Nonlocal Gravity Simulates Dark Matter

A nonlocal generalization of Einstein's theory of gravitation is constructed within the framework of the translational gauge theory of gravity. In the linear approximation, the nonlocal theory can be interpreted as linearized general relativity but in the presence of"dark matter"that can be simply expressed as an integral transform of matter. It is shown that this approach can accommodate the Tohline-Kuhn treatment of the astrophysical evidence for dark matter.

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): 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).
Wave phenomena tend to violate the locality postulate (unless we consider the eikonal limit).
2. Nonlocal theory of special relativity Acceleration induces nonlocality into SR.
Mashhoon (1993) proposed a nonlocal theory of accelerated systems of the type (memory effect) 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): The nonloc. kernel turned out to be (see, however, Mash. 2007) Connection Γ µ γ with respect to the accelerating frame.
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.
How can we then extend 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.
(3)  y) is the world-function of Ruse-Synge (half the square of the geodesic distance connecting x and y); furthermore and Ω ai (x, y ) = Ω ia (x, y ) with lim y→x Ω ai (x, y ) = −g ai (x) .
Causal scalar kernel K (x, y ) indicates 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, as, e.g., of (1) We chose the simplest nonlocal constitutive model involving a scalar kernel. The physical origin of this kernel will be discussed below.
Indices become the same. Grav. field strength C ij k = 2ψ k [j,i] . Moreover, K(x, y ) = K (x − y ). Then, the constitutive relation reads, In linear approximation, ∂ k C ij k = 0 G i k (= linearized Einstein tensor of the Riemannian space Fierz-Pauli for spin 2). X ijk is, as we saw, a certain linear combination of the irreducible torsion pieces. In linear approximation, Linearized field equations, cf. Mashhoon, NLG, Eqs.(7.21), (7.22): ∂ i t k i = 0 follows therefrom. Let us define T ik in terms of the nonlocal parts of the field equation: Then, Fredholm integral eq. of 2nd kind. Solve by the Liouville-Neumann method of successive substitutions. Infinite series in terms of iterated kernels K n (x − y ), n = 1, 2, 3, ...: K 1 (x − y ) := K (x − y ) , K n+1 (x − y ) := − K (x − z)K n (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 ) := − ∞ n=1 K n (x − y ) . Then the solution of the linearized field equation can be written as 0 G ik (x) = κt ik (x) + S ik (x) −pU ik (x)+ R(x−y) [κt ik (y) + S ik (y ) −pU ik (y)] d 4 y .