Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space, $\textrm{d}s^2 = -\textrm{d}t^2 + \textrm{d}x^2 + \textrm{d}y^2 + \textrm{d}z^2$,
$$\xi = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}.$$
How do I find the associated diffeomorphism, if one exist? I believe in this example the Lorentz boost in the x-direction is associated to the diffeomorphism, but I am having trouble understanding how to arrive at that answer.
Additionally, how do you tell when they might not have an associated diffeomorphism? I am lead to believe that this vector field cannot have a diffeomorphism translating points forward
$$ \xi = e^{x}\frac{\partial}{\partial x}.$$
Hi this is a great question! You should be able to take the matrix exponential of $\xi$. The vector field that you gave is an element in the Lie algebra which determines the symmetries of the space. In spacetime when you talk about metrics invariant under infinitesimal transformations these vectors are called Killing vectors and the set of all Killing vectors of a space defines a Lie algebra. This in turn can be exponentiated (See Lie' Third Theorem: https://en.wikipedia.org/wiki/Lie%27s_third_theorem) to a Lie Group which contains the diffeomorphism you want. In this case, a 4x4 matrix representation of this group element (call it $\Xi\in M_{4,4}(\mathbb{R})$), which results from the matrix exponential is given by:
$\Xi=\begin{bmatrix} \cosh(\theta) & \sinh(\theta) & 0 & 0\\ \sinh(\theta) & \cosh(\theta) & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix}$.
This matrix $\Xi$ will act on a 4 vector and boost it with rapidity $\theta$ (https://en.wikipedia.org/wiki/Rapidity). Hopefully this helps!