the modern interpretation gravity is not a force, but what we
attribute to the force of gravity is really a non-Euclidean space-
time in the neighborhood of massive objects. The only real force
is the one at your feet that is accelerating you upwards at 1 g.
When you are stationary on the surface of the earth you are not
in an inertial frame, but in one that is accelerating upwards at
1 g. It is the free falling objects that are in an inertial frame and
not accelerating.

Rich L.

[Stefan Ram]

Richard Livingston <richali...@gmail.com> writes:
>In the modern interpretation gravity is not a force, but what we
>attribute to the force of gravity is really a non-Euclidean space-
>time in the neighborhood of massive objects. The only real force
>is the one at your feet that is accelerating you upwards at 1 g.

"Accelerating" means "changing the speed",
and "speed" means "change of position", right?

a = dv/dt, v = dx/dt, a = 1 g ==> dv/dt = 1 g.
v = g t + v0, x = (1/2) g t^2 + v0 t + x0.
Assuming: v0 = 0 and x0 = 0: x = (1/2) g t^2.

I assume that the Australians are my antipodes.

So, I am accelerated since a while with 1 g upwards, as you
say above, (=change of my velocity upwards (=change of my
location upwards)) and the Australians with 1 g downwards.

Shouldn't I then be moving further and further away from
Australians?

[[Mod. note -- In order to add/subtract an acceleration vector "here"
to/from an acceleration vector "there", we need a common inertial reference
frame that contains both "here" and "there". If were in a flat spacetime,
that would be easy, since in a flat spacetime inertial reference frames
are of infinite extent. But we don't live in a flat spacetie, we live
in a curved spacetime, and in a curved spacetime an inertial reference
frame is only an approximation valid in a small region. (The precise
definition of "small" depends on how good of an approximation you want,
i.e., how small you want an acceleration to be in order to call it
"negligable".)

As you've just observed, the opposite sides of the Earth are too far
apart to be contained in a common inertial reference frame (assuming
that we're not willing to treat +/- 1 g accelerations as "negligable").

That is your 1 g "up" acceleration is measured with respect to a
*different* inertial reference frame from the the Australian's "1 g down",
and hence you can't just add/subtract them without taking into account
the non-trivial transformation (induced by spacetime curvature) between
those two different inertial reference frames.
-- jt]]

[Luigi Fortunati]

Richard Livingston marted=EC 19/04/2022 alle ore 15:28:38 ha scritto:
> With respect to this inertial frame it is the earth that is=20
> accelerating upward.=20

This is incomprehensible.

Acceleration occurs in the presence of a force (F=3Dma).

The force existing between the 300 kg lift and the Earth is worth 300=20
kg-weight.

This force justifies the downward acceleration of the elevator but=20
could never justify the acceleration of the entire earth mass upward!

[Tom Roberts]

On 4/21/22 2:24 AM, Luigi Fortunati wrote:
> Richard Livingston marted=EC 19/04/2022 alle ore 15:28:38 ha
> scritto:
>> With respect to this inertial frame it is the earth that is

>> accelerating upward.
>
> This is incomprehensible.

Not really. But one must be thinking in terms of General Relativity
(GR), not Newtonian mechanics (NM).

> Acceleration occurs in the presence of a force (F=ma).

But there quite clearly is a force: for an object sitting at rest on the
surface of the earth, there is an upward force on it, which we call
"weight".

> This force justifies the downward acceleration of the elevator but

> could never justify the acceleration of the entire earth mass
> upward!

You need more precision in your thoughts and words. "Acceleration" by
itself is insufficiently defined -- use either "proper acceleration" or
specify a (locally) inertial frame relative to which it is measured.
"The entire earth mass" is likewise ill defined -- consider just a small
portion of its surface. A small object at rest on earth's surface has a
proper acceleration of 9.8 m/s^2 (directed upward); in GR this is in
response to the (upward) force exerted on the object by the earth's surface.

[In physics, "proper" means "relative to the instantaneously
co-moving inertial frame of the object in question".]

In NM, near the surface of the earth, we generally use coordinates in
which that surface is at rest. This hides the underlying issue -- these
coordinates hide the force that the surface exerts on such objects. NM
then adds a gravitational force to cancel the force the surface exerts,
yielding net zero force -- this is CLEARLY WRONG as we humans can feel
the force from the surface on our bodies, and it is clearly not zero. GR
corrects this conceptual error:

In GR, near the surface of the earth, locally inertial frames are all
accelerating downward at 9.8 m/s^2, so an object at rest on the surface
is accelerating (upward) relative to them -- responding to the force
that the surface exerts on such objects.

Tom Roberts

[Luigi Fortunati]

You say that for Newton it is we who exert a downward force on the
earth's
surface (reacting), while for Einstein it is the earth's surface
exerting
an upward force on us (reacting).

And you say Einstein is right and Newton is wrong.

But action and reaction are INTERCHANGEABLE!

The two opposing forces are both actions and they are both reactions.

And there is nothing INERTIAL in either.

Accelerating force is one and accelerating force is the other.

Neither is privileged.

Just think of two bodies of equal mass: how would you determine who is
acting
and who reacts?

Luigi Fortunati

[[Mod. note -- Assuming a person standing on (at rest with respect to)
the Earth's surface: In Newtonian mechanics
(a) Newtonian gravity exerts a downward force on the person, AND
(b) The person's feet exert a downward force on the Earth's surface, AND
(c) the Earth's surface exerts an upwards reactive force (reacting
against (b)) on the person's feet.
The net vertical force acting on the person (= the sum of (a) and (c))
is zero
[(b) is not included in the sum because it's not acting
on the person, but rather on the Earth's surface]
, and hence the person has zero vertical acceleration with respect to
the Earth's surface.

In general relativity,
(a) isn't there, AND
(c) is still true, AND
(b) is now categorized as a downwards reactive force on the Earth's
surface, reacting against (c).
The net vertical force acting on the person is now just (c), and is
upwards. Thus the person accelerates upwards at 1 g acceleration
relative to an inertial reference frame. But in GR, an inertial
reference frame is *free-falling*, so near the Earth's surface an
inertial reference frame must have a 1 g accelreation downwards
relative to the Earth's surface. Thus the person's acceleration with
respect to the Earth's surface is zero (= same as the Newtonian mechanics
analysis).

It's not that "Einstein is right and Newton is wrong". More accurately,
both descriptions are internally consistent ways of describing physics.
Newtonian mechanics is the slow-motion weak-gravitational-field limit
of general relativity, so if you only look at weak gravitational fields,
and you move much slower than the speed of light, then you'll see only
tiny difference between the two, and it's reasonable to continue using
Newtonian mechanics.

But if you make very precise measurements (atomic clocks & suchlike),
and/or you measure things in strong gravitational fields (neutron stars,
black holes, & suchlike), then these theories are distinguishable, and
you need general relativity to accurately describe observations.
-- jt]]

[Luigi Fortunati]

> [[Mod. note -- Assuming a person standing on (at rest with respect to)
> the Earth's surface: In Newtonian mechanics
> (a) Newtonian gravity exerts a downward force on the person, AND
> (b) The person's feet exert a downward force on the Earth's surface, AND
> (c) the Earth's surface exerts an upwards reactive force (reacting
> against (b)) on the person's feet.

> The net vertical force acting on the person (=3D the sum of (a) and (c))

> is zero
> [(b) is not included in the sum because it's not acting
> on the person, but rather on the Earth's surface]
> , and hence the person has zero vertical acceleration with respect to

> the Earth's surface. =20

>
> In general relativity,
> (a) isn't there, AND
> (c) is still true, AND
> (b) is now categorized as a downwards reactive force on the Earth's
> surface, reacting against (c).
> The net vertical force acting on the person is now just (c), and is
> upwards. Thus the person accelerates upwards at 1 g acceleration
> relative to an inertial reference frame. But in GR, an inertial
> reference frame is *free-falling*, so near the Earth's surface an
> inertial reference frame must have a 1 g accelreation downwards
> relative to the Earth's surface. Thus the person's acceleration with

> respect to the Earth's surface is zero (=3D same as the Newtonian mechanics
> analysis).
> -- jt]]

You and Einstein say that the reference frames in free fall are
inertial.

Ok.

The elevator in free fall (relative to the Earth) is an inertial
reference frame.

And why is the Earth in free fall (relative to the elevator) NOT an
inertial reference frame?

Still, both of them are in free fall!

[Richard Livingston]

On Tuesday, April 26, 2022 at 2:21:54 PM UTC-5, Luigi Fortunati wrote:
...

> The elevator in free fall (relative to the Earth) is an inertial
> reference frame.
>
> And why is the Earth in free fall (relative to the elevator) NOT an
> inertial reference frame?
>
> Still, both of them are in free fall!

You need to be more precise about what frame you are talking about.
The center of gravity of the earth is in free fall (around the sun), but
a reference frame tied to the surface of the earth is not, due to the
distortion of space-time by the mass of the earth. An elevator on
one side of the earth in free fall is an inertial frame, but an elevator
in free fall on the opposite side of the earth is a different inertial
frame. Each of these inertial frames will see the other as accelerating.
That doesn't mean either of these frame are not inertial. The property
of being an inertial frame is a local thing.

The earth is not "in free fall relative to the elevator". Relative to the
elevator the earth is accelerating upwards. It should not be
considered an inertial frame because in that frame it is accelerating.
That is, an object in the surface-of-the-earth frame can only be
stationary in that frame if it has a force accelerating it upwards.

Rich L.

[Luigi Fortunati]

Richard Livingston giovedě 28/04/2022 alle ore 09:05:51 ha scritto:
> You need to be more precise about what frame you are talking about.

> The center of gravity of the earth is in free fall (around the sun)...

The center of gravity of the elevator is also in free fall around the
sun.

And both (Earth and elevator) are in free fall also with respect to
Jupiter,
Mars and all the other planets.

We want to talk only about the Earth and the elevator without third
party
inconveniences which, moreover, act on both and not on just one?

And therefore, in ALL references the free-fall elevator does not move
at random
but accelerates exactly in the direction that goes towards the center
of the Earth.

In ALL references the free-falling Earth does not move haphazardly but
accelerates
exactly in the direction that goes towards the center of the elevator.

They are two opposite free falls where the center of gravity of each
mass goes exactly towards the center of gravity of the other mass.

The surface of the Earth has nothing to do with it just as the surface
of the elevator has nothing to do with it.

The interaction is between two masses (whose centers of gravity tend
to approach each other) and not between two surfaces.

[[Mod. note -- The fundamental difference between the elevator and
the Earth is that the Earth is a self-gravitating system -- different
parts of the Earth have a non-trivial gravitational interaction with
each other. That means that (a) an inertial reference frame (IRF)
on one side of the Earth (right next to the elevator), (b) an IRF at
the center of mass of the Earth, and (c) an IRF on the other side of
the Earth, are three DISTINCT IRFs.

As measured with respect to IRF (a), the free-falling elevator is
unaccelerated (stationary or moving uniformly).

If we were to try to extend the Earth-center-of-mass IRF (b) to cover
the entire Earth and its immediate neighbourhood, we'd find that with
respect to the extended IRF (b), IRF (a) and the free-falling elevator
are both accelerating at 1 g in the (vector) direction from the elevator
towards the center of the Earth, while IRF (c) is accelerating at 1 g
in the (vector) direction from the center of the Earth towards the
elevator.

As Richard Livingston said in a previous article in this thread,

> Each of these inertial frames will see the other as accelerating.
> That doesn't mean either of these frame are not inertial. The property
> of being an inertial frame is a local thing.

So, one reasonable answer to the question you asked in a previous posting
in this thread,

> And why is the Earth in free fall (relative to the elevator) NOT an
> inertial reference frame?

is that the center of mass of the Earth (and its corresponding IRF (b))
*is* in free-fall with respect to the elevator. But no part of the Earth's
surface is in free-fall (it's all supported in a non-free-fall state by
the solid body of the Earth).
-- jt]]

[Luigi Fortunati]

Luigi Fortunati giovedì 28/04/2022 alle ore 19:59:24 ha scritto:
> [[Mod. note -- The fundamental difference between the elevator and
> the Earth is that the Earth is a self-gravitating system -- different
> parts of the Earth have a non-trivial gravitational interaction with
> each other. That means that (a) an inertial reference frame (IRF)
> on one side of the Earth (right next to the elevator), (b) an IRF at
> the center of mass of the Earth, and (c) an IRF on the other side of
> the Earth, are three DISTINCT IRFs.

If so, then there are not only 3 inertial reference frame but there are
infinite of them and they are all directed radially towards the center
of the Earth, right?

[Luigi Fortunati]

Luigi Fortunati giovedì 28/04/2022 alle ore 19:59:24 ha scritto:

> [[Mod. note -- The fundamental difference between the elevator and
> the Earth is that the Earth is a self-gravitating system -- different
> parts of the Earth have a non-trivial gravitational interaction with
> each other. That means that (a) an inertial reference frame (IRF)
> on one side of the Earth (right next to the elevator), (b) an IRF at
> the center of mass of the Earth, and (c) an IRF on the other side of
> the Earth, are three DISTINCT IRFs.

If each free-fall elevator becomes a distinct inertial reference frame
from all other free-fall elevators, then there are not only 2 or 3
distinct inertial reference frame but there are infinite, one for each
possible elevator.

And all of these distinct inertial reference frame converge towards the
center of the Earth.

> As Richard Livingston said in a previous article in this thread,
>> Each of these inertial frames will see the other as accelerating.
>> That doesn't mean either of these frame are not inertial. The property
>> of being an inertial frame is a local thing.
>
> So, one reasonable answer to the question you asked in a previous posting
> in this thread,
>> And why is the Earth in free fall (relative to the elevator) NOT an
>> inertial reference frame?
> is that the center of mass of the Earth (and its corresponding IRF (b))
> *is* in free-fall with respect to the elevator. But no part of the Earth's
> surface is in free-fall (it's all supported in a non-free-fall state by
> the solid body of the Earth). -- jt]]

How is it possible?

Whoever is in the center of the Earth is stationary at the exact point
of maximum inertia, where there is no force (Newton) and where there is
no "curvature" (Einstein).

Whoever stands in the center of the Earth (where all the free-falling
elevators converge) sees the elevators accelerating towards him.

How does he (totally inert) accelerate to all the elevators?

And if he actually accelerated toward the elevator coming down from the
north pole, how would he accelerate toward the elevator coming from the
south pole as well?

[Richard Livingston]

[Moderator's note: Too much quoted text deleted. -P.H.]

I'm afraid you are missing the concept that in a gravitational field the
concept of an inertial reference frame is very localized. Yes, there
ARE an infinite number of possible reference frames. And for each one
if you get very far from the origin of that frame (i.e. the location
where a free object floats without acceleration) then you are no longer
in an inertial frame FROM THE POINT OF VIEW OF THAT FRAME. That is,
while an object inside your freely falling elevator will not accelerate
when released, if you place that object a short distance outside your
elevator it will begin to accelerate when released.

And from the point of view of each of these localized inertial frames,
objects falling freely in other frames are accelerating, even though in
their local inertial frames those objects are "weightless".

Rich L.

[Luigi Fortunati]

I asked the moderator to cancel my last two posts because I wanted to
improve them but something went wrong and they were published anyway: I
apologize to everyone.

[Moderator's note: Posts are distributed among the active moderators.
So such a request sent as if it were a post will probably not go to the
same moderator. Even if it did, he could really "cancel" it only if he
hadn't already posted it, as many NNTP servers no longer honor requests
to cancel posts. Note that there is another address which reaches all
active moderators, which should be used for such requests. See
http://www.astro.multivax.de:8000/spr/spr.html -P.H.]

After some thought, my final answer (which replaces the previous two)
is the following.

> [[Mod. note -- The fundamental difference between the elevator and
> the Earth is that the Earth is a self-gravitating system -- different
> parts of the Earth have a non-trivial gravitational interaction with
> each other. That means that (a) an inertial reference frame (IRF)
> on one side of the Earth (right next to the elevator), (b) an IRF at
> the center of mass of the Earth, and (c) an IRF on the other side of
> the Earth, are three DISTINCT IRFs.
>
> As measured with respect to IRF (a), the free-falling elevator is
> unaccelerated (stationary or moving uniformly).

> ...

> So, one reasonable answer to the question you asked in a previous posting
> in this thread,
>> And why is the Earth in free fall (relative to the elevator) NOT an
>> inertial reference frame?
> is that the center of mass of the Earth (and its corresponding IRF (b))
> *is* in free-fall with respect to the elevator. But no part of the Earth's
> surface is in free-fall (it's all supported in a non-free-fall state by
> the solid body of the Earth).
> -- jt]]

Newton says that the reference of the center of the Earth is inertial
and those of the 2 free-falling elevators from the north and south
poles are accelerated.

Instead, Einstein argues that all three motions are inertial.

If one of the two is right, the other is wrong: it is obvious.

How can we determine who is right and who is wrong?

In my opinion, a good way to judge is the following.

Any measurement of their reciprocal speeds (that of one elevator
relative to the other and of each of the 2 elevators relative to the
center of the Earth) guarantees us that ALL their reciprocal motions
are accelerated and that there is no mutual velocity that is uniform.

This mutual acceleration is justified if the motion of the elevators is
accelerated (as Newton argues) but it is not at all justified if the
motion of the elevators and the center of the Earth are all inertial
(as Einstein argues).

In fact, if all reciprocal motions were truly inertial, where would the
mutual acceleration we measure come from?

[Tom Roberts]

On 5/2/22 3:37 AM, Luigi Fortunati wrote:
> Newton says that the reference of the center of the Earth is inertial
> and those of the 2 free-falling elevators from the north and south
> poles are accelerated.
>
> Instead, Einstein argues that all three motions are inertial.

No. GR says that the three are LOCALLY inertial. You cannot omit
"locally", and that is the crux of your confusion.

> If one of the two is right, the other is wrong: it is obvious.

Not true, because they are really saying different things, in different
contexts. Both are true within their respective contexts. But your
imprecise and ambiguous language hides that.

> How can we determine who is right and who is wrong?

By making precise, unambiguous statements. Binary logic applied to
ambiguous statements is useless, as is logic applied to statements
belonging to different contexts.

Statements containing ambiguous words like "acceleration" can be
ambiguous: neither true nor false. Correct statements must avoid all
such words, and be precise enough to be adjudged true.

"Proper acceleration" and "coordinate acceleration" are precise enough
here, while unqualified "acceleration" is not. Newtonian mechanics does
not have the concept of proper acceleration; GR introduced it to avoid
the ambiguity that is confusing you.

> In my opinion, a good way to judge is the following. [... useless
> method using speeds]

The correct way is to distinguish proper acceleration from coordinate
acceleration (which you failed to do).

The proper acceleration of a (pointlike) object is its acceleration
relative to its instantaneously co-moving locally inertial frame; it is
invariant (independent of coordinates -- all observers agree on its
value), while coordinate accelerations are not invariant. This is true
independent of whether the object is in freefall (zero proper
acceleration), or not (nonzero proper acceleration).

In the first paragraph quoted above, objects at rest in the center of
the earth and at rest in each elevator have zero proper accelerations.
When one uses the coordinates of their locally-inertial frame, each has
zero coordinate acceleration. When one uses the coordinates of one of
those frames to describe an object at rest in a different one, the
coordinate acceleration is nonzero.

(In general, the coordinates of a locally inertial frame
might not be valid far away -- they are LOCAL.)

Bottom line: complicated and subtle subjects like modern physics require
precision in thought and word. You need to make more precise statements
that are not ambiguous.

Tom Roberts

[Luigi Fortunati]

Tom Roberts lunedì 02/05/2022 alle ore 17:24:10 ha scritto:
> ....
> These can be statements containing ambiguous words such as "acceleration".

> ambiguous: neither true nor false.

The accelerations of the free-falling elevator towards the center of
the
Earth and of the center of the Earth towards the free-falling elevator
are not ambiguous, because they are observable and measurable in all
references.

>....
> In the first paragraph cited above, objects at rest in the center of
> the ground and at rest in each lift have their own zero accelerations.

The acceleration of the object in the elevator is really (obviously)
null as all the accelerations with respect to themselves are null but
what does it have to do with gravity?

The acceleration of gravity of the object in the elevator is directed
towards the center of the Earth and not towards the elevator!

[Luigi Fortunati]

Richard Livingston luned=EC 02/05/2022 alle ore 10:36:50 ha scritto:
> I'm afraid you are missing the concept that in a gravitational field the

> concept of an inertial reference frame is very localized.

Gravity is not very localized because it does not go from man to
elevator.

Gravity goes from the man-elevator to the center of the earth!

[Moderator's note: Even if gravity is not localized, the concept of an
inertial frame can be. -P.H.]

[Luigi Fortunati]

Luigi Fortunati martedì 03/05/2022 alle ore 10:17:57 ha scritto:
> [Moderator's note: Even if gravity is not localized, the concept of an inertial frame can be. -P.H.]

Ok, so let's ask ourselves who is at rest and who is not in the "local"
reference.

Let us ask ourselves: if the man in the elevator stopped at the floor
drops the ball he is holding, is it the ball that falls towards the
floor (Newton) or is it the floor that falls towards the ball
(Einstein)?

It is entirely reasonable to imagine that there may be a force capable
of accelerating the ball downwards but it takes a lot of faith to be
able to accept that there may be a force capable of accelerating the
entire Earth towards the ball.

[Richard Livingston]

On Thursday, May 5, 2022 at 2:26:53 AM UTC-5, Luigi Fortunati wrote:

I think this will be my last post on this issue:

-"Who is at rest?" is the wrong question. The relevant question is "who
is in an inertial frame?" If you are in an inertial frame you can let go
of an object and it will float where you left it. If you let go of an
object and it accelerates away, then you are not in an inertial frame.

-In the paradigm of General Relativity there are no "forces" due to
gravity, only curvature of space-time. The result is that a reference
frame that is at a fixed position away from the center of mass of
a large massive object is no longer an inertial frame. That is, if you
release an object that is initially stationary in that frame it will start
to accelerate away from you. In this paradigm the released object
has no forces on it, it is merely following its normal world line through
space-time. You, on the other hand, feel a force on your feet that is
accelerating you upwards relative to the inertial frame that is
accelerating downwards wrt you.

-You can choose to ignore this point of view and say the released
object is experiencing a force downwards, but if that is the case
why does someone in free fall feel "weightless"? If you were inside
an elevator far from any mass you would feel weightless. If you
were in an elevator in free fall near a large mass you would again
feel weightless. In one case you would say there is no force, in the
other you would say there is. What difference does it make?

-If you don't want to think in terms of the curvature of space-time and
the effect that has on an object's world line, you will not progress
much in understanding General Relativity.

Rich L.

[Luigi Fortunati]

Richard Livingston gioved=EC 05/05/2022 alle ore 22:05:20 ha scritto:
> -"Who is at rest?" is the wrong question. The relevant question is "wh=
o
> is in an inertial frame?" If you are in an inertial frame you can let =

go
> of an object and it will float where you left it. If you let go of an
> object and it accelerates away, then you are not in an inertial frame.

And, therefore, the free-fall elevator is not an initial frame because
if you let go of an object, it accelerates toward the floor if the
gravity is that of a black hole and if the object is below the center
of the elevator where gravity acting on the object is greater than
gravity acting on the entire elevator!

[[Mod. note -- No, the correct conclusion is to observe that inertial
frames are always of limited size, with the actual size limit depending
on your tolerance (threshold) for how small an acceleration difference
(a.k.a tidal acceleration) is "negligable".

If your freely-falling elevator is big enough that you notice the
acceleration differences between different free-falling objects in
the elevator (all of which were initially at rest with respect to
the elevator), and/or between these and the elevator itself, that's
a statement that your elevator is too big for any one inertial frame
to cover the entire elevator. If you want to apply the concept of
"inertial frame" in the elevator, then you need a smaller elevator
and/or a looser tolerance for acceleration differences.
-- jt]]