Lapse function and spacetime splicing
Hi, I have a question regarding the foliation of space-time by slices of constant "time".
I know that such a foliation is possible given a globally hyperbolic manifold, and one can define a "time function" t, the level sets of which are 3-D Cauchy surfaces which foliate the spacetime. My question is, if we define a "time vector" to this foliation by the requirement that [itex]t^\mu\nabla_\mu t=1[/itex], why is this "time vector" not (in general) orthogonal to the space-like Cauchy surfaces? The lapse function and shift vector measure the amount by which this time vector fails to be orthogonal to the Cauchy surfaces, but it seems to me that due to that requirement above, the vector should always be orthogonal shouldn't it?
My logic is such: since t is a scalar function, the covariant derivative of it is nothing other than its (one form) gradient, and the (vector) gradient of a function is always orthogonal to the level sets of the function is it not?