A BEEMIAN SAMPLER: 1966-2002
23
On a very different theme, the many useful warped product curvature and
connection formulae were exploited by Beem, Ehrlich and Powell [26] to derive an
interesting result on the null geodesic incompleteness of Robertson-Walker space-
times.
For a Lorentzian warped product of the form
(M
=
(a, b)
x
f
H, -dt2
EB
fh),
where
(H, h)
is Riemannian, a given future directed null geodesic "Yo in
M =(a, b)
x
JH may be reparametrized in the form
"Y(t)
=
(t,
c(t)), where /' is a smooth null
pregeodesic in ( M,
g).
The expression of the connection '\7 on M in terms of the
corresponding objects on the base and fiber leads directly to the equation
-V--n
I
1
I
[ ]
I
It=
2f(t)
I'
(t),
relating the warping function
f
to the scaling factor necessary for reparametrizing
the pregeodesic /' into a null geodesic. Defining the monotone increasing (hence
invertible) function
p(t)
=it
VJ[;)
ds,
wo
the authors invoked a result from the classical differential geometry of curves to
conclude that the curve
f'Op- 1
(t)
=
(p-
1
(t),cop-
1
(t)) is a null geodesic. Defining
A=
lim
p(t),
and
B
=
lim
p(t),
t-+a+ t-+b-
one has a bijection
p :
(a, b)
---
(A, B)
and the reparametrized null geodesic /' o
p-
1
:
(A, B)
---
M.
Finiteness of
A
(respectively,
B)
thus ensures that
M
is past
(respectively, future) null geodesically incomplete. The authors stated the result as
follows in
[26],
and in pp. 109-110 of
[24]:
THEOREM
7. 5. Let M
=
(a, b) x
f
H be a warped product with Lorentzian metric
g
=
-dt2
EB
fh where -oo
::=:;
a b
::=:;
+oo, (H, h) is an arbitrary Riemannian
manifold, and f : (a,b)--- (O,oo). Set S(t)
= /TfJ5.
Then if
lim
jwo
S(s)ds
t-+a+ t
[respectively,
limt_,.b-
J~ 0
S(s) ds] is finite, every future directed null geodesic in
(M, g) is past {respectively, future} incomplete.
Recall that the Robertson-Walker solutions model the stress-energy content
of the universe as a perfect fluid
(U
=
Bt,
p,
p) with energy density
p
and pressure
p. One relationship between the geometry (warping function
f)
and the physical
content of the model
(p
and p) is expressed through the equation ( cf.
[78],
p. 346])
S" ( t)
(7.1) 3
S(t)
=
-41r(p
+
3p),
where
S(t)
=
/TfJ5,
as above. Taken together, Equation (7.1) and Theorem
7.5 provide a direct connection between null geodesic completeness (and incom-
pleteness) and the
physical
parameters which characterize the Robertson-Walker
models.
During his time at the University of Missouri, John Beem guided the research
of six doctoral students in mathematics; we conclude this section with a brief dis-
cussion of their work. The inclusion of this material in this section is particularly
appropriate since five of the six students worked on research topics involving warped
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