H is the Hubble constant. Suppose that there is a 3 dim reference frame and a 4 dim reference frame. The 4 dim reference frame consists of 3 dim space + 1 dim motion (light motion).
The ratio of dimensions 4: 3 might account for why astronomical observation reveals the expansion of space: energy per dimension for 3 dim space is 4/3 that of energy per dimension for 4 dim space. If lengths are proportional to energy, then 3 dim space would expand by 4/3 relative to 4 dim space.
The definition of H implicitly assumes a one dimensional space. How assured is the definition of H?
RS: Suppose that there is a 3 dim reference frame and a 4 dim reference frame. The 4 dim reference frame consists of 3 dim space + 1 dim motion (light motion).
If the 3D frame is space, the 4th dimension would be time. Motion is the ratio of spatial displacement to time, not a dimension itself.
RS: The definition of H implicitly assumes a one dimensional space. How assured is the definition of H?
The variable H is valid in the Friedmann equations which are dependent on the Cosmological Principle which includes that space must be isotropic hence H can be defined in terms of 1D radial distance within 3D space. If the Cosmological Principle is not valid, then the whole model falls but to date it has survived all observational tests.
George Dishman.
Thank you for your reply.
As follow-up questions to your points:
At cosmologically large scales, can 4D space be isotropic? homogeneous?
Can the cosmological principle apply to 4D space notwithstanding big bang cosmology?
Does the cosmological principle preclude a 4 dim (space + time) system co-existing with a 3 dim (space only) system? Why (or why not)?
Bob Shour
Hi Robert,
RS: At cosmologically large scales, can 4D space be isotropic? homogeneous?
Space is 3D, not 4D, as we experience it. String theory postulates higher dimensions but they are "compact" so not directly observable. The cosmological principle is that the distribution of matter is isotropic and homogeneous throughout 3D space at any given time. The density can vary with time though hence it falls as the universe expands.
RS: Can the cosmological principle apply to 4D space notwithstanding big bang cosmology?
If the cosmological principle is correct then the "big bang" model follows, specifically the Robertson-Walker Metric.
RS: Does the cosmological principle preclude a 4 dim (space + time) system co-existing with a 3 dim (space only) system? Why (or why not)?
A "3 dim (space only) system" would have no extent in time, it would exist only for an infinitesimal time, a single instant of zero duation, so as far as we are concerned, it would not exist at all. That follows regardless of the cosmological principle.
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