Page 6 - Nature Of Space And Time

p
S (t)

every timelike curve
S
intersects (t)

The physical signi cance of global hyperbolicity comes from the fact that it implies
that there is a family of Cauchy surfaces (t)for U. A Cauchy surface for U is a space

like or null surface that intersects every time like curve in U once and once only. One can
predict what will happen in U from data on the Cauchy surface, and one can formulate a
well behaved quantum eld theory on a globally hyperbolic background. Whether one can
formulate a sensible quantum eld theory on a non globally hyperbolic background is less

clear. So global hyperbolicity may be a physical necessity. But my view point is that one
shouldn't assume it because that may be ruling out something that gravity is trying to
tell us. Rather one should deduce that certain regions of spacetime are globally hyperbolic
from other physically reasonable assumptions.

The signi cance of global hyperbolicity for singularity theorems stems from the fol-
lowing.

q

geodesic of
maximum length p

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