Conformal diagrams: stationary black holes - Revision history

Cosmo All at 00:15, 4 February 2014

2014-02-04T00:15:08Z

2014-02-04T00:13:32Z

‎Problem 8: Examples.

2014-02-04T00:12:09Z

‎Problem 8: Examples.

2014-02-04T00:11:07Z

2014-02-03T23:37:22Z

‎Problem 6: Extension across horizons.

2014-02-03T23:36:05Z

‎Problem 5: Killing horizons.

2014-02-03T23:35:26Z

2014-02-03T22:56:05Z

2014-02-03T22:55:22Z

‎Problem 4: Piecing the puzzle.

2014-02-03T22:54:44Z

‎Problem 4: Piecing the puzzle.

@@ Line 370: / Line 370: @@
 {|class="wikitable" style="margin: 1em auto 1em auto;"
-|[[File:Conf-BH-ExampleGEN_vec.png|center|thumb|365px|Conformal diagram for the Reissner-Nordstrom-de Sitter black hole solution. T-regions are shaded with darker grey. The diagram is infinitely continued up and down.]]
+|[[File:Conf-BH-ExampleGEN.png|center|thumb|365px|Conformal diagram for the Reissner-Nordstrom-de Sitter black hole solution. T-regions are shaded with darker grey. The diagram is infinitely continued up and down.]]
 |}
 </p>

@@ Line 370: / Line 370: @@
 {|class="wikitable" style="margin: 1em auto 1em auto;"
-|[[File:{Conf-BH-ExampleGEN_vec.png|center|thumb|365px|Conformal diagram for the Reissner-Nordstrom-de Sitter black hole solution. T-regions are shaded with darker grey. The diagram is infinitely continued up and down.]]
+|[[File:Conf-BH-ExampleGEN_vec.png|center|thumb|365px|Conformal diagram for the Reissner-Nordstrom-de Sitter black hole solution. T-regions are shaded with darker grey. The diagram is infinitely continued up and down.]]
 |}
 </p>

@@ Line 327: / Line 327: @@
 </div>
 </div>
-<!--
 <div id=""></div>
-=== Problem 1 ===
+=== Problem 8: Examples. ===
 <div class="NavFrame collapsed">
    <div class="NavHead">solution</div>
    <div style="width:100%;" class="NavContent">
-     <p style="text-align: left;"></p>
+     <p style="text-align: left;">1) In the generic case there are two positive roots $+\infty>r_+ >r_- >0$, which give horizons, while $r=0$ is a singularity. This gives us three regions between the roots: (a) $r\in(r_+ ,+\infty)$ between the horizon and asymptotically flat infinity, thus having the same structure as the Schwarzschild exterior region and the shape of a "diamond" $\Diamond$; (b) $r\in (r_- ,r_+)$ is a T-region between the horizons (the shape is the same  $\Diamond$) and (c) $r\in (0,r_-)$ is again an R-region between the timelike singularity and the horizon. The timelike singularity turns the latter conformal diagram into a triangle, $\triangleright$ or $\triangleleft$. Gluing all the blocks together gives us the left diagram of Figs.
 </div>
 </div>
+<!--
 <div id=""></div>
 === Problem 1 ===

@@ Line 233: / Line 233: @@
 We see that $x\to -\infty$ at the horizon. It converges for $q<1$, but in this case one can show that the Kretschmann scalar $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ diverges, so this is not a horizon but a singularity.
 <br/>
-'''(c)''' $ds^2 =f \; dV\;dW$; the horizon is at $x\to -\infty$ while $t$ is fixed, so in terms of $V$ and $W$ it is split into two parts: the future horizon at $V\to -\infty$ while $W$ is fixed, and the past horizon at $W\to +\infty$ while $V$ is fixed.
+'''(b)''' $ds^2 =f \; dV\;dW$; the horizon is at $x\to -\infty$ while $t$ is fixed, so in terms of $V$ and $W$ it is split into two parts: the future horizon at $V\to -\infty$ while $W$ is fixed, and the past horizon at $W\to +\infty$ while $V$ is fixed.
 <br/>
 '''(c)''' Near the past horizon in terms of $(v,W)$ the metric takes form
@@ Line 261: / Line 261: @@
 <div id=""></div>
 === Problem 7: Infinities, horizons, singularities. ===
 Draw the parts of conformal diagrams near the boundary that correspond to the limiting process

@@ Line 197: / Line 197: @@
      <p style="text-align: left;">The formulas derived for Schwarzschild (\ref{BH-energy}) and (\ref{BH-lambda}) still work, just now $f(\rho)$ is not fixed.
 <br/>
-'''(a)'''As for almost all particles $d\tau /d\rho$ stays bounded, the proper time of reaching the horizon candidate at $\rho=\rho^\star$ is finite if and only if $\rho^\star$ itself is finite. Otherwise we would have the ``horizon'' at infinite proper distance. It can be called a remote horizon, but extension across such a surface is not possible and not needed, as spacetime in that direction is already geodesically complete.<br/>
+'''(a)'''As for almost all particles $d\tau /d\rho$ stays bounded, the proper time of reaching the horizon candidate at $\rho=\rho^\star$ is finite if and only if $\rho^\star$ itself is finite. Otherwise we would have the "horizon" at infinite proper distance. It can be called a remote horizon, but extension across such a surface is not possible and not needed, as spacetime in that direction is already geodesically complete.<br/>
 '''(b)''' The Killing vector that forms the horizon is $\xi_t =\partial_t$, its norm $|\xi_t |^2 =g_{tt}=f$. Thus $\rho=\rho^\star$ is indeed a Killing horizon by definition.</p>
 </div>
@@ Line 203: / Line 203: @@
 <div id=""></div>
 === Problem 6: Extension across horizons. ===
 Let us shift the $\rho$ coordinate so that $\rho^\star =0$, and assume that

@@ Line 171: / Line 171: @@
 {|class="wikitable" style="margin: 1em auto 1em auto;"
-|[[File:Conf-BH-SchwFull.png|center|thumb|365px|{The full conformal diagram for the (maximally extended) Schwarzschild solution.}
+|[[File:Conf-BH-SchwFull.png|center|thumb|365px|The full conformal diagram for the (maximally extended) Schwarzschild solution.]]
-\end{figure]]
 |}
 </p>

@@ Line 158: / Line 158: @@
 {|class="wikitable" style="margin: 1em auto 1em auto;"
-|<gallery widths=300px heights=300px perrow=2 caption="The puzzle pieces: two for the exterior region (above) and two for the interior region (below).}
+|<gallery widths=300px heights=300px perrow=2 caption="The puzzle pieces: two for the exterior region (above) and two for the interior region (below).">
-\end{figure">
 File:Conf-BH-SchwExtA.png
 File:Conf-BH-SchwExtB.png