1 - Volumes with/without mass
 

Elasticity of spacetime:
This property of elasticity of spacetime is defined by Einstein Field Equations (EFE).

M and V variables:
In closed volumes, we can switch from the mass variable to the volume variable and conversely applying a simple conversion parameter, the density of nuclear matter: 5.92E-17 kg/m3 for the proton, from CODATA 2010. However, a more precise formulation is given in the Mathematical section of the main Webpage. This allow us to replace the mass variable "m" of closed volumes with its expression m = f(x,y,z,t). Strictly speaking, it is a 3D expression, but it is better to remain in 4D because SR, GR and many constants such as c or G use 4D.

Examples of closed volumes:
Sure: Leptons. Probably: some fermions, bosons, black holes and other similar objects. Some objects such as atoms with halo are "apparent volumes", i.e. combinations of closed and open volumes. See the explanation of apparent volumes in the main Webpage.

Spacetime curvature and mass:
Contrary to what we think, it is not the mass that produces the curvature of spacetime but the reverse. See our Webpage "Energy-Momentum Tensor" in the Mathematical section of the main Webpage.

About the principle:
The principle discussed here is nothing but an application of the Bulk Modulus Theory from the 1850's Fluids Mechanics, which connects curvature to pressure: ΔP = -KB ΔV/V. See our Webpages "Newton Law" and "Energy-Momentum Tensor" in the Mathematical section of the main Webpage.

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2 - Atoms
 

Binding energy:
The given example has been simplified because, strictly speaking, we should also take into account the equivalent mass of various binding energies (m = E/c²). Solving this problem requires to understand the equivalence E=mc² and the wave-particle duality. Please see our Webpage E=mc² in the Applications section of the main Webpage and our website www.wave-particle-duality.com.

Schrödinger Probability
As we know, the probability density of presence of an electron at one location and at a given time is calculated from the normalized square of Ψ(r,t) from the Schrödinger Equation. As a result, orbitals look more like a cloud of probability than like a perfect line, as shown on the figure of the atom drawn in the main Webpage. This figure is only a pedagogical drawing that must be interpreted with great care.

Orbitals:
Orbitals are not elliptic, as shown on this figure. Their form depends of their level (1s, 2s...).

Special atoms:
Some atoms, such as super-fluids, have particular characteristics not mentioned here.

Exceptions:
We also have exceptions in some nuclei. For example, closed and open volumes may be mixed in the few nuclei having a halo. However, all these exceptions can be explained and do not question the theory proposed here.

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3 - Apparent volumes
 

We can continue to use apparent volumes in traditional physics. The separation closed/open volumes, as described here, is only necessary to explain the curvature of spacetime and the origin of mass and gravitation, and perhaps in astrophysics to solve some enigmas such that of black holes or black matter.

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4 - What is Gravitation?
 

This figure is not exact because an amount of spacetime always exists between the two (or more) closed volumes. To understand what happens, let's consider the following demonstration.

1/ If the distance between the two spheres were null as shown in the figure, the internal spacetime wouldn't exist. So, the pressure would come exclusively from external spacetime. This is shown by the arrows.

2/ Now, let's increase the space between the closed volumes from 0 to "d". A small pressure on the internal side of each closed volume will appear. This pressure will increase from 0 to "P".

So, there is a relation between "d" and "P". Whatever the distance "d", this distance will be always smaller than the infinite. Since the external side of each closed volume are oriented toward the infinite, the external pressure on each closed volume will be always greater than the internal pressure.

Note: This demonstration also applies to the following section "Split Principle".

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5 - Split Principle
 

Grossmann and Einstein developed the EFE from the constraint (Hooke-Riemann) Tensor of the Fluid Mechanics, assuming that spacetime was a fluid. The above "Split Principle" also comes from the Fluid Mechanics. Therefore, there is no objection to use the Split Principle for demonstrations. Remember also that in Fluid Mechanics, the fluid exerts a pressure force, not an attractive force, on the surface of the object (the trace of the tensor is an isostatic pressure).

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6 - Earth-Moon Spacetime Curvature
 

Here, we don't take into account the curvature of spacetime made by the sun and other planets or galaxies. So, contrary to the drawing in the main Webpage, we are not in a pure flat spacetime between the Earth and the Moon. Moreover, this figure is a kind of "snapshot" because the Moon would crash on the Earth if gravity were the only force in action. The centripetal force due to the movement of the Moon around the Earth is not considered here because it concerns Newtonian physics.

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7 - Mathematics
 

All the Webpages about mathematics concern a static object with spherical symmetry in spherical coordinates. On the other hand, the EFE development, in particular with approximation in a weak field, has not been described here because these calculations already exist in many books related to GR. Anyway, result is the same. Instead to have "- -" signs, we have "+ +" signs, as explained in the "EFE and the energy-momentum tensor" webpage.

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