Our current understanding of the deeper aspects of our universe in terms of the fundamental forces and fundamental matter was described in Chapter 2. As illustrated in Figs. 2.3 and 2.6, the smallest bits of matter that have been established experimentally are the quarks , leptons , and force particles , while their interactions are governed by the strong, weak, electromagnetic, and gravitational forces.
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- 1.
The reason for the weakness of gravity inside the atom can be understood by estimating the gravitational force between two massive bodies, by using Newton’s formula \(F = -G\frac{m_{1}m_{2}}{r^{2}}.\) Here G = 6.67259 × 10−11m3/kg1/s2 is Newton’s gravitational constant, m1, m2 are the masses of the two bodies, and r is the distance between them. For two protons of tiny masses m proton = 1.6726231 × 10−27 kg within the nucleus, at a distance r nucleus = 10−15 m, the magnitude of the gravitational force is \(\left\vert F_{{\rm nucleus}}\right\vert \simeq 1.9 \times 10^{-34}\) N. For two up-quarks of mass \(m_{u} \simeq 0.002\,{\rm GeV}\) at 1000th the size of the proton, this becomes roughly \(\left\vert F_{{\rm inside\,proton}}\right\vert \simeq 8 \times 10^{-34}\,{\rm N}\). Compared to the electromagnetic force on the same quarks this is weaker by a factor of about 10−41 as indicated in Fig. 9.1, and therefore there is no chance of observing the effects of gravity within the atom or the nucleus at the current level of experimentation with small particles. However, at much smaller distances, such as at the Planck scale \(r_{{\rm Plank}} \simeq 1.61605 \times 10^{-35}\,{\rm m}\), the magnitude of the gravitational force between two quarks grows dramatically by a factor of 1040 to \(\left\vert F_{{\rm Planck}} \right\vert \simeq 10^{-5}\,{\rm N}\). At that point Newton’s formula is no longer the correct description and should be replaced by general relativity with all of its consequences, so the naive estimate is not accurate. In any case, the main point is that at the Planck length, which is probed with energies of about 1019 GeV, the gravitational force becomes as important as the other forces as shown in Fig. 9.2.
- 2.
When gravity is also included, the global SO(4, 2) symmetry is promoted to a local gauge symmetry in “tangent space”. This is part of the 2T-physics approach to General Relativity. I should emphasize that 2T-gravity, or its 1T shadow, has no relation to what is known in the literature as the problematic “conformal gravity”.
- 3.
In flat space the Sp(2, R) constraints discussed in Fig. 7.6 were
$$X \cdot X = 0,\,X \cdot P = 0,\,P \cdot P = 0$$In curved space, in the presence of gravity, these constraints get modified to
$$W\left(X\right) =0,\;V^{M}\left(X\right) P_{M}=0,\;G^{\rm MN}\left(X\right) P_{M}P_{N}=0.$$Here V M(X) is given by \(V^M (X) = G^{{\rm{MN}}} (X)\frac{{\partial W(X)}}{{\partial X^N }}\), while the background fields W(X) and the metric G MN(X) are required to satisfy certain geometric properties called “homothety conditions,” so that the Lie algebra for the Sp(2, R) gauge symmetry is satisfied.
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Bars, I., Terning, J. (2010). Fundamental Universe as a Shadow from 4 + 2 Dimensions. In: Nekoogar, F. (eds) Extra Dimensions in Space and Time. Multiversal Journeys. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77638-5_9
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