I have been reading a book about General Relativity, and the term "grad $t$" appears as the gradient of a time-function. From the little differential geometry knowledge that I have, the gradient can be written as $\nabla t=g^{ij}\partial_it$ given there is a coordinate frame $\{\partial_i\}_{i\in\{1,...,\text{dim}M\}}$.
Question: How can one write the expression above for a gradient without having defined a local coordinate frame? (if it is even possible)
In any set-up with an inner product (e.g., a Riemannian or semi-Riemannian manifold), the gradient $\text{grad}\, f$ is defined by the equation $$\langle \text{grad}\,f (x),v \rangle = df_x(v),$$ where $v$ is in the tangent space at $x$, and $df_x$ is the derivative of $f$ at $x$. If you know about differential forms, you can think of the latter expression as evaluating the $1$-form $df(x)$ on the tangent vector $v$.