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*coordinate-radius*`:

[tex]Coordinate-r = circumference/2\pi[/tex]

Under relative motion, i.e. special relativity, the observed spatial length is said to contract in the direction of motion. However, a comparable change in the observed rate of time is understood to keep the speed of light [c] invariant in both the moving and stationary frames of reference. Time also dilates under gravity, but now we appear to have to resolve the implications of an expansion of space in the radial direction towards [Rs] rather than a contraction. The relativistic factor by which time dilates and space expands under gravity is defined by:

[tex]\gamma = \frac {1}{\sqrt{1-Rs/r}}[/tex]

Therefore, this would suggest that the true radius or `

*Spatial-radius*` would be defined as follows:

[tex]Spatial-r = \gamma(coordinate-r)[/tex]

So the basis of my questions relate to whether this spatial expansion of radius is real and what implication follow from it. Now it is assumed that time dilation is real in the sense that if 2 twins (A & B) are initially located at a distance from [Rs], but twin (B) then approaches the event horizon and returns, (B) would be physically younger than (A). Now on this journey, the implication is also that `

*coordinate-r`*would become increasingly smaller than

*`spatial-r`,*as the curvature of space increases under the effect of gravity.

So does light have to travel an ever-longer physical distance between the twins as (B) approaches [Rs]?

If we assume that twin (B) approaches [Rs] and then stops, then only the effects of gravity need to be considered. We might also assume that both twins are firing a laser pulse every second, which is then reflected back off the other craft:

Can the round trip distance be inferred from the speed of light [c]?

What distance [d] would each twin inferred from [d=ct/2]?

At one level, irrespective of the expansion, it would seem reasonable to assumed that the distance between A-B is the same as B-A, at least to the photons, but the implication of time dilation is that the rate of time in (A) and (B) is running at different rates. So

Can we assume the value of [c] is constant with respect to spatial-radius?

Continuing this process all the way to the black hole event horizon, the relative time in (B) with respect (A) seems go to zero and the spatial-r separation would become infinite.

So is it valid to ask exactly when and where does a black hole exists in spacetime?

*Footnote:*

In part, this thread was raised after coming across the formal definition of a black hole as “a region of spacetime that is not in the causal past of the infinite future.” Not sure that I really understand what this implies either!

In part, this thread was raised after coming across the formal definition of a black hole as “a region of spacetime that is not in the causal past of the infinite future.” Not sure that I really understand what this implies either!