Re: The Elevator in Free Fall

In article <vk0k8g$2p4uk$1@dont-email.me>, Luigi Fortunati wrote:

The cables break and the elevator goes into free fall.

[[...]]

for the first law, Einstein says that a body in the elevator
in free fall is at rest with respect to the elevator itself.

Not quite: a body *sufficiently close to the elevator's center of mass*
is *unaccelerated* with respect to the elevator itself.  "Unaccelerated"
means constant velocity, but that velocity need not be zero.

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?

Assuming this elevator is near the Earth's surface, the answer to
your question is "tidal forces".  In more detail...

[For the following, I'll use the sign convention that the Newtonian
"little g" vector point down, and has *positive* magnitude.]

We observe that the Newtonian "little g vector" (as measured by observers
stationary with respect to (wrt) the Earth's surface; let's call this
g_wrt_Earth) varies with position.  In this case (measuring near to,
and above, the Earth's surface), g_wrt_Earth always points roughly
down, but decreases in magnitude as we go up in altitude away from the
Earth's surface.

If we now go to a free-falling local inertial reference frame (FFLIRF),
say one accelerating down with acceleration g_elevator wrt the Earth's
surface -- we'll measure a little-g vector (as measured wrt the FFLIRF
-- let's call this g_wrt_FFLIRF) of
  g_wrt_FFLIRF(position) = g_wrt_Earth(position) - g_elevator         (1)
where I've explicitly shown which terms are position-dependent.

Since g_elevator is (by the definition of FFLIRF) precisely g_wrt_Earth
at the elevator's center-of-mass position, by equation (1), g_wrt_FFLIRF
must be zero at this position.  But since g_wrt_Earth varies with position,
g_wrt_FFLIRF must also vary with position.  That is, g_wrt_FFLIRF will in
general be nonzero for positions away from the elevator center-of-mass.

In this case (elevator near to, and above, the Earth's surface),
g_wrt_Earth points down everywhere in the elevator, but at the TOP of the
elevator
  |g_wrt_Earth(top of elevator)|
            < |g_wrt_Earth(elevator center of mass)|                  (2)
where I'm using | | to denote the magnitude of a vector.
By equation (1), this means that g_wrt_FFLIRF(top of elevator) points UP.
Similarly, at the BOTTOM of the elevator,
  |g_wrt_Earth(bottom of elevator)|
            > |g_wrt_Earth(elevator center of mass)|                  (3)
so equation (1) tells us that g_wrt_FFLIRF(bottom of elevator) points
DOWN.

This means that test masses placed in different parts of the elevator
have a "tidal" acceleration wrt each other.  Alternatively, if we hold
on to test masses in different parts of the elevator (i.e., we hold them
at fixed positions wrt the elevator, so they are NOT in free-fall), then
we'll feel "tidal" forces acting on the masses.  Tidal forces/accelerations
represent a deviation of our supposed FFLIRF from actually being inertial.


In article <vk8fh7$h93j$1@dont-email.me>, Luigi Fortunati wrote:

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?

Tha answer to all of these questions is the same, namely, "It depends
on your accuracy tolerance for measurements".

That is, it's easy to see that |g_wrt_FFLIRF(top of elevator)| and
|g_wrt_FFLIRF(bottom_of_elevator)| both vary with the size of the
elevator.  In particular, if you make the elevator 10 times smaller,
both of these numbers will also be about 10 times smaller.  So, if you
make the elevator small enough, then both of these numbers will be tiny
enough that we can approximate them as zero.  That is, if the elevator
is small enough, then tidal accelerations/forces are negligible, i.e.,
the elevator is (to within our accuracy tolerance) an inertial reference
frame.

Similarly, the effect of any given nonzero g_wrt_FFLIRF compounds over
time, i.e., if we say that our experiments are only going to last for
some finite duration, then the shorter that time, the less the effect of
any given nonzero g_wrt_FFLIRF.  Thus if our duration is short enough,
then we can neglect the effects of g_wrt_FFLIRF being zero, and the
elevator is (to within our accuracy tolerance) an inertial reference
frame.

More precisely, for any fixed accuracy tolerance, if we make the elevator
small enough and/or make our measurements for a short enough duration,
then the tidal accelerations/forces across the elevator will be small
enough to be negligible.

As Tom Roberts explained earlier in this thread, just how small the
elevator must be and/or just how short our measurement duration must be,
depends on how fast the local little-g field varies with position.  For
example, if instead of our elevator being near to (and having a little-g
vector dominated by) the Earth, our elevator is instead near to the
event horizon of a billion-solar-mass black hole, then little-g varies
much less strongly with position than it does near the Earth.  So, in
this case (near a huge black hole) the elevator can be larger, and/or
we can make measurements for a longer duration, and still stay within
the same fixed accuracy tolerance.

Are tidal forces real?

They can do work, so they must be real.

To see this, fix a stick running from floor to ceiling in the elevator,
and place two beads on it, which can slide up and down with a (small)
bit of friction.  If the friction is small enough (but still nonzero),
then the beads will move under the influence of the tidal forces, and
the friction will cause the beads and the stick to heat up.  This
heating shows that the tidal forces are doing work, and hence that
tidal forces must be real.

[Digression: the same question arose in general relativity,
when well into the 1950s many researchers were unsure whether
or not gravitational waves were real.  The two-beads-on-a-stick
gedanken experiment for gravitational waves was proposed by
Feynman and analyzed in detail by Bondi in 1957, proving that
gravitational waves could do work and must thus be real.]

See
  https://en.wikipedia.org/wiki/Tidal_power
for some real-world examples of work done by tidal forces, with (in some
cases) power output measured in the hundreds of megawatts.  These systems
all ultimately exploit the fact that g_wrt_Earth is NOT constant from one
edge of the tidal power basin to the other edge, i.e., that the (entire)
tidal power basin as NOT a local inertial reference frame.

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