Page 8 - Nature Of Space And Time

ought to be a globally hyperbolic region in which there ought to be conjugate points on
every geodesic between two points. This establishes a contradiction which shows that the
assumption of geodesic completeness, which can be taken as a de nition of a non singular
spacetime, is false.
The reason one gets conjugate points in spacetime is that gravity is an attractive force.

It therefore curves spacetime in such a way that neighbouring geodesics are bent towards
each other rather than away. One can see this from the Raychaudhuri or Newman-Penrose
equation, which I will write in a uni ed form.

Raychaudhuri - Newman - Penrose equation

d 2 ij 1 a b
= + ij + R abl l
dv n

where n = 2 for null geodesics
n = 3 for timelike geodesics

Here v is an a ne parameter along a congruence of geodesics, with tangent vector l a
which are hypersurface orthogonal. The quantity is the average rate of convergence of
a b
the geodesics, while measures the shear. The term R abl l gives the direct gravitational
e ect of the matter on the convergence of the geodesics.

Einstein equation

1
R ab − g abR =8 T ab
2

Weak Energy Condition

a b
T abv v ≥ 0

a
for any timelike vector v .

a
By the Einstein equations, it will be non negative for any null vector l if the matter obeys
the so called weak energy condition. This says that the energy density T 00 is non negative
in any frame. The weak energy condition is obeyed by the classical energy momentum
tensor of any reasonable matter, such as a scalar or electro magnetic eld or a
uid with

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