Difference between revisions of "Particles' motion in general black hole spacetimes"
From Universe in Problems
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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 | + | 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 | + | 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 | + | 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 | + | 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 | + | We call it a fiducial observer (FidO) in accordance with [1]. This |
| − | notion is applied in | + | 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 | + | 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 | + | 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 | + | 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 | + | 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 18:50, 1 May 2013
In this section we use the $(-+++)$ signature, Greek letters for spacetime indices and Latin letters for spatial indices.
