Schwarzchild metric - rescaled coordinates

In summary: Therefore, in summary, the Schwarzschild metric can be expressed to first order in ##M/\tilde{r}## in a simplified form by re-scaling the radial coordinate.
  • #1
radiogaga35
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Schwarzschild metric - rescaled coordinates

Hi,

I've been working through a problem (no. 14 in ch. 9) of Alan Lightman's book of GR problems. I can't understand one of the results that are stated without proof. Basically it amounts to a rescaling of coordinates.

I know that to first order in [tex]{\textstyle{M \over r}} \ll 1[/tex] (weak gravitational field), the standard Schwarzschild metric can be written

[tex]d{s^2} = - (1 - {\textstyle{{2M} \over r}})d{t^2} + (1 + {\textstyle{{2M} \over r}})d{r^2} + {r^2}d{\Omega ^2}[/tex]

in units where [tex]G = c = 1[/tex], and where [tex]d{\Omega ^2} = d{\theta ^2} + {\sin ^2}\theta d{\phi ^2}[/tex].

Lightman states that, in an "appropriate" coordinate system, the Schwarzschild metric (again to lowest order in [tex]{\textstyle{M \over r}} \ll 1[/tex]) can be written

[tex]d{s^2} = - (1 - {\textstyle{{2M} \over r}})d{t^2} + (1 + {\textstyle{{2M} \over r}})(d{x^2} + d{y^2} + d{z^2})[/tex],

where [tex]{r^2} = {x^2} + {y^2} + {z^2}[/tex].

I've been pulling my hair out trying to derive Lightman's form from the standard form I gave above. Apparently it's just a simple rescaling of the radial coordinate but I've had no luck. Any help would be appreciated!

Thanks in advance.
 
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  • #2
Indeed, for this problem it's convenient to introduce a re-scaling ##r \rightarrow \tilde{r}## which satisfies ##r = \tilde{r} (1+M/2\tilde{r})^2##, which can also be re-arranged for ##1-2M/r = (1-M/2\tilde{r})^2/(1+M/2\tilde{r})^2##. Take the derivation,\begin{align*}
dr = (1+M/2\tilde{r})(1-M/2\tilde{r}) d\tilde{r}
\end{align*}Consider the classic Schwarzschild metric\begin{align*}
g &= -(1-2M/r) dt^2 + (1-2M/r)^{-1} dr^2 + r^2 d\Omega^2 \\ \\
&= - \dfrac{(1-M/2\tilde{r})^2}{(1+M/2\tilde{r})^2} dt^2 + \dfrac{(1+M/2\tilde{r})^2}{(1-M/2\tilde{r})^2} dr^2 + r^2 d\Omega^2 \\ \\
&= - \dfrac{(1-M/2\tilde{r})^2}{(1+M/2\tilde{r})^2} dt^2 + (1+M/2\tilde{r})^4 d\tilde{r}^2 + \tilde{r}^2 (1+M/2\tilde{r})^4 d\Omega^2 \\ \\
&= - \dfrac{(1-M/2\tilde{r})^2}{(1+M/2\tilde{r})^2} dt^2 + (1+M/2\tilde{r})^4 \left\{ d\tilde{r}^2 + \tilde{r}^2 d\Omega^2 \right\}
\end{align*}To first order in ##M/\tilde{r}##, one has simply\begin{align*}
g = (1-2M/\tilde{r}) dt^2 + (1+2M/\tilde{r})\{ d\tilde{r}^2 + \tilde{r}^2 d\Omega^2 \}
\end{align*}Defining new Cartesian-ish coordinates ##(x,y,z) = (\tilde{r} \sin{\theta} \cos{\phi}, \tilde{r} \sin{\theta} \sin{\phi}, \tilde{r}\cos{\theta})## puts the metric into this familiar form\begin{align*}
g = (1-2M/\tilde{r}) dt^2 + (1+2M/\tilde{r})\{ dx^2 + dy^2 + dz^2 \}
\end{align*}
 
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1. What is the Schwarzchild metric?

The Schwarzchild metric is a mathematical representation of the spacetime curvature around a non-rotating, spherically symmetric mass. It describes the effects of gravity on the trajectory of objects in the vicinity of the mass.

2. How is the Schwarzchild metric related to Einstein's theory of general relativity?

The Schwarzchild metric is a solution to Einstein's field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy. It is a key prediction of general relativity and has been extensively tested and confirmed through observations.

3. What are rescaled coordinates in the context of the Schwarzchild metric?

Rescaled coordinates are a mathematical transformation applied to the original coordinates in the Schwarzchild metric, which allows for a more intuitive interpretation of the spacetime curvature. They also help to simplify the equations and make them easier to solve.

4. How does the Schwarzchild metric explain the phenomenon of gravitational time dilation?

The Schwarzchild metric predicts that time slows down in regions of strong gravitational fields, such as near a massive object. This is due to the warping of spacetime by the mass, which causes time to pass more slowly for an observer in that region compared to someone in a region with weaker gravity.

5. What are some practical applications of the Schwarzchild metric?

The Schwarzchild metric has been used to accurately predict the orbits of planets and other celestial bodies in our solar system. It is also crucial for understanding and studying black holes, as it provides a mathematical framework for describing their properties and behavior. Additionally, it has practical applications in fields such as astrophysics, cosmology, and space travel.

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