Physics Letters B 769 (2017) 281–288
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Gravitational microlensing in Verlinde’s emergent gravity
Lei-Hua Liu
∗
, Tomislav Prokopec
Institute for Theoretical Physics, Spinoza Institute and the Center for Extreme Matter and Emergent Phenomena (EMME), Utrecht University,
Buys
Ballot Building, Princetonplein 5, 3584 CC Utrecht, The Netherlands
a r t i c l e i n f o a b s t r a c t
Article history:
Received
6 December 2016
Received
in revised form 29 March 2017
Accepted
29 March 2017
Available
online 31 March 2017
Editor:
M. Trodden
We propose gravitational microlensing as a way of testing the emergent gravity theory recently proposed
by Eric Verlinde [1]. We consider two limiting cases: the dark mass of maximally anisotropic pressures
(Case I) and of isotropic pressures (Case II). Our analysis of perihelion advancement of a planet shows
that only Case I yields a viable theory. In this case the metric outside a star of mass M
∗
can be modeled
by that of a point-like global monopole whose mass is M
∗
and a deficit angle =
(2GH
0
M
∗
)/(3c
3
),
where H
0
is the Hubble rate and G the Newton constant. This deficit angle can be used to test the theory
since light exhibits additional bending around stars given by, α
D
≈−π/2. This angle is independent on
the distance from the star and it affects equally light and massive particles. The effect is too small to be
measurable today, but should be within reach of the next generation of high resolution telescopes. Finally
we note that the advancement of periastron of a planet orbiting around a star or black hole, which equals
π per period, can be also used to test the theory.
© 2017 Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Global monopole metric
In a recent paper Eric Verlinde [1] has proposed a novel emer-
gent
gravity theory. The most important claim of the theory is that
dark matter has no particle origin but instead it is an emergent
manifestation in modified gravity. Assuming spherical symmetry
Verlinde obtains,
r
0
GM
D
(r
)
2
r
2
dr
=
cH
0
M
B
(r)r
6
, (1)
where H
0
= 2.36 × 10
−18
s
−1
√
/3is the current Hub-
ble
parameter, is the cosmological constant (whose value is
determined by the current dark energy density), G = 6.674 ×
10
−11
m
3
/(kg s
2
) is the Newton’s constant, c ≈3 ×10
8
m/sis the
speed of light, M
B
(r) (M
D
(r)) is the baryonic mass (dark mass)
inside a sphere of radius r.
Eq. (1) implies that for a star of uniform density ρ
∗
, M
∗
=
4π
3
r
3
ρ
∗
, inside the star,
M
D
(r) =
2cH
0
M
∗
r
5
3GR
3
∗
∝r
5/2
, r < R
∗
, (2)
*
Corresponding author.
E-mail
addresses: L.Liu1@uu.nl (L.-H. Liu), t.prokopec@uu.nl (T. Prokopec).
where R
∗
denotes star’s radius. On the other hand, outside the star,
M
D
∝r, and we have,
M
D
(r) =
cH
0
M
∗
6G
×r , r ≥ R
∗
. (3)
The main goal of this paper is to construct the metric tensor
that consistently incorporates (1) within the Verlinde’s emergent
gravity theory and to investigate how that metric can be used to
test the theory. The fundamental assumption we make is that the
theory admits metric formulation that can be obtained by solv-
ing
suitably modified Einstein’s equations (23). In the Appendix we
perform a detailed analysis of such a theory. Unfortunately, we do
not have all of the information needed to fully specify the metric.
A reasonable assumption is that the modified stress energy tensor
is diagonal, T
μ
ν
=diag[−ρ, P
r
, P
θ
, P
ϕ
], see (24). In the weak grav-
itational
field regime (which is of our principal concern here) that
should be justified. This leaves us with four unknown functions:
energy density ρ (which we can determine from (1)) and three un-
known
pressures: P
r
, P
θ
, P
ϕ
. For spherically symmetric mass dis-
tribution
the two angular pressures must be equal, P
θ
= P
ϕ
≡ P
⊥
.
The remaining pressures are unknown, but are nevertheless tightly
constrained by the TOV equation (41), however not enough to be
completely specifiable. Rather than attempting to extend Verlin-
de’s
theory to obtain a relationship between the energy density
and pressures, here we consider two simple and plausible Ansätze:
http://dx.doi.org/10.1016/j.physletb.2017.03.061
0370-2693/
© 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
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