On the trajectories of null and timelike geodesics in different wormhole geometries
Eur. Phys. J. C
On the trajectories of null and timelike geodesics in different wormhole geometries
Anuj Mishra 0 1
Subenoy Chakraborty 0
0 Department of Mathematics, Jadavpur University , Kolkata 700032 , India
1 National Institute of Technology , Rourkela, Odisha 769008 , India
The paper deals with an extensive study of null and timelike geodesics in the background of wormhole geometries. Starting with a spherically symmetric spacetime, null geodesics are analyzed for the MorrisThorne wormhole (WH) and photon spheres are examined in WH geometries. Both bounded and unbounded orbits are discussed for timelike geodesics. A similar analysis has been done for trajectories in a dynamic spherically symmetric WH and for a rotating WH. Finally, the invariant angle method of Rindler and Ishak has been used to calculate the angle between radial and tangential vectors at any point on the photon's trajectory.

1 Introduction
In general relativity, a wormhole (WH) is considered to be
a tunnel through which two distant regions of spacetime can
be connected [
1
]. Long back in 1916, Flamm [
2
] introduced
the idea of wormhole, analyzing at that time the recently
discovered Schwarzschild solutions. In 1935, Einstein and
Rosen [
3
] constructed a WH type solution considering an
elementary particle model as a bridge connecting two
identical sheets. This mathematical representation of space being
connected by a WH type solution is known as an “Einstein–
Rosen bridge”. Wheeler [
4,5
] in the 1950s considered WHs
as objects of quantum foam connecting different regions of
spacetime and operating at the Planck scale. Subsequently,
using this idea, Hawking [6] and collaborators introduced
the idea of Euclidean wormholes. But these types of WHs
are not traversable and, in principle, would develop some
type of singularity [
7
]. However, these hypothetical shortcut
paths, i.e., traversable WHs, have been rekindled by the
pioneering work of Morris and Thorne [
8
] which is considered as
the modern renaissance of WH physics. Subsequently, it was
claimed that there is no strong ground [
9,10
] for the energy
conditions and hence one considered a WH, with two mouths
and a throat, to be an object of nature, i.e., an astrophysical
object.
On the other hand, in general relativity, WH physics is a
specific example where the matter stressenergy tensor
components are evaluated from the spacetime geometry by
solving Einstein’s field equations. But for a traversable WH, the
stressenergy tensor components so obtained always violate
the null energy condition [
1,8
]. As the null energy condition
(NEC) is the weakest of all the classical energy conditions,
its violation signals that the other energy conditions are also
violated. In fact, they violate all the known pointwise energy
conditions and averaged energy conditions, which are
fundamental to the singularity theorems and theorems of classical
black hole thermodynamics. Generally, it is believed that a
classical matter obeys energy conditions [11] but, in fact, it
is known that they also get violated by some quantum fields
(namely as regards the Casimir effect and Hawking
evaporation [
12
]). Further, for a quantum system in classical gravity,
it is found that the averaged weak or null energy condition
(ANEC), which states that the integral of the energy density
as measured by a geodesic observer is nonnegative, could
also be violated by a small amount [
13,14
].
Finally, it is worth to mention a few important
dynamical WH solutions. Hochberg and Visser [
15
] and Hayward
[
16
] independently formulated the dynamical WH solutions,
choosing a quasilocal definition of the WH throat in a
dynamical spacetime. Accordingly, the WH throat is a
trapping horizon [
17
] of different kind, but again matter in both of
them violates the NEC. On the other hand, Maeda et al. [
18
]
have developed another class of dynamical WHs
(cosmological WHs) which are asymptotically FRW spacetimes with big
bang singularity at the beginning. This class of WHs contain
matter which not only obey the NEC but also the dominant
energy condition everywhere. These two classes of
dynamical WHs are distinct from the geometrical point of view.
and on the other to a Schwarzschild manifold with negative
mass parameter ‘m¯ ’, with the condition −m¯ > m. As a
consequence, there is attraction of particles on one side, while
there is repulsion on the other side (with higher strength).
The present work presents a detailed investigation of both
timelike and null geodesics both for static and dynamical
WHs. The paper is organized as follows: Sect. 2 deals with
static spherical WHs in which null and timelike geodesics
are studied in great detail. A similar geodesic analysis is
presented for a dynamical WH in Sect. 3 and for a rotating
WH in Sect. 4. Section 5 uses the invariant angle method
of Rindler and Ishak to calculate the angle between radial
and tangential vectors at a point on the photon’s trajectory.
Finally, the paper ends with a short discussion and concluding
remarks in Sect. 6. Throughout our analysis, we have chosen
to work with wormholes whose material extends all the way
from the throat out to infinity.
2 Trajectories in a spherically symmetric and static
geometry
The metric for a general spherically symmetric and static
metric can be written as [
39,40
]
ds2 = − A(r )dt 2 + B(r )dr 2 + C (r )d 2
where
lim A(r ) = rl→im∞ B(r ) = 1 and
r→∞
lim C (r ) = r 2.
r→∞
An important relation between momenta oneforms of a
freely falling body and the background geometry is given
by the geodesic equation [39],
For the former one, the WH throat is a 2D surface of
nonvanishing minimal area of a null hypersurface, while for the
latter one, there is no past null infinity due to the initial
singularity. Hence, the WH throat is defined only on a spacelike
hypersurface and the spacetime is trapped everywhere
without any trapping horizon [
19
]. Recently, Lobo et al. [
20–22
]
formulated wormhole solutions which are dynamically
generated using a single charged fluid. Also, dynamical WHs are
considered with a twofluid system [
23,24
], for a matter
distribution relevant to present day observations [25] and using
the mechanism of particle creation [
26
]. Then for evolving
WH,1 one may refer to Refs. [
27–31
].
The particle motion in wormhole spacetimes is an
important issue related to traversable WHs. It is interesting to
examine whether a timelike or null geodesic can tunnel
through the throat in the case Cataldo et al. [
32
] studied,
of the motion of test particles in the background of zero
tidal force Schwarzschildlike WH spacetime. They showed
that particles moving along the radial geodesics reach the
throat with zero tidal velocity in finite time, while the
particle velocity reaches maximum at infinity if it travels along
a radially outward geodesic. For nonradial geodesics on the
other hand, the particles may cross the throat with some
restrictions. Olmo et al. [
33
] carried out a detailed
investigation of the geodesic structure for three possible WH
configurations, namely: the Reissner–Nordströmlike WH, the
Schwarzschildlike WH and the Minkowskilike WH. They
have shown that it is possible to have geodesically complete
paths for all these WH spacetimes. Culetu [
34
] examined
both timelike and null geodesics for a WH belonging to the
Planck world (WHs whose throat size is of the order of the
Planck length l P ) where quantum fluctuations are supposed
to exist and the spacetime smoothness seems to break down.
Muller [
35
] also studied null and timelike geodesics in the
WH configuration using elliptic and Jacobian integral
functions. He showed that it is possible to connect two distant
events geodesically. Regarding a geodesic study in nonstatic
WHs, recently Chakraborty and Pradhan [
36
] have studied
the geodesic structure of the rotating traversable Teo WH.
Also, Nedkova et al. [
37
] discussed the shadow of a class
of rotating traversable WH in the framework of general
relativity. They showed that the images depend on the
angular momentum of the WH and the inclination angle of the
observer. Finally, it is worthy to mention the work of Ellis
[
38
]. He constructed a static, spherically symmetric,
geodesically complete, horizonless spacetime manifold with a
topological hole (drainhole) at its center by coupling the geometry
of Schwarzschild spacetime to a scalar field. It is found that
on one side of the drainhole the manifold is asymptotic to a
Schwarzschild manifold with positive mass parameter ‘m’,
1 These are not as popular as static WHs and also they are not well
understood.
(1)
(2)
(3)
d pβ
dλ
1
= 2 gνα,β pν pα
where λ is some affine parameter. This relation tells us
immediately that if all the components of gαν are independent of
x β for some fixed index β, then pβ is a constant along any
particle’s trajectory, i.e., a constant of motion. Now, if we
work in the equatorial plane by setting θ = π/2, then, in Eq.
(1), all the gαβ become independent of t, θ , φ (cyclic
coordinates). That means that we can find the respective Killing
vector fields δαν ∂ν with α as cyclic coordinates. Now, since
pt and pφ are constants of the motion, we will set them as
pt = −E , pφ = L
where E is the energy and L is the angular momentum of the
photon or a particle as measured by observers at
asymptotically flat regions far from the source. Thus, we get
pt = t˙ = gtν pν = AE(r ) ,
pφ = φ˙ = gφν pν = C L(r )
and we let pr = ddλr = r˙
where the dot represents the derivative w.r.t. some affine
parameter λ.
2.1 Null geodesics
Now, for null geodesics, we have pα pα = 0. Thus,
Using Eqs. (4) and (5), we can write the equation of the
photon trajectory in terms of the impact parameter, μ =
L/E , as
dr
dφ
2
C 2(r ) 1 μ2
= μ2 B(r ) A(r ) − C (r )
.
If we assume that the geometry is caused by a source of
radius rs , then the photon coming from infinity will not hit
the surface if there exists a solution ro > rs for which r˙2 = 0.
We then call ro the distance of closest approach or the turning
point. In that case,
L2 C (ro)
E 2 = A(ro)
{if B−1(r ) = 0 for any r > rs }.
The impact parameter then becomes
L
μ = E = ±
C(ro)
A(ro)
Now, if a photon coming from the polar coordinate
limr→∞(r, −π/2 − α/2) passes through a turning point at
(ro, 0) before approaching the point limr→∞(r, π/2 + α/2),
then this α, which is a function of ro, is what we refer to as
the deviation/deflection angle, given by Ref. [
41
],
∞
ro
√B(r )dr
√C(r )√[ A(ro)/ A(r )][C(r )/C(ro)] − 1
However, it is possible that a photon might get trapped
in a sphere of constant r and thus may not approach
limr→∞(r, π/2+α/2). In that case, the integral will diverge.
Such spheres are called photon spheres; they are discussed
in Sect. 2.3.
2.2 Morris–Thorne wormhole
The Morris–Thorne wormhole metric [
8
] is given by
where (r ) is the redshift function and b(r ) is the shape
function of the wormhole for which b(r ) ≤ r and equality
holds only at the throat. Both the functions are such that they
also satisfy asymptotic flat conditions. Thus, the equation of
the trajectory for null geodesics, Eq. (6), becomes
However, note that the coordinate r cannot be used for
describing the whole spacetime since it accounts for a
coordinate singularity at the throat and is therefore valid for
describing geometry only at one side of the throat. Thus,
for geodesics that actually reach and pass through the throat,
one should not use this formula for the trajectory equation.
Instead, one can always work with the proper distance (l)
which must be valid everywhere and throughout the
wormhole. As an example, for the metric given in Eq. (1), we have
dl = √B(r )dr , thus
r
bo
l(r ) = ±
dr
B(r )
where, by definition, this proper radial distance is positive
for the upper universe, negative for the lower universe and is
zero at the throat. Using this, Eq. (6) can be generalized for
wormholes as
(4)
(5)
(6)
(7)
(8)
(9)
where we have substituted r in terms of l, which, in principle,
could be obtained by inverting Eq. (13) to get r ≡ r (l). In this
paper, however, we will mostly be interested in the behavior
of trajectories on one side of a throat and so we will mostly
work with r for our convenience. Now, using Eq. (10), the null
geodesics coming from infinity and not reaching the throat
get deflected by the angle
∞
ro
ro dr
r [r − b(r )][exp{2 (ro) − 2 (r )}r 2 − ro2]
as given in Fig. 1. For the Schwarzschild Metric, the deviation
angle becomes
It turns out that, for stationary observers in the r, θ , φ
system, the radial tidal forces can be made to vanish if we
have (r ) = 0, which we can do by simply choosing (r ) ≡
0, say. This condition gives us a simple class of solutions and
corresponds to precisely zero tidal forces. Using Eq. (8), it
can also be deduced that, for these wormholes, light can reach
the throat only if μ < bo, where μ is the impact parameter
and bo is the radius of throat. Thus, for these ultrastatic
wormholes, the light deflection angle becomes
Now, for an asymptotically flat geometry, a good choice
for the function b(r ) is,
b(r ) = bo
bo
r
n−1
= b0nr 1−n, n > 0
where bo = b(rt ) = rt corresponds to the throat radius and
n=2 gives us the famous Ellis wormhole [
38
]. We will call
this parameter ‘n’ the shape exponent.
Now, the deflection angle for this choice of b(r ) in terms of
ro and n becomes
α(ro, n) = −π + 2
∞
ro
r ( n2 −1)ro dr
(r n − bon)(r 2 − ro2)
.
(18)
(15)
(16)
(17)
⇒ α(ro) = −π + 2
∞
ro
(ro/r )dr
2M
r 2 1 − ro
2M
− r02 1 − r
It turns out that, for the Ellis wormhole, we can write the
exact expression for α [
42
] as
α(ro) = π
∞
n=1
(2n − 1)!! 2 bo
(2n)!! ro
2n
where we have written μ = ro. In the weakfield regime
where μ << bo, the deflection angle becomes
π bo 2
α(ro) ≈ 4 ro
9π bo 4
+ 64 ro
+ O
bo 6
ro
Now, we wish visualize the geometry of such a wormhole,
i.e. Eq. (11) [
8
]. Since the geometry is spherically symmetric
and static, we can confine our attention to an equatorial slice
through our wormhole at any instant in time. Then the metric
becomes
ds2 =
1− r
dr 2 +r 2dφ2 {∵ θ = π/2, t = const.}
We can see how the deflection angle depends upon the
value of shape exponent and the distance of closest approach
Now to visualize this 2D geometry, we can embed it into a
higher dimensional space of IR3, i.e., ordinary 3D Euclidean
Thus, after integrating, the embedding function becomes
[
43
]
z(r, n) = i r 2 F1
1 1
,
2 n
n + 1
,
n
, (r/bo)n
where 2 F1 represents the hypergeometric function. Note that
due to spherical symmetry, there is no dependence on the
coordinate φ. Thus, we can easily visualize the geometry
through a 2D plot as in Fig. 2 which represents the embedding
function for various values of the shape exponent and the
throat radius.
We can also study how proper length l depends upon the
radial coordinate r and the shape exponent n. If we keep
bo = 1, then we have
l(r, n) = ±
1
r
1
1 − r n
2.3 Photon spheres
A photon sphere is a location where the curvature of
spacetime is such that even null geodesics can travel in circles. In
other words, a photon sphere is a region where both l˙ and l¨
vanish for a photon. We will calculate the possibility for such
a region considering a general static metric as given in Eq.
(1). Rewriting Eq. (5) in terms of l gives
Thus, a photon sphere will exist at a location where
L2
l˙2 = A(r ) − C (r ) .
dC/dl
C
=
d A/dl
A
Another way of looking at it is that it is the region for which
the deviation angle of a photon diverges. Thus, for a Morris–
Thorne wormhole, Eq. 15 implies that the throat itself is a
photon sphere which is evident from Fig. 1a. Also, by Eq.
30, any other place which satisfies the following condition
contains a photon sphere:
r (r) = 1 {condition for photon sphere to exist at any r > bo}.
2.4 Timelike geodesics
For timelike geodesics, we have pμ pμ = −m2 where m is
the mass of the particle. If we define the quantities E and L as
the energy per unit mass (E /m) and the angular momentum
per unit mass (L/m), respectively, then
(29)
(30)
(31)
increases. Note that the throat radius has been set to bo = 1. a The plot
shows how the proper length depends upon the radial coordinate r and
the shape exponent n. b The plot shows how dr/dl varies with r
L 2
A(r ) − C (r ) − 1 .
Notice that a timelike particle always reaches the throat
with zero radial velocity, independent of the value of the
impact parameter μ (∵ B(r )−1 = 0 at throat). Now, the
general equation of the trajectory becomes
dl
dφ
2
where μ = L /E and l is the proper length. If we differentiate
Eq. (32) with respect to the affine parameter, we obtain for
the second derivative of the radial coordinate
B (r ) 2 1 L 2 E 2
r¨ = − 2 B(r ) r˙ + 2 B(r ) C (r )2 C (r )− A(r )2 A (r ) . (34)
The dependence on L 2 is a consequence of spherical
symmetry, as it tells that the orientation of the angular momentum
does not affect the radial acceleration. For a Morris–Thorne
wormhole, it becomes
1
r¨ = 2
1 −
+ 1 −
b(r ) −1 b(r ) − r b (r )
r r 2
b(r ) L 2 E 2
r r 3 − e2 (r )
r 2
˙
(r )
For a particle with zero initial velocity (r˙ = L = 0),
r¨ ∝ − (r ). Thus in ultrastatic wormholes, a particle stays
at the same position if not given any initial velocity. Also, at
the throat, r˙ = 0 also implies r¨ = 0. Thus, a particle reaching
the throat not only attains a zero radial velocity but also has
vanishing radial acceleration. The expression for the radial
acceleration in an ultrastatic wormhole with shape exponent
reduces to
nbon
r¨ = 2r n+1
bon
1 − r n
L 2
The case n = 1 is studied in detail in Ref. [
32
]. However,
note that if L = 0, then r¨ ≡ 0 for E = 1, while on the
other hand, r¨ > 0 for E > 1. Thus, this family of geometries
correspond to repulsive gravity.
2.4.1 Unbounded orbits
If the particle falling from infinity does not hit the throat, it
will get deflected after approaching a closest distance of ro,
where ro is then the real solution of the equation
Using Eqs. (33) and (37), we can then write
[μ/C (r )]√B(r )
Now, if the particle does not fall into the throat, the total
deflection angle (α) for a particle falling from infinity will
be,
1 −
For ultrastatic wormholes, we can simply write
dl 2
dλ
= E 2 −
L2
r 2 + 1
= E 2 − V 2(r (l))
where dl is the differential proper length and V 2(l) can be
thought of as the effective potential. This case is studied in
detail in Ref. [
43
]. However, we will choose a different form
of e2 (r) and will try to study the trajectories it allows for.
Let us define
e2
= 1 −
where (r ) is a continuous function which is significant only
near the throat and is vanishingly small otherwise. Now for
this choice, we can write a simplified form of Eq. (41), for
distances far from the throat, thus:
r˙2 = E 2 − 1 −
Note that we should not choose (r ) ≡ 0, because then the
throat of the wormhole will be a horizon which will make
the wormhole nontraversable.
Now, we can define an effective potential,
Therefore
r˙2 = (E + V )(E − V ),
(46)
(48)
which tells us immediately that the allowed region for a
particle with energy E (as measured at infinity) can be determined
from the inequality
V (r ) < E
{∵ V (r ) = V (r ) as V (r ) > 0 ∀ r > bo}.
(47)
In other words, the radial range of a particle, depending
upon its conserved energy E , is bounded within those radii
for which V is smaller than E .
Also note that, since r > bo, we must have
lim V 2(r ) = 0 and
r→ bo
lim V 2(r ) = 1.
r→∞
It is important to note that any bound orbit that exists
around a spherically symmetric source can be of only two
types. It can be either a circular orbit (stable or unstable) or
an orbit that oscillates around the radius of a stable circular
orbit [
44
]. So, let us study the possibility of circular orbits in
our geometry.
Circular orbits
Now for circular orbits, we require that both r˙ and r¨ vanish
for at least some r. Therefore
Condition I:
Condition II:
r˙ = 0 ⇒ E = V ,
d
r¨ = 0 ⇒ dr V 2(r ) = 0.
It means, for circular orbits, that the energy of a particle
should be an extremum of the effective potential. Precisely, if
the conserved energy corresponds to a maximum or a saddle
point of the potential, then it will be an unstable orbit, while
if it corresponds to a minimum of the potential, it will be a
stable orbit. Now, if we choose b(r ) = bonr 1−n as described
in Eq. (17), we can write
d
0 = dr V 2(r ) = (nbonr −n−1)
L2
r 2 + 1
+
where we have defined
f (r )
V 2 (r ) = r n+3 .
Also note that
n nr 2 n
f (bo) = −bo 2L2 + 2
< 0.
f (r ) = r 2 1 − Lbo22
If L > bo, then
rc =
√
2bo
1 − bo2/L2
> bo.
Now if rc is some real root of Eq. (49), then a circular orbit
is possible only when
Condition III: rc > bo.
Now, let us study what kind of solutions Eq. (49), i.e. f (r ) =
0, has. First we write,
f (r ) = nr r n−2 − Lbon2
, f (r ) = n(n − 1)r n−2 − L2
Let r p be the point where the first derivative vanishes.
Then
f (r p) = 0
f (r p) = nbon (n − 2).
nbon
(53)
(54)
(55)
(56)
Thus the behavior of f (r) is such that it will start from a
negative value at r = 0 and will grow further negative with
the increase in r until it hits a turning point at r = r p, after
which it increases monotonically. Thus, it can be inferred
that f (r ) will have only one positive real root, rc say. Then
it is clear that f (r ) > 0 ∀ r > rc. Thus, from Eq. (51), we
can say that this root must also satisfy condition III. Since
there is only one turning point of f (r ), it is obvious from Eq.
(48) that this corresponds to the maximum of the potential,
in which case it will always lead to an unstable orbit.
Hence, for n > 2, there will be only one unstable circular
orbit for a particular value of E and L.
Case II: n = 2 For n=2, f (r ) becomes
(57)
(58)
(59)
As we can see, if the conserved angular momentum L
is larger than the throat radius, then we definitely have one
root, rc, which satisfies condition III. By the same argument
as above, it is clear that it must correspond to a maximum
of the potential which can only lead to an unstable circular
orbit.
Hence, for n = 2, there is a possibility of only one unstable
circular orbit depending upon L and bo and E .
Case III: 0 < n < 2 For this case, we have
lim f (r ) < 0 , and lim f (r ) < 0.
r→0 r→∞
Thus, it can be seen that only when f (r p) ≥ 0 we have
real roots. Precisely, when f (r p) = 0 we have one positive
real root, while if f (r p) ≥ 0 we have two positive real roots.
Now, we can write
n
f (r p) = r np−2 1 − 2
2
r p − L
Since we want f (r p) ≥ 0, we can write
where equality holds for f (r p) = 0 and inequality for
f (r p) > 0. Note that when f (r p) = 0, then, by using Eqs.
(50) and (54), we also have d2V 2(r p)/dr 2 = 0. Thus, it will
correspond to a saddle point of the potential at r = r p. For
the condition given in Eq. (59), we always have r p > bo
which means condition III is also satisfied. So, we will have
the possibility of one unstable circular orbit for this case.
Meanwhile if f (r p) > 0, we will have two real roots.
Now, from Eq. (48), it is clear that the smaller of these roots
will correspond to a local maximum and the larger root will
correspond to a local minimum. Using Eq. (51) and the
condition in Eq. (59), it can be inferred that both of these roots
will also satisfy condition III.
Hence, for 0 < n < 2, there is a possibility for one
unstable circular orbit or a combination of one unstable circular
orbit and a stable circular orbit. Note that this is the only case
where we have the possibility of stable circular orbits. It is
also interesting to note that the Schwarzschild geometry, for
which n = 1, lies in this case.
As mentioned before, any bound orbit, which is not circular,
is possible only when it oscillates around the radius of a
stable circular orbit. Thus, we can say that we surely have no
noncircular bound orbits when n ≥ 2 for any E and L of
the particle.
For the Schwarzschild case (n = 1), we can substitute bo =
2M (Schwarzschild radius) with r satisfying r > bo. Then
the condition for circular orbit, Eq. (59), becomes
L ≥ (2M )
1−2
2 − 1 2(1)
2 + 1
⇒
L ≥
√12M.
(60)
which we know is the correct limit for the Schwarzschild
case. Thus, what we have done in this section is a general
treatment for any shape exponent. But physically, we can
say that [
45
]
Vg(r ) = −(bo/r )n
where Vg(r ) is the gravitational potential for a
Newtonianlike gravitational force given by
Fg(r ) = mr¨ = −(nbon )r −(n+1) = −kr −(n+1)
(61)
So, all our conclusions are valid for this interesting
analogy as well. Hence, we have proved, using GR, that in a
universe where “Newtonianlike gravity” dies out as r −3 or
faster, no stable orbits are possible. In other words, the
existence of planets will itself be almost impossible.
Time period of circular orbits
We have seen that there is at least one unstable circular orbit
possible for any value of the shape exponent. So now, we will
try to calculate the time period of a circular orbit of a particle
at a distance rc in terms of n and bo. Using Eq. (49) and the
fact that E = V (r ), we can write
Thus, the total time period of a revolution for a circular orbit
becomes
t
˙
φ˙ ≈ rc
2rcn
nbon
.
T ≈ 2πrc
2rcn 1/2
nbon
.
It means that the velocity required for a satellite to be set
in an orbit of radius rc around a wormhole is given by
v ≈
As we can see, the time period is always proportional to the
radius of the orbit but is inversely proportional to the radius
of the throat. The latter condition signifies that increasing the
throat radius can be thought of as keeping the throat radius
fixed but decreasing the radius of the circular orbit itself; in
which case it is logical that its time period will decrease.
Again, we can recognize Eq. (65) as the generalization of
Kepler’s third law for an attractive force law given by Eq.
(61). It can be proved immediately by scaling arguments if
we put, say, r = λr and t = μt in Eq. (61), to get μ2 ∝
λn+2 (Ref. [
46
]). So, we can retrieve Kepler’s third law in its
original form by putting n=1, so that
T 2 ∝ rc3.
T ≈
2πrc2
bo
.
Choice of (r )
For n=2, the time period becomes
(62)
(63)
As we have mentioned, Fig. 4 is not valid near the throat as
it does not consider the significance of (r ) in that region.
Now, we shall try to guess a physically reasonable form for
(r ). First, let us consider the problem of tidal forces. For a
spaceship whose one end is at r = a and the other end is at
r = b, the magnitude of the tidal force experienced by the
ship would just be the difference between the forces at r = a
and r = b. If the spaceship is far away from the throat where
(r ) << 1, then the tidal force can be written in terms of the
effective potential as
τ =
dV 2(r )
dr
r=b
−
dV 2(r )
dr
r=a
where τ represents the tidal force. The above expression
for the tidal force is obtained by considering the gradient
of the potential throughout the spaceship and using the fact
that the gradients, except at the two ends of the spaceship,
(64)
(65)
(66)
(67)
(68)
should almost vanish for it to not deviate much from its rigid
structure. Pictorially, it is just the absolute difference between
the slopes at the two points on V 2(r ) vs. r curve in Fig. 4. It
can also be noted that without considering the significance
of (r ), we get a discontinuity in the slopes of the potential
at the throat. That would correspond to an impulse of force
which will be experienced by a particle at the throat while
traversing through the wormhole. It would be like hitting
a thin membrane of a tough material. However, it should
be noted that even in the presence of infinite tidal forces,
causal contact is never lost among the elements making up
the observer; this suggests that curvature divergences may
not be as pathological as traditionally thought [
47
].
We would definitely want to remove this problem by
choosing a reasonable (r ). If we want a particle to smoothly
traverse through the wormhole without any impulse of force,
then we can do so by demanding that the slope of V 2(r ) goes
to zero as it reaches the throat. It would imply that
lim e2 (r )
r → bo
E 2b2
o
≈ L 2 + bo2 = o {using Eq. (41).}
And since we want it to vanish for large r, a simple choice
of (r ) might be
(r ) = oe1−(r/bo)κ ,
κ > 0
(69)
(70)
The larger κ , the faster it will die out. We can sketch the
plot for r˙2 vs. r as shown in Fig. 5.
It is clearly visible that we have removed the discontinuity
in the slope at the throat. Also note that there exists a local
minimum of r˙2 at the throat. Such a local minimum will
correspond to an unstable bound orbit. Thus, for our choice
of (r ), the throat will correspond to a region of unstable
circular orbits.
3 Trajectories in a dynamic spherically symmetric
wormhole
The metric for a spherically symmetric and dynamic Morris–
Thorne wormhole can be written as [
26
]
ds2 = −e2 (r)dt 2 + a2(t )
1 −
which corresponds to a 3geometry with a timedependent
scale factor a(t ).
3.1 Null geodesics
Due to spherical symmetry, we should expect to find the same
answer for the deflection angle as that of the static case. For
geodesics in equitorial plane, θ = π/2 and pθ = pθ = 0.
And since φ is a cyclic coordinate, pφ is a constant of motion.
So let,
pφ = L
⇒ pφ = gφν pν = a2Lr 2 = φ˙
For null geodesics, ds2 = 0
ds 2
Now, the timecomponent of the geodesic equation for this
metric becomes
t¨ + rt r r˙2 + φtφ φ˙ 2 + rt t r˙t˙ = 0
aa˙ 1 −
t¨ + t
˙
r˙t˙ = 0
Since φ˙ = L/a2r 2, the equation of trajectory becomes
where μ = L/E . Note that the time independence of this
equation is just an artifact of our poorly chosen coordinate
system. This is because r is itself a comoving coordinate.
We should define a new coordinate, r (r, t ) = a(t ).r , so that
any surface r = const., t = const. is a twosphere of area
4πr 2 and circumference 2πr . This coordinate r can then
be called the ’curvature coordinate’. In this coordinate, the
equation of the trajectory becomes
a2(t )
du 2
dφ
However, the total deflection angle can be calculated using
the coordinate r by the following equation:
,
which is the same as Eq. (16). Also, light will reach the throat
only if μ < bo as in the static case. We can make above
conclusions due the fact that the geometry, inspite of a
timedependent scale factor, is always spherically symmetric.
3.2 Timelike geodesics
For simplicity, we will work for the ultrastatic case, i.e., in
which (r ) = 0. Then, for timelike geodesics, we have
−t˙2 + a2(t ) 1 −
r
b(r ) −1r˙2 + a2(t )r 2φ˙ 2 = −1.
The timecomponent of geodesic equation becomes
aa˙ 1 −
t¨ + t
˙
b(rr ) −1r˙2 + at˙a˙ r 2φ˙ 2 = 0.
Now, from the above two equations, we get
t˙t¨ a˙
t˙2 − 1 + a = 0
which upon integration gives
4 Trajectories in a rotating wormhole
The metric for a rotating wormhole can be written as [
28
]
ds2 = −N 2dt 2 +
1 −
where N , K , ω and μ are functions of r and θ , and ω(r, θ )
may be interpreted as the angular velocity dφ/dt of a particle
that falls freely from infinity to a point (r, θ ). Assume that
K (r, θ ) is a positive, nondecreasing function of r that
determines the proper radial distance R, i.e., R ≡ r K . We also
require this metric to be asymptotically flat, which implies
= rl→im∞ K (r ) = 1 , lim ω(r ) = 0. (87)
r→∞
However, the metric (86) was initially derived for slowly
rotating stars [
48
] and hence it implicitly assumes the absence
of effects due to centrifugal forces [
49
]. Now, at the equatorial
plane, the metric becomes
ds2 = −(N 2 − r 2 K 2ω2)dt 2 +
1 −
+r 2 K 2dφ2 − 2r 2 K 2ωdφdt.
We can compare it with the metric far from a rotating
source of mass M and angular momentum S as given by Ref.
[
40
],
In cylindrical coordinates, x 1 = r cos φ, x 2 = r sin φ, x 3 =
z. Assuming axial symmetry, only the S3 term survives. Let
us call it J . Then
Sk xl
4 jkl r 3
=
4 J [x 1d x 2 − x 2d x 1]dt
r 3
=
.
Thus, in the asymptotically flat limit, comparing it with the
gtφ metric term of Eq. (88), we get
2 J
ω(r ) = r 3 + O
1
r 4 .
Now since the metric terms in Eq. (88) are independent of t
and φ, the corresponding momenta oneforms are conserved.
Thus, we can write [
50
]
E = − pt = At˙ + Bφ˙ ,
L = pφ = −Bt˙ + C φ˙ ,
where A = (N 2 − r 2 K 2ω2), B = r 2 K 2ω, C = r 2 K 2.
Let = AC + B2 = (N 2 −r 2 K 2ω2)(r 2 K 2)+(r 2 K 2ω)2 =
N 2r 2 K 2 .
Thus, the expressions for E and L in terms of t˙ and φ˙ become
t˙ =
C E − B L
, φ˙ =
B E + AL
V± = ωL ± Nr KL .
Now, a photon will make its closest transit from the wormhole
at a distance ro if at that point the condition E = V±(ro)
is satisfied. If there is no such point, then the photon will
definitely fall into the throat. Without loss of generality, we
can assume J > 0, where J is the angular momentum of the
rotating wormhole. This assumption also implies that ω(r ) >
0. Now, we have two possibilities for the conserved angular
momentum (L) of the photon: it can be either L > 0 or
So, considering the above inequality and using the fact that
E > 0, the condition E = V±(ro) becomes
E =
L ωo + roNKoo , if L ≥ 0,
L − ωo + roNKoo , if L < 0,
where ωo = ω(ro), No = N (ro) and so on. If we denote by
ro the distance of closest approach when L < 0 and by ro
when L > 0, then from the above equation, we can write
ro Ko
No
ro Ko
No
⇒
L
= E − Lωo
L
= E + Lωo
ro Ko > ro Ko .
No No
If N (r ) is a smooth decreasing function, then this equation
proves that the distance of closest approach is greater when
the light ray is moving in the direction of frame dragging than
that of light moving opposite to it. Now, from Eqs. (93) and
(94), we can write the equation of motion of photon trajectory
as
(1 − b(r )/r )(C E2 − 2B EL − AL2)
(B E + AL)2
(1 − b(r )/r )(C E2 + 2B EL − AL2)
(B E − AL)2
, if L ≥ 0,
, if L < 0.
(102)
⎧
dr 2 = ⎨⎪⎪⎪
dφ
⎧
dφ 2 = ⎨⎪⎪⎪
dr
As we can see, the equation of motion of a photon along
the direction of framed dragging is different from that of the
opposite direction.
Now, if a photon does not fall into the throat, it will get
deflected according to
(B + Aμ>)2
[1 − b(r )/r ][C − 2Bμ> − Aμ2>] , if L ≥ 0,
(B − Aμ<)2
[1 − b(r )/r ][C + 2Bμ< − Aμ2<] , if L < 0,
(103)
(98)
(99)
(100)
(101)
1
C
L < 0. This will determine whether the light ray is traversing
along the direction of frame dragging or opposite to it.
Also, since we have assumed that there are no horizons, the
gtt term of the metric can never change sign. Thus,
−gtt = (N 2 − r 2 K 2ω2) > 0
where
μ> = EL =
μ< = EL =
According to the above equation, the deflection angle for a
photon moving along the direction of the frame dragging will
be larger than that of a photon coming the other way [
51
].
4.2 Timelike geodesics
For timelike geodesics, we have
[C E 2 − 2B L E − ( AL2 + )] (104)
where for simplicity, we have denoted E as simply E and L
as L. Now, we can rewrite the above equation as
V± = ωL ± Nr KL
1 +
r 2 K 2
L2 .
(B + Aμ>)2
[1 − b(r)/r][C − 2Bμ> − (Aμ2> + /E2)] , if L ≥ 0,
(B − Aμ<)2
[1 − b(r)/r][C + 2Bμ< − (Aμ2< + /E2)] , if L < 0,
(108)
Now, the equation of a particle’s trajectory can be written
(1 − b(r)/r)[C E2 − 2B EL − (AL2 + )] , if L ≥ 0,
(B E + AL)2
(1 − b(r)/r)[C E2 + 2B EL − (AL2 + )] , if L < 0.
(B E − AL)2
So, for timelike geodesics, if it does not reach the throat, it
will be follow a trajectory given by the equation
μ> =
L
E =
5 Invariant angle method of Rindler and Ishak
In this section, we will calculate the angle between radial
and tangential vectors at a point on the photon’s trajectory by
Invariant angle Method which was proposed by Rindler and
Ishak [
52
]. Let δ represent the radial direction and d represent
the tangential direction at any point on the photon’s trajectory.
Let ψ be the angle between them. Then the invariant formula
for cos ψ becomes
(gi j di δ j )
cos ψ = (gi j di d j )1/2(gi j δi δ j )1/2
For a photon coming from far left of a source and heading
toward the far right while being deflected, the directions d
and δ in the (r, φ) basis can be written as
d ≡ (±dr, −dφ) = (∓dr/dφ, 1)dφ = (∓ A, 1)dφ, where
A = dr/dφ δ ≡ (dr, 0) = (1, 0)dr
⇒
grr dr δr + gφφ dφ δφ
cos ψ = (grr δr δr )1/2(grr dr dr + gφφ dφ dφ )1/2
Rewriting this in the form of tan ψ (Ref. [
53
]), we get
(109)
(110)
(111)
(112)
the case of dynamical and rotating wormhole geometries as
well.
The expression for ψ in the Schwarzschild geometry takes
the form
tan ψ =
1
r3(ro−2M)
ro3(r−2M) − 1
6 Short discussion and concluding remarks
A detailed study of particle and photon trajectories has been
conducted in the background of wormhole geometry.
Starting with the Morris–Thorne wormhole, null geodesics and
photon spheres have been analyzed, while for particle
trajectories both bounded and unbounded orbits are considered.
Subsequently, both null and timelike geodesics are analyzed
in the geometry of dynamic spherically symmetric WH and
rotating WH. Finally, using the invariant angle method of
Rindler and Ishak, the angle between radial and tangential
vectors on the photon’s trajectory has been evaluated.
Based on the above study, we have found that in a Morris–
Thorne wormhole and its dynamic and rotating counterparts,
the throat itself is a photon sphere. We have also seen that
in such geometries, the angle between tangential and radial
vectors at any point on a photon’s trajectory is independent of
the shape function b(r ). The geodesics in ultrastatic
wormholes with shape exponents have already been studied in great
detail in Ref. [
43
]. Also, in Ref. [
32
] Cataldo et al. studied
a Schwarzschildlike traversable WH which is obtained by
putting n = 1 with some slight modification. For geodesics,
they showed that a test particle which is radially moving
towards the throat always reaches it with zero velocity and
at a finite time, while for radially outward geodesics the
particle velocity tends to a maximum value, reaching infinity.
However, in this paper we have shown that it is true for all
possible n. Also, general conditions for nonradial geodesics
were derived which are required to be satisfied in order for
it to cross the throat. These results are in agreement with
our study and can, roughly, be obtained by putting n = 1
in our general equations for arbitrary n. Similarly, in Ref.
[
38
], the Ellis wormhole (n = 2) is studied in great detail
including the behavior of geodesics in such geometry. For
the Ellis wormhole, the particles are attracted on one side
and are repelled on the other and so the throat is of a saddle
nature. In our paper, we have mainly stressed the geodesics
that remain on one side of the wormhole, unlike the
abovementioned references where geodesics through the throat are
studied in detail.
Furthermore, we have analyzed the possibility of bounded
timelike orbits for different shape exponents in a different
tan ψ =
tan ψ =
tan ψ =
 A√grr
A2grr + gφφ
.
gφφ dφ
grr dr
.
For a general Morris–Thorne wormhole, it becomes
ro
[exp{ (ro) −
(r )}r 2 − ro2]
.
In the ultrastatic limit, it simplifies to
ro
(r 2 − ro2)
It is interesting to note that the expression for ψ is
independent of the shape function b(r ). This independence is true in
WH geometry, which can be regarded as the generalization
of the Schwarzschild geometry far from the throat. There, we
used the fact that any bounded timelike orbit in a spherically
symmetric geometry is either a circular orbit or an orbit that
oscillates around the radius of a stable circular orbit. For this
geometry, we found that, for a wormhole with shape
exponent n > 2, there always exists the possibility of one unstable
circular orbit while for n = 2, there exists one unstable
circular orbit only when L > bo and no bound orbits otherwise.
That means, no noncircular bound orbits exist when shape
exponent n ≥ 2 for any value of the impact parameter. For
0 < n < 2, we found that depending upon the value of L it
can either have the possibility of one unstable circular orbit
or a combination of one unstable circular orbit and a stable
circular orbit. While studying trajectories in a rotating
wormhole geometry, we have seen that the equations of motion of
both photon and particle depend upon whether it is traveling
in the direction of frame dragging or opposite to it.
Acknowledgements The authors are thankful to the Inter University
Centre for Astronomy and Astrophysics (IUCAA), Pune (India) for
their hospitality as the initiation of this work was taken during a visit
there. Anuj is also thankful to the library facility at the department of
mathematics of Jadavpur University.
Open Access This article is distributed under the terms of the Creative
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