Choosing diffeomorphisms for the pullback metric in the Weak Field approximation

In the weak field approximation of the EFEs $$G_{\mu\nu}=\kappa T_{\mu\nu}$$ we take $$g_{\mu\nu}\approx \eta_{\mu\nu}+h_{\mu\nu}$$. The $$\eta_{\mu\nu}$$ term is just the flat space Minkowski metric and $$h_{\mu\nu}$$ is a small perturbation of Minkowski space such that it satisfies the condition $$|h_{\mu\nu}| \ll 1.$$ The weak field perturbation metric $$h_{\mu\nu}$$ is defined to be $$h_{\mu\nu}=(\phi^*g)_{\mu\nu}-\eta_{\mu\nu}$$ where $$(\phi^*g)_{\mu\nu}$$ is the pullback metric given by $$(\phi^*g)_{\mu\nu}=\frac{\partial y^{\alpha}}{\partial x^{\mu}}\frac{\partial y^{\beta}}{\partial x^{\nu}}g_{\alpha\beta}$$ where $$x^{\mu}$$ is a coordinate chart defined on a differentiable manifold $$M$$ and $$y^{\alpha}$$ a coordinate chart defined on another differentiable manifold $$N$$, thus making $$\phi$$ a diffeomorphism $$\phi:M \to N$$. We have to choose diffeomorphisms $$\phi$$ such that $$|h_{\mu\nu}| \ll 1$$. Forgetting about the weak field approximation, we thus look at an example. From this set of lecture notes by Sean Carroll we consider $$x^{\mu}=(\theta,\phi)$$ on $$M=S^2$$ with metric $$d \theta^2+\sin^2\theta d \phi^2$$and $$y^\alpha=(x,y,z)$$ on $$N=R^3$$ with metric $$dx^2+dy^2+dz^2$$, then our map $$\phi:M \to N$$ can be defined as $$\phi=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).$$ This is obvious because this is just the components of spherical coordinates. My question is, how can we choose what our diffeomorphism will be given the coordinate charts and metric? Note: $$\eta_{\mu\nu}$$ has a coordinate chart $$x^\alpha=(t,x,y,z)$$ with line element $$ds^2=-dt^2+dx^2+dy^2+dz^2$$. The question is for our pullback metric $$g$$. Furthermore, here M and N are the physical spacetime with $$\eta_{\mu\nu}$$ defined on it and the other manifold is our background spacetime.

Also, I apologize if my question seems pretty vauge, I am new to the notion of pullbacks and pushfowards.

• I'm not sure I understand the question: the map $\phi$ is the diffeomorphism, and as Carroll explains, this is essentially just the coordinate transformation between the two sets of coordinates. Is your question just, how to find a coordinate transformation between two line elements? May 12 at 13:40
• Kind of, but in his lectures and from other articles, I have seen not to think of pullback like coordinate transformations. The pullback looks very familiar to the transformation law of a second rank covariant tensor. But as I understand it, they arnt the same thing. But yes, my question is, how do I find the coordinate transformation between two line elements/metric? Sorry for the lack of clarity
– aygx
May 12 at 14:44