# The effect of a fifth large-scale space-time dimension on the conservation of energy in a four dimensional Universe

## TL;DR

Introducing a fifth large-scale space-time dimension modifies energy conservation laws, providing explanations for galaxy rotation curves, the Pioneer Anomaly, and potentially deriving gravity's inverse square law from energy conservation principles.

## Contribution

This paper restates energy conservation within a five-dimensional framework, offering a novel method to estimate an acceleration parameter using cosmological constants and proposing new insights into gravity and cosmic relationships.

## Key findings

- Good agreement with galaxy rotation curves and Pioneer Anomaly
- Estimation of acceleration parameter ar from cosmological constants
- Inverse square law of gravity may stem from energy conservation at a boundary

## Abstract

The effect of introducing a fifth large-scale space-time dimension to the equations of orbital dynamics was analysed in an earlier paper by the authors. The results showed good agreement with the observed flat rotation curves of galaxies and the Pioneer Anomaly. This analysis did not require the modification of Newtonian dynamics, but rather only their restatement in a five dimensional framework. The same analysis derived a acceleration parameter ar, which plays an important role in the restated equations of orbital dynamics, and suggested a value for ar. In this companion paper, the principle of conservation of energy is restated within the same five-dimensional framework. The resulting analysis provides an alternative route to estimating the value of ar, without reference to the equations of orbital dynamics, and based solely on key cosmological constants and parameters, including the gravitational constant, G. The same analysis suggests that: (i) the inverse square law of gravity may itself be due to the conservation of energy at the boundary between a four-dimensional universe and a fifth large-scale space-time dimension; and (ii) there is a limiting case for the Tulley-Fisher relationship linking the speed of light to the mass of the Universe.