Tangent vector to a geodesic

Question

What exactly is a Tangent vector to a geodesic? I see this term is used often but I just can't find a definition for that.

I got stuck at this place in a book:

Answer 1

I am afraid this is a purely notational problem; notation is, as you may already know, sometimes a tricky issue in geometry and related fields. Apologies in advance if it feels like I am going round the houses in my answer.

The first paragraph defines the exponential map at a point $p\in\Sigma$ as $$\exp:(-\epsilon,\epsilon)\times \Sigma\to M;\, (r,p)\mapsto c_p(r). $$

This gives us, for each point $p$ in $\Sigma$ a path $c_p(r)$ that passes through $p$ and is contained in $\Sigma$. That path has one parameter ($r$), which ranges from $-\epsilon$ to $\epsilon$. The vector field $\frac{\partial}{\partial r}$, when applied to a point $c_p(r_0)$ gives us a vector tangent to $c_p(r_0)$.

Let me rephrase what we know so far: $c_p(\cdot)$ is, as the first paragraph says, a null geodesic. The vector field $\frac{\partial}{\partial r}$, when applied to a point on the geodesic line, will give us a vector that is tangent to the geodesic. That is the reason why we can speak of $\frac{\partial}{\partial r}$ as a vector tangent to the geodesic.

With regards to the 'null' part, that only means that the vector is lightlike, i.e. that $g_{rr} = 0$. You can find more about space-, time-, and lightlike vectors on Wikipedia, and probably on your book.