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.

Close the window

 
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.

Close the window

 
3 - Apparent volumes
 

In reality three categories exist:

1/ Standard apparent volumes. These volumes are combinations of volumes with mass (closed volumes) and massless volumes (open volumes). In atoms for example, the nucleus and electrons curve spacetime and have mass, whereas the orbitals are transparent regarding spacetime and are massless.

2/ Hermetic apparent volumes. These volumes are also combinations of closed and open volumes but their global volume is hermetic regarding spacetime. For example, a nucleus is made of nucleons (closed volumes), separated by empty space (open volume). The behaviour of this combination of volumes is that of a closed volume regarding spacetime. Consequently, the whole volume of the nucleus (closed volumes of nucleons + enclosed open volume) deforms spacetime and gets mass since spacetime curvature = mass (Einstein, GR). Note: Open volumes enclosed inside the nucleus may have energy (gluons). However, this does not conflict with Quantum chromodynamic (QCD).

3/ Special apparent volumes. These volumes are "special" because we don’t know their behaviour regarding spacetime. This is the case with 6He, 8He, 14Be... For example, 11Li has a core with 3p6n and a halo of 2n. Since we don’t know exactly the structure of such nuclei or the penetration of spacetime inside them, it is not possible to classify these volumes into a particular category.

To summarize, we must always bear in mind that the word “volume” without any qualification is meaningless. It is important to understand the difference between:

• Volumes (undefined),
• Closed Volumes (with mass),
• Open Volumes (massless),
• Standard Apparent Volumes,
• Hermetic Apparent Volumes,
• Special Apparent Volumes.

Since these volumes have different behaviours regarding the curvature of spacetime, i.e. mass, they must be differentiated.

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.

Close the window

 
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. Here we say "infinite" but in reality, the universe is probably finite. It means that if the distance between two object exceeds the radius of the universe, we could have a repulsive force between the two objects. In this case, the pressure force comes from the internal spacetime between the two objects. As a result, we can observe an expansion of the universe greater than that expected.

Close the window

 
5 - Earth-Moon Spacetime Curvature
 

Here, we do not 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.

Close the window

 
6 - 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.

Close the window