Abstract
Higher dimensional theories, wherein our four dimensional universe is immersed into a bulk ambient, have received much attention recently, and the directions of investigation had, as far as we can discern, all followed the ordinary Euclidean hypersurface theory’s isometric immersion recipe, with the spacetime metric being induced by an ambient parent. We note, in this paper, that the indefinite signature of the Lorentzian metric perhaps hints at the lesser known equiaffine hypersurface theory as being a possibly more natural, i.e., less customized beyond minimal mathematical formalism, description of our universe’s extrinsic geometry. In this alternative, the ambient is deprived of a metric, and the spacetime metric becomes conformal to the second fundamental form of the ordinary theory, therefore is automatically indefinite for hyperbolic shapes. Herein, we advocate investigations in this direction by identifying some potential physical benefits to enlisting the help of equiaffine differential geometry. In particular, we show that a geometric origin for dark energy can be proposed within this framework.
Highlights
There are a number of different theories that were constructed to describe the extrinsic geometry of a hypersurface1 immersed into a bulk ambient
One is called the equiaffine2 differential geometry, wherein a symmetric bilinear form called equiaffine metric is laid onto the immersed hypersurface, but which is conformal to the second fundamental form of the immersion, in the more familiar ordinary Euclidean terminology
Any smooth change in metric signature will inevitably lead us to a transition boundary, that can be identified with the big bang, where at least one of the eigenvalues of IIab vanishes, so it becomes degenerate, and the machinery of equiaffine geometry breaks down
Summary
There are a number of different theories that were constructed to describe the extrinsic geometry of a hypersurface immersed into a bulk ambient. If nature had intended to grant us the convenience of an ambient metric, it is only doing so half-heartedly It is at the very least prudent to consider the possibility that there is no such intention, and our intrinsic Lorentzian metric is not induced by an ambient metric, but instead bespeaks a different concept in some non-metric extrinsic theory. At the very least prudent to consider the possibility that there is no such intention, and our intrinsic Lorentzian metric is not induced by an ambient metric, but instead bespeaks a different concept in some non-metric extrinsic theory Such theories do exist, and one is called the equiaffine differential geometry, wherein a symmetric bilinear form called equiaffine metric is laid onto the immersed hypersurface, but which is conformal to the second fundamental form of the immersion, in the more familiar ordinary Euclidean terminology.
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