Physics Letters B 728 (2014) 268–273
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Physics Letters B
www.elsevier.com/locate/physletb
Spin–spin interactions in massive gravity and higher derivative
gravity theories
˙
Ibrahim Güllü, Bayram Tekin
∗
Department of Physics, Middle East Technical University, 06800 Ankara, Turkey
article info abstract
Article history:
Received 25 September 2013
Received in revised form 29 November 2013
Accepted 29 November 2013
Available online 4 December 2013
Editor: S. Dodelson
We show that, in the weak field limit, at large separations, in sharp contrast to General Relativity
(GR), all massive gravity theories predict distance-dependent spin alignments for spinning objects. For
all separations GR requires anti-parallel spin orientations with spins pointing along the line joining
the sources. Hence total spin is minimized in GR. On the other hand, while massive gravity at small
separations (m
g
r 1.62) gives the same result as GR, for large separations (m
g
r > 1.62) the spins become
parallel to each other and perpendicular to the line joining the objects. Namely, the potential energy is
minimized when the total spin is maximized in massive gravity for large separations. We also compute
the spin–spin interactions in quadratic gravity theories and find that while at large separations GR result
is intact, at small separations, spins become perpendicular to the line joining sources and anti-parallel to
each other.
© 2013 The Authors. Published by Elsevier B.V.
1. Introduction
Consider two widely separated spinning massive objects (for
example two galaxies or galaxy clusters) that interact via gravity:
What is the minimum energy configuration for their spin orien-
tations, and how does the result depend on whether the graviton
is massive or not? In this work we will compute the spin–spin
interactions of point-like objects in massive gravity. We will show
that introducing a small graviton mass gives the highly unexpected
result of changing the spin orientations of sources from the one
predicted in GR. Arguably, massive gravity is the most natural
modification of GR that has implications in the overall dynamics
– accelerated expansion – of the universe and hence a detailed
study of gravitomagnetic effects such as the one done in this work
is needed.
Before we give a detailed derivation of the results in the next
section
in D-dimensional spacetimes and higher curvature the-
ories, let us summarize our findings here for the case of D
=
3 + 1 for GR and massive gravity. Consider two localized spinning
point-like sources described with the components of the energy-
momentum tensor
*
Corresponding author.
E-mail addresses: ibrahimgullu2002@gmail.com (
˙
I. Güllü), btekin@metu.edu.tr
(B. Tekin).
T
00
=m
a
δ
(3)
(
x −
x
a
),
T
i
0
=−
1
2
J
k
a
ikj
∂
j
δ
(3)
(
x −
x
a
), (1)
where a = 1, 2. Here m
a
is the mass and
J
a
is the spin of the
particle. Then, working in a flat background, from the tree-level
diagram of one graviton exchange, we can calculate the potential
energy as
U =−
4π G
t
d
4
xd
4
x
T
μν
(x)G
μναβ
x, x
T
αβ
x
,
(2)
where G
μναβ
(x, x
) is the Green’s function of the theory at hand
and t is a large time that will drop at the end. In GR this compu-
tation gives
U
GR
=−
Gm
1
m
2
r
−
G
r
3
J
1
·
J
2
−3
J
1
·
ˆ
r
J
2
·
ˆ
r
,
(3)
where
r = r
ˆ
r is the distance between the two sources. Spin–spin
part can be attractive or repulsive depending on the spin orienta-
tions. Maximum value of
J
1
·
J
2
−3
J
1
·
ˆ
r
J
2
·
ˆ
r, that is the minimum
of the potential energy is achieved when
J
1
and
J
2
are anti-
parallel and point along
ˆ
r as depicted in Fig. 1. That means in GR,
for any given r, potential energy is minimized for anti-parallel spin
orientations, if we neglect the tidal and orbital angular momentum
effects. (The computation here is of course not a good approxima-
tion for close binary systems, such as two neutron stars etc.,but
it is a valid approximation for two widely separated galaxies or
galaxy clusters.) Let us give the results of the same computation
0370-2693 © 2013 The Authors. Published by Elsevier B.V.
http://dx.doi.org/10.1016/j.physletb.2013.11.064
Open access under CC BY license.
Open access under CC BY license.