Nonlocal Gravity Simulates Dark
Matter
1
Friedrich W. Hehl1,2 and Bahram Mashhoon2
Univ. zu Köln and 2 Univ. of Missouri, Columbia, MO
Modified Gravity Approaches to the Dark Sector
Observatoire Astronomique de Strasbourg
28 June to 1 July 2010
We thank Benoit Famaey and his coorganizers for giving us the
possibility to present our results.
• FWH, B. Mashhoon, Phys. Rev. D 79, 064028 (2009)
• H.-J. Blome et al., Phys. Rev. D 81, 065020 (2010)
file nlGrav2010Strasb1.tex
Contents
1. Postulate of locality and its limits
2. Nonlocal theory of special relativity
3. Analogy between (local) vacuum electrodynamics
and ‘Einsteinian’ teleparallelism
4. Nonlocal gravitation generalizes general relativity
5. Linear approximation
6. Reciprocal kernel
7. Simulating dark matter
8. Digression: Motion of stars in a spiral galaxy
9. Modified Poisson equation
10. Recovering the Tohline-Kuhn system
11. Modified Nonlinear Poisson Equation
1. Postulate of locality and its limits
I
In SR, generalize Poincaré transformations from inertial
observers to accelerated observers. For a Newtonian point
particle, the initial configuration of which is specified by position
and velocity, this is obvious (point coincidences):
I
Postulate of locality: An accelerated observer (measuring
device) along its worldline is at each instant physically
equivalent to a hypothetical inertial observer (measuring device)
that is otherwise identical and instantaneously comoving with
the accelerated observer (measuring device).
I
Acceleration lengths for an Earth-bound laboratory for
translational acceleration `tr := c 2 /g⊕ ≈ 1 light year and for
rotational acceleration `rot := c 2 /Ω⊕ ≈ 28 AU. Dimension of
experiment λ; then locality postulate fulfilled if λ/` 1. Usually
fulfilled (for point coincidences); however, becomes problematic
for (extended) waves.
I
Decaying muon in storage ring: a ∼ γ vr ∼ 1022 g; ` ∼ 10−6 m:
"
2 #
λ
2
Comp
τµ = γτµ0 1 +
,
corrections ∼ 10−16 , negligible.
3
`
2
I
Radiating electron: classical charged particle accelerated by
external force f; typical wave length of radiation λ, thus λ ∼ ` or
λ/` ≈ 1, locality violated: Abraham-Lorentz equation
m
dv 2 e2 d 2 v
+ · · · = f;
−
dt
3 c 3 |{z}
dt 2
x and v no longer sufficient.
“jerk”
Wave phenomena tend to violate the locality postulate (unless
we consider the eikonal limit).
I
Accelerated system and in-coming electromagnetic wave:
Accelerated frame eα = ei α dx i (with i as coordinate and α as
frame index, both = 0, 1, 2, 3) obeys
dei α
= Φβ α ei β with acc. tensor Φαβ = (−g, Ω) = −Φβα .
dτ
Plane electromagnetic wave with wave vector k i = (ω, k).
Observer rotates with Ω0 around the wave. Then Bahram
Mashhoon’s analysis yielded ω̂ = γ(ω ∓ Ω0 ) = ωDop (1 ∓ Ω0 /ω).
For a wave with spin s, one finds ∼ ∓γs · Ω0 . This spin-rotation
coupling, which has been experimentally verified (GPS), is of a
non-local origin. When waves are involved, we are beyond point
coincidences and nonlocality sets in.
2. Nonlocal theory of special relativity
I
Acceleration induces nonlocality into SR.
I
Mashhoon (1993) proposed a nonlocal theory of accelerated
systems of the type (memory effect)
Z t
Faccelerated (t) = Finertial (t) +
K (t, t 0 )Finertial (t 0 )dt 0
0
I
In 4d, for electrodyn., we have for the excitation H = (D, H) and
the field strength F = (E, B) in comp. (fwh + Yuri Obukhov,
Foundations of Classical Electrodynamics, Boston, 2003):
r Z
1 ε0
dτ 0 Kαβ γδ (τ, τ 0 ) Fγδ (τ 0 , ξ) .
Hαβ (τ, ξ) =
2 µ0
{z
}
|
nonlocal kernel
Mashhoon’s nonlocal kernel turned out to be
1
Kαβ γδ (τ, τ 0 ) = αβ µ[δ δµγ] δ(τ − τ 0 ) − ucΓµ γ] (τ 0 ) .
2
Connection Γµ γ with respect to the accelerating frame.
I
Conventionally, SR plus equivalence principle (EP) → GR. Can
we apply the EP to the nonlocal SR? No, this does not seem to
be possible, probably since the EP is a strictly local principle.
I
How can we then generalize general relativity which is a strictly
local theory? Idea: We know electrodynamics is a gauge theory;
we can make it nonlocal, as shown on the last slide. Take gravity
as a gauge theory of translations; generalize it to a nonlocal
theory in a similar way as in electrodynamics.
I
Poincaré gauge theories of gravity, see arXiv:gr-qc/9602013:
Gauging translations → Cartan’s torsion; gauging Lorentz
rotations → curvature; Riemann-Cartan spacetime:
GR
Curvature
↑
I
Poincaré gauge theory
Curvature + Torsion
←− equivalent −→
GR||
Torsion
↑
The subcase of the translational gauge theory GR|| is equivalent
to GR.
3. Analogy between (local) vacuum electrodynamics and
‘Einsteinian’ teleparallelism
Vacuum electrodynamics
Minkowski spacetime
U(1) one generator
charge conserv. ⇒
dJ = 0
dH = J
def. of F ⇒
fα = (eα cF ) ∧ J
dF = 0
∗
GR|| (in Γα β = 0): transl. gauge theory
Weitzenböck spacetime (torsion, flat)
T 4 four generators
dTα = (eα cC β ) ∧ Tβ
dHα − Eα = Tα
⇐ energy conserv.
⇐ field eq. of gravity
fα = (eα cC β ) ∧ Tβ
⇐ def. of C β
α
dC = 0
tensor
F is irreducible
F = dA
r
ε0 ?
H=
F
µ0
α
vector
α
C = de
1 ? (1) α
Hα =
(a1 C + a2 (2)C α + a3 (3)C α )
{z
}
2κ |
Cα
a1 = −1,
elmg
elmg
− T α = eα c V +(eα cF ) ∧ H
ax. vec.
z }| { z }| { z }| {
⇒ C = (1)C α + (2)C α + (3)C α
α red.
a2 = 2,
a3 =
1
2
Eα = eα cV + (eα cC β ) ∧ Hβ
4. Nonlocal gravitation generalizes GR
Introduce nonlocality into the framework of teleparallel gravity:
Z
p
1p
Hab c (x) =
−g(x) [Cab c (x) − Ωai Ωbj Ωck K (x, y ) Cij k (y ) −g(y ) d 4 y] ,
|
{z
}
|κ
{z
}
new nonlocal piece
this is equivalent to GR
where Ω(x, y ) is the world-function of Ruse-Synge (half the square of
the geodesic distance connecting x and y ); furthermore
Ωa (x, y ) =
∂Ω
,
∂x a
Ωi (x, y ) =
∂Ω
;
∂y i
Ω satisfies
2Ω = g ab Ωa Ωb = g ij Ωi Ωj ,
and Ωai (x, y ) = Ωia (x, y ) with limy→x Ωai (x, y ) = −gai (x) .
The causal scalar kernel K (x, y ) indicates the nonlocal deviation from
GR. For K (x, y ) = 0, we recover GR|| and thus, GR. K (x, y ) is in
general a function of coordinate invariants, such as the quadratic
torsion invariants (Weitzenböck)
2
2
2
(1) k
(2) k
(3) k
Cij
,
Cij
,
Cij
.
We chose the simplest nonlocal constitutive model involving a scalar
kernel. The physical origin of this kernel will be discussed below.
5. Linear approximation
ei α = δiα + ψ α i ,
|ψ α i | 1 .
Indices become the same. Metric of the Weitzenböck spacetime:
gij = ηij + hij ,
k
hij = 2ψ(ij)
k
Gravitational field strength Cij = 2ψ [j,i] and the modified one
1 i j
Cij k = −
hk , − hk j ,i + ψ [ij] ,k + δki ψ, j − ψl j, l − δkj ψ, i − ψl i, l ,
2
where ψ = ηij ψ ij . Constitutive relation:
Z
K(x, y)
κHij α (x) = Cij α (x) +
| {z }
| {z }
equiv. to GR
Cij α (y)d 4 y ,
new nonlocal piece
K(x, y ) is scalar kernel K (x, y) evaluated in Mink. spacetime. Then,
Z
∂K(x, y ) ij
∂j Cij k +
C k (y )d 4 y = κTk i .
∂x j
In linear approximation, ∂k Cik j = Gi j (= linearized Einstein tensor l
Fierz-Pauli for spin 2). ∂i Tk i = 0 follows from lin. field eq. in box.
6. Reciprocal kernel
Assume that K(x, y ) is a function of x − y; then
Z
Gij (x) + K(x − y)Gij (y )d 4 y = κTij (x) ,
Fredholm integral equation of the 2nd kind. Solve formally by the
Liouville-Neumann method of successive substitutions. Infinite series
in terms of iterated kernels Kn (x, y), n = 1, 2, 3, ...:
Z
K1 (x, y ) := K(x, y) , Kn+1 (x, y) := − K(x, z)Kn (z, y)d 4 z .
If the resulting infinite series is uniformly convergent, we can define a
reciprocal kernel R(x, y ) given by
R(x, y ) := −
∞
X
Kn (x, y ) .
n=1
Then the solution of the linearized field equation can be written as
Z
Gij (x) = κTij (x) + κ R(x, y )Tij (y )d 4 y .
7. Simulating dark matter
Assume that the constitutive kernel is of the form
K(x −y) = δ(x 0 −y 0 )p(x−y) .
Thus, R(x −y) = δ(x 0 −y 0 )q(x−y) ,
where p and q are reciprocal spatial kernels; the ansatz δ(x 0 − y 0 )
should be sufficient for a non-rel. approx.; rewrite field eq. as
Gij = κ Tij + Πij .
Interpret the new source for linearized GR as “dark matter”. Πij is the
symmetric energy-momentum tensor of “dark matter”:
Z
Πij (x) = R(x, y)Tij (y )d 4 y .
Dark matter is the integral transform of matter by the reciprocal kernel
R(x, y). Thus, dark matter should be similar to actual matter. For
example, dark matter associated with dust would be pressure-free,
while Πij is traceless for radiation with Tk k = 0. For dust of density ρ,
we find
Z
ρD (t, x) = q(x − y) ρ(t, y)d 3 y ,
so that the density of dark matter ρD is in effect the convolution of ρ
and q. In linear approximation, the kernel is universal.
8. Digression: Motion of stars in spir. galaxies
I
Spiral galaxies, missing mass: Oort (1932), Zwicky (1933)...
I
Structure of a spiral galaxy: bulge, disk, globular clusters.
According to Kepler’s third law (planets around the Sun):
r
GM
4π 2
GM
2πr
⇒ v=
Kepler:
= 3 , v=
T2
r
T
r
I
However, Rubin + Ford (1970) found flat rotation curves for stars
√
of spiral galaxies: v (r ) −→ v0 = constant, instead of ∼ 1/ r
Review: Sofue + Rubin, Ann. Rev. Astr. Ap. 39 (2001) 137
I
If Newton-Einstein theory is OK:
M = M(r ),
From g =
v02
,
r
M(r ) ∼ r ⇒ dark matter
find ρDark = −
Z
Hence
MDark |{z}
∼
(large r )
v2 1
1 ~
∇ · ~g ⇒ ρDark = 0 2 .
4πG
4πG r
ρDark 4πr 2 dr =
v02
r.
G
I
Consider circular motion of star in a spiral. Outside the bulge,
the Newtonian acceleration of gravity for each star at radius |x|
is toward the galactic center with magnitude v02 /|x|, where v0 is
the (approximately) constant speed of stars.
I
We neglect the dimensions of the galactic bulge,
ρ(t, y) = Mδ(y) ,
where M is the effective mass of the galaxy. We derived on the
last slide,
v2 1
ρD (t, x) = 0
.
4πG |x|2
I
Substitute both equations into
Z
ρD (t, x) = q(x − y) ρ(t, y)d 3 y ,
⇒
q(x) =
1 1
,
4πλ |x|2
where λ := GM/v02 is a constant length parameter of order 5
kpc.
9. Modified Poisson equation
I
With this reciprocal kernel q(x) = 1/(4πλ|x|2 ), the Newtonian
limit of our linearized nonlocal theory is given by
Z
1
ρ(t, y)d 3 y
2
∇ Φ = 4πG ρ(t, x) +
4πλ
|x − y|2
(Φ = Newtonian pot.).
I
For a point mass m with ρ(t, x) = mδ(x), we find
1 |x|
1
+ ln
Φ(t, x) = Gm( −
),
|x|
|λ {z λ}
| {z }
Newton
‘dark matter’
the logarithmic dark matter term can be neglected in the solar
system.
10. Recovering the Tohline-Kuhn system
We recover the Tohline-Kuhn scheme that—apart from a
disagreement with the empirical Tully-Fisher law M ∝ v04 —has
been quite successful in dealing with dark-matter issues in
galaxies and clusters. However, the universality of our kernel
q(x) implies M ∝ v02 ; therefore, the Tully-Fisher relation is
violated (Bekenstein, private communication). We recall:
v2
I Joel Tohline (1983): g = GM
1 + λr = GM
+ r0
r2
r2
~ , Φ = − GM + GM ln( r )
In general: ~g = −∇Φ
I
r
λ
λ
λ = GM
= fixed constant ∼ 1 to 10 kpc (based on rotation
v02
curves of spiral galaxies)
I
Jeffrey Kuhn et al. (1987)
h
Modified Poisson’s equation: ∇2 Φ = 4πG ρ +
1
4πλ
R
ρ(~
y )d 3 y
|~
x −~
y |2
i
OK for spiral galaxies + cluster of galaxies
Note: Integro-differential eq. relates Φ and ρ: nonlocal gravity
I
Can this Tohline-Kuhn scheme be obtained from first principles?
11. Modified Nonlinear Poisson Equation
In the infrared 2.2 µm K band, the Tully-Fisher relation is
v0 = 220
L
L∗
0.22
km s−1 ,
where L∗ is a characteristic galaxy luminosity.
Combined with empirical results regarding mass–to–light ratios for
spiral galaxies, one gets roughly M ∝ v04 .
More basic treatment of Newtonian limit of nonlocal gravity:
Z
2
3
∇ Φ = 4πG ρ + ρΦ + q(x − y) [ρ(y) + ρΦ (y)] d y ,
Z
∂χ ∂Φ(y) 3
1 X
d y,
ρΦ = −
4πG
∂x i ∂y i
i
|∇y Φ|
χ = χ x − y;
|∇x Φ|
.
We hope that this nonlinear and nonlocal equation will help resolve
discrepancy of linear equation with Tully-Fisher relation.
————
——