# Active Diffeomorphisms of Schwarzschild Metric

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• Prathyush
In summary: 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.
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.

Last edited:
Prathyush said:
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.

Prathyush said:
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?

PeterDonis said:
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

Prathyush said:
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.

PeterDonis said:
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 ?

Prathyush said:
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.

Prathyush

## 1. What are active diffeomorphisms?

Active diffeomorphisms refer to transformations of a spacetime manifold that preserve the underlying structure and coordinates of the manifold. In other words, they are changes in the coordinates of a spacetime without altering the geometry of the spacetime itself.

## 2. What is the Schwarzschild metric?

The Schwarzschild metric is a mathematical description of the curvature of spacetime around a non-rotating, spherically symmetric mass. It is used to describe the spacetime outside of a black hole, and is a solution to Einstein's field equations in general relativity.

## 3. How are active diffeomorphisms applied to the Schwarzschild metric?

Active diffeomorphisms can be used to transform the coordinates of the Schwarzschild metric, while keeping the metric itself unchanged. This allows for different observers to describe the same spacetime using different coordinates, depending on their frame of reference.

## 4. What is the significance of active diffeomorphisms in studying black holes?

Active diffeomorphisms are important in understanding the properties of black holes, as they allow for the description of the same spacetime using different coordinates. This can help in visualizing the effects of gravity on spacetime and understanding the behavior of matter and light near a black hole.

## 5. How do active diffeomorphisms relate to the principle of general covariance?

The principle of general covariance states that the laws of physics should be the same for all observers, regardless of their frame of reference. Active diffeomorphisms play a key role in this principle, as they allow for the transformation of coordinates while preserving the underlying physical laws and geometry of the spacetime.

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