Firstly, I know this is a physics-related problem, and I have posted here, but the physics forum seems so much more empty then this one, so here it goes:
I was in the process to find the Killing vectors for the Minkowski Metric and I stumbbled into a material that does a different procedure at the very end of the process, in comparisson to usual books and articles I've seen. The reasoning goes as follows:
For example, suppose we have found the Killing vector $$K=x \frac{\partial}{\partial_t} + t\frac{\partial}{\partial_x}$$
The way I would check this is a generator for the boost in the x direction is by acting these vectors onto t and x and check they give, respectively, x and t.
The way this material I've found does is, they suddenly deffine:
$$ \Lambda=\exp[\lambda(x \frac{\partial}{\partial_t} + t\frac{\partial}{\partial_x})]=\sum_{n=0}^{\infty}\frac{1}{n!}\lambda^n (x \frac{\partial}{\partial_t} + t\frac{\partial}{\partial_x})^n $$
then he proceeds to find the explicit form of the boost:
$$ \Lambda t = x\sinh \lambda +t\cosh \lambda $$
$$ \Lambda x = t\sinh \lambda +x\cosh \lambda $$
I understand the steps in this process. What I don't get is where did the motivation for the exponentiation come. What does that mean? It seems to me to have something to do with the application of the diffeomorfisms, but I'm not sure.
Also, would this be a more correct way to proceed then what I've done?I really would appreciate any comments on this, as well as reccomended material.
On a manifold $M$, every complete vector field $V\in\mathfrak{X}M$ has a flow $\Theta:\mathbb{R}\times M\to M$, such that, for fixed $p\in M$, $\Theta_\lambda(p)$ is the integral curve of $V$ starting at $p$. That is, $$ \Theta_0(p)=p\ \ \ \ \ \ \ \frac{d}{d\lambda}\Theta_\lambda(p)=V(\Theta_\lambda(p))\ \ \ \ \ \ \ \forall p\in M $$ $\Theta_\lambda $ form a one-parameter subgroup of diffeomorphims, which means that each $\Theta_\lambda$ is a diffeomorphism and $\Theta_{\lambda'}\circ\Theta_\lambda=\Theta_{\lambda+\lambda'}$. This is generally what is meant when one says that a vector field "generates" a family of diffeomorphisms: the diffeomorphisms are the flow of the vector field. The flow is sometimes written as $\exp(\lambda V):=\Theta_\lambda$. To find the flow generated by a vector field, we must solve this differential equation. Conversely, if given a one-parameter subgroup of diffeomorphisms, we can differentiate w.r.t. $\lambda$ to obtain the vector field that generates it.
Using the standard coordinates in Minkowski space, any Killing vector field $V$ is linear, in the sense that each component is a linear function of the coordinates. $$ V^i(x^0,x^1,x^2,x^3)=A^i_jx^j $$ Where $A^i_j$ is some fixed matrix. This means that the ODE corresponding to the flow is also linear: $$ \Theta_0^i(x^0,x^1,x^2,x^3)=x^i \\ \frac{d}{d\lambda}\Theta_\lambda^i(x^0,x^1,x^2,x^3)=A^i_jx^j $$ Where $\Theta_\lambda^i$ is the $i$th coordiante of $\Theta_\lambda$. The solution is therefore given by the matrix exponential $$ \Theta_\lambda^i(x^0,x^1,x^2,x^3)=(\exp(\lambda A))^i_jx^j $$