Kruzkal Coordinates Inside Horizon: Defining u', v

In summary, the conversation discusses the definition of coordinates u and v in the Kruskal representation of the Schwarzschild metric. It is established that the coordinates u' and v' are defined differently for r>2GM and r<2GM, but the line element remains the same for both regions. The conversation also mentions that Carroll's notes and Wald's book have slightly different definitions and that there may be some errors in Carroll's notes. It is suggested to look up Kruskal coordinates on Wikipedia for further clarification.
  • #1
maxverywell
197
2
We start by defining two coordinates ##u=t+r^*## and ##v=t-r^*##. Then we define another two coordinates ##u'=e^{u/4GM}## and ##v'=-e^{-v/4GM}##. But from what I have understood this is true for ##r>2GM##. How do we define ##u'## and ##v'## for ##r<2GM##?

I think it's ##u'=e^{u/4GM}## and ##v'=e^{-v/4GM}## for ##r<2GM## so that the line element is the same for ##r<2GM## and ##r>2GM##.
 
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  • #3
Yes, I looked at it. But I asked something specific and the wiki article doesn't answer it.
I'm reading Carroll's notes and there he defines u' and v' in the similar way, but only for r>2GM.
Same in Wald's book.
 
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  • #4
The extended Kruskal representation of the Schwarzschild metric ##ds^{2} = \frac{32M^{3}e^{\frac{-r}{2M}}}{r}(-dT^{2} + dX^{2}) + r^{2}d\Omega^{2}## is valid for all ##r > 0## which from the definition ##(\frac{r}{2M} - 1)e^{\frac{r}{2M}} = X^{2} - T^{2}## yields ##-1 < X^{2} - T^{2}##. This is made clear in both the Wiki article and in Wald (page 154). Compare this with the Rindler space-time.
 
  • #5
In Carroll's notes on page 186 he defines u' and v' only for r>2GM. We need also to make transformation for r<2GM and than write the line element for both regions.

Please check the page 186. I found there another mistake.
 
  • #6
Ah ok. Yes Carroll isn't being complete. The way he wrote it down, the chart has to be defined piecewise for ##r > r_g## and ##r < r_g## (it will agree continuously at ##r = r_g## of course).
 

1. What are Kruzkal coordinates and how are they used?

Kruzkal coordinates are a type of coordinate system used in general relativity to describe the geometry of spacetime. They are based on the concept of spacetime curvature, and are useful for studying black holes and other highly curved regions of spacetime.

2. How are u' and v coordinates defined inside the horizon?

Inside the horizon, u' and v coordinates are defined using the Kruskal-Szekeres coordinates, which are a type of Kruzkal coordinate system specifically designed for black holes. They are defined as a transformation of the standard Schwarzschild coordinates, and are useful for studying the properties of black holes.

3. Can u' and v coordinates be used to describe events outside the horizon?

No, u' and v coordinates are only valid for events inside the horizon. Outside the horizon, different coordinate systems such as Schwarzschild or Kerr coordinates are used to describe events in spacetime.

4. How do u' and v coordinates relate to other coordinate systems?

U' and v coordinates are related to other coordinate systems through coordinate transformations. For example, they can be transformed into Schwarzschild coordinates using specific equations. These transformations allow for the conversion of coordinates between different coordinate systems.

5. Are u' and v coordinates unique to black holes?

No, u' and v coordinates can also be used to describe the geometry of other highly curved regions of spacetime, such as the region around a rotating black hole. They are not exclusive to black holes, but they are particularly useful for studying these objects due to their unique properties.

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