p
r
point conjugate to p
One can now consider the second variation of the length of a geodesic γ. One can show that γ can be varied to a longer curve if there is an infinitesimally neighbouring geodesic from p which intersects γ again at a point r between p and q. The point r is said to be conjugate to p. One can illustrate this by considering two points p and q on the surface of the Earth. Without loss of generality one can take p to be at the north pole. Because the Earth has a positive definite metric rather than a Lorentzian one, there is a geodesic of
minimal length, rather than a geodesic of maximum length. This minimal geodesic will be
a line of longtitude running from the north pole to the point q. But there will be another geodesic from p to q which runs down the back from the north pole to the south pole and then up to q. This geodesic contains a point conjugate to p at the south pole where all the geodesics from p intersect. Both geodesics from p to q are stationary points of the length under a small variation. But now in a positive definite metric the second variation of a
geodesic containing a conjugate point can give a shorter curve from p to q. Thus, in the example of the Earth, we can deduce that the geodesic that goes down to the south pole
and then comes up is not the shortest curve from p to q. This example is very obvious.
However, in the case of spacetime one can show that under certain assumptions there
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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 definition 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 unified form.
Raychaudhuri – Newman – Penrose equation
dρ
1
=
ρ 2
+ σijσij
+
Rablalb
dv
n
where n = 2 for null geodesics
n = 3 for timelike geodesics
Here v is an affine parameter along a congruence of geodesics, with tangent vector la which are hypersurface orthogonal. The quantity ρ is the average rate of convergence of the geodesics, while σ measures the shear. The term Rablalb gives the direct gravitational effect of the matter on the convergence of the geodesics.
Einstein equation
Rab − 1 gabR = 8 πTab
2
Weak Energy Condition
Tabvavb
≥ 0
for any timelike vector va.
By the Einstein equations, it will be non negative for any null vector la 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 field or a fluid with
8
a reasonable equation of state. It may not however be satisfied locally by the quantum
mechanical expectation value of the energy momentum tensor. This will be relevant in my
second and third lectures.
Suppose the weak energy condition holds, and that the null geodesics from a point p
begin to converge again and that ρ has the positive value ρ 0. Then the Newman Penrose equation would imply that the convergence ρ would become infinite at a point q within an affine parameter distance 1 if the null geodesic can be extended that far.
ρ 0
If ρ = ρ 0 at v = v 0 then ρ ≥
1
. Thus there is a conjugate point
ρ− 1+ v 0 −v
before v = v 0 + ρ− 1.
γ inside (p)
+
I
crossing region
q
of light cone
future end point
of in (p)
+
γ
I