Case 2: Unit speed spacelike curve [alpha] e[member of] [E.sup.4.sub.1] with the characterization of spacelike vectors T, [E.sub.2], [E.sub.3] and
timelike vector [E.sub.1] on any fixed interval [[[gamma].sub.1], [[gamma].sub.2]] is defined in the Equation 42:
Indeed, by "static" it is meant that the metric tensor satisfies the Killing equations for the
timelike vector field [[chi].sub.(0)]
If [alpha]' = t is a unit spacelike vector, then [eta] is a unit
timelike vector. In this case, the curve [alpha] is called a spacelike curve and we have an orthonormal Sabban frame {[alpha](s), t(s), [eta](s)} along the curve [alpha], where [eta](s) = [alpha](s) [conjunction] t(s) is the unit spacelike or
timelike vector.
Let D be the Levi-Civita connection on [right arrow] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be a
timelike vector. If [V.sub.1] moves along [alpha], then a timelike ruled surface which is given by the parameterization
(iii)
timelike vector is never orthogonal to a null vector.
We say that a
timelike vector is future pointing or past pointing if the first compound of the vector is positive or negative, respectively.
Observe that [e.sub.n+1] = (0,..., 0,1) is a unit
timelike vector field globally defined on [L.sup.n+1], which determines a time-orientation on [L.sup.n+1].
[[xi].sub.2] is then the normalized
timelike vector field corresponding to d[[xi].sub.1].
(c) Gravity is always attractive; that is, for any
timelike vector [V.sup.a], the energy momentum tensor of matter satisfies the inequality ([T.sub.ab]-1/2[g.sub.ab]T)[V.sup.a][V.sup.b][is greater than or equal to]0.
To keep generality, instead of looking for a
timelike vector [u.sub.i], we will concentrate on a particular vector [l.sup.a], with norm [l.sub.a][l.sup.a] [equivalent to] [l.sup.2].
Then we have orthonormal Sabban frame {[alpha](s), T(s), [xi](s)} along the curve [alpha], where [xi](s) = -[alpha](s) x T(s) is the unit
timelike vector. The corresponding Frenet formulae of [alpha], according to the Sabban frame, read
We remark that B(s) is a unit
timelike vector. The corresponding Frenet equations are