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.

Frames, time intervals and distances

In the next several problems we again consider the procedure of measuring time and space intervals by different observers, but in a different, more formal and powerful approach.

Problem 1

Let a particle move with the four-velocity $U^{\mu }$. It can be viewed as some observer carrying a frame attached to him. Locally, it defines the hypersurface orthogonal to it. Show that \begin{equation} h_{\mu \nu }=g_{\mu \nu }+U_{\mu }U_{\nu } \label{h} \end{equation} is (i) the projection operator onto this hypersurface, and at the same time (ii) the induced metric of the hypersurface. This means that (i) for any vector projected at this hypersurface by means of $h^\mu_\nu$, only the components orthogonal to $U^{\mu}$ survive, (ii) the repeated application of the projection operation leaves the vector within the hypersurface unchanged. In other words, $h_{\mu \nu}$ satisfies \begin{align} &h^{\mu}_{\nu}U^{\nu }=0 ; \label{1} \\ &h^{\mu}_{\nu }h^{\nu}_{\lambda}=h^{\mu }_{\lambda}. \label{2} \end{align}

Problem 2

Let us consider a particle moving with the four-velocity $U^{\mu }$. The interval $ds^{2}$ between two close events is defined in terms of differentials of coordinates, \begin{equation} ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }. \end{equation}

For given $dx^{\mu }$, what is the value of the proper time $d\tau _{obs}$ between the corresponding events measured by this observer? How can one define locally the notions of simultaneity and proper distance $dl$ for the observer in terms of its four-velocity and the corresponding projection operator $h^\mu_\nu$ ? How is the interval $ds^{2}$ related to $d\tau _{obs}$ and $dl$?


Problem 3

Let our observer measure the velocity of some other particle passing in its immediate vicinity. Relate the interval to $d\tau _{obs}$ and the particle's velocity $w$.


Problem 4

Analyze the formulas derived in the previous three problems applied to the case of flat spacetime (Minkovskii space) and compare them to the known formulas of special relativity.


Problem 5

Consider an observer being at rest with respect to a given coordinate frame: $x^{i}=const$ ($i=1,2,3$). 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 Landau and Lifshitz [1], where they are derived in a different way.

[1] Landau L.D., Lifshitz E.M. Vol. 2. The classical theory of fields, 4ed., Butterworth-Heinemann, 1994; ISBN 0-7506-2768-9.


Problem 6

Consider two events at the same point of space but at different values of time. Find the relation between $dx^{\mu}$ and $d\tau _{obs}$ for such an observer.


Fiducial observers

Problem 7

Consider an observer with \begin{equation} 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 [1]. This 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 $t=const$.

[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 .


Problem 8

Find the explicit form of the metric coefficients in terms of the components of the FidO's four-velocity. Analyze the specific case of axially symmetric metric in coordinates $(t,\phi ,r,\theta )$ with $g_{0i}=g_{t\phi }\delta _{i}^{\phi }$.


Problem 9

Consider a stationary metric with the time-like Killing vector field $\xi^\mu =(1,0,0,0)$. Relate the energy $E$ of a particle with four-velocity $u^\mu$ as measured at infinity by a stationary observer to that measured by a local observer with 4-velocity $U^\mu$.


Problem 10

Express $E_{rel}$ and $E$ in terms of the relative velocity $w$ between a particle and the observer (i.e. velocity of the particle in the frame of the observer and vice versa).


Problem 11

Show that in the flat spacetime eq. (\ref{ep}) is reduced to the usual formula of the Lorentz transformation.


Problem 12

Find the expression for $E$ for the case of a static observer ($U^{i}=0$).


Problem 13

Find the expression for $E$ for the case of the ZAMO observer and, in particular, in case of axially symmetric metric.


Collision of particles: general relationships

Problem 14

Let two particles collide. Define the energy in the center of mass (CM) frame $E_{c.m.}$ at the point of collision and relate it to $E_{rel}$ and the Lorentz factor of relative motion of the two particles.


Problem 15

Let us consider a collision of particles 1 and 2 viewed from the frame attached to some other particle 0. How are different Lorentz factors related to each other? Analyze the case when the laboratory frame coincides with that of particle 0.


Problem 16

When can $\gamma $ as a function of $\gamma _{1}$ and $\gamma _{2}$ grow unbounded? How can the answer be interpreted in terms of relative velocities?


Problem 17

A tetrad basis, or the orthonormal tetrad, is the set of four unit vectors $h_{(a)}^\mu$ (subscripts in parenthesis $a=0,1,2,3$ enumerate these vectors), of which one, $h_{(0)}^\mu$, is timelike, and three vectors $h_{(i)}^\mu$ ($i=1,2,3$) are spacelike, so that \begin{equation} g_{\mu\nu}h_{(a)}^\mu h_{(b)}^\nu =\eta_{ab},\qquad a,b=0,1,2,3. \end{equation} A vector's tetrad components are \begin{equation} u_{(a)}=u_\mu h_{(a)}^\mu,\qquad u^{(b)}=\eta^{ab}u_{(b)}. \end{equation}

Define the local three-velocities with the help of the tetrad basis attached to the observer, which would generalize the corresponding formulas of special relativity.


Problem 18

Derive the analogues of formulas (\ref{e}), (\ref{ep}) for massless particles (photons). Analyze the cases of static and ZAMO observers.


Problem 19

The ergosphere is a surface defined by equation $g_{00}=0$. Show that it is the surface of infinite redshift for an (almost) static observer.


Problem 20

Consider an observer orbiting with a constant angular velocity $\Omega $ in the equatorial plane of the axially symmetric back hole. Analyze what happens to redshift when the angular velocity approaches the minimum or maximum values $\Omega _{\pm }$.


Problem 21

Let two massive particle 1 and 2 collide. Express the energy of each particle in the centre of mass (CM) frame in terms of their relative Lorentz factor $\gamma (1,2)$. Analyze the limiting cases of ultra-relativistic $\gamma (1,2)\rightarrow \infty $ and non-relativistic $\gamma (1,2)\approx 1$ collisions.


Problem 22

For a stationary observer in a stationary space-time the quantity $\alpha =(U^0)^{-1}$ is the redshifting factor: if this observer emits a ptoton with frequency $\omega_{em}$, it is detected at infinity by another stationary observer with frequency $\omega_{det}=\alpha \omega_{em}$. For a generic observer this interpretation is invalid, however, $\alpha =ds/dt$ still determines the time dilation for this observer, and thus can still be called the same way. Express the redshifting factor of the center of mass frame $\alpha _{c.m.}$ through the redshifting factors of the colliding particles $\alpha_{1}$ and $\alpha_2$.


Problem 23

Relate the energy of a particle at infinity $E_{1}$, its energy at the point of collision in the C.M. frame $(E_{1})_{c.m.}$ and $\mu$.


Problem 24

Solve the same problem when both particles are massless (photons). Write down formulas for the ZAMO observer and for the C.M. frame.