Reissner Nordstrom Metric: Extreme Case Analysis

[latentcorpse]

Show that the extreme Reissner Nordstrom metric in isotropic coordinates is
$ds^2=-(1+\frac{M}{\rho})^{-2}dt^2 + ( 1+ \frac{M}{\rho})^2 ( d \rho^2 + \rho^2 d \Omega^2)$

I have done this using the substitution

$r=\rho + M + \frac{M^2-Q^2}{4 \rho}=\rho + M$ since $M= | Q |$ for the extreme case

The next part of the question asks me to verify that $\rho=0$ is an infinite proper distance from any finite $\rho$ along any curve of constant $t$.

Now I had a bit of a problem here. I had to also assume that $\theta, \phi$ are constant as well. This then gave

$ds=( 1 + \frac{M}{\rho} ) d \rho$
$s=\int_0^\rho (1 + \frac{M}{\rho} ) d \rho = \rho + M \ln{\rho} -M \ln{0} = \infty$ since $\ln{0}=-\infty$

Does anyone know how to do this without having to make the extra assumption that $d \Omega=0$?

Then I am asked to verify that $|t| \rightarrow \infty$ as $\rho \rightarrow 0$ along any timelike or null curve but that a timelike or null ingoing radial geodesic reaches $\rho=0$ in finite affine parameter.
I have no idea how to do this. What do I take as the form of my curve?

Thanks.

 

[mark726]

To verify that \rho=0 is an infinite proper distance from any finite \rho along any curve of constant t, we can use the fact that the proper distance d\rho is defined as the length of a curve on the surface of a sphere. This means that we can consider a curve on the surface of a sphere with constant t, \theta, and \phi. In this case, we can use the formula for the circumference of a circle on the surface of a sphere to calculate the proper distance:

d\rho = \rho d\phi

Since \phi is a constant along this curve, we can integrate from 0 to 2\pi to find the proper distance:

s = \int_0^{2\pi} \rho d\phi = 2\pi\rho

As \rho approaches 0, the proper distance s approaches infinity. This shows that \rho=0 is an infinite proper distance from any finite \rho along a curve of constant t, \theta, and \phi.

To verify that |t| \rightarrow \infty as \rho \rightarrow 0 along any timelike or null curve, we can use the fact that the metric is invariant under coordinate transformations. This means that we can choose a curve with a simple form, such as t=\rho, to calculate the limit. In this case, as \rho approaches 0, t also approaches 0. However, since we are dealing with an extreme Reissner Nordstrom metric, where M=|Q|, we know that t cannot be equal to 0. This means that t must approach either positive or negative infinity as \rho approaches 0.

For a timelike or null ingoing radial geodesic, we can use the geodesic equation to show that it reaches \rho=0 in finite affine parameter. The geodesic equation is given by:

\frac{d^2 x^\mu}{d\lambda^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda} = 0

where \lambda is the affine parameter. For an ingoing radial geodesic, we can set \theta=\pi/2 and \phi=0, and use the fact that the metric is diagonal to simplify the geodesic equation. This gives us:

\frac{d^