_{1}

^{*}

Using Dvoretzky’s theorem in conjunction with Bohm’s picture of a quantum particle inside a guiding quantum wave akin to De Broglie-Bohm pilot wave we derive Einstein’s famous formula
*E = mc*
^{2} as the sum of two parts
*E(O) = mc*
^{2}/22 of the quantum particle and
*E(D) = m c*
^{2} (21/22) of the quantum wave where m is the mass, c is the speed of light and E is the energy. In addition we look at the problem of black holes information in the presence of extra dimensions where it seems initially that extra dimensions would logically lead to a hyper-surface for a black hole and consequently a reduction of the corresponding information density due to the dilution effect of these additional dimensions. The present paper argues that the counterintuitive opposite of the above is what should be expected. Again this surprising result is a consequence of the same well known theorem on measure concentration due to I. Dvoretzky. We conclude that there are only two real applications of the theorem and we expect that many more applications in physics and cosmology will be found in due course.

The main purpose of the present paper is to show that Dvoretzky’s theorem is not only pure mathematics but also has profound applications in physics and cosmology. For instance it would be intuitively reasonable to suppose that in the case of black holes [

In the present short work however we show that due to the above mentioned well known theorem of Dvoretzky on measure concentration the above conclusion is fallacious [

We will start in the present work with a derivation of Dvoretzky’s theorem and will follow two converging roads to show the counterintuitive results of measure concentration due to very high dimensionality. The theorem is derived first in a conventional mathematical fashion [

The moral which we can learn, in fact relearn from this theorem is a well known wisdom from many counterintuitive results of geometry in higher dimensions, namely that we should in general never generalize an obvious conclusion from a low dimensional space to a higher one [

Working in the usual way to find the volume of this n dimensional ball we arrive via gamma function and Stirling formula to [

That means for V = 1 the radius is a very large one equal approximately to

Now we proceed to the distribution of the mass, i.e. how the “volume” of this ball is distributed. To do that we estimate first the

then the volume of the slice

Using the Stirling formula again we find that the slice has the volume

Since r is approximately [

one finds [

That means we obtain “mass” distribution that is almost Gaussian, with variance which surprisingly does not depend upon n:

That way we conclude the following remarkable result, namely that almost all the “volume” stays within a flab of fixed width and our result announced in the introduction of the present paper follows that about 96% of the “mass”, i.e. the volume lies in the slab [

That means 96% is concentrated near the subspace

The second derivation of our theorem is more or less based on physics.

In E-infinity theory the pre-quantum particle as well as the pre-quantum wave follows from the fundamental equation fixing the invariants of the noncommutative E-infinity spacetime [

where

which models the quantum wave. Transferring this result to Kaluza-Klein “quantum” spacetime we note that the “inner” volume must be correlated, i.e. intersectional which is appropriate for a volume and leads to [

where D(Kaluza-Klein) = 5. The outer surface, i.e. the quantum wave on the other hand is additive and non- correlated so that the union operation is what leads to the volume [

In the above we tacitly assumed the validity of the Bohm picture of a quantum particle surrounded by a guiding quantum wave. A typical volume representative for both would be clearly the arithmetic mean

In turn looking at the above as energy density we see that

for m = c = 1 while

In other words E(O) is our familiar ordinary measurable energy density of the quantum particle [

while E(D) is our dark energy density of the quantum wave which we cannot measure [

Adding both together we obtain the celebrated result [

Now remembering that energy and information are directly related via entropy, the preceding result is confirmation of what we obtained earlier on using Dvoretzky’s theorem, namely that 96% of the information is drawn to the surface higher dimensionality rather than “diluted” by it. Needless to say, the preceding results remain valid for a rotating Kerr black hole [

The present work is in the first place a vivid demonstration of the power of pure mathematics, in the present case the power of Dvoretzky’s theorem of measure concentration, in solving problems in physics and cosmology. The mathematical literature abounds with examples demonstrating the failure of our low dimensional intuition to extrapolate from low dimensional results to higher dimensional ones. [

tein’s E = mc^{2} and explain that way the measure ordinary energy density of the cosmos ^{2}

could account for a quantum relativity formula

Mohamed S. ElNaschie, (2015) Application of Dvoretzky’s Theorem of Measure Concentration in Physics and Cosmology. Open Journal of Microphysics,05,11-15. doi: 10.4236/ojm.2015.52002