Igor at 06:26, 2 May 2013

2013-05-02T06:26:34Z

Igor at 18:50, 1 May 2013

2013-05-01T18:50:36Z

Igor at 08:40, 1 May 2013

2013-05-01T08:40:27Z

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Igor: moved Help:Editing to Particles' motion in general black hole spacetimes: !

2013-05-01T08:27:39Z

moved Help:Editing to Particles' motion in general black hole spacetimes: !

Igor: General Black Holes

2013-05-01T08:25:50Z

General Black Holes

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@@ Line 4: / Line 4: @@
 In this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.
 ==Frames, time intervals and distances==
@@ Line 1,002: / Line 1,001: @@
    </p></div>
 </div>

@@ Line 4: / Line 4: @@
 In this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.
 ==Frames, time intervals and distances==
@@ Line 140: / Line 141: @@
 $x^{i}=const$ ($i=1,2,3$). Find $h_{\mu \nu }$, $d\tau _{obs}$, the
 condition of simultaneity and $dl^{2}$ for this case. Show that the
-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.
@@ Line 171: / Line 172: @@
 h_{ij}&=g_{ij}-\frac{g_{0i}g_{0j}}{g_{00}},  \label{hi}
 \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
@@ Line 177: / Line 178: @@
 dt=-dx^{i}\frac{g_{0i}}{g_{00}}
 \end{equation}
-which coincides with eq. (84.14) of \cite{lan}.
+which coincides with eq. (84.14) of [1].
    </p></div>
 </div>
@@ Line 198: / Line 200: @@
 d\tau _{obs}=\sqrt{-g_{00}}\;dt
 \end{equation}
-which coincides with eq. 84.1 of \cite{lan}.
+which coincides with eq. 84.1 of Landau and Lifshitz$^*$.</p>
    </p></div>
 </div>
@@ Line 212: / Line 216: @@
 U_{\mu }=-N\delta _{\mu }^{0}=-N(1,0,0,0)\text{.}  \label{uz}
 \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
 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 $t=const$.
 <div class="NavFrame collapsed">
@@ Line 259: / Line 265: @@
 \begin{equation}
 =U_{i}=\frac{g_{i 0}}{N}+\frac{g_{ij}N^{j}}{N},
-\end{equation}%
+\end{equation}
 we find
 \begin{equation}
@@ Line 403: / Line 409: @@
 E=E_{rel.}\sqrt{-g_{00}}=\frac{m}{\sqrt{1-w^{2}}},
 \end{equation}
-which coincides with eq. (88.9) of \cite{lan}.
+which coincides with eq. (88.9) of Landau and Lifshitz$^*$.</p>
    </p></div>
 </div>
@@ Line 542: / Line 550: @@
 \vec{n}_{2}).
 \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></div>
 </div>
@@ Line 557: / Line 567: @@
    <div style="width:100%;" class="NavContent">
      <p style="text-align: left;">
 ) 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
 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>
 ) Let $\gamma _{1}\rightarrow \infty $ but $\gamma _{2}$ remain finite.
 Then,
@@ Line 571: / Line 581: @@
 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
-light.
+light.</p>
-) Let $\gamma _{1}\rightarrow \infty $, $\gamma _{2}\rightarrow \infty $.
+<p style="text-align: left;">
 a) If $\varepsilon \neq +1$,
 \begin{equation}
 \gamma \approx \gamma _{1}\gamma _{2}(1-\varepsilon )\rightarrow \infty ,
 \qquad w\rightarrow 1.
-\end{equation}
+\end{equation}</p>
 b) Let $\varepsilon =+1$. Then
 \begin{equation}
@@ Line 814: / Line 827: @@
 &w(1,CM)\approx w\frac{m^{2}}{(m_{1}+m_{2})},
 \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></div>
 </div>
@@ Line 915: / Line 930: @@
 K^{\mu }=\left( k^{\mu }\right) _{1}+\left( k^{\mu }\right) _{2}  \label{K}
 \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}
 \mu ^{2}=-K^{\mu }K_{\mu }=-2k_{1\mu }k^{2\mu },
@@ Line 972: / Line 987: @@
 \end{equation}
 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>
 ZAMO frame
 \begin{equation}
@@ Line 986: / Line 1,002: @@
    </p></div>
 </div>

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Particles' motion in general black hole spacetimes - Revision history

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Igor: moved Help:Editing to Particles' motion in general black hole spacetimes: !

Igor: General Black Holes