Difference between revisions of "Conformal diagrams: stationary black holes"

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(Problem 6: Extension across horizons.)
 
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The resulting diagram can turn out to be either finite, as for Schwarzshild, when singularities and infinities form a closed curve enclosing the whole spacetime, or not.</p>
 
The resulting diagram can turn out to be either finite, as for Schwarzshild, when singularities and infinities form a closed curve enclosing the whole spacetime, or not.</p>
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=== Problem 1 ===
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=== Problem 8: Examples. ===
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Draw the conformal diagrams for the following spacetimes:
  
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1) Reissner-Nordstrom charged black hole:
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\begin{equation}
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f(r)=1-\frac{r_g}{r}+\frac{q^2}{r^2}, \qquad 0<q<r_{g},\qquad r>0;
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2) Extremal Reissner-Nordstr\"{o}m charged black hole
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\begin{equation}
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f(r)=\Big(1-\frac{q}{r}\Big)^2, \qquad q>0,\qquad r>0;
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\end{equation}
  
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3) Reissner-Nordstrom-de Sitter charged black hole with cosmological constant (it is not asymptotically flat, as $f(\infty)\neq 1$)
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\begin{equation}
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f(r)=1-\frac{r_g}{r}+\frac{q^2}{r^2}-\frac{\Lambda r^2}{3}\qquad q,\Lambda>0;
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analyze the special cases of degenerate roots.
 
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     <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.
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|<gallery widths=300px heights=300px perrow=2 caption="Conformal diagrams for the generic (with no degenerate horizons) Reissner-Nordstrom black hole solution on the left and the extremal (with double horizon) Reissner-Nordstrom on the right. Blocks I and III are R-regions, blocks II are T-regions. Horizons are denoted by dashed lines, infinities by solid thick lines, singularities by wriggling curves. Both diagrams are infinitely continued up and down.">
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File:Conf-BH-ExampleRNv3.png
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File:Conf-BH-ExampleRNeXv3.png
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2) Due to degeneracy the T-region is absent, instead the two kinds of R-regions are separated by double horizons; the III-blocks on the right vanish, while on the left the singularity merges into a solid vertical line.
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3) There can be up to three positive roots of $f$ in case $\Lambda>0$: $r_{-}<r_+<r_\Lambda$. Then there are four different conformal blocks:<br/>
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a) block IV for the R-region between the timelike singularity and the inner horizon $r\in (0,r_-)$: $\triangleleft,\triangleright$;<br/>
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b) block III for the T-region between the two horizons $r\in(r_- ,r_+)$; it is the same as for Reissner-Nordstr\"{o}m: $\Diamond$;<br/>
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c) block II for the R-region between the horizons $r\in (r_+ ,r_\Lambda)$; it differs from the exterior region of Reissner -Nordstrom or Schwarzschild by replacement of asymptotic infinity with another pair of horizons. The shape is the same, $\Diamond$, but the boundary now allows continuation across it in all directions;<br/>
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d) block I for the T-region between the external (cosmological) horizon and the de-Sitter-like infinity. The structure of infinity is determined by $f(r)$ for large $r$, where the $\sim 1/r$ and $1/r^2$ terms can be neglected, so effectively we have the de Sitter spacetime. Thus the infinity is spacelike and represented by one horizontal line. The block is the triangle: $\triangle,\nabla$ (the same as the upper and lower sectors of the exact full de Sitter spacetime).
  
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The full diagram is shown on Fig.
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|[[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.]]
 
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Latest revision as of 03:15, 4 February 2014


In the context of black hole spacetimes there are many subtly distinct notions of horizons, with the most useful being different from those used in cosmology. In particular, particle horizons do not play any role. The event horizon is defined not with respect to some selected observer, but with respect to all external observers: in an asymptotically flat spacetime* a future event horizon is the hypersurface which separates the events causally connected to future infinity and those that are not. Likewise the past event horizon delimits the events that are causally connected with past infinity or not. Another simple but powerful concept is the Killing horizon: in a spacetime with a Killing vector field $\xi^\mu$ it is a (hyper-)surface, on which $\xi^\mu$ becomes lightlike. We will see explicitly for the considered examples that the Killing horizons are in fact event horizons by constructing the corresponding conformal diagrams. In general, in the frame of GR a Killing horizon in a stationary spacetime is (almost) always an event horizon and also coincides with most other notions of horizons there are.

This section elaborates on the techniques of constructing conformal diagrams for stationary black hole solutions. The construction for the Schwarzschild black hole, or its variation, can be found in most textbooks on GR; for the general receipt see \cite{BrRubin}.

* Meaning the spacetime possesses the infinity with the same structure as that of Minkowski, which is important.

Schwarzschild-Kruskal black hole solution

Problem 1: Schwarzschild exterior.

The simplest black hole solution is that of Schwarzschild, given by \begin{equation} ds^{2}=f(r)dt^2 -\frac{dr^2}{f(r)}-r^2 d\Omega^2,\qquad f(r)=1-\frac{r_g}{r}, \label{SchwBH} \end{equation} where $r_g$ is the gravitational radius, and $d\Omega^2$ is the angular part of the metric, which we will not be concerned with. The surface $r=r_g$ is the horizon. Focus for now only on the external part of the solution, \[\big\{-\infty<t<+\infty,\;\; r_g<r<+\infty)\big\}.\] The general procedure of building a conformal diagram for the $(t,r)$ slice, as discussed in the cosmological context earlier, works here perfectly well, but needs one additional step in the beginning:
(a) use a new radial coordinate to bring the metric to conformally flat form; (b) pass to null coordinates; (c) shrink the ranges of coordinate values to finite intervals with the help of $\arctan$; (d) return to timelike and spacelike coordinates.

Identify the boundaries of Schwarzschild's exterior region on the conformal diagram and compare it with Minkowski spacetime's.

Problem 2: Schwarzschild interior.

The region $r\in (0,r_g)$ represents the black hole's interior, between the horizon $r=r_g$ and the singularity $r=0$. Construct the conformal diagram for this region following the same scheme as before.
(a) Which of the coordinates $(t,r)$ are timelike and which are spacelike? (b) Is the singularity spacelike or timelike? (c) Is the interior solution static?
The Schwarzschild black hole's interior is an example of the T-region, where $f(r)<0$, as opposed to the R-region, where $f(r)>0$.

Problem 3: Geodesic incompleteness and horizon regularity.

Consider radial motion of a massive particle and show that the exterior and interior parts of the Schwarzschild are not by themselves geodesically complete, i.e. particle's worldlines are terminated at the horizon at finite values of affine parameter. Show that, on the other hand, the horizon is not a singularity, by constructing the null coordinate frame, in which the metric on the horizon is explicitly regular.

Problem 4: Piecing the puzzle.

Geodesic incompleteness means the full conformal diagram must be assembled from the parts corresponding to external and internal solutions by gluing them together along same values of $r$ (remember that each point of the diagram corresponds to a sphere). Piece the puzzle.

Note that a) there are two variants of both external and internal solutions' diagrams, differing with orientation and b) the boundaries of the full diagram must go along either infinities or singularities.


Other spherically symmetric black holes

The Schwarzschild metric (\ref{SchwBH}) is the vacuum spherical symmetric solution of Einstein's equation. If we add some matter, we will obtain a different solution, e.g. Reissner-Nordstr\"{o}m for an electrically charged black hole. In all cases, spherical symmetry means that in appropriately chosen coordinate frame the metric takes the same form (\ref{SchwBH}), \begin{equation} ds^2 =f(\rho)dt^2 -\frac{d\rho^2}{f(\rho)}-r^2 (\rho)d\Omega^2 , \end{equation} but with some different function $f(\rho)$: one can always ensure that the metric functions have this form by choosing the appropriate radial coordinate\footnote{It is sometimes called the ``quasiglobal coordinate} $\rho$. The angular part $\sim d\Omega^2$ does not affect causal structure and conformal diagrams. The zeros $\rho=\rho^\star$ of $f(\rho)$ define the surfaces, which split the full spacetime into R- and T-regions and are the horizon candidates.


Problem 5: Killing horizons.

Consider radial geodesic motion of a massive or massless particle. Make use of the integral of motion $E=-u^\mu \xi_\mu$ due to the Killing vector and find $\rho (t)$ and $\rho(\lambda)$, where $\lambda$ is the affine parameter (proper time $\tau$ for massive particles)
(a) When is the proper time of reaching the horizon $\tau^\star =\tau (\rho^\star)$ finite? (b) Verify that surface $\rho=\rho^\star$ is a Killing horizon

Problem 6: Extension across horizons.

Let us shift the $\rho$ coordinate so that $\rho^\star =0$, and assume that \begin{equation} f(\rho)=\rho^{q}F(\rho),\qquad q\in\mathbb{N}, \end{equation} where $F$ is some analytic function, with $F(0)\neq 0$. Suppose we want to introduce a new radial "tortoise" coordinate $x$, such that the two-dimensional part of the metric in terms of $(t,x)$ has the conformally flat form: \begin{equation} ds^2_2 =f(\rho(x))\big[dt^2 -dx^2 \big]. \end{equation}

(a)What is the asymptotic form of relation $\rho (x)$?
(b) Rewrite the metric in terms of null coordinates \begin{equation} V=t+x,\qquad W=t-x. \end{equation} Where is the horizon in terms of these coordinates? Is there only one?
(c) Suppose we pass to new null coordinates $V=V(v)$ and $W=W(w)$. What conditions must be imposed on functions $V(v)$ and $W(w)$ in order for the mixed map $(v,W)$ to cover the past horizon and the map $(w,V)$ to cover the future horizon without singularities?

Problem 7: Infinities, horizons, singularities.

Draw the parts of conformal diagrams near the boundary that correspond to the limiting process \begin{equation} \rho\to\rho_0 ,\qquad |\rho_0|<\infty, \end{equation} under the following conditions:
(a) spacelike singularity: $f(\rho_0)>0$, $|x_0 |<\infty$;
(b) timelike singularity: $f(\rho_0)<0$, $|x_0 |<\infty$;
(c) asymptotically flat infinity: $\rho_0 =\pm \infty$, $f(\rho)\to f_0>0$;
(d) a horizon in the R region: $f(\rho)\to +0$, $|x_0 |\to \infty$;
(e) a horizon in the T region: $f(\rho)\to -0$, $|x_0 |\to \infty$;
(f) remote horizon in a T-region: $\rho_0 =\pm \infty$, $f(\rho)\to -<0$;

Remember that any spacelike line can be made "horizontal" and any timelike one can be made "vertical" by appropriate choice of coordinates.

Problem 8: Examples.

Draw the conformal diagrams for the following spacetimes:

1) Reissner-Nordstrom charged black hole: \begin{equation} f(r)=1-\frac{r_g}{r}+\frac{q^2}{r^2}, \qquad 0<q<r_{g},\qquad r>0; \end{equation}

2) Extremal Reissner-Nordstr\"{o}m charged black hole \begin{equation} f(r)=\Big(1-\frac{q}{r}\Big)^2, \qquad q>0,\qquad r>0; \end{equation}

3) Reissner-Nordstrom-de Sitter charged black hole with cosmological constant (it is not asymptotically flat, as $f(\infty)\neq 1$) \begin{equation} f(r)=1-\frac{r_g}{r}+\frac{q^2}{r^2}-\frac{\Lambda r^2}{3}\qquad q,\Lambda>0; \end{equation} analyze the special cases of degenerate roots.