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PLuz
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I've been learning the Israel formalism (see original article here) for thin shells. I think I understand the formalism well and how to do the matching given two manifolds (that are solutions of the Einstein's field equations - EFE).
I've been studying several articles that use the matching formalism to construct new solutions of the EFE and I've stumbled on the article: "Tension shells and tension stars" by D. Lynden-Bell and J. Katz (see original article here). The article is only accessible through payment so I'll try to explain my doubt.
The authors consider the following setup: A static spherical shell of surface ##4\pi R^2## placed between two Schwarzschild spacetimes, with mass ##m## and ##\bar{m}<m##. The metrics on each side of the shell, in Schwarzschild coordinates, are
$$ds^{2}=-\left(1-\frac{2\, m}{r}\right)dt^{2}+\left(1-\frac{2\, m}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$$
where ##d\Omega ^2 =d\theta^2+\sin(\theta)d\varphi^2##, and
$$ds^{2}=-\left(1-\frac{2\, \bar{m}}{\bar{r}}\right)dt^{2}+\left(1-\frac{2\, \bar{m}}{\bar{r}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$$
Following Israel formalism the authors find the following expression for the surface energy density of the shell:
$$8\pi \sigma=-\frac{2}{R}\left(\bar{\epsilon}\bar{A}+\epsilon A \right),$$
where
$$A^2=1-\frac{2\, m}{r}$$
and
$$\bar{A}^2=1-\frac{2\, \bar{m}}{\bar{r}}.$$
As seen from the ##\bar{m}##-side, if ##\bar{\epsilon}=1##: ##\bar{r}>R## near the shell; or ##\bar{\epsilon}=-1##: ##\bar{r}<R## near the shell. Similarly, ##\epsilon=1##: ##r>R## near the shell; or ##\epsilon=-1##: ##r<R## near the shell.
Now the authors claim that in order to have a positive surface energy density ##\bar{\epsilon}=-1##. If they consider that "inside" the shell the spacetime is flat - ##\bar{A}=1## then by considering ##\epsilon=-1## they will have a shell on the other side of the Einstein-Rosen bridge. Now, I understand they argument and I accept it. My problem is, how can they relate the sign of the ##\bar{\epsilon}## and ##\epsilon## with the condition that the Schwarzschild radial coordinate is greater or smaller than the radial coordinate of the shell.
From my calculations their ##\epsilon## variables appear in the expression for the normal to the shell, this is, for example, as seen from the ##m##-side I get:
$$n_\mu = \epsilon (0, u^0),$$
where ##u^0## is the time coordinate of the velocity of a particle comoving (well in this case just chilling out while time passes) with the shell.
Can somebody help me?
I've been studying several articles that use the matching formalism to construct new solutions of the EFE and I've stumbled on the article: "Tension shells and tension stars" by D. Lynden-Bell and J. Katz (see original article here). The article is only accessible through payment so I'll try to explain my doubt.
The authors consider the following setup: A static spherical shell of surface ##4\pi R^2## placed between two Schwarzschild spacetimes, with mass ##m## and ##\bar{m}<m##. The metrics on each side of the shell, in Schwarzschild coordinates, are
$$ds^{2}=-\left(1-\frac{2\, m}{r}\right)dt^{2}+\left(1-\frac{2\, m}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$$
where ##d\Omega ^2 =d\theta^2+\sin(\theta)d\varphi^2##, and
$$ds^{2}=-\left(1-\frac{2\, \bar{m}}{\bar{r}}\right)dt^{2}+\left(1-\frac{2\, \bar{m}}{\bar{r}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}.$$
Following Israel formalism the authors find the following expression for the surface energy density of the shell:
$$8\pi \sigma=-\frac{2}{R}\left(\bar{\epsilon}\bar{A}+\epsilon A \right),$$
where
$$A^2=1-\frac{2\, m}{r}$$
and
$$\bar{A}^2=1-\frac{2\, \bar{m}}{\bar{r}}.$$
As seen from the ##\bar{m}##-side, if ##\bar{\epsilon}=1##: ##\bar{r}>R## near the shell; or ##\bar{\epsilon}=-1##: ##\bar{r}<R## near the shell. Similarly, ##\epsilon=1##: ##r>R## near the shell; or ##\epsilon=-1##: ##r<R## near the shell.
Now the authors claim that in order to have a positive surface energy density ##\bar{\epsilon}=-1##. If they consider that "inside" the shell the spacetime is flat - ##\bar{A}=1## then by considering ##\epsilon=-1## they will have a shell on the other side of the Einstein-Rosen bridge. Now, I understand they argument and I accept it. My problem is, how can they relate the sign of the ##\bar{\epsilon}## and ##\epsilon## with the condition that the Schwarzschild radial coordinate is greater or smaller than the radial coordinate of the shell.
From my calculations their ##\epsilon## variables appear in the expression for the normal to the shell, this is, for example, as seen from the ##m##-side I get:
$$n_\mu = \epsilon (0, u^0),$$
where ##u^0## is the time coordinate of the velocity of a particle comoving (well in this case just chilling out while time passes) with the shell.
Can somebody help me?