Lagrangian (11) is that of the 4-dimensional minimal variety in the 4+” dimen- sional space with the given metric tensor das. It is obvious, therefore, that condition (B) is satisfied. If we perform the é integration first in Eq. (8) we obtain where di, 62 are constants and a@=det das with daz being the 4+” dimensiona metric tensor defined through is the conventional Robertson-Walker metric implied by the cosmological princi- ple.* The form [6(¢%)]?ga(t) is implied by the form of Eq. (17)’ which in turn is a direct consequence of the equivalence of space-time and field. Equation (24) clearly shows the collapse rather than the compactification of the extra-dimen- sional coordinates. Mathematically minded readers may feel uneasy about the appearance of the square of delta functions. A rigorous treatment must presumably employ some limiting procedure which we do not digress to do in this paper. We proceed here in a purely formal way treating d(x) as if it were an ordinary function. Miraculously enough, we find that any relevant gauge invar- iant quantity is free of [6(¢)]? or of more complicated functions of 6(¢) or 8’(¢). For example, the gauge non-invariant affine tensor is given by With this preparation we can calculate the scalar curvature which appears in our basic Lagrangian. Leaving the details of the calculation to Appendix A, we immediately write down the final result: Let us now discuss what we find numerically. First of all, we find that there exists no solution if the extra-dimensions are less than 10. There exists a set of solutions to Eqs. (29)’ and (31) but none of them satisfy consistency condition (55) if <9. The minimal internal symmetry group therefore is SO(10). We must of course bear in mind that we have reached this conclusion neglecting the fermion contribution entirely. We now solve the vacuum equations numerically as was done in § 4 for the On the Structure of Space, Time and Field
![is the conventional Robertson-Walker metric implied by the cosmological princi- ple.* The form [6(¢%)]?ga(t) is implied by the form of Eq. (17)’ which in turn is a direct consequence of the equivalence of space-time and field. Equation (24) clearly shows the collapse rather than the compactification of the extra-dimen- sional coordinates. Mathematically minded readers may feel uneasy about the appearance of the square of delta functions. A rigorous treatment must presumably employ some limiting procedure which we do not digress to do in this paper. We proceed here in a purely formal way treating d(x) as if it were an ordinary function. Miraculously enough, we find that any relevant gauge invar- iant quantity is free of [6(¢)]? or of more complicated functions of 6(¢) or 8’(¢). For example, the gauge non-invariant affine tensor is given by](https://figures.academia-assets.com/71743264/figure_001.jpg)
