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naima
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Hi Pfs
Is there a method to draw a Penrose diagram when the spacetime metric is known?
Thanks.
Is there a method to draw a Penrose diagram when the spacetime metric is known?
Thanks.
Shyan said:The method is to transform the coordinates in a way that the metric becomes conformally flat in the new coordinates. Then you transform the coordinates somehow that the new coordinates in the flat part of the metric have a finite range. Then you simply draw the flat part of the metric in the new coordinates.
bcrowell said:I don't think this is quite right. Conformal flatness is an intrinsic property of a spacetime, so a coordinate transformation doesn't change this property.
Conformal flatness is a hypothesis that must hold for the original spacetime (or it has to be spherically symmetric).
Shyan said:Yeah, I should have been more careful. Actually only one part of the metric becomes conformally flat. For example when you write the Schwarzschild metric in Kruskal coordinates, you have ## ds^2=\frac{4r_s^3}{r}e^{-r/r_s}(dT^2-dR^2)-r^2 d\Omega^2 ##. As you can see, the ## \Omega=const ## part of the metric is conformally flat and that's the part of the metric we'll use for the Penrose diagram.
naima said:I am beginning to read an introduction to BH evaporation
when i am at eq 2.5 How can i get the Penrose diagram on the next page?
But e.g. the ## \Omega=const## part of the Schwarzschild metric (##ds^2=(1-\frac{r_s}{r})dt^2-(1-\frac{r_s}{r})^{-1} dr^2 ##) is not manifestly conformally flat.bcrowell said:Hmm...well, any 2-dimensional space is conformally flat, so I don't think it's necessary to display the 2-dimensional metric explicitly, in a particular coordinate system, in order to show that it's conformally flat.
stedwards said:What does it mean, that a metric is "conformally flat"? somewhere in my notes I have the metric about a schwarzschild singularity where the light cones everywhere run off at 45 degrees in transformed (r,ct) Would this be it?
naima said:Penrose diagrams are in a square.
naima said:Once i changed the coordinates and got a conformally flat metric on a plane
naima said:how can i map in in the square?
naima said:What is the new change of coordinates which will map this plane to the square ##\mathcal{R} ## (look at fig 9)?
naima said:I get "not found" for the links. Are they still valid?
A Penrose diagram is a type of spacetime diagram that represents the geometry and causal structure of a specific spacetime metric. It is a 2-dimensional projection of a higher-dimensional spacetime, typically used in general relativity to visualize black hole solutions and other complex spacetimes.
To create a Penrose diagram, you first need to determine the spacetime metric of the system you want to represent. This can be done using mathematical equations or numerical simulations. Then, you use a coordinate transformation to project the higher-dimensional spacetime onto a 2-dimensional plane, while preserving the geometry and causal structure. This projection is the Penrose diagram.
Penrose diagrams are useful for visualizing and understanding complex spacetimes, as they provide a simplified representation that is easier to interpret. They also allow for easier calculation of certain properties, such as the trajectory of particles or the behavior of light rays, which can be difficult to analyze in the higher-dimensional spacetime.
No, Penrose diagrams are limited to spacetimes that can be described by a metric. This means they cannot accurately represent spacetimes with quantum effects or those that do not follow the laws of general relativity.
Penrose diagrams are commonly used to represent the spacetime around black holes, as they provide a visual understanding of the event horizon and the singularity. They can also be used to study the behavior of light and particles near a black hole, such as the effects of strong gravitational lensing.