Generating new solutions of the Einstein equation by active transformation, and the physical interpretation of the new ones

Question

Given a manifold $\mathcal{M}$ with coordinates $\psi : \mathcal{M} \rightarrow \mathbb{R^4}$ , $\Psi(p)= (r,\theta ,\phi,t)$ for $ p \in \mathcal{M}$

Suppose we have the active transformation $F : \mathcal{M} \rightarrow \mathcal{M}$ in coordinates given by

$$\Psi \circ F\circ \Psi^{-1}(r,\theta ,\phi,t)=(f(r),\theta ,\phi,t)$$

where is $f$ a function in $\mathbb{R}$

Given the Schwarzschild metric $$ \Psi \circ g\circ \Psi^{-1}(r,\theta ,\phi,t) =-\left(1-\frac{1}{ r}\right)dt^2+\left(1-\frac{1}{ r}\right)^{-1}dr^2+r^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \;. $$ we can induce another metric $h$ by

$$\Psi \circ h\circ \Psi^{-1}(r,\theta ,\phi,t)=\Psi \circ F^*g\circ \Psi^{-1}(r,\theta ,\phi,t)=-\left(1-\frac{1}{ f(r)}\right)dt^2+\left(1-\frac{1}{ f(r)}\right)^{-1}f'(r)^2dr^2+f(r)^2\left(d \theta^2 +\sin^2 \theta d \phi^2\right) \;.$$

where $f'(r)=\frac{df(r)}{dr }$

Then for a curve $\gamma(\tau )=(2,0,0,\tau) \in \mathcal{M} $ we would have

then $h(\gamma'(\tau), \gamma'(\tau))=1-\frac{1}{ f(2)}$ so we have
$$\int_{[0, 1]} h(\gamma'(\tau), \gamma'(\tau))d\tau = -\int_{[0, 1]} \left(1-\frac{1}{ f(2)}\right)d\tau $$

where $\gamma'(\tau)=\frac{d\gamma}{d\tau }$

and $$\int_{[0, 1]} g(\gamma'(\tau), \gamma'(\tau))d\tau = \frac{1}{ 2} $$

Since they both satisfy the Enstein equation and they do not give the same line element, how to know which solution should we chose for a given problem.

Answer 1

Assuming that $f$ is well behaved, so that the transformation is a diffeomorphism, then this is the same spacetime described in different coordinates.

Since they both satisfy the Enstein equation and they do not give the same line element, how to know which solution should we chose for a given problem.

They don't give the same line element because they're expressed in different coordinates. Similarly, the line element for the Euclidean plane in Cartesian coordinates is different from the line element expressed in polar coordinates. Which coordinate system to use is a matter of convenience.

These would not normally be considered distinct solutions to the field equations. They're the same solution. The choice of coordinates for a given problem could be based on what coordinates make the metric look simpler, or which ones respect the symmetry of the spacetime.

If the map isn't a diffeomorphism, then things can get more nontrivial. For example, the external Schwarzschild spacetime can be extended through the coordinate singularity at the horizon, and can also be extended to the maximal extension, which includes an Einstein-Rosen bridge.