Solving the Transformation to Eddington-Finkelstein in Schwarzschild Geometry

In summary, the conversation is about a student struggling with understanding the transformation to Eddington-Finkelstein in the Schwarzschild geometry. They provide the relevant equations and attempt at a solution, but are unsure where they went wrong. They are seeking help to understand and complete the transformation.
  • #1
Astrofiend
37
0

Homework Statement



I'm having problems seeing how the transformation to Eddington-Finkelstein in the Schwarzschild geometry works. Any help would be great!

Homework Equations



So we have the Schwarzschild Geometry given by:

[tex]
ds^2 = -(1-2M/r)dt^2 + (1-2M/r)^-^1 dr^2 + r^2(d\theta^2+sin^2\theta d\phi^2)
[/tex]

and the Edd-Fink transformation assigns

[tex]
t = v - r -2Mlog\mid r/2M-1 \mid
[/tex]

The textbook says it is straight-forward to simply sub this into the S.G line element to get the transformed geometry, but I can't seem to get it.

The Attempt at a Solution



OK, so differentiating the expression for the new t coordinates above, I get:

[tex]
dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]
[/tex]
[tex]
dt = dv - dr -2M. \left(\frac{1}{r-2m}\right)
[/tex]
[tex]
dt = dv - dr - \left(\frac{2m}{r}-1\right)
[/tex]


...but the book says the answer is

[tex]
dt = -\frac{dr}{1-\frac{2M}{r}} + dv
[/tex]

>where am I going wrong here? Given this last expression, it is fairly easy to sub it into the SG line element to get the transformed coordinates. Problem is, I can't seem to get that far! Any help much appreciated...
 
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  • #2


Astrofiend said:

Homework Statement


OK, so differentiating the expression for the new t coordinates above, I get:

[tex]
dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]
[/tex]

Ermm... shouldn't this be

[tex]dt = dv - dr -2M. \frac{d}{dr} [log(r-2m)-log(2m)]dr[/tex]

?:wink:

Also, [tex]2m\left(\frac{1}{r-2m}\right)
\neq \left(\frac{2m}{r}-1\right)[/tex]
 
  • #3


Damn it! Still not seeing it...
 
  • #4


[tex]dt=dv-dr-2m\left(\frac{1}{r-2m}\right)dr=dv-\left[1+2m\left(\frac{1}{r-2m}\right)\right]dr[/tex]

Just simplify...it's basic algebra from here.
 

1. What is the Eddington-Finkelstein transformation in Schwarzschild geometry?

The Eddington-Finkelstein transformation is a mathematical transformation used in general relativity to convert the Schwarzschild metric, which describes the geometry outside a non-rotating, spherically symmetric mass, into a form that is more suitable for solving problems involving light rays. It transforms the coordinates from the standard Schwarzschild coordinates (t, r, θ, φ) to advanced or retarded Eddington-Finkelstein coordinates (v, r, θ, φ), where v is the advanced or retarded time coordinate. This transformation is particularly useful for studying the behavior of light near black holes.

2. What is the significance of solving the transformation to Eddington-Finkelstein in Schwarzschild geometry?

Solving the transformation to Eddington-Finkelstein coordinates is important because it allows us to better understand the behavior of light near massive objects, such as black holes. It also helps us to study the gravitational lensing effect caused by these objects, which can give us valuable information about their mass and structure. Additionally, the Eddington-Finkelstein coordinates are particularly useful for studying the event horizon of a black hole, as they avoid the problem of coordinate singularities that occur in the standard Schwarzschild coordinates.

3. How is the transformation to Eddington-Finkelstein in Schwarzschild geometry derived?

The Eddington-Finkelstein transformation is derived by solving the Einstein field equations for the Schwarzschild metric and then applying a coordinate transformation. This results in a new metric that is equivalent to the Schwarzschild metric, but is expressed in terms of advanced or retarded time instead of the standard time coordinate. This transformation is a type of null coordinate transformation, meaning that it follows the path of a light ray. It is also possible to derive the transformation using the Kruskal-Szekeres coordinates, which are a different set of coordinates used to describe the Schwarzschild geometry.

4. Can the Eddington-Finkelstein transformation be applied to other geometries?

While the Eddington-Finkelstein transformation is most commonly used with the Schwarzschild geometry, it can also be applied to other geometries, such as the Kerr geometry (which describes the geometry of a rotating black hole). In general, it can be applied to any spacetime that is described by a metric that can be written in a form similar to the Schwarzschild metric. However, the transformation may look different in these cases and may require different approaches to solve.

5. What are the practical applications of solving the transformation to Eddington-Finkelstein in Schwarzschild geometry?

One major practical application of solving the Eddington-Finkelstein transformation is in the field of gravitational lensing. This effect, predicted by Einstein's theory of general relativity, causes light to be bent by massive objects such as galaxies and black holes. By using the Eddington-Finkelstein coordinates, we can accurately calculate the path of light rays near these objects and study the resulting lensing effects. This can give us valuable information about the mass and structure of these objects, as well as test the validity of general relativity. Additionally, the Eddington-Finkelstein transformation is used in numerical simulations of black hole mergers and other astrophysical phenomena, allowing us to model and understand these events more accurately.

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