Trajectory Around a Spherically Symmetric Non-Rotating Black Hole
Trajectory of a test particle or a photon around a general spherical black hole is studied and bending of light trajectory is investigated. Pseudo-Newtonian gravitational potential describing the gravitational field of the black hole is determined and is compared with the related effective potential for test particle motion. As an example, results are presented for Reissner-Nordström black hole.
pacs:04.50.-h, 04.40.Dg, 97.60.Lf
Motion of a test particle around a black hole is a very old topic
of investigation to know the behavior of the gravitational field
around black hole(for example [1-3] and any other book on general
relativity). Such investigation was started long back when within
a year of publication of general theory of relativity by
Einstein(1915)[4-7], Schwarzchild  gave a vacuum solution to
the Einstein field equations. The solution describes the geometry
of vacuum space-time outside a spherical massive body and is known
today as Schwarzchild black hole solution. At present in all text
books on general relativity there is an exhaustive study of the
motion of a test particle around the Schwarzchild black hole(see
for example [1-3]). In the present work, an investigation of the
motion of a test particle is done around a general static
non-rotating black hole. A general formula for determining bending
of light is evaluated and is tested for Schwarzchild black
hole.Gravitational field outside the black hole is approximated by
Pseudo-Newtonian(PN) gravitational potential and is compared with
the corresponding effective potential for test particle motion.
Finally all results are verified for
Reissner-Nordström black hole solution.
The paper is organized as follows: Section II deals with the
motion of a massive test particle around a general spherically
symmetric non-rotating black hole. Also effective potential,energy
and condition for circular orbit are determined for the black
hole. Further some comments are presented point wise. In section
III trajectory of a photon and bending of light is studied.
Pseudo-Newtonian gravitational potential is determined and it is
compared with effective potential in section IV. As an example all
results are deduced for
Reissner-Nordström black hole in section V. The paper ends with a short conclusion in section VI.
Ii Motion Of a Test Particle around A General Black Hole
The line element of a static spherically symmetric space time which describes a black hole can be written as,
where is at least a function and it should satisfy the following conditions so that line element (eq.1) describes a black hole solution: i) must have a zero at some positive r (say ) so that time dilation is infinite at , ii) The Kretschmann scalar () should be finite at but it diverges at i.e. the space time described by equation (1) has curvature singularity only at . Here
is the metric on unit two sphere. Suppose we consider the motion of a test particle of rest mass m around the black hole. The corresponding Lagrangian will be
where is any affine parameter.
As the Lagrangian has two cyclic co-ordinates and so the corresponding momenta must be constant. This leads to
a constant and
is also a constant.
As the space-time is spherically symmetric so the motion is always confined to a plane which for convenience chosen to be the equatorial plane . The explicit form of the momentum components are
Using the above expressions for momentum components in the energy momentum conservation relation
is called the (square of the) effective potential.
Now differentiating both sides of equation (7) we have
Also from (5) the momentum in the direction gives
So eliminating the affine parameter between and the differential equation of the trajectory of the particle in the equatorial plane is given by
Which can be written as
We can make the following conclusions on the trajectory of the
The energy of the particle should not be less than
the potential i.e. for a given the trajectory should be
such that the radial range is restricted to those radii for which
is smaller than .
If for all values of then the
particle comes from infinity and moves directly to the origin.
This is called the terminating escape orbit.
If has one positive zero then the particle
either starts from finite distance moves directly to the origin
(known as terminating bound orbit) or it may move on an escape
orbit with a finite impact parameter .
If has two positive zeros then we have two
possible cases: I. if between the two zeros then the
trajectory is called periodic bound orbit like planetary orbit or
II. if between these two zeros then the trajectory is
either an escape orbit or a terminating bound orbit.
The points where are known as turning points of the trajectory i.e. the value of r which satisfies
are turning points and Eq.(13) determines the potential
For circular orbit we have from (9) i.e. circular orbits are possible for those radial co ordinates which correspond to maximum (unstable) or minimum(stable) of the potential. Thus for circular orbit we must have
Iii Trajectory of a photon: Bending of Light
To determine the photon trajectory we shall proceed as before. Here from the energy momentum conservation relation we have
i.e. is the (square of the) effective potential. So differentiating both sides we get
Thus the differential path of a light ray is given by
where . Now for photon circular orbit we have
i.e. the radius of the circular orbit satisfies
One may note that the radius of photon circular orbit is independent of the angular momentum of photon. In particular, for a Schwarzchild black hole the radius of photon circular orbit is .For an ingoing photon choosing we have
Note that if is a constant then has the solution
(choosing the constant to be unity) a straight line. This is expected as f(r)=constant means the space time is minkowskian (having no gravitational effect) and photon trajectory will be straight line. Further we see that at large distance (small ) the gravitational field due to the black hole will be negligible so we may expand in a power series of u i.e.
So keeping up to first order in we have from
Let then . Then the above differential equation becomes
So on integration,
If we choose as the initial incoming direction of light i.e. as and as in the approximation corresponds to the smallest that photon can travel then
So the angle of deflection is , as the photon comes from infinity to the point of closest approach. Hence the total deflection would be . Therefore, considering the straight line path the net amount of deflection will be
If we take schwarzschild solution then we have and . Further, if it so happen that then we choose . Using the transformation
the net amount of deflection will be given by
Iv Pseudo-Newtonian gravitational and effective potentials
Based on a general heuristic method() ,the PN gravitational potential can be defined as
where, r is the usual radial co-ordinate and is the
general relativistic specific angular momentum i.e.
is the ratio of the conserved
angular momentum and energy per particle mass, related to the
circular geodesic in the equatorial plane.
In newtonian theory, the gravitational potential is given by
with , the newtonian angular momentum per mass of the particle moving in the circular orbit. The motivation of choosing PN potential (27) is to match the newtonian angular momentum per particle mass on a circular orbit with the general relativistic angular momentum. In the present study, the general relativistic conserved angular momentum and energy per particle mass for circular orbit are given by (from equations (13) and (14))
Hence from (27) the PN gravitational potential is
where the constant of integration ’c’ is determined from the known
result of the Schwarzchild black hole as follows:(note that c has
For Schwarzchild black hole, the well known Paczyński-Witta gravitational potential() is
substituting in eq.(31) we get and hence the PN gravitational potential for a general spherically symmetric black hole described by eq.(1) is
As for static radius gravitational potential should be zero so from (33) we have . Hence from equations (28),(29) the circular orbit of the test particle exist for radius in the range()
i.e. is the
photon circular orbit. Thus all circular orbits of the test
particle are lower bounded by the photon circular orbit and are
extended upto the static radius.
Further, from eq.(33) we see that the PN potential diverges at the event horizon (i.e.) reaches its maximum value at (where) and then decreases for i.e. the gravitational field corresponding to PN potential become repulsive for . Also if the metric(1) becomes asymptotically flat(i.e. as ) then asymptotically(,). Moreover, for central gravitational fields, if we assume that the motion of the test particle is confined to the equatorial plane then for Keplarian motion along the radial direction gives
where e stands for total PN energy per particle mass and stands for PN effective potential per particle mass having explicit form ()
where is the PN gravitational potential given by eq.(33) and is the PN angular momentum per particle mass. Thus circular Keplarian orbits are characterized by the extrema of (i.e.) and we have
which can be written as (using eq.s (28) and (29))
The corresponding expression for energy is
using it can be written as
It is to be noted that angular momentum per particle mass is same
for both General Relativity as well as for PN effective potential
We shall now examine the stability of circular orbits studied above. As stability criteria is determined by the extrema of the effective potential , so we have
for stable circular orbit and
for unstable circular orbit. Due to identical nature of both GR and PN potential theory have same criteria for stability. Further, inner and outer marginally stable circular orbits corresponds to extrema of and we have
Hence for stability one should have
Finally, from the above expressions (eq.s (36) and (38)) of
and we have the following observations:
At the event horizon() both and
diverge while and exist if
is positive definite. For example in case of
Schwarzchild-de Sitter space time() they become
finite while for our Reissner-Nordström black hole (see next section) they do not exist.
At the static radius () both
and vanish while and depend on choice of
At the photon circular orbit () both
and diverge while and
is finite there.
Thus we conclude that both in PN approach and in relativistic
approach the circular orbits are bounded from above by static
radius while bound from below by event horizon in PN approach and
that by the radius of photon orbit in relativistic approach. This
is due to the fact that we do not get photon circular orbit in PN
V An Example: Reissner-Nordström Black Hole
For Reissner-Nordström(R-N) black hole solution we have
where M being the mass and q the charge of black hole. So for horizon we have i.e.
Black hole solution exists for
and we have event horizon at
corresponds to black hole Cauchy horizon. When
then both the horizons coincide and it is the case of extremal black hole.
The static radius is given by ()
The radius at which gives the solution
One may note that
So both the static radius and lie inside the event horizon. Hence they have no physical significance for R-N black hole.
Also the radius of photon circular orbits are given by
For circular orbit the conserved angular momentum and energy per particle mass are
Variation of with respect to the variation of and for has been shown in figure 1 [variables in the figures are in any standard units].
with effective potential
PN gravitational potential and effective potentials are
The graph of for variation of both q and r with is presented in figure 2. Also in figure 3, we have shown the dependence of by drawing the graphs of for three different values of q.
Also we have
The graphical presentation of for the variation of both q and r with has been shown in figure 4. Also graphically a comparative study of (given by eq.(46)) and (given by eq.(52)) has been presented in figure 5(a) and 5(b) for two different values of q.
The marginally stable circular orbits are given by the positive root of the equation
If the cubic eq has three positive real roots()then and are respectively the radii of the outer and inner marginally stable circular orbits.
Finally, for stable circular orbit we have
In this work we give a general formulation of the trajectory of a test particle (or a photon) around any spherically symmetric black hole in four dimensional space time. We also classify the trajectories by studying the possible positive zeros of the function
is given for a given black hole we can immediately tell the
trajectories of a test particle or a photon around it. In the PN
approach physical parameters for circular orbits are evaluated and
are compared with the corresponding quantities in relativistic
treatments. Stability condition for the circular orbits are
determined and bounds of the marginally stable circular orbits are
compared in both formalism. As an example we have applied our
results for R-N black hole solution.
It is to be noted that the above analysis of the trajectory is not restricted to Einstein gravity, it can also be applied to any black hole solution in a modified gravity theory. Further our analysis can be extended to any higher dimension. For example, the metric ansatz for a dimensional black hole is written in the form
Where is the metric on unit sphere and is given by
Then due to spherical symmetry of the space time the motion of a test particle can be restricted to the equatorial plane defined by . As before energy and the angular momentum are two conserved quantities and the differential equation for the path of the test particle becomes
where or for photon or massive particle. Finally, one may note that throughout our calculations we have not used any specific gravity theory. So if we have a black hole solution given by equation (1) not only in Einstein gravity but also in any other gravity theory, the above analysis of the trajectory of a test particle is valid. Moreover from equations (56) and (57), we may conclude that above analysis of particle trajectory can be extended to any dimension of space time.
For future work, an extension of the above approach to non-spherical systems (particularly axi-symmetric) would be interesting.
The first author is thankful to Dr.Prabir Kr. Mukherjee,
Department of Physics, Presidency College, Kolkata, for valuable
help in preparing the manuscript.The first author is also thankful
to DST, Govt.of India for awarding KVPY fellowship.The authors
greatly acknowledge the warm hospitality at IUCAA, Pune where
apart of the
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