Active Diffeomorphisms of Schwarzschild Metric

[Prathyush]

TL;DR Summary

Active diffeomorphism of a Schwarzschild metric under r ->c r

I am trying to understand active diffeomorphism by looking at Schwarzschild metric as an example.

Consider the Schwarzschild metric given by the metric

$$g(r,t) = (1-\frac{r_s}{r}) dt^2 - \frac{1}{(1-\frac{r_s}{r})} dr^2 - r^2 d\Omega^2 $$

We identify the metric new metric at r with the old metric at $c r$

this gives
$$g(r,t) = (1-\frac{r_s}{ c r}) dt^2 - \frac{1}{(1-\frac{r_s}{c r})} dr^2 - c^2 r^2 d\Omega^2 $$

We can do a passive transformation to re-define the metric so that we have a Minkowski metric as $r->\infty$ We can define $\tilde{r} = c r$

and we get the metric
$$g(r,t) = (1-\frac{\tilde{r_s}}{r}) dt^2 - \frac{1}{c^2 (1-\frac{\tilde{r_s}}{r})} d\tilde{r}^2 - \tilde{r}^2 d\Omega^2 $$

This does not look like the Schwarzschild metric and should not be a solution of GR.

This is not surprising because we simply dragged the metric numerically and hence introduced distortions to the original manifold.

Now I can define active diffeomorphism such that distance between $(r, r+dr)$ is actually the same as the distance between $(c r, c r + cdr)$ at fixed time then I won't introduce distortions(the $c^2$ in the denominator just cancels) but that is just an ordinary passive transformation. Maybe this is what duality between active and passive transformations means?

Please correct me if I am making a mistake or I have misunderstood something. I heard a claim that an active diffeomorphism is a symmetry in GR and generates new solutions of GR, I don't think it is correct.

 

[PeterDonis]

Consider the Schwarzschild metric given by the metric

You have written this metric down incorrectly. Where you have the factor $1 - \frac{r}{r_s}$, it should be $1 - \frac{r_s}{r}$.

You need to correct this error and then rethink your post.

 

[PeterDonis]

We identify the metric new metric at r with the old metric at cr

What does "identify" mean? Can you write down the explicit transformation equations for the diffeomorphism you are describing?

 

[Prathyush]

What does "identify" mean? Can you write down the explicit transformation equations for the diffeomorphism you are describing?

As I understand that the concept of active diffeomorphism means to drag the P to another P'. I can understand what it means to drag things on a Minkowski space, but its unclear to me what It means to drag in a general spacetime manifold.

So the active diffeomorphism that I am interested in drags the point $(r,t)$ to $(c r,t)$ keeping the angular co-ordinates fixed

 

[PeterDonis]

its unclear to me what It means to drag in a general spacetime manifold

If your active diffeomorphism is not an isometry, the manifold itself changes. So the notion of "dragging" something in a fixed background manifold doesn't really work for a general active diffeomorphism.

 

[Prathyush]

If your active diffeomorphism is not an isometry, the manifold itself changes. So the notion of "dragging" something in a fixed background manifold doesn't really work for a general active diffeomorphism.

So active diffeomorphisms are only well defined when there are isometries ?

 

[PeterDonis]

So active diffeomorphism only well defined when there are isometries ?

I'm not saying that. I'm just saying that the view of an active diffeomorphism as "dragging points" does not work if the active diffeomorphism is not an isometry.