daniel rey m. said:
As you say, conclusions follow observations, which was how they inferred the existence of atoms in those days (B.C.), but contrary to what you say, it was a case in which, surprisingly, observing implied no mathematical reckoning, and they managed to discover the microstructure of matter, because the proof was there for all to see.
Doesn't work. The problem here is that without mathematics, you have no way of knowing how good an observation is. While it's obviously possible to reach some very limited but correct conclusions in this manner, you have no way of ensuring the validity of the result.
Even if you don't like this, I'll just point out that with math you might potentially be able to make the very limited deduction that atoms exist, but there's no way you could discover even part of the periodic table.
daniel rey m. said:
Oh yes it does, because for Bigbangers their "singularity" IS "the Beginning".
This is just false, though. As I said, while there is a singularity in the math, the theory doesn't include that singularity.
One way of understanding it is this: the big bang theory assumes General Relativity. But we are quite sure that General Relativity cannot be valid above the Planck density. So we cannot trust the theory in any regime where it states there is a density near or above the Planck density. And the region around the singularity is one such regime.
So the singularity is excluded from the theory, meaning the big bang theory says nothing whatsoever about any beginning of our universe.
daniel rey m. said:
By "simply stating" that you're decreeing nothingness into existence, which is an illegitimate leap of logic. The magical "manifold", like the "singularity", is merely a creature of the mathematical equations.
No. It's just a statement that space-time is defined on the manifold. There is no concept of "outside" the manifold to talk about. The manifold, by the way, is hardly magical, but a very specific mathematical construct.
daniel rey m. said:
…just an approximation, yes, because the loss of matter in chemical reactions (vs. NUCLEAR ditto) is infinitesimal and can be conveniently ignored, but we live in a strangely dual world. Physics has a set of rules for the macroworld and another one for the nanoworld. First you mention the Law of Conservation of Energy, then the Law of Conservation of Matter, but since matter and energy are the twin manifestations of one same thing and so are interchangeable (cp. E = mc^2) what we have is matter/energy and one law implies the other. If a theory throws one of them out the window then surely it must be a window of the house where the theory dwells temporarily, and from which it can be thrown out by the next lodger, not the window of the Universe.
***
But as I stated, the energy is not a conserved quantity in General Relativity. Instead, General Relativity conserves the stress-energy tensor, which includes energy density, momentum density, pressure, and twisting stresses. Conservation of the stress-energy tensor forces, under specific conditions, the energy component to
not be conserved. General Relativity is still quite well-behaved, and does lead to energy conservation under many situations. But the expanding universe is not one of them.
One way this can be understood is by the fact that under General Relativity, gravitational potential energy is not considered. For example, if I have two masses that begin stationary some distance from one another, then their gravitational attraction will pull them together, giving them more and more kinetic energy. Under the typical formulation of General Relativity, this is just an example of the two masses gaining energy.
Under classical mechanics, we usually consider the kinetic energy of the particles as stemming from gravitational potential energy, so that the overall energy of the system remains the same. If we so choose, we can also define a potential energy in General Relativity. This formulation is known as the Hamiltonian formulation of GR. With this definition, energy is conserved overall, and in fact the energy of a closed universe is always identically zero. However, no matter how you describe the energy, whether using the normal formulation or the Hamiltonian one, the fact still remains that you can have a universe that changes the amount of energy in matter fields quite dramatically with time, and one which changes the amount of matter as well.
Finally, a small nitpick. The equation:
E=mc^2
is not complete. This is only the energy of a non-moving particle. The full equation is:
E = \gamma mc^2
Where \gamma is the usual velocity-dependent factor in General Relativity. This means, for instance, that you can slam together two low-mass particles with very large kinetic energy, and end up with two different high-mass particles with lower kinetic energy.