Can Dimensional Anisotropy Satisfy Mach's Principle? A Topological Approach to Variable Dimensions of Space using the Borsuk-Ulam Theorem

Abstract

In general relativity, Einstein's equations relate the geometry of space-time to the distribution of matter. Nevertheless, the equations are in contradiction with quantum mechanics and even possibly our experience of physical reality. We propose a thought experiment to investigate a compact wave function (WF) insulated by an information-blocking horizon. The WF can produce entanglement independent of distance, but interaction with the horizon evolves the quantum state (frequency) and the topology (curvature) of the horizon in an orthogonal relationship. Their mutual evolution satisfies the Borsuk-Ulam Theorem and the Page and Wootters mechanism of static time. Therefore, the field curvature measures the particle's evolution as time and fine-tunes the cosmos' parameters. The interaction of the field and the compact WF give rise to poles with dimensionality transformations, and it formulates global self-regulation. Because field strength generates pressure, culminating in two-dimensional black hole horizons (infinite time), whereas vacuum gives rise to four-dimensional cosmic voids (time zero). The four-dimensional cosmic voids can produce accelerating expansion without dark energy on the one hand, and pressure gives the impression of dark matter on the other. The verifiable and elegant hypothesis satisfies Mach's principle.