Difference between revisions of "Particles' motion in general black hole spacetimes"

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In this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.
 
In this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.
  
 +
<!--
 
==Frames, time intervals and distances==
 
==Frames, time intervals and distances==
  
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$x^{i}=const$ ($i=1,2,3$). Find $h_{\mu \nu }$, $d\tau _{obs}$, the
 
$x^{i}=const$ ($i=1,2,3$). Find $h_{\mu \nu }$, $d\tau _{obs}$, the
 
condition of simultaneity and $dl^{2}$ for this case. Show that the
 
condition of simultaneity and $dl^{2}$ for this case. Show that the
corresponding formulas are equivalent to eqs. (84.6), (84.7) of \cite{lan},
+
corresponding formulas are equivalent to eqs. (84.6), (84.7) of Landau and Lifshitz [1],
 
where they are derived in a different way.
 
where they are derived in a different way.
  
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h_{ij}&=g_{ij}-\frac{g_{0i}g_{0j}}{g_{00}},  \label{hi}
 
h_{ij}&=g_{ij}-\frac{g_{0i}g_{0j}}{g_{00}},  \label{hi}
 
\end{align}
 
\end{align}
which coincides (up to the choice of the overall signature) with the spatial metric $\gamma_{ij}$ as defined in eq. (84.7) of \cite{lan}. Then, $dl^{2}=h_{ij}dx^{i}dx^{j}$ turns into eq. (84.6).
+
which coincides (up to the choice of the overall signature) with the spatial metric $\gamma_{ij}$ as defined in eq. (84.7) of [1]. Then, $dl^{2}=h_{ij}dx^{i}dx^{j}$ turns into eq. (84.6).
  
 
The condition of simultaneity (\ref{sim}) with (\ref{uco}) taken into account reads
 
The condition of simultaneity (\ref{sim}) with (\ref{uco}) taken into account reads
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dt=-dx^{i}\frac{g_{0i}}{g_{00}}
 
dt=-dx^{i}\frac{g_{0i}}{g_{00}}
 
\end{equation}
 
\end{equation}
which coincides with eq. (84.14) of \cite{lan}.
+
which coincides with eq. (84.14) of [1].
 
   </p></div>
 
   </p></div>
 
</div>
 
</div>
 +
[1] Landau L.D., Lifshitz E.M. Vol. 2. The classical theory of fields, 4ed., Butterworth-Heinemann, 1994; ISBN 0-7506-2768-9.
  
  
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d\tau _{obs}=\sqrt{-g_{00}}\;dt
 
d\tau _{obs}=\sqrt{-g_{00}}\;dt
 
\end{equation}
 
\end{equation}
which coincides with eq. 84.1 of \cite{lan}.
+
which coincides with eq. 84.1 of Landau and Lifshitz$^*$.</p>
 +
 
 +
<p style="text-align: left;">$^*$ Landau L.D., Lifshitz E.M. Vol. 2. The classical theory of fields. 4ed., Butterworth-Heinemann, 1994; ISBN 0-7506-2768-9.
 
   </p></div>
 
   </p></div>
 
</div>
 
</div>
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U_{\mu }=-N\delta _{\mu }^{0}=-N(1,0,0,0)\text{.}  \label{uz}
 
U_{\mu }=-N\delta _{\mu }^{0}=-N(1,0,0,0)\text{.}  \label{uz}
 
\end{equation}
 
\end{equation}
We call it a fiducial observer (FidO) in accordance with \cite{mb}. This
+
We call it a fiducial observer (FidO) in accordance with [1]. This
notion is applied in \cite{mb} mainly to static or axially symmetric rotating
+
notion is applied in [1] mainly to static or axially symmetric rotating
 
black holes. In the latter case it is usually called the ZAMO (zero angular
 
black holes. In the latter case it is usually called the ZAMO (zero angular
 
momentum observer). We will use FidO in a more general context.  
 
momentum observer). We will use FidO in a more general context.  
  
 
Show that a FidO's world-line is orthogonal to hypersurfaces of constant time $t=const$.
 
Show that a FidO's world-line is orthogonal to hypersurfaces of constant time $t=const$.
 +
 +
[1] Black Holes: The Membrane Paradigm. Edited by Kip S. Thorne, Richard H. Price, Douglas A. Macdonald. Yale University Press New Haven and London, 1986; ISBN 978-030-003-769-2 .
  
 
<div class="NavFrame collapsed">
 
<div class="NavFrame collapsed">
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\begin{equation}
 
\begin{equation}
 
0=U_{i}=\frac{g_{i 0}}{N}+\frac{g_{ij}N^{j}}{N},
 
0=U_{i}=\frac{g_{i 0}}{N}+\frac{g_{ij}N^{j}}{N},
\end{equation}%
+
\end{equation}
 
we find
 
we find
 
\begin{equation}
 
\begin{equation}
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E=E_{rel.}\sqrt{-g_{00}}=\frac{m}{\sqrt{1-w^{2}}},
 
E=E_{rel.}\sqrt{-g_{00}}=\frac{m}{\sqrt{1-w^{2}}},
 
\end{equation}
 
\end{equation}
which coincides with eq. (88.9) of \cite{lan}.
+
which coincides with eq. (88.9) of Landau and Lifshitz$^*$.</p>
 +
 
 +
<p style="text-align: left;">$^*$Landau L.D., Lifshitz E.M. Vol. 2. The classical theory of fields, 4ed., Butterworth-Heinemann, 1994; ISBN 0-7506-2768-9.
 
   </p></div>
 
   </p></div>
 
</div>
 
</div>
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\vec{n}_{2}).
 
\vec{n}_{2}).
 
\end{equation}
 
\end{equation}
Then, eqs. (\ref{ga}), (\ref{gw}) turn into those listed in problem 1.3 of \cite{li}.
+
Then, eqs. (\ref{ga}), (\ref{gw}) turn into those listed in problem 1.3 of Lightman et al $^*$.</p>
 +
 
 +
<p style="text-align: left;">$^*$ A. P. Lightman, W. H. Press, R. H. Price, and S. A. Teukolsky, Problem book in Relativity and Gravitation (Princeton University Press, Princeton, New Jersey, 1975); ISBN 069-108-160-3.  
 
   </p></div>
 
   </p></div>
 
</div>
 
</div>
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   <div style="width:100%;" class="NavContent">
 
   <div style="width:100%;" class="NavContent">
 
     <p style="text-align: left;">
 
     <p style="text-align: left;">
 
 
1) If $\gamma _{1}$ and $\gamma _{2}$ are finite, it follows from (\ref{ga})
 
1) If $\gamma _{1}$ and $\gamma _{2}$ are finite, it follows from (\ref{ga})
 
that $\gamma $ is also finite. It means that if $w_{1}$ and $w_{2}$ are
 
that $\gamma $ is also finite. It means that if $w_{1}$ and $w_{2}$ are
 
finite, the relative velocity $w$ of particles 1 and 2 is also finite (in
 
finite, the relative velocity $w$ of particles 1 and 2 is also finite (in
the sense that it is separated from $c$).
+
the sense that it is separated from $c$).</p>
  
 +
<p style="text-align: left;">
 
2) Let $\gamma _{1}\rightarrow \infty $ but $\gamma _{2}$ remain finite.
 
2) Let $\gamma _{1}\rightarrow \infty $ but $\gamma _{2}$ remain finite.
 
Then,
 
Then,
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velocity of particles, one of which moves with some finite speed and the
 
velocity of particles, one of which moves with some finite speed and the
 
other one almost with the speed of light is always close to the speed of
 
other one almost with the speed of light is always close to the speed of
light.
+
light.</p>
  
3) Let $\gamma _{1}\rightarrow \infty $, $\gamma _{2}\rightarrow \infty $.
+
<p style="text-align: left;">
 +
3) Let $\gamma _{1}\rightarrow \infty $, $\gamma _{2}\rightarrow \infty $.</p>
  
 +
<p style="text-align: left;">
 
3a) If $\varepsilon \neq +1$,
 
3a) If $\varepsilon \neq +1$,
 
\begin{equation}
 
\begin{equation}
 
\gamma \approx \gamma _{1}\gamma _{2}(1-\varepsilon )\rightarrow \infty ,
 
\gamma \approx \gamma _{1}\gamma _{2}(1-\varepsilon )\rightarrow \infty ,
 
\qquad w\rightarrow 1.
 
\qquad w\rightarrow 1.
\end{equation}
+
\end{equation}</p>
  
 +
<p style="text-align: left;">
 
3b) Let $\varepsilon =+1$. Then  
 
3b) Let $\varepsilon =+1$. Then  
 
\begin{equation}
 
\begin{equation}
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&w(1,CM)\approx w\frac{m^{2}}{(m_{1}+m_{2})},
 
&w(1,CM)\approx w\frac{m^{2}}{(m_{1}+m_{2})},
 
\end{align}
 
\end{align}
which agrees with formulas of nonrelativistic mechanics -- see Ch. 3, Sec. 13 of \cite{lan}.
+
which agrees with formulas of nonrelativistic mechanics -- see Ch. 3, Sec. 13 of Landau and Lifshitz$^*$.</p>
 +
 
 +
<p style="text-align: left;">$^*$ Landau L.D., Lifshitz E.M. Vol. 2. The classical theory of fields, 4ed., Butterworth-Heinemann, 1994; ISBN 0-7506-2768-9.
 
   </p></div>
 
   </p></div>
 
</div>
 
</div>
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K^{\mu }=\left( k^{\mu }\right) _{1}+\left( k^{\mu }\right) _{2}  \label{K}
 
K^{\mu }=\left( k^{\mu }\right) _{1}+\left( k^{\mu }\right) _{2}  \label{K}
 
\end{equation}
 
\end{equation}
is \emph{time-like} (unless the two photons are collinear, but then they would not collide). Dividing by the energy $\mu$ in the center of mass frame
+
is ''time-like'' (unless the two photons are collinear, but then they would not collide). Dividing by the energy $\mu$ in the center of mass frame
 
\begin{equation}
 
\begin{equation}
 
\mu ^{2}=-K^{\mu }K_{\mu }=-2k_{1\mu }k^{2\mu },
 
\mu ^{2}=-K^{\mu }K_{\mu }=-2k_{1\mu }k^{2\mu },
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\end{equation}
 
\end{equation}
 
the same as for the case of massive particle. The difference is only that for photons we
 
the same as for the case of massive particle. The difference is only that for photons we
cannot express $\nu _{loc}$ in terms of mass and velocity as $\frac{m}{\sqrt{1-w^{2}}}$.
+
cannot express $\nu _{loc}$ in terms of mass and velocity as $\frac{m}{\sqrt{1-w^{2}}}$.</p>
  
 +
<p style="text-align: left;">
 
ZAMO frame
 
ZAMO frame
 
\begin{equation}
 
\begin{equation}
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   </p></div>
 
   </p></div>
 
</div>
 
</div>
 +
-->

Revision as of 21:50, 1 May 2013


In this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.