My calculations show that until about 3.8 Gy ago (z ~ 1.7) the volume of space defined by our light cone had a high enough density to constitute a black hole, in the sense that 2GM > c(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}r. The further back in time, the more our light cone exceeded that threshold, because the density increases and the proper radius of our light cone decreases as we go back in time before that point. After that time, the volume of space defined by our light cone has not been dense enough to constitute a BH because both the density and the radius of our light cone have decreased, the latter as an inevitable result of the light travel time becoming increasingly constrained.

I'm not sure whether to attach any particular significance to this fact. In theory no light should have been able to escape (or even move one iota outward) from its emission point at our light cone. Since our light cone is calculated to have expanded for the first 6 Gy or so after the big bang, this seems to be inconsistent with being a bona fide BH. For example, in proper distance coordinates light emitted from distant sources (such as the CBR) is supposed to have moved away from us during that initial time period, and begun moving toward us only after our light cone crossed the Hubble sphere at around z ~ 1.6.

Setting aside the theoretical imponderables, I find this situation to be annoying because I want to calculate the instantaneous gravitational time dilation at our light cone at various historical values of z. But the Schwarzschild metric (both the interior and exterior versions) "blows up" when 2GM > c^{2}r. (Square root of a negative number.) Is there any reasonable alternative means for calculating a "pure" instantaneous gravitational time dilation (i.e., no Doppler effects mixed in) under these circumstances?

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# Our light cone was a black hole

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