# Time in a black hole and Weyl curvature

#### Naty1

Kip Thorne says (Lecture in 1993 Warping Spacetime, at Stephan Hawking's 60th birthday celebration, Cambridge, England,)

The flow of time slows to a crawl near the horizon, and beneath the horizon time becomes so highly warped that it flows in a direction you would have thought was spacial: it flows downward towards the singularity. That downward flow, in fact, is why nothing can escape from a black hole. Everything is always drawn inexorably towards the future, and since the future inside a black hole is downward, away from the horizon, nothing can escape back upward, through the horizon.

I thought classical time was always symmetric ....apparently not. Is this same description applicable to a "big crunch" as well? Apparently Weyle curvature at the big bang and black holes seems to go to infinity while at the big crunch it's essentially zero....how does that relate to this "direction" of time??

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#### alle.fabbri

Hi all!!
I think the keypoint of Thorne's statement can be found in the form of the Schwarzchild metric
$$ds_S^{2} = \left(1-\frac{2M}{r} \right) dt^2 - \left(1-\frac{2M}{r} \right)^{-1} dr^2 - r^2 d \Omega^2$$
as you can see at the event orizon, located at $$r_S = 2 M$$ the metric becomes ill defined, i.e. singular. This peculiarity divides the whole space-time in two regions. The outer one which is asymptotically flat, i.e. $$ds_S^2 \rightarrow dt^2 - $$d \vec{r}$$^2$$, describing the space surrounding the black hole. And the inner region, which is properly the black hole, a region in which the relative signs of $$dt^{2}$$ and $$dr^{2}$$ change. This is the fact pointed out by Thorne. That inside the black hole the space coordinate acts as the temporal one and viceversa, from a causal point of view. More specifically you have that that all the trajectories pointing outward the center can only reach, asymptotically the event horizon.
Hope this helps.....forgive my english.....

#### Naty1

Atyy...Kip specifically refers to black holes and I should have made that clear...my quote above is from page 93, THE FUTURE OF THEORETICAL PHYSICS AND COSMOLOGY, 1993
I'll have to check the white hole reference tomorrow...

ALLE: GREAT INSIGHT....Thorne happens to state the Schwarschild solution you posted (in a slightly different form) earlier in his talk.....I'll bet THAT IS what he is referring to....makes sense....

#### atyy

I meant that a white hole is the time reversal of a black hole.

#### alle.fabbri

I meant that a white hole is the time reversal of a black hole.
Mmh....i don't think it is so simple.....i mean, a black hole is a physical entity (or, at least, we hope so...) while its white counterpart it's only a mathematical tool needed to cover the entire space-time manifold with the fewest possible number of charts. Indeed I think that Wheeler prooved (sorry but no references....) that a collapsing star cannot create a white hole since the time is not symmetric in the collapsing process. So handle with care....

@Naty: Can you post or link the form of the metric choosen by Thorne? Thnx

#### atyy

There are comments regarding this in the first paragraph of
The arrow of time, black holes, and quantum mixing of large N Yang-Mills theories
Guido Festuccia, Hong Liu
http://arxiv.org/abs/hep-th/0611098

I know in the Schwarzschild solution, a white hole is the formal time reverse of a black hole, but I wonder whether it is true that for every black hole solution, there is a corresponding white hole solution (in theory, although it may not be realised in nature)?

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