I think that problem would be that in reality GR (General Relativity)
is not full 4-dimensional.

GR is really (3+1)-dimensional and time dimension is different kind
that three
space dimensions as mentioned reference below.

Arbitrary 4-dim coordinate transformations are not allowed in GR..

For example in reference below it is mentioned that
it is not allowed to transform "meter stick" to "clock" and vice
versa.

Reference 1.
Tolman,R.C, 1962.
Relativity, Thermodynamics and Cosmology.

Please take a look.

Best Regards,

Hannu Poropudas
Vesaisentie 9E
90900 Kiiminki
Finland
hanporop
@
luukku.com

Reference 2.

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From:
Newsgroups: sci.physics.foundations
Subject: Schwarzchild Inside the Event Horizon
Date: Wed, 2 Jun 2010 18:21:28 CST
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Mike_Fontenot wrote:

A few months ago, I spent some time studying the Schwarzchild metric
OUTSIDE the event horizon (EH). Lately, I've been studying that metric
INSIDE the EH. The more I get into it, the stranger it seems.

It might help to know that the Schwarzschild metric inside the horizon
is equivalent to a collapsing Kantowski-Sachs cosmology. At a fixed
time r, the universe is an infinite cylinder RxS^2; the radius of that
cylinder descreases in time, and the singularity at r=0 is essentially
a "crunch."

There's a nice paper by Krasnikov at arxiv.org/abs/0804.3619,
published
in Gravitation and Cosmology 14 (2008) 362, that you may find helpful.

Steve Carlip