Penrose Diagram for Schwarzschild

In summary, the conversation discusses the use of Penrose diagrams for the Schwarzschild space-time in standard (t,r,θ,ϕ) coordinates and the issue of closing light cones at r=2M. The solution is to use tortoise coordinates and advanced/retarded Eddington-Finkelstein coordinates to avoid this issue. However, the question is raised on how to construct a Penrose diagram using only (u,r,θ,ϕ) or (v,r,θ,ϕ) coordinates. The answer can be found by transforming Kruskal-Szekeres coordinates, as shown in the second link provided.
  • #1
coalquay404
217
1
Hi. I've got a quick question on Penrose diagrams for the Schwarzschild space-time that I'd appreciate some comments on. In standard [tex](t,r,\theta,\phi)[/tex] coordinates the Schwarzschild metric is

[tex]ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \left(1-\frac{2M}{r}\right)^{-1}dr^2 + r^2 d\Omega^2.[/tex]

The immediate difficulty with this is that if one looks at radial null curves one finds that the light cones appear to close up as [tex]r\to 2M[/tex]. The standard way of getting around this is to define a tortoise radial coordinate according to

[tex]r^* \equiv r + 2M\log|r-2M|.[/tex]

Then the metric in [tex](t,r^*,\theta,\phi)[/tex] coordinates becomes simply

[tex]ds^2 = \left(1-\frac{2M}{r}\right)(-dt^2 + dr^{*2}) + r^2 d\Omega^2.[/tex]

The thing about this radial coordinate is that the light cones don't close up anywhere. The downside, however, is that the surface [tex]r=2M[/tex] has now been pushed to [tex]r^*\to-\infty[/tex]. So, we introduce advanced and retarded Eddington-Finkelstein coordinates by defining

[tex]v \equiv t + r^*,[/tex]
[tex]u \equiv t - r^*.[/tex]

Then, for example, the metric in [tex](u,r,\theta,\phi)[/tex] coordinates becomes

[tex]ds^2 = -\left(1-\frac{2M}{r}\right)du^2 - 2dudr + r^2d\Omega^2.[/tex]

(I think this is correct.) Now to my question. Is there any way that I can conformally compactify this metric so that I can draw a Penrose diagram of it? I know that I can go to Kruskal-Szekeres coordinates and construct a Penrose diagram there, but surely it's possible to construct Penrose diagrams using just [tex](u,r,\theta,\phi)[/tex] or [tex](v,r,\theta,\phi)[/tex] coordinates. My problem is that I can't readily see what coordinate transformations I should take so as to get to a finite coordinate range from the infinite ranges of [tex]u,v,r[/tex]. Can anyone shed some light on this for me?
 
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  • #2
  • #3
pervect said:
You don't need that much detail for a penrose diagram.

You do if you want to do it correctly. I'm not talking about simply drawing the bare Penrose diagram. What I'm after is a coordinate transformation which will compactify the spacetime so that it can be drawn as a Penrose diagram. I'm *not* looking to draw the entire Penrose diagram for the Kruskal extension that you quoted in the first link. What I want to be able to do is to draw a Penrose diagram that arises from using advanced *or* retarded Eddington-Finkelstein coordinates. This would correspond to the right-hand region in the Penrose diagram for the Kruskal extension, but there would be only one singularity (the singularity is on the bottom for the retarded coordinate and on the top for the advanced coordinate).
 
  • #4
While I've never gone through it in the level of detail you want to, the second reference I quoted appears to do the job, i.e.

http://casa.colorado.edu/~ajsh/schwp.html#penrose

has the equations that transform Kruskal-Szekeres into a penrose diagram, also it also has a GIF graphic that shows one morphing into the other.
 

1. What is a Penrose Diagram for Schwarzschild?

A Penrose Diagram for Schwarzschild is a spacetime diagram used to represent the geometry of a black hole. It is a two-dimensional projection of the four-dimensional spacetime, with the horizontal axis representing space and the vertical axis representing time. It is named after the physicist Roger Penrose who first introduced the concept.

2. How is a Penrose Diagram constructed?

A Penrose Diagram is constructed by transforming the coordinates of the four-dimensional spacetime using a mathematical transformation called a conformal transformation. This transformation preserves angles and light cones, allowing for a visual representation of the spacetime curvature near a black hole. The resulting diagram is a diamond shape, with the center representing the black hole singularity.

3. What information can be gleaned from a Penrose Diagram for Schwarzschild?

A Penrose Diagram can provide insights into the causal structure of the black hole, the location of the event horizon, and the paths of light rays and particles near the black hole. It also allows for a comparison between the spacetime of a black hole and that of a flat, uncurved spacetime.

4. How does a Penrose Diagram illustrate time dilation near a black hole?

In a Penrose Diagram, the closer a path is to the event horizon, the steeper it appears, indicating a greater time dilation. This means that time appears to pass slower for an observer near the event horizon compared to one far away from the black hole. This effect is due to the intense gravitational pull of the black hole.

5. Can a Penrose Diagram be used for other types of black holes?

Yes, a Penrose Diagram can be used for any stationary, spherically symmetric black hole, not just the Schwarzschild black hole. It can also be used to represent the spacetime of rotating black holes, such as the Kerr black hole, with some modifications to the conformal transformation.

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