A&A 437, L39-L42 (2005)
C. Chicone1 - B. Mashhoon2
1 - Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, USA
2 - Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, Missouri 65211, USA
Received 29 March 2005 / Accepted 27 May 2005
We investigate the motion of free relativistic particles relative to the ambient medium around a gravitationally collapsed system. If the relative speed exceeds a critical value given by , the gravitational tidal effects exhibit novel features that are contrary to Newtonian expectations. In particular, ultrarelativistic jet clumps moving freely outward along the rotation axis strongly decelerate with respect to the ambient medium, while ultrarelativistic particles strongly accelerate in directions normal to the jet axis. The implications of these direct consequences of general relativity for jets in microquasars and the origin of the high-energy cosmic rays are briefly mentioned.
Key words: gravitation - acceleration of particles - black hole physics
The purpose of this Letter is to discuss some of the observable consequences of the motion of relativistic particles in the field of gravitationally collapsed configurations. In particular, we point out novel features of general relativity regarding the tidal influence of a neutron star or a black hole on the motion of nearby free particles moving faster than the critical speed relative to the ambient medium surrounding the central mass. Such ultrarelativistic particles originating near the poles of the collapsed system suffer significant tidal deceleration within a cone of angle around the axis of rotation. Here represents half the angle at the vertex of the cone. Outside the cone, however, an ultrarelativistic particle experiences tidal acceleration. The situation is depicted schematically in Fig. 1 for the north pole of a Kerr black hole. Similar outflows are expected near the south pole.
|Figure 1: Schematic diagram of the critical velocity cone.|
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The solutions of the geodesic equation as well as the Lorentz force equation in black hole fields in standard Schwarzschild-like systems of coordinates have been extensively studied and many useful results have thereby been obtained (see, e.g., Chandrasekhar 1983). For the study of certain phenomena, however, it is more useful to study relative motion, i.e. motion of one particle with respect to another. This is in keeping with the spirit of the theory of relativity. It also corresponds to the physical interpretation of some astronomical situations. Consider, for instance, the Chandra X-ray images of the Crab Nebula: to reveal the accelerating rings in the equatorial plane, different images have to be compared with each other keeping certain central features "fixed'' (http://chandra.harvard.edu/photo/2002/0052/index.html). To study relative motion in an invariant setting corresponding to actual observations, it is useful to establish a Fermi normal coordinate system along a reference geodesic . The motion of a nearby free particle with respect to is then given by the geodesic equation in the quasi-inertial Fermi coordinate system (Synge 1960).
be the proper time and
be the four-velocity vector of
A triad of ideal
orthonormal gyro directions
i=1,2,3, can be
parallel propagated along
's worldline such that
is a local orthonormal tetrad frame.
Based on these local axes, the Fermi coordinates
simply provide a geodesic normal coordinate
system along the worldline of
The reference worldline is then the temporal axis
and the metric tensor in the Fermi system is given by
The geodesic equation for a free test particle
in the Fermi system can be
reduced to the general tidal equation (Mashhoon 1975,1977); however,
if we limit our considerations to the terms
given explicitly in Eq. (1),
we get the generalized Jacobi equation (Chicone & Mashhoon 2002)
Equation (4) follows from the timelike character of particle motion; that is, the requirement that leads to Eq. (4) when we use the terms in the expansion of the metric tensor given explicitly in Eq. (1). The modified Lorentz factor is positive and approaches infinity if the timelike geodesic approaches a null geodesic. It follows from Eq. (4) that V2 < 1 along the reference geodesic; elsewhere in the Fermi frame, however, V2 is simply constrained by the requirement that the right-hand side of Eq. (4) be positive.
The quasi-inertial Fermi coordinate system has been widely used in the theory of general relativity; for example, it is employed by Ehlers & Rindler (1997) to study the local bending of light in a gravitational field.
The generalized Jacobi Eq. (3) in the case
under consideration takes the form
|Figure 2: Plot of the Lorentz factor versus T/(GM) based on integration of Eq. (14) with initial data X=0 at T=0 with corresponding respectively to , 10, 15, 20, 25, 40, 70 and 100. In this plot a/(GM)=1and r0/(GM)=15. The graph illustrates acceleration of the particle such that essentially approaches infinity at , 9, 7, 5, 3 and 2.6 for , 20, 25, 40, 70 and 100, respectively.|
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The breakdown of our simple theoretical scheme as implies that the influence of such a particle on the background spacetime cannot be neglected. A more complete treatment should take this influence into account, thereby moderating the singularity that we have encountered. Moreover, the tidal energy exchange resulting in the deceleration/acceleration phenomena must be augmented to take due account of gravitational radiation energy emitted by the particles. This energy, as well as the corresponding electromagnetic radiation energy emitted by the electrically charged particles, is in the form of long-wavelength radiation of frequency proportional to . The tidal acceleration of charged particles could be strongly affected by the electromagnetic field configuration around the source.
It is interesting to consider the exchange of energy between the particle and the gravitational field. Einstein's principle of equivalence prevents a local (i.e. pointwise) description of gravitational energy. From the standpoint of observers at rest with the ambient medium, however, an initially ultrarelativistic particle would lose energy to the gravitational field along the jet direction, but would gain tidal energy propagating outward in the equatorial plane; thus, the gain in gravitational energy by the accelerating particles could be essentially balanced with the loss of gravitational energy by the particles decelerating along the jets.
For an invariant characterization of these tidal phenomena, consider the
class of fundamental observers
that are at rest with
with respect to the Fermi coordinate system.
W =(-g00)-1/2. Let
be the Lorentz factor of
measured by the static observers
which are in general accelerated.
It turns out that
Ultrarelativistic particles are expected to be produced near a highly magnetized rapidly rotating neutron star or as a consequence of the complicated accretion phenomena in the vicinity of an active black hole (Punsly 2001; Guthmann et al. 2002). For our purposes, we imagine an abundance of such particles near the poles of the collapsed system. Extensive numerical studies of Eqs. (10)-(12) indicate that the deceleration along the rotation axis extends to a cone with an opening half-angle given by ; moreover, motion outside the cone corresponds to tidal acceleration away from the collapsed system.
Tidal acceleration of ultrarelativistic particles is maximum in the equatorial plane and decreases away from it, turning to deceleration at an inclination angle of about . Our results regarding tidal acceleration appear to be consistent with recent Chandra X-ray studies of four neutron stars: Crab Pulsar, Vela Pulsar, PSR B1509-58 and SNR G54.1+0.3 (http://chandra.harvard.edu/photo/chronological.html). In each case, there is considerable activity near the equatorial plane of the central source. Moreover, a detailed analysis of the Chandra X-ray images of the Crab Nebula suggests that the equatorial activity is somewhat more energetic than the activity along the jets (Mori et al. 2004), which is consistent with the tidal acceleration/deceleration of ultrarelativistic particles studied in this work. The same appears to be the case for the Vela Pulsar (http://chandra.harvard.edu/photo/2000/vela/index.html).
The deceleration/acceleration phenomena occur relative to the ambient medium. That is, the geodesic orbits of the exterior Kerr spacetime under consideration in this work would not appear to have any extraordinary features when considered from the standpoint of the static inertial observers at spatial infinity. The situation is different, however, in the Fermi system.
The ultrarelativistic particles that are created near a gravitationally collapsed system or in the accretion process around the system are tidally decelerated in a cone around the rotation axis of the collapsed system and appear as confined relativistic outflows from the system. These correspond to jets from some neutron stars (http://chandra.harvard.edu/photo/chronological.html) or X-ray binaries (microquasars) in our galaxy that have been extensively studied (Guthmann et al. 2002; Fender 2004). On the other hand, the extremely relativistic particles that result from tidal acceleration outside the cone may interact with the ambient medium within the Fermi system thereby transferring their tidal energy to the ambient particles that may escape from the system altogether and can appear as extremely energetic cosmic rays. Such neutron star or microquasar particles may account for the inferred ultrahigh energy primary cosmic rays with energies above , since the corresponding extragalactic particles would collide with the cosmic microwave background photons resulting in photopion production and pair creation (Greisen 1966; Zatsepin & Kuzmin 1966). It would therefore be interesting to search for any correlation between the directionality associated with the highest energy cosmic rays and the distribution of certain neutron stars and microquasars in our galaxy using the Pierre Auger Observatory.