Bitensorial Formulation in Gravity and Stokes Flow

• Bitensorial formulation is a framework where tensors are defined on two distinct spacetime points, with indices attached separately to local and source spaces.
• In gravity, it replaces local connections with bitensorial ones that encode nonlocal quantum correlations, modifying Einstein’s equations and impacting cosmological models.
• In Stokes flow, the formulation constructs a bi-invariant Green function using cylindrical harmonic expansions to capture singular and regular flow features under varying boundary conditions.

Searching arXiv for the papers on arXiv to ground the article in current literature. Tool call: arxiv_search(query="id:(Morales et al., 15 Dec 2025) OR id:(Procopio, 9 Jul 2025)", max_results=5, sort_by="submittedDate") Bitensorial formulation denotes a class of constructions in which the fundamental objects are defined on a pair of points rather than a single point, with one set of indices attached to the tangent or cotangent space at the field point and another set attached to the corresponding space at the source, pole, or second spacetime point. In the recent literature, this idea appears in two distinct settings. In gravity, it is used to promote the affine connection to an independent bitensorial field that couples to a bitensorial energy–momentum source, with the metric kept classical and the nonlocal structure carried by the connection (Morales et al., 15 Dec 2025). In Stokes flow, it is used to write the Green function and all derived singularities as true tensors in two independent points, so that coordinate changes, pole differentiation, and orientation changes remain tensorially well defined (Procopio, 9 Jul 2025).

1. General definition and geometric content

A bitensor is multilinear in vectors or covectors at two points. In the gravitational formulation, a bitensor $$T^{a_1\ldots a_k}{}_{b_1\ldots b_\ell}{}^{a'_1\ldots a'_{k'}}{}_{b'_1\ldots b'_{\ell'}}(x,x')$$ has unprimed indices at $x$ and primed indices at $x'$, so the bilocal character is explicit in both its argument structure and its index structure (Morales et al., 15 Dec 2025). In the hydrodynamic formulation, the central object is the Green function

$G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$

where the upper index $b$ is a contravariant index at the field point $\mathbf{x}$, and the lower index $\beta$ is a covariant index at the pole point $\boldsymbol{\xi}$ (Procopio, 9 Jul 2025).

Two geometric objects recur in both settings. One is Synge’s world function, denoted $\sigma(x,x')$ in curved spacetime and represented in the Stokes setting through $r^2/2$, with $x$0 the Euclidean distance between field and pole. The other is the parallel propagator, denoted $x$1 or $x$2, which transports tensorial data between the two points. In the gravitational paper, the parallel propagator is used to convert primed and unprimed indices in the definition of the bitensorial Einstein tensor (Morales et al., 15 Dec 2025). In the Stokes paper, it makes the Green function bi-invariant under separate coordinate changes at the field and pole points, including Cartesian–Cartesian and cylindrical–cylindrical representations (Procopio, 9 Jul 2025).

This bilocal index structure is not merely notational. It determines how differentiation at the second point is performed, how divergences are interpreted, and how symmetry under interchange of the two points is imposed. A plausible implication is that the bitensorial formulation is best understood as a geometric framework for retaining the full two-point structure of a theory rather than collapsing it to a local coincidence limit.

2. Bitensorial gravity and the independent connection

In the gravitational construction, the starting point is the observation that semiclassical gravity couples the classical Einstein tensor to $x$3, but for a Klein–Gordon field the stress tensor expectation value is defined by point-splitting and depends fundamentally on the two-point function $x$4. The coincidence limit $x$5 is described as an unnatural step, and Bell-inequality violations are taken to indicate that nonlocal correlations should appear in the gravitational description (Morales et al., 15 Dec 2025).

The proposed remedy is to keep $x$6 classical and encode quantum and nonlocal features in an independent bitensorial connection,

$x$7

This field is independent of the Levi-Civita connection. The theory is formulated directly at the level of field equations rather than from an explicit action. The classical decomposition

$x$8

is generalized by replacing $x$9 with its bilocal counterpart $x'$0, and a bitensorial Einstein tensor $x'$1 is defined to be symmetric under exchange $x'$2 (Morales et al., 15 Dec 2025).

The central field equation is

$x'$3

The source is decomposed as

$x'$4

where $x'$5 is the conventional local stress tensor for the classical or coherent part of matter and $x'$6 is the genuine bitensorial part encoding quantum correlations (Morales et al., 15 Dec 2025).

Matching the purely tensorial part yields the usual Einstein equation,

$x'$7

while the bitensorial remainder determines the nonlocal sector through an equation for the bilocal connection contribution $x'$8. When all matter is classical, $x'$9, the simplest solution is $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$0, and the formalism reduces to standard General Relativity. The paper therefore characterizes the bitensorial connection as a structure that is turned on only in the presence of genuinely quantum sources (Morales et al., 15 Dec 2025).

3. Nonlocality, two-point functions, and coincidence structure

The gravitational formalism incorporates nonlocality by placing both geometry and matter directly on pairs of points. The source $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$1 is constructed from two-point functions via point-splitting, schematically as

$G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$2

without taking the coincidence limit (Morales et al., 15 Dec 2025).

For a real, massive Klein–Gordon field smeared with a function $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$3, the field operator is written in terms of a wave-packet $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$4, and the two-point function takes the form

$G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$5

After renormalization, the bitensorial stress tensor in the static Newtonian case is approximated by

$G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$6

so the energy density depends bilocally on the positions of both arguments through products of wave-packets (Morales et al., 15 Dec 2025).

The same emphasis on bilocality appears in the Stokes-flow formulation, though in a mathematically different context. There, the Green function is represented as a bitensor that transforms tensorially under separate coordinate changes at $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$7 and $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$8. This bi-invariance is specifically motivated by the need to differentiate with respect to the pole position and to rotate the forcing direction arbitrarily without solving separate problems for each orientation (Procopio, 9 Jul 2025).

In both cases, the coincidence structure is central. In gravity, the usual Hadamard subtraction and coincidence limit are replaced by the requirement that the bitensorial Einstein tensor and the bitensorial matter source share the same short-distance divergence structure as $G^b_{\;\beta}(\mathbf{x},\boldsymbol{\xi}),$9, treated as a boundary condition for the connection (Morales et al., 15 Dec 2025). In confined Stokes flow, the singular behavior at the pole is retained explicitly in the free-space Stokeslet, while the regular part is constructed to enforce the boundary conditions on cylindrical walls (Procopio, 9 Jul 2025).

4. Cosmological and Newtonian consequences in the gravitational formulation

The gravitational paper develops two applications: late-time cosmology and the Newtonian limit. For cosmology, the classical sector gives a spatially flat FRW metric,

$b$0

and the bitensorial source consistent with homogeneity and isotropy is taken as

$b$1

A symmetry-compatible ansatz for the bitensorial connection is

$b$2

with $b$3 (Morales et al., 15 Dec 2025).

The resulting equations are

$b$4

Assuming a bitensorial equation of state $b$5, the requirement that $b$6 diverge in the coincidence limit as $b$7 forces $b$8, and the solution becomes

$b$9

An effective local contorsion is then defined by integrating over the second point within a convex normal neighborhood, restricted to points spacelike to $\mathbf{x}$0,

$\mathbf{x}$1

At late times this leads to

$\mathbf{x}$2

so the bitensorial structure naturally gives rise to a positive effective cosmological constant (Morales et al., 15 Dec 2025).

In the Newtonian regime, the theory is expanded around Minkowski spacetime with small bitensorial corrections. The source is a single massive particle in a superposition of two spatially separated Gaussian wave-packets centered at $\mathbf{x}$3, with amplitudes $\mathbf{x}$4 and $\mathbf{x}$5. The dominant source term remains

$\mathbf{x}$6

and the geometric side is reduced to an equation for the antisymmetric part of a parametrization of the spatial contorsion (Morales et al., 15 Dec 2025).

After a simplifying choice $\mathbf{x}$7, an explicit bilocal function $\mathbf{x}$8 is obtained, leading to an effective local connection

$\mathbf{x}$9

The motion of a test particle of mass $\beta$0 is governed by

$\beta$1

In the Newtonian limit the force components are

$\beta$2

and

$\beta$3

Because these terms depend explicitly on the test-particle velocity, the effective force is nonconservative. In the limit $\beta$4, equivalently $\beta$5 through $\beta$6, one has $\beta$7 and the additional force disappears (Morales et al., 15 Dec 2025).

5. Green functions, cylindrical harmonics, and Stokes flow

In confined Stokes flow, the bitensorial formulation is built around the Green problem

$\beta$8

The domains considered are the annular region between two coaxial infinite cylinders, the interior of a single cylinder, and the exterior of a single cylinder, all with no-slip boundary conditions on the cylindrical walls (Procopio, 9 Jul 2025).

The construction begins with the decomposition

$\beta$9

where $\boldsymbol{\xi}$0 is the free-space Stokeslet and $\boldsymbol{\xi}$1 is a regular correction solving a homogeneous Stokes problem with inhomogeneous boundary data. In generic coordinates the free-space Stokeslet is

$\boldsymbol{\xi}$2

with pressure kernel

$\boldsymbol{\xi}$3

Here $\boldsymbol{\xi}$4 is the Euclidean distance between field and pole, $\boldsymbol{\xi}$5, and $\boldsymbol{\xi}$6 (Procopio, 9 Jul 2025).

The scalar kernel $\boldsymbol{\xi}$7 is expanded in cylindrical harmonics. For $\boldsymbol{\xi}$8,

$\boldsymbol{\xi}$9

with the usual interchange $\sigma(x,x')$0 for $\sigma(x,x')$1. Substituting this into the bitensorial Stokeslet produces cylindrical–cylindrical components of the form

$\sigma(x,x')$2

where the phase shifts encode the required parity and the coefficients $\sigma(x,x')$3 are Bessel combinations (Procopio, 9 Jul 2025).

The regular part $\sigma(x,x')$4 is expanded using Brenner–Happel’s representation of Stokes solutions in cylindrical coordinates in terms of three harmonic scalar potentials $\sigma(x,x')$5. This yields a harmonic representation

$\sigma(x,x')$6

with coefficients fixed by the boundary conditions on the cylindrical walls through a $\sigma(x,x')$7 linear system (Procopio, 9 Jul 2025).

The resulting formulation remains explicitly bitensorial throughout: one index at the field point, one at the pole point, and full dependence on $\sigma(x,x')$8. This is the basis for all subsequent higher-order singularities and applications.

6. Pole differentiation, reciprocity, and derived singularities

A major advantage of the Stokes-flow formulation is that higher-order singularities are generated by covariant differentiation with respect to the pole coordinates rather than by solving new boundary-value problems. For the dipole,

$\sigma(x,x')$9

where the Christoffel-symbol term appears because $r^2/2$0 is a covariant index at the pole (Procopio, 9 Jul 2025).

The free-space dipole is

$r^2/2$1

From this derivative one obtains the antisymmetric and symmetric-trace-free decompositions associated with the couplet and the stresslet. The paper identifies the couplet as the antisymmetric part of the dipole in source indices and the stresslet as the symmetric traceless part (Procopio, 9 Jul 2025).

A distinct construction is required for the Sourcelet, because it is associated with a mass source rather than a momentum source. The key step is reciprocity. By interchanging field and pole and using the reciprocal properties

$r^2/2$2

the paper derives a dual Green system and shows that the dual pressure vector solves the Sourcelet system up to a constant factor. The result is

$r^2/2$3

and, equivalently,

$r^2/2$4

with $r^2/2$5 the regular pressure part from the Green solution (Procopio, 9 Jul 2025).

This reciprocity-based method means that all source multipoles are generated from the pressure kernel of the standard Green function, while all momentum multipoles are generated from derivatives of $r^2/2$6. The two operations—covariant pole differentiation and pressure reciprocity—constitute the core of the bitensorial machinery for singular Stokes solutions.

7. Applications, scope, and limitations

The two formulations lead to sharply different applications. In gravity, the late-time FRW analysis yields a positive effective cosmological constant after the bitensorial connection is averaged over spacelike-separated points in a convex normal neighborhood. In the Newtonian limit, a source mass in a spatial superposition produces an additional velocity-dependent, nonconservative force on a test particle. The paper also states that for a single packet the force points toward the packet, for a symmetric superposition it is antisymmetric under $r^2/2$7 and vanishes at the origin for motion in the plane $r^2/2$8, and in the far region $r^2/2$9 it depends only on the total mass (Morales et al., 15 Dec 2025).

In cylindrical Stokes flow, the bitensorial Green function is applied to passive and active colloidal hydrodynamics. A sedimenting particle in the annular region is modeled by a Stokeslet directed along the cylinder axis, and the corresponding axial Green-function component shows strong backflow, with the flow in the direction of the Stokeslet confined to a relatively small region near the particle while most of the annular cross-section experiences opposite flow. This is used to argue that two sedimenters at opposite sides of the annulus radially will tend to slow each other down (Procopio, 9 Jul 2025).

For microswimmers modeled in the far field by a stresslet, the regular part of the confined stresslet determines the radial hydrodynamic force due to the cylindrical walls. The cited results distinguish three orientations. A radially oriented swimmer is repulsive with respect to the nearest wall. An angularly oriented swimmer is attracted toward the closer wall. An axially oriented swimmer is also attracted toward the nearest wall. The curvature dependence is analyzed by comparison with the planar-wall limits $x$00 for normal swimmers and $x$01 for parallel swimmers, and finite curvature is reported to enhance or reduce attraction depending on orientation and distance (Procopio, 9 Jul 2025).

Both papers also emphasize limitations. The gravitational model is underdetermined: the bitensorial equation gives $x$02 equations for $x$03 components of $x$04, and an additional equation for the connection is said to be needed, analogous to the torsion–spin relation in Einstein–Cartan theory. Its explicit calculations also rely on perturbative expansions, late-time cosmology, homogeneity and isotropy, nonrelativistic velocities, static sources, and small bitensorial contributions (Morales et al., 15 Dec 2025). The Stokes formulation assumes the Stokes regime, rigid smooth infinite cylinders, and Euclidean geometry, and it involves Bessel-function expansions, infinite sums, and integrals whose numerical evaluation requires truncation and convergence checks (Procopio, 9 Jul 2025).

Taken together, these works show that bitensorial formulation is not a single theory but a geometric strategy: retain the two-point structure, attach indices separately to the two points, and use world functions, parallel propagators, and covariant derivatives to keep bilocal objects manifestly tensorial. In gravity this strategy is used to encode quantum nonlocality in an independent connection while preserving a classical metric (Morales et al., 15 Dec 2025). In confined Stokes flow it is used to construct a bi-invariant Green function from which all higher singularities and source-type solutions follow systematically by pole differentiation and reciprocity (Procopio, 9 Jul 2025).