Gravitational deflection of relativistic massive particles by Kerr black holes and Teo

wormholes viewed as a topological effect

Kimet Jusufi1, 2, ∗
1
Physics Department, State University of Tetovo, Ilinden Street nn, 1200, Tetovo, Macedonia.
2
Institute of Physics, Faculty of Natural Sciences and Mathematics,
Ss. Cyril and Methodius University, Arhimedova 3, 1000 Skopje, Macedonia.
(Dated: August 31, 2018)
We consider the problem of gravitational deflection of a propagating relativistic massive particles
by rotating black holes (Kerr black holes) and rotating wormholes (Teo wormholes) in the weak limit
approximation. In particular we have introduced an alternative way to calculate the deflection angle
for massive particles based on the refractive index of the optical media and the Gauss-Bonnet theorem
arXiv:1806.01256v2 [gr-qc] 30 Aug 2018

applied to the isotropic optical metrics. The refractive index governing the propagation of massive
particles is calculated by considering those particles as a de Broglie wave packets. Finally applying
the Gauss-Bonnet theorem leads to an exact result for the deflection angle in both geometries. Put
in other words, the trajectory of light rays as well as the trajectory of massive particles in a given
spacetime background can be viewed as a global spacetime effect, namely as a topological effect.

PACS numbers: 04.20.Gz, 03.75.-b, 04.20.Cv, 04.70.Bw

Keywords: Deflection angle; Relativistic massive particles; Refractive index; Gauss-Bonnet theorem; de Broglie
wave packets; Black holes; Wormholes

I. INTRODUCTION deflection angle can be viewed as global effect, or to put

in physics words, by integrating on a domain outside
General theory of relativity provides an elegant math- the light ray. A step forward was made by Werner who
ematical relation of the spacetime geometry in one hand, extended this method for asymtotically flat stationary
and matter described by the energy-momentum tensor metrics such as the Kerr black hole [9]. This approach
on the other hand. This theory explains many astro- is more complicated and involves the use of Finsler-
physical phenomena and has been tested many times in Randers geometry. Furthermore, one has to apply the
the past. The experimental results strongly suggest an Nazım’s method to construct a Riemannian manifold
amazing agreement with theoretical predictions. Black osculating the Randers manifold [9]. It is interesting to
holes and wormholes are mathematically predicted by not that Werner’s method has been extended to asymp-
general relativity. A traversable wormhole is a hyper- totically non-flat statonary fields such as a presence of
space tunnel, connecting together two distant regions topological defects, namely a rotating cosmic string and
within our universe or different univereses [1–3]. Nowa- a rotating global monopole [10–14]. Note that GBT was
days there is a lot of interesting research towards the applied also to the strong limit and a finite distance
quantum description of these objects in terms of quan- corrections in the presence of the cosmological constant
tum entanglement. [17–19].
The bending of light is one of the most famous clas- Another important question which naturally arises is
sical experiments confirming the curved nature of the whether one can calculate the gravitational deflection of
spacetime geometry, furthermore this phenomenon is massive particles using the GBT. As we shall see indeed
well explained in almost all the text books covering gen- this is the case. Very recently, Crisnejo and Gallo [20]
eral relativity. The standard explanation of light deflec- were able to compute the deflection angle for massive
tion is very simple; due to the presence of a massive particles in a static spacetime geometry.
body the light ray is deflected with an angle of deflection Our main motivation in this paper is to address and
proportional to the mass of the system enclosed within solve the problem of computing the deflection angle for
a certain region usually known as the impact parameter. massive particles in a rotating spacetime geometry in
Gibbons and Werner, discovered yet another method the weak limit. Towards this purpose, we shall con-
offering a different perspective on the deflection of light. sider an alternative way to compute the deflection of
Namely, it shows the importance of topology on the tra- light and massive particles in a Kerr black hole space-
jectory of light rays in presence of a static and sperically time and Teo wormhole spacetime. In doing so, we
symetric gravitational field [8]. This method involves will use an isotropic type metrics for a linearized rotat-
the application of Gauss-Bonnet theorem (GBT) in order ing gravitational field which drastically simplifies the
to compute the deflection angle. In this approach, the problem. Such a procedure for instance was applied
in Refs. [21–24] where authors investigated the deflec-
tion of light in terms of another method known as the
optical-mechanical analogy. The importance of isotropic
∗ Electronic address: kimet.jusufi@unite.edu.mk metric relies in the fact that one can easily find the refrac-
 2

tive index of the corresponding optical media. In order The last equation represents an isotropic metric, so
to study the case of massive particles we shall consider that the line element can be written in the form
those particles as a de Broglie wave packets introduced
in Ref. [22]. In fact, this will be crucial since a modifica- ds2 = −F 2 (ρ)dt2 + G 2 (ρ)|d~
ρ|2 . (7)
tion of the refractive index in a stationary gravitational
The isotropic coordinate speed of light v(ρ) can be
field is needed.
found from the relation
This paper is organized as follows. In Section II we
use an isotropic metric form for the Kerr spacetime, af-
q
4Ma dϕ
d~
ρ (1 − M 2
2ρ ) + ρ dt
ter which we derive the refractive index of the Kerr op- v(ρ) = | | = . (8)
tical media. We also compute the deflection angle for dt (1 + M2ρ )
3

light rays using the GBT. In Section III, we consider the

problem of the gravitational deflection of massive parti- Using n(ρ) = c/v(ρ) with [c = 1] the last equation
cles. In Section IV, we study deflection of massive parti- yields the effective refractive index for light in the Kerr
cles in a Teo wormhole geometry. In Section V, we com- gravitational field
ment on our results. In this paper we use natural unites
(1 + M2ρ )
3
G = c = ~ = 1. n(ρ) = q , (9)
(1 − M )2 + 4Ma dϕ
2ρ ρ dt

II. DEFLECTION OF LIGHT IN KERR SPACETIME In this way the optical metric reads

A. The refractive index dt2 = n(ρ)2 dρ2 + ρ2 n(ρ)2 dϕ2 . (10)

Note that the expression dϕ/dt can be computed
Let us start by writing the Kerr solution in the Boyer by using the relativistic action function S and the
Lindquist form given by Hamilton-Jacobi equation. Without going into details
one can show that the following relation holds
Σ2 2
 
2M r
ds2 = − 1 − dt2 + dr + Σ2 dθ2 + sin2 θdϕ2
Σ 2 ∆ dϕ 2M raE + (Σ2 − 2M r)J
= (11)
dt Σ2 (r2 E − 2M raJ]

2M ra2 4M ra sin2 θ where Σ2 = r2 in the equatorial plane. Note that E is

 
2
× r 2 + a2 + sin θ − dϕdt (1) the energy of the particle (photon) given by E = pc the
Σ2 Σ2
angular momentum J = p r0 , r0 being the distance of
with the closest approach which can be approximated with
the impact factor, i.e. r0 = b in the weak limit. Using the
Σ2 = r2 + a2 cos2 θ, (2) definition for the impact factor
J
2 2 = b, (12)
∆ = r − 2M r + a . (3) E
we find
To simplify the problem we can consider the deflec-
tion in the equatorial plane in a linearized Kerr metric dϕ 2M a + (r − 2M )b
= . (13)
in a. Such a metric can also describe the gravitational dt r3 − 2M ab
field around a rotating star or planet given by [21]
By inverting the coordinate relation (5) one finds
!
2 2M 4M a dϕ
dt dr2 1 p 
ds ≃ − 1 − + dt2 + + r2 dϕ2 . ρ= r − M + r(r − 2M ) , (14)
r r 1 − 2M
r
2
(4) which suggests that in leading order terms
Use the following coordinate transformation
ρ ≃ r − M + O(M 2 ). (15)
 2
M
r =ρ 1+ , (5) Hereinafter, we shall use the approximation ρ ≃ r −
2ρ M . In particular we find the following relation for the
refractive index
after which the metric takes the following form [21]
2M 2M ab
M 2 4Ma dϕ 4 n(r) = 1 + − + O(M 2 , a2 ). (16)
(1 − 2ρ ) + r r 3

ρ dt M
ds2 = − 2 2 2 2


M 2
dt + 1 + dρ + ρ dϕ .
(1 + 2ρ )
2ρ This result clearly indicates that the refractive index is
(6) modified due to the angular momentum parameter a.
 3

B. The GBT theorem and deflection of light Without loss of generality, in the large radial limit we
can set CR := r(ϕ) = R = const, implying the radial
Let us continue by rewriting the optical metric (10) in component relation
terms of a new coordinates. Namely introducing  r    2
ϕ r ϕ
∇ĊR ĊR = ĊR ∂ϕ ĊR + Γ̃rϕϕ ĊR . (26)
dr⋆ = n(ρ) dρ, (17)
f (r⋆ ) = n(ρ) ρ. (18) A direct computation reveals that

Now the optical metric reads

 
lim κ(CR ) = lim ∇ĊR ĊR  ,
 
R→∞ R→∞
dt2 = g̃ab dxa dxb = dr⋆ 2 + f 2 (r⋆ )dϕ2 . (19) 1
→ , (27)
R
In terms of the coordinates the Gaussian optical cur-
vature K is expressed as: with a similar simplification from the optical metric

1 d2 f (r⋆ ) lim dt → R dϕ. (28)

K = − (20) R→∞
f (r⋆ ) dr⋆2
"    2 2 # If we combine the last two equations, we find that in
1 dρ d dρ df dρ d f fact our optical geometry is asymptotically Euclidean
= − ⋆ ⋆ ⋆
+ ⋆
.
f (r ) dr dρ dr dρ dr dρ2
κ(CR )dt
lim =1 (29)
Furthermore we can express the last equation in terms R→∞ dϕ
of the refraction index resulting with
The deflection angle then is found by solving the inte-
gral
n(ρ)n′′ (ρ)ρ − (n′ (ρ))2 ρ + n(ρ)n′ (ρ)
K =− . (21) Zπ Z∞
n4 (ρ)ρ
g̃ dr⋆ dϕ.
p
α̂ = − K (30)
With these results in hand, we can chose a non- 0 rγ
singular region DR with boundary ∂DR = γg̃ ∪ CR .
The GBT provides a connection between geometry (in Using (16) and (20) for the Gaussian optical curvature
a sense of optical curvature) and topology (in a sense of we find
Euler characteristic number) stated as follows
2M 18 aM b
K ≃− + + O(M 2 , a2 ). (31)
r2 r5
ZZ I X
K dS + κ dt + θi = 2πχ(DR ), (22)
DR ∂DR i From this equations it is possible to show that
Zπ Z∞ 
with κ being the geodesic curvature, and K being the

2M 18 aM b p
Gaussian optical curvature. Following [8] we can choose α̂ = − − 2 + g̃ dr⋆ dϕ, (32)
r r5
a non-singular domain with Euler characteristic number 0 b
sin ϕ
χ(DR ) = 1. The geodesic curvature is defined
where we have used the light ray equation
κ = g̃ (∇γ̇ γ̇, γ̈) , (23)
b
but also keeping in mind an additional unit speed con- r= . (33)
sin ϕ
dition g̃(γ̇, γ̇) = 1, with γ̈ being the unit acceleration
vector. The two corresponding jump angles in the limit Note that we have also used
R → ∞, reads θO + θS → π. Therefore the GBT now is
dS = g̃ dr⋆ dϕ = n(ρ)2 ρdρdϕ ≃ rdrdϕ,
p
simplified as follows (34)

ZZ I ZZ π+
Z α̂ Solving the last integral we find the total deflection
K dS +
R→∞
κ dt = K dS + dϕ = π. (24) angle
DR CR D∞ 0 4M 4M a
α̂ ≃ ± 2 . (35)
b b
The geodesic curvature κ can be easily computed.
Taking into consideration that κ(γg̃ ) = 0 (remember γg̃ is This is exactly the same result found by Werner using
a geodesic), we are left with the following contribution the Rander-Kerr optical geometry [9]. We also note that
the signs of positive and negative stand for a retrograde
κ(CR ) = |∇ĊR ĊR |. (25) and a prograde light rays, respectively.
 4

III. DEFLECTION OF MASSIVE PARTICLES IN KERR constant everywhere in the optical medium, i.e. λN =
SPACETIME const. The refractive index relation for massive particles
in that case reads
A. Index of refraction for massive-particle de Broglie r
waves m2
N (r) = n(r) 1 − 2 F 2 (r), (44)
E
Recall from geometrical optics that the refractive in- We can apply this expression to our Kerr optical me-
dex can be used to find the trajectory of light by varying dia which results with
the path between two fixed points in space
M 2 4Ma dϕ
(1 − 2r ) + r dt
Z x2 F 2 (r) = M 2
, (45)
δ ndl = 0 (36) (1 + 2r )
x1
in which the approximation ρ ≃ r is used. One can write
where dl = |d~r| is the element of the path of integration the above relation in leading order terms
in the three-dimensional space. The orbits of a relativis-
tic particles one the other hand are obtained by requir- 2M 4aM bw
F 2 (r) = 1 − + + O(M 2 , a2 ). (46)
ing that they be geodesics: r r3
Z x2 ,t2 On the other hand the particles energy measured at
δ ds = 0 (37) infinity is given by
x1 ,t1
m
Alternatively, the last equation can be written in ac- E= 1/2
, (47)
(1 − w2 )
cordance with the Hamilton’s principle as follows
Z t2 in which m is the rest mass of the particle and w is
δ L(xi , wi )dt = 0, (38) the relativistic velocity. The angular momentum can be
t1 written as
where the componentes of the three relativistic velocity mwb
J= 1/2
, (48)
are given by wi = dxi /dt and w2 = 3i wi2 . The effective (1 − w2 )
P
Lagrangian is written as [22]
where b is the impact parameter. Following the defini-
tion of the impact parameter we can write
p
L(xi , w) = −mF 1 − w2 n2 (39)
J
The Hamiltonian is found to be = w b, (49)
E
2 −1/2
H = mF 1 − w2 n (40)


. which reduces to b in the case of light w = c = 1. There-
fore, the quantity dϕ/dt in the case of massive particles
In Ref. [22] authors have shown that from Hamilton’s
is modified as follows
principle one can derive an analogous relation to Fer-
mat’s principle written as dϕ 2M a + (r − 2M )wb
= , (50)
Z x2 Z x2 dt r3 − 2M awb
δ pdl = δ (Hn2 w)dl = 0. (41) yielding
x1 x1

Furthermore, using p = ~k and H = E = ~ω, (note wM (r2 w2 − 2abw + r2 )

N (r) = w + + O(M 2 , a2 ). (51)
that we have temporarily introduced ~) leads to the fol- w2 r3
lowing de Broglie wavelength of a given massive parti-
cle
B. Gravitational Deflection angle of massive-particles
hc
λ= q . (42)
nH 1− m2 c 4 F 2 We can, therefore, express the Gaussian optical cur-
H2
vature for massive particles in terms of the modified re-
This equations that can be easily rearranged as fractive index as follows
N (ρ)N ′′ (ρ)ρ − (N ′ (ρ))2 ρ + N (ρ)N ′ (ρ)
(52)
r
m2 c4 F 2 hc K =− .
λn 1 − = = cosnt. (43) N 4 (ρ)ρ
H 2 H
In particular, in leading order terms we find
In other words, this equation is a generalisation of a
well known result in wave-optics. For massive parti- M (r2 w2 − 18abw + r2 )
cles, in a given isotropic metric, the expression should be K ≃− + O(M 2 , a2 ). (53)
r5 w4
 5

Furthermore we can apply exactly the same proce- As in the Kerr spacetime, we simplify the problem by
dure as in the case of light deflection. For instance the considering a linearized rotating wormhole in a. We,
geodesic curvature results with therefore, in the equatorial plane find
dr2
 
 
2 4a dϕ
lim κ(CR ) = lim ∇ĊR ĊR  , ds ≃ − 1 + dt2 + + r2 dϕ2 . (63)
 
R→∞ R→∞ r dt 1 − br0
1
→ . (54) Introducing a further coordinate transformation
wR  2
For an observer located at a very large distance we can b0
r =ρ 1+ , (64)
also write 4ρ
yielding an isotropic Teo wormhole metric
lim dt → wR dϕ. (55)
R→∞ b0 2 dϕ
) + 4a
4
(1 + 4ρ

2 ρ dt 2 b0
dρ2 + ρ2 dϕ2 .


Putting together these results, from the GBT we find ds = − b0 2
dt + 1 +
(1 + 4ρ ) 4ρ
the following expression for the deflection angle
(65)
Zπ Z∞  Without going into details, the expression dϕ/dt is
M (r2 w2 − 18abw + r2 ) p

α̂ = − − g̃ dr⋆ dϕ. computed as follows
r5 w4
0 b dϕ 2a + bwr
sin ϕ = 3 (66)
(56) dt r − 2abw
We can approximate the surface element in terms of r With this result in hand, in the case of the wormhole
as follows optical media we find the following result for the refrac-
tive index
dS = g̃ dr⋆ dϕ = N (ρ)2 ρdρdϕ ≃ w2 rdrdϕ,
p
(57)
b0 w 2abw
N (r) = w + − 3 + O(b20 , a2 ). (67)
This integral can easily be evaluated, yielding 2r r
  Utilizing the expression for the Gaussian optical cur-
2M 1 4M a 1 vature and the the refractive index we find
α̂ ≃ 1+ 2 ± 2 . (58)
b w b w b0 18ab
K ≃ − 3 2 − 5 3 + O(a2 , b20 ) (68)
Interestingly, we recovered the gravitational deflec- 2r w r w
tion angle of massive particles in a Kerr geometry which Setting w = c = 1 we recover the case massless case.
is in perfect agreement with the result reported in Ref. The geodesic curvature is modified as
[25, 26]. Moreover in the special case, w = c = 1 the  
lim κ(CR ) = lim ∇ĊR ĊR  ,
 
light deflection angle is recovered. R→∞ R→∞
1
→ , (69)
IV. DEFLECTION OF MASSIVE PARTICLES BY TEO wR
WORMHOLE SPACETIME together with the relation
lim dt → wR dϕ. (70)
R→∞
The Teo wormhole metric represents a stationary
wormhole solution given by the following metric [4] This implies κ(CR )dt = d ϕ. Finally is we substitute
the Gaussian optical curvature the gravitational deflec-
dr2 h
2 2
i
tion angle is recast in the following form
ds2 = −N 2 dt2 + b0

+r 2 2
K dθ 2
+ sin θ (dϕ − ωdt) ,
1− r Zπ Z∞  
(59) b0 18ab p
α̂ = − − 3 2− 5 3 g̃ dr⋆ dϕ. (71)
where 2r w r w
0 b
sin ϕ
2
(4a cos θ)
N = K =1+ , (60) Evaluating the above integral we find
r
2a b0 4a
ω = 3. (61) α̂ ≃ ± 2 . (72)
r b b w
Where a is referred to the spin angular momentum, b0 In the special case, letting w = c = 1, the above re-
represents the shape function with the conditions r ≥ b0 . sult reduces to the deflection angle of light reported in
The throat of the wormhole is located at the coordinate [16]. It is interesting to observe that the geometric con-
r = b0 . The flare-out condition reads [5] tribution to the deflection angle remains invariant by
the nature the particles. Similarly, the signs of positive
b0 − b0,r r and negative stand for a retrograde and a prograde light
> 0. (62)
2b20 rays, respectively.
 6

V. CONCLUSIONS in the case of Teo wormhole spacetime. It is interest-

ing to point out that the geometric contribution to the
In the present paper we have studied the gravita- deflection angle in the wormhole geometry remains in-
tional deflection of massive and massless particles in variant and not affected by the mass of the particles i.e.
a linearized Kerr and Teo spacetime backgrounds. We b0 /b. In contrast, the spin angular contribution as well
have introduced a new approach to compute the grav- as the mass of the black hole is affected by the relativis-
itational deflection angle based on the refractive index tic velocity of the particle w. From the last two equations
and the GBT applied to an isotropic type metrics. The we observe that an apparent singularity appears when
importance of this method relies in the fact that one w → 0, therefore an additional constraint should be im-
can compute the deflection angle of relativistic parti- posed, namely 0 < w ≤ 1. In principle this apparent
cles in terms of the refractive index N (r) by assuming singularity can be resolved, for instance there should be
the propagating massive particles as a de Broglie wave some critical value wmin < w which solves the problem
packets, resulting with an important constant quantity of this apparent singularity. In a very recent article [27]
λN = const, in a given optical media. This is different, authors discuss such a critical value in the framework of
say, to the Werner’s approach which involves the use the geodesic approach for the charged black hole. More-
of Finsler geometry. The refractive index governing the over they find wmin2
< (M/b)3/2 which removes the sin-
propagating of massive particles in is found by consid- gularity only for the Schwarzschild case, however there
ering those particles as a de Broglie wave packets. is no satisfactory solution for a more general case. It will
Consequently, we have shown that the refractive in- be interesting to see if the deflection angle of a slowly
dex for massive particles is affected by the angular mo- moving particles can be found in terms of the GBT.
mentum parameter a, mass of the black hole, the worm-
hole shape function b0 , and finally the relativistic veloc- Finally our results clearly suggest the importance of
ity of the particle w. Applying the GBT to the isotropic the spacetime topology on the lensing effect, namely
metrics we have found the following results for the the gravitational deflection of massive particles can be
gravitational deflection angles: viewed as a global effect. Lensing of particles might be
an important tool in astrophysics in order to distinguish
  black holes from wormholes by their deflection angles of
2M 1 4M a 1 massive particles, such as the lensing of massive neutri-
α̂Kerr ≃ 1+ 2 ± 2 ,
b w b w nos. We plan to extend this method by adding addition
fields, such as scalar and electromagnetic fields. In ad-
in the case of Kerr spacetine confirming [25, 26], and dition it will be also interesting to study finite distance
b0 4a corrections on the deflection angle where the source and
α̂T eo ≃ ± 2 observer are located at a finite distance from each other.
b b w

[1] J. A. Wheeler, Phys. Rev. 97, 511 (1955); R. W. Fuller and J. [13] K. Jusufi, A. Övgün, J. Saavedra, P. A. Gonzalez and
A. Wheeler, Phys. Rev. 128, 919 (1962). Y. Vasquez, Phys. Rev. D 97, 124024 (2018)
[2] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988); [14] K. Jusufi and A. Övgün, Phys. Rev. D 97, no. 6, 064030
M. S. Morris K. S. Thorne and U. Yurtsever, Phys. Rev. D (2018).
61, 1446 (1988). [15] K. Jusufi, Int. J. Geom. Methods Mod. Phys. 14 (2017)
[3] M. Visser, Lorentzian Wormholes: From Einstein to 1750179.
Hawking (American Institute of Physics, New York, [16] K. Jusufi and A. Övgün, Phys. Rev. D 97, no. 2, 024042
1995). (2018).
[4] Edward Teo, Phys. Rev. D 58, 024014 (1998). [17] A. Ishihara, Y. Suzuki, T. Ono and H. Asada, Phys. Rev. D
[5] Naoki Tsukamoto, Cosimo Bambi, Phys. Rev. D 91, 084013 95, no. 4, 044017 (2017).
(2015) [18] T. Ono, A. Ishihara and H. Asada, Phys. Rev. D 96, no. 10,
[6] H. G. Ellis and J. Math. Phys. 14, 104 (1973). 104037 (2017).
[7] L. Chetouani and G. Clement, Gen. Rel. Grav. 16, 111-119 [19] A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura and H. Asada,
(1984). Phys. Rev. D 94, no. 8, 084015 (2016).
[8] G. W. Gibbons and M. C. Werner, Class. Quant. Grav. 25, [20] G. Crisnejo and E. Gallo, 10.1103/PhysRevD.97.124016
235009 (2008). [21] Saswati Roy, A.K. Sen, Astrophys Space Sci (2015) 360: 23
[9] M. C. Werner, Gen. Rel. Grav. 44, 3047 (2012). [22] J.C. Evans, P.M. Alsing, S. Giorgetti, K.K. Nandi,
[10] K. Jusufi and A. Övgün, Phys. Rev. D 97, no. 2, 024042 Am.J.Phys. 69 (2001) 1103-1110
(2018) [23] Kamal K. Nandi, Anwarul Islam, Amer. J. Phys, 63, 251-
[11] K. Jusufi, M. C. Werner, A. Banerjee and A. Övgün ,Phys. 256 (1995)
Rev. D 95, no. 10, 104012 (2017). [24] Paul M. Alsing, American Journal of Physics 66, 779
[12] K. Jusufi, I. Sakalli, A. Övgün, Phys. Rev. D 96, no. 2, (1998)
024040 (2017). [25] Guansheng He, Wenbin Lin, Class. Quantum Grav. 33
 7

(2016) 095007 [27] Xiankai Pang, Junji Jia, arXiv:1806.04719

[26] Guansheng He, Wenbin Lin, International Journal of
Modern Physics D, 23, 4 (2014) 1450031