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