Khronometric theories of modified Newtonian dynamics

TL;DR

This paper analyzes a relativistic theory of modified Newtonian dynamics (MOND) involving a khronon field, demonstrating its stability in slow motion and its impact on non-stationary systems.

Contribution

It establishes the non-relativistic limit of the khronometric MOND theory and examines the stability of stationary solutions and corrections in non-stationary regimes.

Findings

The theory has a consistent slow motion limit reproducing MOND solutions.

Stationary solutions are stable under khronon perturbations in low acceleration regimes.

Non-stationary systems experience order unity corrections from the khronon field.

Abstract

In 2011 Blanchet and Marsat suggested a fully relativistic version of Milgrom's modified Newtonian dynamics (MOND) in which the dynamical degrees of freedom consist of the spacetime metric and a foliation of spacetime, the khronon field. This theory is simpler than the alternative relativsitic formulations. We show that the theory has a consistent non-relativistic or slow motion limit. Blanchet and Marsat showed that in the slow motion limit the theory reproduces stationary solutions of modified Newtonian dynamics. We show that these solutions are stable to khronon perturbations in the low acceleration regime, for the cases of spherical, cylindrical and planar symmetry. For non-stationary systems in the low acceleration regime we show that the khronon field generally gives an order unity correction to the modified Newtonian dynamics.

Equations

$$n_a=\frac{-{\nabla }_{a}T}{\sqrt{-g^{ab}{\nabla }_{a}T{\nabla }_{b}T}},$$

$${\nabla }_a{n}_b=-n_a{a}_b+\frac{1}{3}\theta (g_{ab}+n_a{n}_b)+{\sigma }_{ab},$$

$$S=\frac{c^3}{16\pi G}\int d^{4}x\sqrt{-g}\left[R-2f(a)\right]+S_{\mathrm{m}}[g_{ab},\Psi ]$$

$$f(a)={\Lambda }_0-a^2+\frac{2c^2{a}^3}{3a_0}+O(a^4),a\ll a_0$$

$$f(a)\to {\Lambda }_{\infty },a\gg a_0.$$

$$G^{ab}=\frac{8\pi G}{c^4}\left[T_{\mathrm{m}}^{ab}+T_T^{ab}\right],$$

$$T_T^{ab}=-\frac{c^4}{8\pi G}\left[f{g}^{ab}+2n^a{n}^b{\nabla }_c(\chi a^c)-2\chi a^a{a}^b\right]$$

$$\chi (a)=\frac{f^{\prime }(a)}{2a}.$$

$${\nabla }_a\left[n^a{\nabla }_b(\chi a^b)\right]=\frac{1}{2}{n}^a{\nabla }_{a}f+\chi a^a{a}^b{\nabla }_a{n}_b.$$

$$\hat{G}=\frac{G\mathcal{M}{\mathcal{T}}^2}{{\mathcal{L}}^3},\hat{c}=\frac{c\mathcal{T}}{\mathcal{L}},{\hat{a}}_0=\frac{a_0{\mathcal{T}}^2}{\mathcal{L}}.$$

$$d{s}^2$$

$$T(t,x^i)=t+\frac{1}{c^2}\pi (t,x^i)+O(c^{-4}).$$

$$T^{tt}=O(c^0),T^{ti}=O(c^0),T^{ij}=O(c^0),$$

$$\Psi =\Phi .$$

$$\bar{a}=c^{2}a$$

$$\bar{f}(\bar{a})=c^{4}f(\bar{a}/c^2).$$

$$\mathbf{n}$$

$${\bar{a}}^{2}=({\mbox{\boldmath$\nabla$}\Xi})^{2}+O(c^{-2}),$$

$$\Xi=\Phi-{\dot{\pi}}+\frac{1}{2}({\mbox{\boldmath$\nabla$}}\pi)^{2}$$

$$S=-\int dt\int d^{3}x\left[\frac{(\mbox{\boldmath$\nabla$}\Phi)^{2}}{8\pi G}+\rho_{\rm m}\Phi+\frac{1}{8\pi G}{\bar{f}}({\bar{a}})\right]+O(c^{-2}),$$

$${\dot{\rho}}_{T}+\mbox{\boldmath$\nabla$}\cdot(\rho_{T}{\bf v}_{T})=0.$$

$$\rho_{T}=-\frac{1}{4\pi G}\mbox{\boldmath$\nabla$}\cdot\left[{\bar{\chi}}({\bar{a}})\mbox{\boldmath$\nabla$}\Xi\right]$$

$$\bar{\chi }(\bar{a})=\frac{{\bar{f}}^{\prime }(\bar{a})}{2\bar{a}}=\chi (\bar{a}/c^2),$$

$${\bf v}_{T}=-{\mbox{\boldmath$\nabla$}}\pi.$$

$$\nabla^{2}\Phi+\mbox{\boldmath$\nabla$}\cdot\left[{\bar{\chi}}({\bar{a}})\mbox{\boldmath$\nabla$}\Xi\right]=4\pi G\rho_{\rm m}.$$

$$\mbox{\boldmath$\nabla$}\cdot\left[(1+{\bar{\chi}}({\bar{a}}_{\rm b}))\mbox{\boldmath$\nabla$}\Phi\right]=4\pi G\rho_{\rm m},$$

$${\bar{a}}^{i}_{\rm b}=\partial_{i}\Phi,\ \ \ \ {\bar{a}}_{\rm b}=|\mbox{\boldmath$\nabla$}\Phi|.$$

$$t\to \bar{t}=t+\pi /c^2,$$

$$T(\bar{t},x^i)=\bar{t}+O(c^{-4}).$$

$$g_{\bar{t}i}={\pi }_{,i}+O(c^{-2}),$$

Full text

1 Introduction

In 1983 Milgrom suggested a modification of Newtonian gravity in order to provide a better fit to Galactic rotation curves (Milgrom, 1983a, b, c), now known as Modified Newtonian Dynamics (MOND). On galactic scales this theory fits the data well and provides an explanation for a number of otherwise mysterious regularities (Famaey & McGaugh, 2012; McGaugh, 2015; Dainotti et al., 2023). Conventional Newtonian dynamics within the $\Lambda$CDM framework has difficulty explaining these regularities and has other difficulties when confronted with data on galactic structure (Peebles & Nusser, 2010).

On the other hand MOND is clearly inconsistent with the data on cluster scales, and in particular with observations of the bullet cluster which exhibits a spatial separation between the dark and baryonic matter in the aftermath of a collision (Clowe et al., 2006). The MOND theory requires the assumption of neutrino dark matter on these scales to be consistent with these observations (Famaey & McGaugh, 2012; McGaugh, 2015; Dainotti et al., 2023). These neutrinos need not contribute appreciably on galactic scales due to the Tremaine-Gunn bound (Tremaine & Gunn, 1979), i.e. Fermi degeneracy pressure.

The original formulation of MOND was valid only in the non-relativistic limit, and it is necessary to have a fully relativistic formulation for a number of reasons. The first such theory was suggested by Bekenstein (2004), but this theory has since been ruled out by LIGO observations which show that gravitational and electromagnetic waves propagate at very nearly the same speed (Boran et al., 2018). A more recent theory that is compatible with LIGO data is that of Skordis & Zlosnik (2021, 2022). However this theory is quite complicated with the dynamical fields consisting of a metric, a scalar and a unit-norm vector field. In 2011 Blanchet and Marsat (henceforth BM) suggested a fully relativistic version of MOND in which the dynamical degrees of freedom consist of the spacetime metric and a foliation of spacetime, the khronon field (Blanchet & Marsat, 2011, 2012; Sanders, 2018). Their theory is a special case of a general class of theories called khronometric theories which includes Horava gravity (Horava, 2009; Blas et al., 2010; Blas & Lim, 2015; Blas et al., 2011). They can be obtained as a limiting case of a class of theories with a dynamical unit timelike vector field, which includes the Einstein-Aether theory (Jacobson, 2010, 2014).

In this paper we show that the BM theory has a consistent non-relativistic or slow motion limit, of a kind different to that previously considered in the literature in the context of a more general class of theories in which the limit to the BM theory is pathological (Bonetti & Barausse, 2015).

We also study the dynamics of the khronon field in the BM theory. Blanchet & Marsat (2011) showed that in the slow motion limit the theory reproduces stationary solutions of modified Newtonian dynamics with the khronon perturbation set to zero in a certain coordinate system. We show that these solutions are stable to khronon perturbations in the low acceleration regime, for the cases of spherical, cylindrical and planar symmetry. We also show that for non-stationary systems in the low acceleration regime the khronon field generally gives an order unity correction to the modified Newtonian dynamics.

2 The khronon theory of Blanchet and Marsat

In khronometric theories of gravity the dynamical fields are a metric $g_{ab}$ and a spacelike spacetime foliation. The foliation is encoded in a scalar field $T$, called the khronon field, and the theory is invariant under $T\to h(T)$ where $h$ is any monotonic function. The unit, future directed vector field ${\vec{n}}$ normal to the foliation is given

[TABLE]

and we can decompose its derivative as

[TABLE]

where $a^{a}=n^{b}\nabla_{b}n^{a}$ is the acceleration, $\theta=\nabla_{a}n^{a}$ is the expansion and $\sigma_{ab}$ satisfying $n^{a}\sigma_{ab}=0$, $\sigma_{[ab]}=0$ is the shear. The action of the theory is a function of $a=\sqrt{a^{a}a_{a}}$, $\theta$ and $\sigma=\sqrt{\sigma_{ab}\sigma^{ab}}$, all of which have dimensions of inverse length.

We focus on the particular theory suggested by Blanchet & Marsat (2011) and Blanchet & Marsat (2012) as a fully relativistic version of MOND. The action for the theory is

[TABLE]

where $S_{\rm m}$ is the matter action, $G$ is Newtons constant and $c$ the speed of light. Here the function $f(a)$ satisfies

[TABLE]

where $a_{0}=1.2\times 10^{-2}{\rm m}{\rm s}^{-2}$ is the MOND acceleration scale, and

[TABLE]

for some constants $\Lambda_{0}$ and $\Lambda_{\infty}$. The equations of motion for this theory are (Blanchet & Marsat, 2011)

[TABLE]

where $T_{\rm m}^{ab}$ is the matter stress energy tensor, and

[TABLE]

is the stress energy tensor of the khronon field, with

[TABLE]

The equation of motion for the khronon field is

[TABLE]

3 Slow motion limit

We now specialize to the slow motion limit, which we define as follows. Consider an isolated source of gravity characterized by a mass scale ${\cal M}$, a lengthscale ${\cal L}$ and a timescale ${\cal T}$. We define the dimensionless quantities

[TABLE]

The slow motion limit is the limit ${\hat{c}}\to\infty$ at fixed ${\hat{G}}$ and fixed ${\hat{a}}_{0}$. In particular the slow motion limit encompasses both the regime ${\hat{a}}_{0}\ll 1$ where normal Newtonian gravity is recovered, and the regime ${\hat{a}}_{0}\gg 1$ where modified Newtonian dynamics applies111The timescale ${\cal T}$ is of course not independent of the other parameters when the source is self-gravitating. In the Newtonian regime ${\hat{a}}_{0}\ll 1$ this occurs when ${\hat{G}}\sim 1$ or ${\cal T}\sim G^{-1/2}{\cal L}^{3/2}{\cal M}^{-1/2}$. In the MOND regime ${\hat{a}}_{0}\gg 1$ the condition is instead ${\hat{G}}{\hat{a}}_{0}\sim 1$ or ${\cal T}\sim G^{-1/4}a_{0}^{-1/4}{\cal L}{\cal M}^{-1/4}$.. Note that this limit is usually called the Newtonian limit in the context of general relativity; we will generally avoid that terminology here since conventional Newtonian dynamics are not recovered in the limit ${\hat{c}}\to\infty$ unless ${\hat{a}}_{0}\ll 1$.

To examine the slow motion limit we specialize to conformally Cartesian harmonic gauge, with a metric expansion of the standard form

[TABLE]

with the leading order potential $\Phi$ and subleading potentials $\Psi$, $\psi$ and $\zeta_{i}$. We assume an expansion of the khronon field of the form

[TABLE]

where the khronon perturbation $\pi$ has dimensions of $[L]^{2}[T]^{-1}$. The motivation for the ansatz (12) is that it guarantees that the khronon stress energy tensor (7) has scalings with $c$ of the standard post-Newtonian form (see the appendix)

[TABLE]

consistent with the metric ansatz (11). This in turn, when combined with the Einstein equation (6), implies the equality of the potentials $\Phi$ and $\Psi$ in the metric (11):

[TABLE]

To explore the dynamics in the slow motion limit it is easiest to work with the action (3) rather than the equations of motion (6) and (9). It is convenient to define a rescaled acceleration variable

[TABLE]

which has units of acceleration, and a rescaled function of acceleration

[TABLE]

By inserting the ansatzes (11) and (12) and the result (14) into the definitions (1), (2) and (15) we obtain the expansions (see the appendix)

[TABLE]

and

[TABLE]

where

[TABLE]

and where $\nabla$ is the spatial gradient. The nonrelativistic limit of the action (3) becomes222The general form of the dependence of this action on $\pi$ is dictated by Galilean invariance, and the same class of actions is obtained for a superfluid coupled to Newtonian gravity (Khoury, 2021).

[TABLE]

where $\rho_{\rm m}$ is the mass density.

Varying with respect to the khronon perturbation $\pi$ gives the khronon equation of motion in the slow motion limit in the form of a continuity equation

[TABLE]

Here the khronon mass density is

[TABLE]

where

[TABLE]

and its velocity is

[TABLE]

Varying the action (20) with respect to the Newtonian potential $\Phi$ and using Eqs. (8), (18) and (19) gives

[TABLE]

This reduces to the MOND equation for $\Phi$ when $\pi=0$, as shown by BM:

[TABLE]

where we have defined the background acceleration

[TABLE]

3.1 Unitary gauge

Starting from the conformally Cartesian harmonic gauge used here, it is possible to set the khronon perturbation to zero by a change of gauge of the form

[TABLE]

yielding from (12) that

[TABLE]

In this so-called unitary gauge the form (11) of the metric expansion is altered, and in particular the ${\bar{t}}i$ component of the metric has a term at $O(c^{0})$,

[TABLE]

rather than being of order $O(c^{-2})$ as is normally the case.

It should be possible to rederive all the results of this paper using the unitary gauge (29). However, it is somewhat awkward to use this gauge, since it is incompatible with the standard post-Newtonian expansion framework and so requires generalizing this framework. This awkwardness is why we chose conformally Cartesian harmonic gauge. However we emphasize that there is no issue of principle with the use of unitary gauge.

3.2 Dynamics of the khronon field

Some insight into the dynamics of the khronon field can be obtained by linearizing the equation of motion (21) in $\pi$. We obtain

[TABLE]

Here the tensor $h^{ij}$ is given by

[TABLE]

with {\hat{a}}_{\rm b}^{i}={\bar{a}}_{\rm b}^{i}/{\bar{a}}_{\rm b}=\partial_{i}\Phi/|{\mbox{\boldmath\nabla}\Phi}|. The quantity $\rho_{T0}$ is the khronon mass density (22) evaluated at $\pi=0$:

[TABLE]

We see that the khronon field is generated when there is a time varying potential $\Phi$.

We can use the linearized equation (31) to make an order of magnitude estimate of $\pi$ for self-gravitating sources. In the MOND regime ${\bar{a}}_{\rm b}\ll a_{0}$ we obtain ${\bar{\chi}}\sim 1$, $h^{ij}\sim G^{-1}$, $\nabla\Phi\sim\sqrt{G{\cal M}a_{0}}{\cal L}^{-1}$, $\rho_{T0}\sim\sqrt{{\cal M}a_{0}/G}{\cal L}^{-2}$ and ${\dot{\rho}}_{T0}\sim\rho_{T0}{\cal T}^{-1}$. From footnote 1 we find that the two terms on the left hand side of Eq. (31) are comparable. Comparing the second term to the right hand side then gives the estimate

[TABLE]

Inserting this estimate into the formula (19) for the potential $\Xi$ we see that all three terms are comparable333In particular the term nonlinear in $\pi$ in Eq. (19) is comparable to the linear terms, which implies that the omitted nonlinear terms in the linearized equation (31) should be included. However that linearized equation should be adequate for the order of magnitude estimates derived here.. From Eq. (25) it then follows that the khronon corrections to the MONDian dynamics are of order unity, and that the acceleration of the khronon foliation $\sim\nabla{\dot{\pi}}$ is of order the characteristic acceleration of the system ${\cal L}{\cal T}^{-2}$. This disproves the conjecture by BM that the preferred foliation essentially coincides with the cosmological rest frame with an acceleration {\mbox{\boldmath\nabla}}{\dot{\pi}} small compared to $a_{0}$ and unimportant dynamically when ${\bar{a}}_{\rm b}\sim a_{0}$.

A similar analysis can be carried out in the high acceleration or Newtonian regime ${\bar{a}}_{\rm b}\gg a_{0}$. In this case we have $|{\bar{\chi}}|\ll 1$, $\Phi\sim G{\cal M}{\cal L}^{-1}$, $\rho_{T0}\sim{\bar{\chi}}{\cal M}{\cal L}^{-3}$ and $h^{ij}\sim{\bar{\chi}}/G$. The two terms on the left hand side of Eq. (31) are again comparable and we again find the estimate (34). Although the perturbations to the foliation are large in this regime, the effect of the khronon field on the dynamics is suppressed by the small parameter ${\bar{\chi}}$ in Eq. (25) which goes to zero as ${\bar{a}}_{\rm b}\to\infty$ 444Note that the validity of the approximation $\pi=0$ which we are discussing here is unrelated to the question of whether one can work in the unitary gauge (29), discussed in Sec. 3.1 above. In our analysis the gauge is determined by the conformally Cartesian harmonic gauge condition with the expansion (11), and there is a physical, gauge-invariant quantity which reduces to $\pi$ in this gauge..

The special case of stationary solutions with ${\dot{\rho}}_{\rm m}={\dot{\Phi}}=0$ are consistent with a vanishing khronon field $\pi=0$, from Eq. (31). We will study the stability of these solutions in Sec. 4 below.

3.3 Consistency of the slow motion limit and subleading/post-Newtonian corrections

We next demonstrate that the assumed scalings (11) and (12) that we have used for the slow motion limit give rise to a consistent computational framework to subleading (post-Newtonian) order. Specializing to the gravitomagnetic sector, we find that the potential $\zeta_{i}$ is given by (see the appendix)

[TABLE]

Here ${\bf v}_{\rm m}$ is the matter velocity and we have assumed for simplicity that the matter is a fluid. We have also used the harmonic gauge condition specified before Eq. (11), which implies that $\partial_{i}\zeta_{i}=4{\dot{\Phi}}$. Equation (35) is the standard harmonic gauge equation of general relativity but with the mass current supplemented by a khronon contribution. A similar calculation in the gravitoelectric sector shows that the potential $\psi$ is given by (see the appendix)

[TABLE]

Here the ellipses represent source terms that are independent of $\psi$, and ${\hat{a}}^{i}={\bar{a}}^{i}/{\bar{a}}$ with ${\bar{a}}^{i}=\partial_{i}\Xi$. The subleading equations (35) and (36) yield unique solutions for the potentials $\psi$ and $\zeta_{i}$ and place no constraints on the leading order fields.

3.4 Comparison with other treatments of the slow motion limit in the literature

Blanchet & Marsat (2011) derived the slow motion limit of their theory in the unitary gauge (29). However, they assumed the standard scaling (11) for $g_{ti}$, which as we argued in Sec. 3.1 is inconsistent with the assumption of unitary gauge. Nevertheless, this inconsistency did not affect their derivation of the leading order MOND equation of motion.

A detailed analysis of the slow motion limit of khronometric theories has been given by Bonetti & Barausse (2015). They also assume unitary gauge and the standard post-Newtonian scalings with $c$ of the metric coefficients, in disagreement with Eq. (30). However in their case the assumption is justified, because they work in the context of a more general class of theories obtained by adding to the action (3) the terms

[TABLE]

where $\beta$ and $\lambda$ are dimensionless parameters. In the slow motion limit this action reduces to

[TABLE]

Comparing with the original action (20) we see that the correction (38) is superleading in the limit $c\to 0$, scaling $\propto c^{2}$ rather than $c^{0}$, and thus changing the nature of the post-Newtonian expansion. The slow motion limit $c\to\infty$ does not commute with the limit $\beta,\lambda\to 0$ in which the theory reduces to BM theory. The expansion used by Bonetti & Barausse (2015) is valid in the limit where the dimensionless parameters

[TABLE]

are large compared to unity, as they point out in their Sec. V, and the different expansion derived here is valid in the limit when these parameters are small. Here the parameter ${\hat{c}}$ is defined in Eq. (10).

The expansion method of Bonetti & Barausse (2015) gives results for some of the post-Newtonian fields that diverge as $\beta,\lambda\to 0$. They describe this situation as a “strong coupling” and a breakdown of the post-Newtonian expansion, a pathological limit.555One example given by Bonetti & Barausse (2015) that demonstrates the breakdown of their expansion in the $\lambda,\beta\to 0$ limit is the following. Their gravitomagnetic equation (44) multiplied by $\beta+\lambda$ and then evaluated at $\beta=\lambda=0$ enforces the usual Poisson equation relating the Newtonian potential and the matter mass density. This Poisson equation is inconsistent with the MOND equation for the Newtonian potential, their Eq. (39). In fact the limit is well defined but does require switching to the different kind of post-Newtonian expansion used here (which is consistent at $\beta=\lambda=0$) once the parameters (39) become small.

4 Stability of stationary solutions

As discussed in Sec. 3, stationary solutions666Here by stationary solutions we mean solutions in which the gravitational degrees of freedom $\pi$ and $\Phi$ are independent of time. We do not impose, for example, that fluid velocities should vanish, i.e. that the solutions be static. of modified Newtonian dynamics coincide with stationary solutions of the slow motion limit of BM theory with vanishing khronon perturbation (Blanchet & Marsat, 2011). In this section we show that perturbations to these solutions are stable in the low acceleration regime ${\bar{a}}\ll a_{0}$, for the cases of spherical, cylindrical and planar symmetry.

Consider a stationary solution with $\pi=0$. Expanding the action (20) to second order in the perturbations $\delta\pi$ and $\delta\Phi$ about this solution and making use of Eqs. (18), (19), (22) and (23) gives the quadratic action

[TABLE]

Here the ellipses $\ldots$ represent the matter degrees of freedom, which in the application to cosmology would be the baryons. The interaction term is

[TABLE]

while the khronon action is

[TABLE]

Here the background khronon mass density $\rho_{T0}$ is given by Eq. (33) and the tensor $h^{ij}$ is defined in Eq. (32). The tensor $h^{ij}$ is nonnegative if the function ${\bar{f}}({\bar{a}})$ obeys the conditions

[TABLE]

from Eq. (23). These conditions are satisfied in the deep MOND regime ${\bar{a}}\ll a_{0}$ from Eqs. (4) and (16). Thus the khronon kinetic energy term in the action (42) has the conventional sign in this regime.

We will assume that the background stationary solution is stable in the conventional MOND theory without a khronon field (Milgrom, 1983a). This implies that the dynamics arising from the the term $S_{2,\Phi}$ in the action (40) has no unstable modes.

To show that that the theory (40) has no unstable modes we proceed in two steps. First, we show that the interaction term $S_{2,{\rm int}}$ cannot produce an instability if the khronon term $S_{2,\pi}$ is stable by itself. Second, we analyze the dynamics of the khronon field by itself and show that it has no unstable modes.

To analyze the interaction term $S_{2,{\rm int}}$ we rewrite the action (40) in terms of the schematic Lagrangian

[TABLE]

where ${\cal B}_{AB}={\cal B}_{[AB]}$. Here $Q^{A}$ are configuration space coordinates that encompass $\delta\pi$, $\delta\Phi$ and the perturbations to the matter degrees of freedom, with $Q^{A}=0$ for the stationary solution. The indices $A,B,\ldots$ label these fields and also parameterize the dependence on the spatial coordinates, so sums over these indices contain integrals over the spatial coordinates. The third term in the Lagrangian (44) involving the antisymmetric tensor ${\cal B}_{AB}$ contains the interaction term $S_{2,{\rm int}}$, which from Eq. (41) is a product of two terms, one of which has a time derivative. See the appendix for more details about the schematic Lagrangian (44).

The corresponding Hamiltonian can be written as

[TABLE]

with ${\cal G}^{AB}{\cal G}_{BC}=\delta^{A}_{C}$. A key point now is that the tensors ${\cal G}_{AB}$ and ${\cal V}_{AB}$ are nonnegative. For the non-khronon contributions to these tensors, this follows from our assumption that the background stationary solution is stable in the conventional MOND theory without a khronon field (Milgrom, 1983a). For the khronon contributions, the non-negativity of ${\cal G}_{AB}$ and ${\cal V}_{AB}$ follows from the discussion around Eq. (43) for the kinetic term, and from the form of the second term in Eq. (42) for the potential term (assuming $\rho_{T0}\geq 0$; see below). It follows that $H$ is a non-negative quadratic form on phase space, and so the motion is confined to a compact surface $H=$ constant. This excludes any exponentially growing mode solutions of the form $Q^{A}(t)=Q^{A}_{0}e^{-i\omega t}$.

Turn now to the dynamics of the khronon field by itself, described by the action (42). Consider complex mode solutions of the form

[TABLE]

Substituting this ansatz into the equation of motion obtained from the action (42), multiplying by $\pi_{0}({\bf x})^{*}$ and integrating by parts yields

[TABLE]

It follows that the background MOND solution has no unstable modes if (i) the conditions (43) are satisfied, so that $h^{ij}$ is nonnegative, and (ii) the background khronon mass density given by Eqs. (22) and (33) is nonnegative:

[TABLE]

The khronon mass density is not always nonnegative for stationary solutions, as shown by Milgrom (1986). In particular he showed that if there exists an isolated point where {\mbox{\boldmath\nabla}}\Phi=0, located in a region with $\rho_{\rm m}=0$, then that point must lie on the boundary of a region with with $\rho_{T0}<0$. We now show that $\rho_{T0}\geq 0$ when $\pi=0$ in the cases of spherical, cylindrical and planar symmetry, thus showing that the background solutions are mode stable in the deep MOND regime in these cases. Stability going beyond these special cases is an open question777Note that the fact that $\rho_{T0}$ is sometimes negative does not imply the existence of an instability, since the contributions to the integral in the numerator of Eq. (47) from regions of positive $\rho_{T0}$ may dominate..

In spherical symmetry the magnitude $g(r)$ of the Newtonian acceleration at radius $r$ is related to the enclosed mass $m(r)$ by $g=Gm(r)/r^{2}$. The magnitude ${\bar{a}}$ of the actual acceleration is given in terms of $g$ from Eq. (26) by $\varpi({\bar{a}})=g$, where the function $\varpi$ is given by888The conventional notation for this function is $\varpi({\bar{a}})={\bar{a}}\mu({\bar{a}}/a_{0})$ (Milgrom, 1983a).

[TABLE]

We write this relation as ${\bar{a}}=\lambda(g)$, where $\lambda$ is the inverse of the function $\varpi$. Using the fact that the total effective mass ${\tilde{m}}(r)$ enclosed inside radius $r$ is given by ${\bar{a}}=G{\tilde{m}}(r)/r^{2}$, we obtain

[TABLE]

Now the khronon mass density $\rho_{T0}$ is proportional to ${\tilde{m}}^{\prime}(r)-m^{\prime}(r)$, which from Eq. (50) is given by

[TABLE]

It follows that $\rho_{T0}\geq 0$ if the conditions

[TABLE]

are satisfied. From Eqs. (23) and (49) these conditions are equivalent to

[TABLE]

Both of these conditions are satisfied in the deep MOND regime ${\bar{a}}\ll a_{0}$ from Eqs. (4) and (16).

Similar analyses apply in the cases of cylindrical and planar symmetry. In the cylindrical case the relation (50) is modified by replacing on both sides $Gm(r)/r^{2}$ with $2G\sigma(r)/r$, where $r$ is now distance from the axis of symmetry, and $\sigma(r)$ is mass per unit length along the axis enclosed inside radius $r$. Repeating the analysis gives the same conditions (53) as before. For planar symmetry the argument of the function $\lambda$ becomes $4\pi G\Sigma(r)$, where $r$ is now distance from the plane and $\Sigma(r)$ is enclosed mass per unit area. In this case one obtains the first of the conditions (53) but not the second.

To summarize, we have shown that that stationary solutions are stable if:

We restrict to special configurations of enhanced symmetry. 2. 2.

The solutions are stable in the conventional MOND theory without the khronon field. 3. 3.

The conditions (43) and (53) on the function ${\bar{f}}$ are satisfied.

We can combine the two conditions on ${\bar{f}}$ and write them in terms of the function $f$ that appears in the action (3) using the rescaling (16), to obtain

[TABLE]

Although these conditions are satisfied in the deep MOND regime, the cannot be satisfied for all values of $a$ since they are incompatible with the boundary conditions (4) and (5) at large $a$ and small $a$. Thus there must be a range of values of $a$ with $a\sim a_{0}/c^{2}$ or ${\bar{a}}\sim a_{0}$ where the conditions (54) are violated. Stationary solutions might therefore be unstable in this regime999If instabilities do exist, the growth timescale when ${\bar{a}}\sim a_{0}$ will be of order the gravitational dynamical timescale ${\cal T}$, as can be seen by applying dimensional analysis to the action (42) (see footnote 1 above)..

5 Discussion and Conclusions

The BM theory of modified Newtonian dynamics (Blanchet & Marsat, 2011, 2012) is a minimal fully relativistic version of MOND which is simpler than the alternatives (Skordis & Zlosnik, 2021, 2022). In this paper we have presented arguments in favor of the viability of this theory.

In particular, it was previously shown by BM that stationary solutions of this theory in the slow motion limit coincide with stationary solutions of the conventional nonrelativistic formulation of MOND. The theory would be disfavored if these solutions were unstable to khronon perturbations. We have presented evidence for stability of these solutions in certain limits.

We also showed that predictions of the BM theory generally differ from those of MOND by an amount of order unity for non-stationary solutions, for systems near the MOND acceleration scale. In particular this applies to binary star systems, for which the khronon perturbation is nonzero and is determined by an elliptic equation in the rotating frame. It would be interesting to derive from this equation the form of Kepler’s third law for the BM theory, which will differ from the MOND form (Zhao et al., 2010). This could be useful for current efforts to test MOND with observations of wide binary star systems using GAIA data (Pittordis & Sutherland, 2022; Chae, 2023).

We also note that instabilities arise in the BM theory when perturbed about Minkowski spacetime. These instabilities are generic for all khronometric theories and can be cured by the addition of higher spatial derivative terms to the action which are suppressed by a mass scale, and give small corrections the dynamics at scales of interest (Horava, 2009; Blas et al., 2010; Bonetti & Barausse, 2015).

I thank Enrico Barausse helpful correspondence and for pointing out an error in an earlier version of this paper, Ira Wasserman for helpful discussions, and an anonymous referee for detailed and helpful comments. This research was supported in part by NSF grant PHY 2110463 and by a fellowship from the Simons Foundation.

In this appendix we provide some of the details of the calculations reported in the body of the paper. We obtain the expansion (17) of the normal $n_{a}$ by inserting the ansatz (12) for the khronon field $T$ into the definition (1) and using the metric expansion (11). Inserting the result into the definition (2) of the acceleration and using the expansion (11) again gives

[TABLE]

where $\Xi$ is given by Eq. (19). Using the rescaling (15) now gives the formula (18).

Next, we insert the acceleration (55) and normal (17) into the khronon stress energy expression (7), and make use of the rescalings (16) and (23). This shows that the stress energy components have an expansion with $c$ of the standard post-Newtonian form (13), with the leading order terms being

[TABLE]

where $\rho_{T}$ is the khronon mass density (22) and ${\bf v}_{T}$ its velocity (24).

We next turn to the subleading corrections to the slow motion limit discussed in Sec. 3.3. Substituting the momentum flux (57) together with the fluid momentum flux into the usual post-1-Newtonian harmonic gauge gravitomagnetic Einstein equation gives Eq. (35). In the gravitoelectric sector, the subleading equation for the scalar potential $\psi$ in general relativity in conformally Cartesian harmonic gauge is

[TABLE]

Here on the right hand side $T^{(0)\,\alpha\beta}$ is the $O(c^{0})$ piece of the stress energy tensor, and $T^{(2)\,\alpha\beta}$ is the $O(c^{-2})$ piece. We can apply this equation to the present context by inserting on the right hand side the sum of the fluid stress energy tensor and the khronon stress energy tensor (56) – (58). Bringing all the terms that depend on $\psi$ to the left hand side then results in Eq. (36). Because of how $\psi$ enters into the metric expansion (11), the relevant terms can be computed by taking a variation $\Phi\to\Phi+\delta\Phi$ of the expression (22). Note that the right hand side of Eq. (36) will then depend on subleading corrections to the khronon field expansion (12) beyond the field $\pi$, arising from $T_{T}^{(2)\,tt}$.

Finally we provide more details about the schematic Lagrangian (44) that describes perturbations to stationary solutions. The kinetic energy term (first term) in the Lagrangian (44) has a contribution from the khronon field given by the first term in Eq. (42). It also has a contribution from the matter degrees of freedom, the third term in Eq. (40). The specific form of this term for a fluid, for example, in the framework of Lagrangian perturbation theory, is given in Eq. (6.6.7) of Shapiro & Teukolsky (1983). The potential term (second term) in the Lagrangian (44) has the khronon contribution given by the second term in Eq. (42), and the fluid contribution given by Eq. (6.6.9) of Shapiro & Teukolsky (1983). To get the correct dependence on $\delta\Phi$ for our context we multiply the last term in this equation by two101010This is necessary since $\delta\Phi$ has effectively been integrated out in Shapiro & Teukolsky (1983), and we are restoring it as a dynamical variable., and add the contribution

[TABLE]

Finally the mixed term (third term) in (44) has the khronon contribution (41). It can also have a fluid contribution, for perturbations to stationary systems that are not static (Lynden-Bell & Ostriker, 1967).