If the invariant speed of light (c) were inversely proportional to the local energy density, as you mentioned, it would have significant implications for our understanding of space, time, and the measurements of distances. Such a scenario would contradict the current theories and concepts in physics, including the theory of relativity. However, let's explore the implications hypothetically:
Variable speed of light: In regions of low energy density, such as deep space, the speed of light would be faster than in regions of high energy density. This would mean that light would travel at different speeds depending on the local environment. Consequently, our measurements of distances and the timing of events would need to be adjusted accordingly, as they would be affected by this variable speed of light.
Altered perception of distance: If the speed of light were faster in deep space, it would take less time for light to travel from distant objects to us. As a result, the perceived distances between celestial objects would be greater than currently measured. The vastness of the universe would appear even more expansive, with objects located farther away than previously estimated.
Changes to cosmological models: The variable speed of light would necessitate revisions to our cosmological models and theories. The expansion of the universe, the calculations of cosmic distances, and the understanding of the large-scale structure of the cosmos would all require reassessment to account for the varying speed of light.
Impact on causality and relativity: The variable speed of light would have implications for causality and the theory of relativity. It could potentially challenge the causal order of events and introduce inconsistencies in the temporal sequence of cause and effect, leading to significant conceptual and philosophical questions.
It's important to note that the concept of a variable speed of light, as you described it, goes against the established principles and experimental evidence of modern physics. The speed of light in a vacuum, as we currently understand it, is considered a fundamental constant of nature. While speculative ideas are valuable for pushing the boundaries of knowledge, it is crucial to emphasize that any proposed alternative must align with empirical observations and experimental evidence to gain acceptance in the scientific community.