Relativistic Modified-Dynamics Theories

A low-acceleration force law does not yet say how clocks, light, gravitational waves, or an expanding universe behave. A relativistic completion must specify the metric seen by matter, the dynamics of every additional field, and a solution branch that reduces both to general relativity at high acceleration and to the intended modified Poisson equation in the quasistatic regime. The central design problem is that the same completion must enhance slow-particle dynamics and gravitational lensing without introducing an unhealthy mode.

Required background. Weak-Field Gravity supplies the two weak-field metric potentials and the null-geodesic limit.

Helpful background. Einstein Equations supplies covariant field equations and stress-energy conservation; Gravitational Actions and Boundary Terms supplies variational consistency; Formulations of General Relativity supplies constraint and initial-value language; Gravity Beyond General Relativity supplies mode-counting and branch checks; and Relativistic Matter and Compact Objects supplies the strong-field systems that any proposed completion must eventually confront.

What the nonrelativistic law leaves undetermined

Write a weak physical metric as

$$d\tilde s^2=-(1+2\Psi)dt^2+(1-2\Phi)d\boldsymbol x^2.$$

Slow matter responds primarily to $\nabla\Psi$, whereas light deflection is controlled by the transverse gradient of $\Phi+\Psi$. A modified Poisson equation for one potential therefore does not determine lensing: it neither fixes the second potential nor identifies which metric matter follows.

This explains a classic obstruction. If a scalar merely rescales an Einstein metric, $\tilde g_{ab}=A^2(\varphi)g_{ab}$, unparameterized null geodesics are conformally invariant. The scalar can alter massive-particle dynamics while failing to provide the corresponding extra light bending. The obstruction and the need for a nonconformal matter coupling are developed in Bekenstein 2004, §§ I–II and restated as a construction requirement in Skordis and Zlosnik 2021, “Phenomenological requirements”.

The cancellation can be seen directly. If $A=1+\alpha\,\delta\varphi$ at first order, the physical potentials are

$$\widetilde\Psi=\Psi_E+\alpha\,\delta\varphi, \qquad \widetilde\Phi=\Phi_E-\alpha\,\delta\varphi.$$

Their sum is $\Psi_E+\Phi_E$: the direct conformal contribution cancels from light deflection while remaining in the slow-particle force. This argument holds the Einstein-frame geometry fixed. Scalar stress energy can still backreact on that geometry; conformal invariance does not say that every scalar theory has the same lensing as a baryons-only GR solution.

Tensor–vector–scalar construction

TeVeS introduces an Einstein metric $g_{ab}$, a scalar $\phi$, and a dynamical unit timelike vector $U^a$, with $g_{ab}U^aU^b=-1$. Matter instead couples minimally to

$$\tilde g_{ab} =e^{-2\phi}g_{ab}-2\sinh(2\phi)U_aU_b.$$

The vector makes this relation disformal rather than merely conformal. Around a quasistatic source it permits the scalar contribution that changes massive orbits also to enter the lensing potential. Bekenstein derives the physical metric, field equations, quasistatic limit, and light deflection in Bekenstein 2004, §§ III and VI, Eqs. (21)–(22) and (114)–(116).

That kinematic success is only the beginning of the check. The scalar free function must approach the required low-acceleration law; the vector kinetic couplings must give acceptable spin-0, spin-1, and spin-2 propagation; the unit constraint must be preserved; and the chosen cosmological background must connect to the quasistatic branch. A post-Newtonian limit or a galaxy limit on one background is not a proof of nonlinear stability on all backgrounds.

Later scalar–vector–tensor constructions reorganize these ingredients. In the Skordis–Zlosnik model, matter couples to one metric while a unit timelike vector and a shift-symmetric scalar generate a low-acceleration quasistatic sector. The weak-field calculation gives $\Psi=\Phi$, so a successful dynamical potential also supplies the lensing potential. The cosmological scalar sector can behave approximately like pressureless matter, allowing the additional fields to sustain potentials before recombination (Skordis and Zlosnik 2021, equations (3)–(12)).

Mode and regime checks

A candidate relativistic theory should pass four logically separate tests:

1. Degrees of freedom: solve the constraints and count propagating modes; an auxiliary-looking field may become dynamical on another background.
2. Kinetic and gradient signs: diagonalize the quadratic action and require positive kinetic energy and real characteristic speeds in the parameter regime actually used.
3. Limit matching: derive, rather than assume, the GR, Newtonian, and low-acceleration equations from the same action and matter coupling.
4. Regime coverage: compare the background curvature, frequency, and field gradients with the strong-coupling scale and with every approximation used.

For example, the 2021 construction finds healthy vector modes on its stated Minkowski background when $0\lt K_B\lt2$ and $\lambda_s\gt-1$, and analyzes the full quadratic action there (Skordis and Zlosnik 2021, “Stability and waves,” equations (13)–(14)). This is a background-specific quadratic result. It does not establish a global nonlinear theorem, well-posed evolution for arbitrary matter, or regular compact-object solutions.

Even on that background, the vector conditions are not the whole scalar test. Writing $K_2$ for the curvature of the cosmological scalar kinetic function about its constant-gradient solution, the propagating scalar also requires $K_2\gt0$. The analysis contains a separate zero-frequency sector with a constant solution and a solution linear in time; its long-wavelength Hamiltonian is not generically positive. The authors discuss a Jeans-like interpretation. Thus “the analyzed vector modes are healthy” should not be replaced by “all perturbations have positive energy at all wavelengths” (Skordis and Zlosnik 2021, “Stability and waves,” PDF pp. 4–5).

Results and open questions through 2026-08-19

The literature supports several construction mechanisms, but it does not select a unique relativistic modified-dynamics theory. Different formulations and solution branches answer different parts of the problem.

Formulation or evidence	Supported result	Scope or limitation
Purely conformal scalar coupling	Can change massive-particle dynamics	Null trajectories are conformally invariant, so the same scalar does not supply the required enhanced lensing
Original TeVeS	A dynamical scalar–vector–tensor system has Newtonian and low-acceleration limits and couples lensing to the dynamical potential	Contains free functions and extra modes; success of selected backgrounds does not establish generic nonlinear stability
Generalized vector–tensor families	Broader kinetic terms permit different mode speeds and cosmological behavior	Viability depends on coupling parameters, matter metric, background, and solution branch
Skordis–Zlosnik construction	Produces a low-acceleration quasistatic limit and dustlike linear cosmological behavior, with parameter conditions for healthy propagating modes	The zero-frequency sector needs a separate long-wavelength analysis; no generic nonlinear, compact-object, or ultraviolet theorem follows

Two credible positions therefore remain. A permissive view treats these examples as existence proofs that relativistic lensing and linear cosmology can coexist with low-acceleration dynamics. A conservative view requires one fixed action to pass nonlinear stability, strong-field, cosmological, and cutoff checks before calling the paradigm complete. The unresolved issue is not whether a covariant action can be written, but whether one controlled parameter regime covers all required phenomena.

The conclusions could change after a full nonlinear well-posedness result controls the relevant backgrounds, an independent cosmological calculation tests the same fixed action beyond linear perturbations, or regular compact-object and binary solutions establish that the phenomenological branch survives strong fields.

Diagnostic exercise

Suppose a scalar theory has $\tilde g_{ab}=e^{2\alpha\varphi}g_{ab}$ and changes the potential inferred from circular motion. Explain why that fact alone does not predict an equally enhanced lensing signal, and identify the extra datum a relativistic comparison needs.

Solution

A conformal factor preserves null curves as unparameterized trajectories, so the scalar modification of the massive-particle equation does not automatically add the desired null deflection. In the weak-field metric, circular motion constrains primarily $\Psi$, while lensing depends on $\Phi+\Psi$. The comparison therefore needs the complete physical matter metric, or equivalently field equations that determine both potentials on the same background and branch. A disformal coupling involving a timelike vector is one possible mechanism, but it must then pass its own mode and stability checks. Importing a GR lensing formula with an independently chosen effective mass would not test the proposed relativistic theory.

References

• Bekenstein, Jacob D. (2004), “Relativistic Gravitation Theory for the MOND Paradigm,” Physical Review D 70, 083509; erratum 71, 069901. DOI. Open PDF.
• Famaey, Benoît, and Stacy S. McGaugh (2012), “Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions,” Living Reviews in Relativity 15, 10. DOI. Open PDF.
• Skordis, Constantinos, and Tom Zlosnik (2021), “A New Relativistic Theory for Modified Newtonian Dynamics,” Physical Review Letters 127, 161302. DOI. Open PDF.