2.1 A hint to gravity

Let us imagine we have a sphere of radius R filled with energy E. The sphere initially would have an energy density of:

 (2.0)

then we start compressing the sphere in all directions making the radius R of the sphere smaller, and hence the energy density greater. Now let us assume that there is a maximum energy density that spacetime can have, and let us assume that we reached that density by compressing our sphere to its limits. What we would have is simply a very dense sphere of energy as shown in Figure 2.1, that would be impossible to shrink further. But I am not done with assumptions, and I will assume that I have a much greater force, even greater than the force that spacetime has to keep it from shrinking. Well, if I compress the sphere even further, and spacetime cannot stand more energy than that, the only thing that spacetime can do is to shrink with the energy it contains, hence producing something as shown in Figure 2.2. This figure is the same as Figure 1.5e which was derived from General Relativity as discussed in paragraph 1.3.2.
If this was the case then we would be able to measure the stiffness of space time; we know the force we applied, and we know the displacement we have shrank spacetime. Hence:

 (2.1)

Hugh D. Young in his book University Physics states "all nuclei have approximately the same density". The idea that comes to mind now is; what if what we have described so far is the nucleus of an atom? What if there was a fight between the nuclear force that compress the nucleus together and spacetime that limiting its density prevents it from collapsing into a dimensionless point? Obviously one nucleus on its own would not have enough force to distort spacetime considerably, but many atoms put together as in large masses might just have that force. In this case we would still be able to measure the stiffness of spacetime as we know the force of the nucleus and the displacement. The force is simply the sum of all the forces of all the nuclei that make up the mass, the displacement would be the gravitational redshift (Dl/l) caused by the mass/energy itself, something like:

(2.2)

where N is the number of atoms present in the mass.
Attention must be paid to the fact that the density we would measure is higher than the maximum density that spacetime can hold. That is because the unit we use to measure the density is outside the sphere, and hence not compressed making the volume appear smaller when in fact is the same as the sphere before compression.

Fig 2.1 - Sphere of energy compressed to the limit.

Fig 2.2 - Same as Figure 1.5e, but this time it shows a sphere full of energy compressed with a force greater than the stiffness of spacetime. The volume of the two sphere is the same, and so is the density.

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