A new equivalence principle

Einstein noticed that the gravity force a person is subject to on
Earth was equivalent to the one an astronaut is suject to if his
module accelerates at 9.81 m/s², the acceleration of gravity on Earth.
I noticed another principle of equivalence : when we film an event on
the Moon (a body's fall or a jeep's movement) then the film is shown
in speeding-up (2.5 times the initial speed), we have the feeling that
the event is happening on Earth. But the gravity on Earth is 6 times
stronger than on the Moon. Let y be the acceleration of gravity on
Earth, taken as unit the acceleration on another given planet or
satellit. Let x be the
acceleration we must give to the film turned on this planet, so that
the sequence seems to happen on Earth. We have y = x².
Knowing that d = 1/2 g t²
in order to d is equal whereas g is 6 times weaker, t² must be 6 times
bigger, so the time must be V6 = 2.45 times longer.
It's called a transformation by adimensionalisation.

We find it again when we reduce the space scale. Here is a board
representing the distance oh an object in fall in the empty after 1,
2, 3, 4 seconds on the Moon, on the Earth and on a terrestrial
landscape reduced in 1/6 of its original size, so that trees would be
as small as bedding plants and houses would be as models.

Second Moon Earth Earth 1/6

0 0 0 0
1 0,809 4,905 0,809
2 3,238 19,62 3,238
3 7,286 44,145 7,286
4 12,952 78,48 12,952

So we have 2 ways to reconstitute Moon's gravity on Earth :
1) making so that the observer's time spend 6 times faster than the
time around him, so that he has the feeling the world around him be 6
times slower
2) increasing by 6 times the observer's size, so that he has the
feeling the world around him is 6 times smaller

A 6 times weaker gravity is equivalent to a slower world or to a world
at a bigger scale.

Thus if Earth was 6 times smaller, it would have a diameter of 2126
km. It's the diameter the Moon would have if it had the same density
as the Earth. Is it a coincidence ?

It introduces the idea that gravity has a fractal structure, because
it's invariant according to the space scale.

So a 1 m high hill of dust filmed at a distance of 1 m on the Moon
that would subside by its own weight would give the same scene in the
same time than a 6 m high hill of dust on Earth filmed at a distance
of 6 m (without atmosphere)

The gravity g of a planet or satellit seems to be proportional to its
diameter d when the density is constant. So all celestial bodies whose
average density is the same as the one of Earth, we have g = d /
1.3*10^6 with d in m and g in m/s².

Here is the mathematical demonstration :
The gravity on the ground is proportional to the diameter (or to the
radius)
gravity on the ground :
g=m*G/R^2
mass: m=4/3pi*R^3*density
g=4/3pi*R^3*dens*G/R^2=4/3pi*density*G*R
so
g=1.54E-6*R ou g=d*7.7E-7 or g=d/1.3E6 (with R in m and g in m/s²)

We could in this way express this equivalence principle :
if a planet A has an x times stronger gravity than a planet B, the
fall on A is the same in all steps as the fall on B on an x times
shorter distance.
So it would be impossible to distinguish the fall of an object from
Big Ben's summit on Earth without atmosphere from the fall of another
object from the summit of a 6 times smaller tower on the Moon.

Here is the demonstration : let f be the function that gives the
distance d covered by an object in fall in relation to the time t :
d=f(x)=1/2*gt²
Its derivative is the function that gives the speed v of the object in
relation to the time t :
v=f'(t)=gt
The derivative of its derivative is the fonction that gives the
acceleration a of the object in relation to the time t :
a=f"(t)=g
If we divide g by 6, we alos divide d, v and a by 6.

If we want to reconstitute a 6 times smaller river with its
waterfall's movement, we must put it on the Moon (with an atmosphere),
it would have the same behaviour as its terrestrial original in real
time.

Let be several planets of different sizes and of same density. If we
dig a vertical tunnel through each planet linking some point or other
of its ground and its center, and we drop on each planet an object in
this point, each object in fall will spend the same time to arrive at
the center of each planet. So if we don't know planets' diameters, we
can't know which planet has the strongest gravity.

Jean Peltier