Exam

The vector field

Z : = E^{0} + x_{2} E^{1}

is a time like vector field on the subset

U : = {(x_{0}, x_{1}, x_{2}) : | x_{2} | < 1}

of Minkowski space

R_{1}^{3}

. Show that there is no function

f : U \to R

such that

Z

is normal to the submanifold

[f = 0]

.

Z

and

\nabla f = - \partial_{0} f E^{0} + \partial_{1} f E^{1} + \partial_{2} f E^{2}

must be parallel, i.e. for some function

g

:

g = - \partial_{0} f, g x_{2} = \partial_{1} f, 0 = \partial_{2} f .

The third equation implies that

f

does not depend on

x_{2}

; the first equation then implies that

g

does not depend on

x_{2}

either; finally the second equation shows that

g = \partial_{2} (g x_{2}) = \partial_{2} \partial_{1} f = 0

.