# Bimetric MOND gravity

###### Abstract

A new relativistic formulation of MOND is advanced, involving two metrics as independent degrees of freedom: the MOND metric , to which alone matter couples, and an auxiliary metric . The main idea hinges on the fact that we can form tensors from the difference of the Levi-Civita connections of the two metrics, , and these act like gravitational accelerations. In the context of MOND we can form dimensionless ‘acceleration’ scalars, and functions thereof (containing only first derivatives) from contractions of . I look at a class of bimetric MOND theories governed by the action with a scalar quadratic in the , , the matter action, and allowing for the existence of twin matter that couples to alone. Thus, gravity is modified not by modifying the ‘elasticity’ of the space-time in which matter lives, but by the interaction between that space-time and the auxiliary one. In particular, I concentrate on the interesting and simple choice . This theory introduces only one new constant, ; it tends simply to general relativity (GR) in the limit , and to a phenomenologically valid MOND theory in the nonrelativistic limit. The theory naturally gives MOND and “dark energy” effects from the same term in the action, both controlled by the MOND constant . As regards gravitational lensing by nonrelativistic systems–a holy grail for relativistic MOND theories–the theory predicts that the same potential that controls massive-particle motion also dictates lensing in the same way as in GR: Lensing and massive-particle probing of galactic fields will require the same “halo” of dark matter to explain the departure of the present theory from GR. This last result can be modified with other choices of , but lensing is still enhanced and MOND-like, with an effective logarithmic potential.

###### pacs:

## I Introduction

From the inception of MOND milgrom83 it has been clear that the paradigm needs buttressing by a relativistic formulation. Indeed, efforts to construct such a formulation started shortly thereafter, with the tensor-scalar version sketched in bm84 . This was the first in a chain of theories of increasing force, culminating in the advent of the tensor-vector-scalar theory (TeVeS) of Bekenstein bek04 . Some landmarks along this track are described in sanders97 ; bek04 ; bek06 ; z06 ; z07 ; skordis09 ; see, in particular, the reviews in bek06 ; skordis09 . All these theories involve as independent degrees of freedom an Einstein metric, whose free action is the standard Einstein-Hilbert action, with additional scalar and/or vector degrees of freedom, with their own actions. These scalar/vector degrees of freedom are used to dress up the Einstein metric into the ‘physical’ metric to which matter couples. TeVeS has a version of the nonrelativistic (NR) theory proposed in bm84 as a NR limit.

Another line of relativistic theories that aim to reproduce MOND phenomenology has been propounded in blt08 ; blt09 , based on the omnipresence of a gravitationally polarizable medium proposed in bl07 .

Here I propound a new class of relativistic formulations for the MOND paradigm in the form of bimetric MOND (BIMOND) theories. These came to light as follows: I have recently described milgrom09c a new class of nonrelativistic, bi-potential MOND theories, a subclass of which is governed by a Lagrangian density of the form

(1) |

leading to the field equations

(2) |

with . I also described in detail the requirements from , and that lead to the required MOND and Newtonian limits of these theories. In particular, I discussed at length the interesting case ( then normalizes to be the Newton constant), which leads to the field equations

(3) |

with a function of , such that for ensures the Newtonian limit, and in the MOND regime . This is a particularly tractable MOND theory, as it requires solving only linear differential equations, with the inevitable MOND nonlinearity entering only algebraically. In all the NR theories above, matter couples only to one of the potentials: the MOND potential , while is an auxiliary potential, and in the special case of Eq.(3) their difference is exactly the Newtonian potential of the problem.

These NR MOND theories have inspired the construction of closely
analogous relativistic MOND theories with two metrics as
independent, gravitational degrees of freedom, which I begin to
investigate here.^{1}^{1}1In these theories the two metrics are
independent degrees of freedom. Theories like Brans-Dicke, TeVeS,
etc., are also sometimes described as being bimetric because they
involve two metrics, but those two metrics are a priori related
conformally or disformally via other degrees of freedom such as
scalars or vectors. This new class of BIMOND theories involve only
as a new constant. They tend to general relativity (GR) in the
limit , which is a desirable trait. And, they tend to a
MOND theory compatible with MOND phenomenology in their NR limit.

These theories, like all other relativistic versions of MOND proposed to date, must, I believe, be only approximate, effective theories to be derived from some more fundamental picture that underlies them. This is pointed to by the appearance of an a priori unspecified function in all these theories.

The use of two (or more) metrics to describe gravity has a long history. For example, Rosen rosen74 considered bimetric theories, where the auxiliary metric is forced to be flat. More recently, it was found boulanger01 that ghosts appear in a large class of bimetric theories (apparently not including the present BIMOND). More matter-of-principle questions regarding bimetric gravities are discussed in dk02 ; bdg06 ; bdg07 ; banados09 , but these authors confined themselves to metric couplings that involve only the metrics, not their derivatives, as in the case of BIMOND.

In section II, I present the formalism underlying the BIMOND theories; in section III, I consider the NR limit of these theories, showing how they lead to NR MOND theories; section IV demonstrates how the theories go to GR in the limit ; section V discusses lensing; section VI discusses cosmology briefly, and section VII is a discussion.

## Ii Formalism

The NR theories mentioned above involve two potentials, the MOND potential felt by matter, and an auxiliary one . They point to relativistic BIMOND theories involving the MOND metric , to which matter couples, and which in the NR limit reduces to , and an auxiliary metric .

Working with two metrics enables us to form nontrivial tensors and scalars from the difference in their Levi-Civita connections and ,

(4) |

involving only first derivatives of the metrics, which is not possible with a single metric. This is particularly pertinent in the context of MOND, since connections act like gravitational accelerations. So, without introducing new constants in the relativistic formulation we can write Lagrangian functions of dimensionless scalars constructed from that enable us to interpolate between the GR limit, , and the MOND limit, .

The tensor is related to covariant derivatives of one metric with the connection of the other (more generally, they relate covariant derivatives of tensors with respect to the two connections):

(5) |

(6) |

where the covariant derivative is taken with the connection and with . We can form various scalars out of and the metrics. One scalar that will be of particular use to us is based on the tensor

(7) |

with the same index combination that appears in the expression for the Ricci tensor

(8) |

and in constructed similarly from . One finds

(9) |

Thus, using well known manipulations, the scalar connects the two Ricci scalars and the mixed by

(10) |

Similarly, interchanging the roles of and ,

(11) |

where , , is the Ricci scalar of , and and are minus the determinants of and respectively.

We can construct gravitational Lagrangian densities using the scalars , and scalars constructed by contracting powers of with the two metrics and their inverses (there are also , , etc. that can be used). If we only contract with and , a quadratic scalar is a linear combination (possibly with coefficients depending on scalars such as or ) of the following scalars

(12) |

where
, . The choice of scalars to be used may be forced on
us by various theoretical and phenomenological desiderata (see
below). The main point is that have only first
derivatives of the metrics, that they reduce to derivatives of the
potential difference in the Newtonian limit (in the sense to be
discussed below), and that we can form dimensionless quantities from
them with the MOND acceleration (or a MOND length
). As regards the four curvature scalars, is will be
advantageous to include them in the action only linearly, and eschew
terms such as in the voguish theories. Such nonlinear terms
render the theory a higher derivative one, which I would like to
avoid.^{2}^{2}2For the same reason I avoid scalars that are higher
order in the curvature tensors, such as the different possible
contractions of with itself or with .
These are even less appealing as explained in woodard07 .
Another reason to avoid such terms in the MOND context is that they
do not naturally lead in their NR limit to a single constant
controlling the dynamics.^{3}^{3}3For example, to account for
dimensions correctly, a function of has to be introduced as
, with some length scale. The NR limit of
includes as the dominant term in , and second
order ones such as , and . Thus, in the
argument of the second term will give , with
, which fits well into the MOND frame. But the first,
dominant, term would involve a time scale , not an
acceleration. When appears linearly, the first term becomes
immaterial in the action, as a complete derivative, and we are left
with terms that are welcome in MOND [the term is also
up to a derivative]. Neither obstacle appears if we allow
functions of scalars made of . These contain only
first derivatives of the metrics, and give NR limits in which only
appears (see below). We see from Eqs.(10) that and differ by plus a total derivative so it
is enough to include one of these in the action, as we anyhow permit
functions of . The same is true of the pair and
. Because the number of possible combinations it too
large to explore here, I limit myself to the subclass of actions of
the form

(13) |

where , , ,
and are scalars formed by contracting a product of
(even) , which can be used in principle. In
what follows, I shall confine myself to quadratic
scalars.^{4}^{4}4The MOND constant is normalized so that the
mass-asymptotic-velocity relation is . It defines the
scale length that is used in the coefficient and the argument
of . Any dimensionless factors can be absorbed in the definition
of so that its coefficient is and its
argument is as prescribed here. I have included two matter actions:
The first, , involves the matter degrees of freedom with which
we interact directly, designated symbolically as . It
contains only the MOND metric to which matter is coupled in
the standard way. The other, , involves other matter
degrees of freedom, , and only , to account
for the possibility that controls a matter world of
its own. There are no direct (electromagnetic, etc.) interactions
between the matter and the twin matter.^{5}^{5}5To
obviate possible confusion, note that the twin matter is not to play
the role of the putative dark matter in galactic systems; this is
still fully replaced by MOND effects; see below.

I make two requirements of the action: a. Require that it gives a NR MOND theory in its NR limit. This means the following: given a non relativistic system of slow masses one can express the metrics solution of the relativistic theory in terms of potentials so that the equations of motion for slow particles in the resulting (multi) potential theory are those required by NR MOND, with the appropriate MOND and Newtonian limits; this is a phenomenological requirement (by itself it does not dictate the effects on massless particles–e.g., gravitational lensing–even in NR systems). b. Require that the action gives GR in the limit . This is not a phenomenological necessary (for example, TeVeS does not satisfy it), but I feel that it is highly desirable for various reasons. This automatically causes the theory to agree with the stringent constraints from the solar system and binary pulsars–which are known to agree with GR–because the accelerations in these systems are many orders of magnitude larger than . I also require this limit lest we have to introduce additional constant(s) to the theory, which has to give GR in some limit of its parameters.

When the two metrics are conformally related, which might be the case in certain circumstances, , we have , , , . If we a priori constrain our metrics to be conformally related (i.e. vary the action only over such pairs) we get the Brans-Dicke theory with the choice (and appropriate choice of the constants and , and possibly using in the argument of ). With a more general form of , we then get the relativistic MOND theory sketched in bm84 .

Without the interaction term, the theory separates into two disjoint copies of GR. It is important to note that as a combined structure, the theory then enjoys a larger symmetry involving separate coordinate transformations in the two separate actions. This double symmetry has to be brought to bear when solving the field equations of the theory, which now satisfy two sets of Bianchi identities. So, eight gauge conditions can, and have to, be employed. It is the interaction that breaks this larger symmetry, as, generically, it is only invariant to application of the same coordinate transformation to the two metrics. However, under certain circumstances the interaction is symmetric under a more extended set of coordinate transformation, and we must be careful then to employ the larger gauge freedom. The above mentioned complete decoupling is an example that, as we shall see in section IV, applies in the formal limit of the theory (leading, as we want, to GR). It may also happen, in principle, that the interaction term vanishes only in some limited regions of space-time; for example, if the extreme GR limit applies in some regions. In this case we must allow for gauge freedom involving coordinate transformations that coincide only outside these regions, but not inside them. We shall see another example in section III, where the NR limit of the theory has such a partial double gauge freedom.

### ii.1 Concrete simple example

I shall hereafter concentrate on a simple special case of the class. Some generalizations will be mentioned briefly below, in this section, and in section VI.

In the first place, I take to be a function of only one scalar, quadratic in the . In particular, I find the scalar defined above a natural choice for this argument, as it has the same structure as the first-derivative part of the Ricci curvature scalar (not itself a scalar)

(14) |

It is well known that one can replace in the Einstein-Hilbert action by and still get GR. Here we can also do this, replacing also by the corresponding , and making a function of , which is constructed in the same way from . We shall also see that with this choice of scalar argument the NR limit of the theory is especially simple.

As a further simplification I take . This will yield a particularly interesting and simple subclass of theories, which turn out to have the theory (3) as their NR limit for slowly moving masses in a double Minkowski background. I then take for to be Newton’s constant.

Work in units in which , and use to highlight the connection with MOND. Also, anticipating the expression for NR limit of , I take the argument of to be . The relativistic action I then consider is

(15) |

[Using Eq.(11) we can replace the first two terms by .] I take a mixed volume element for the interaction term, with normalized such that . Note the change of sign in the definition of the twin matter action to match the negative sign for the Hilbert-Einstein action of .

Varying over and over we get, respectively

(16) |

(17) |

where and are the Einstein tensors of the two metrics,

(18) |

and are the matter energy-momentum tensors (EMT); e.g., , and are the functional derivatives (one with an opposite sign) of the interaction term with respect to the two metrics:

(19) |

For the present choice of the scalar argument of , we have

(20) |

(21) |

(22) |

(23) |

(24) |

Here,

(25) |

and signifies symmetrization over the two indices,

The tensor

(26) |

may be viewed as the EMT of the phantom dark matter (DM); whereas the term may roughly be viewed as “dark energy”. Note that the last term in , which contributes may also contribute to the dark energy due to its form. Define in analogy with

(27) |

The Einstein tensors satisfy the usual Bianchi identities
,^{6}^{6}6For each tensor
indices are raise with the corresponding metric; so, e.g.,
. derivable from the invariance of the Einstein-Hilbert
actions to coordinate transformations. In addition, we have here,
for the general action (13), a set of four identities
following from the fact that the mixed term is a scalar; these read

(28) |

Given that the matter EMTs are divergence free (for matter degrees of freedom satisfying their own equations of motion): , the above identities imply four differential identities satisfied by our 20 field equations. If we write these equations as and , respectively, then the four relations

(29) |

hold identically, and, as usual, deprive us of four equations to
account for the fact that the solution can be determined only up to
a coordinate transformation. This seems to leave us with a tractable
Cauchy problem, although this require more careful
checking.^{7}^{7}7As a result of identities (28) and the
Bianchi identities, the four expressions contain only up to first
time derivatives of the metric, and cannot be used to propagate the
problem in time. Instead, the initial conditions have to satisfy the
four equations , and the remaining
sixteen field equations, with the aid of four gauge conditions,
propagate us in time, and insure that these four are always
satisfied.

Of course, for solutions of the field equations we do have separately

(30) |

which can be used as useful constraints of the solutions (only one set is independent).

Note the useful identities

(31) |

(32) |

Identities (32) follow from Eqs.(9)(10). Similar manipulations are possible for , and the field equation (17).

Contracting Eq.(16) with gives

(33) |

Contracting Eq.(17) with gives

(34) |

where . We can thus replace Eq.(16) by

(35) |

and similarly for Eq.(17). We can also use identities (9-11) to write these equations in different forms. Equations (33)(34) can be used to write possibly useful integral (virial) relations by integrating them over space-time, each with its own volume element.

It was deduced in boulanger01 that under certain assumptions about the theory, bimetric theories generically posses ghosts. One of their assumptions was that to lowest order in departure from double Minkowski the theory is a sum of Pauli-Fierz actions for the different metrics, which are quadratic in the metric departures. This, however, leads to a linear theory in this limit, which is at odds with MOND: MOND phenomenology dictates that at any BIMOND theory (or any relativistic MOND theory for that matter) is not even analytic in the squares of the departures (where the argument of in the above version of the theory vanishes, and diverges). It thus remains to be seen if obstacles similar to these are at all relevant to BIMOND, and if they are to what extent they are deleterious.

For conformally related metrics , we have .

### ii.2 Generalizations

Some generalizations of the above simple theory include the following.

1. Instead of using as the argument of , we can use other scalars, or several scalar variables. A quadratic scalar variable can be written, most generally, as

(36) |

where it built from , , their inverses, , and scalars such as and . In this case the terms take a more general form, and so do terms that are second order in the . The only terms in and that survive in the NR limit, which we treat below, are those involving and . For these we now have for example

(37) |

which I shall need in what follows.

For example, taking as the argument of , instead of , would leave us with only the first term in expression (22) for , and with the first three terms in expression (23) for .

2. One can consider more general values.

3. We can increase the symmetry with respect to the two metrics by taking interaction terms of the form , , etc..

4. One can make a function of scalars such as and .

Additional generalizations will be mentioned in section VI.

## Iii Nonrelativistic limit

Consider now the NR limit of the theory derived from the action (15). This limit applies to systems where all quantities with the dimensions of velocities, such as , etc., are much smaller than the speed of light. In the context of GR this limit is attained by formally taking everywhere in the relativistic theory. In the context of MOND one has to be more specific, since system attributes with the dimensions of acceleration, such as , etc., cannot be assumed very small in the limiting process, even though they have velocities in the numerator. We want to consider systems, such as galaxies, in which these are finite compared with the MOND acceleration, which is also a relevant parameter. The NR limit in MOND is thus formally attained by taking everywhere , but at the same time , so that remains finite.

Take a system of quasistatic (nonrelativistically moving) masses, so
that to a satisfactory approximation we can, as usual, neglect all
components of the matter EMT except . I also neglect
here the possible effects of the presence of twin
matter.^{8}^{8}8This is justified if this matter is nonexistent, or
of it is smoothly distributed so its local contribution is
negligible, or if there does not happen to exist a twin body in the
near vicinity of the body under study. First, I consider the
system in a double Minkowski background. This is aesthetically the
most appealing option, which I shall assume. It relies on the
possibility that on cosmological scales the two metrics are,
somehow, maintained the same from some symmetry. There are indeed
versions of BIMOND [made more symmetric in the two metrics than our
simple action (15) is] that have cosmological solutions
with , either at all times, or as vacuum
solutions, which might be appropriate for today (see section
VI). In this case we have for the
cosmological background, and finite values occur
only due to local inhomogeneities. We can then take locally, on
scales much smaller then cosmological ones, a double Minkowski
background. Departures from this assumption will be discussed below.

Write, then, the metrics as slightly perturbed from Minkowski.
Because the source system is time-reversal symmetric in the
approximation we treat it (neglecting motions in the source), we are
looking for a solution for which the mixed space-time elements of
the two metrics vanish.^{9}^{9}9We do not have to assume this a
priori; if we do not, the equations themselves will tell us that
there is a choice of gauge in which the solution satisfies this
ansatz; see the end of this subsection. The ansatz simplifies the
presentation, and is justified a posteriori by our showing below
that such a solution exists. We can then write most generally

(38) |

where . We denote the differences

(39) |

with , . We wish to solve the field equations to first order in the potentials (Roman letters are used for space indices).

Note that there is a subtlety here (as in all metric MOND theories) due to the fact that the NR MOND potential for an isolated mass diverges logarithmically at infinity; so, strictly speaking we cannot formulate a first-order theory for such an isolated mass assuming at all radii. However, we are, in any event, dealing with an effective theory to be understood in the context of the universe at large, and in this context there are no isolated masses, our approach is meant to work only well within the distance from the central mass to the next comparable mass, where we can assume the first-order theory to be a good approximation.

To the required order the only nonvanishing components of
are^{10}^{10}10Because the metric derivatives,
connections, and curvature components are already first order, all
the metrics that are used to contract them can be taken as .

(40) |

These reflect the same relations between the separate connections with their respective potentials.

The only nonvanishing components of the Ricci tensors (shown here for ) are

(41) |

with

(42) |

where is the trace of . The nonvanising components of the Einstein tensor are

(43) |

( is the trace of ). The same expressions exist for the hatted and for the starred quantities. We are now ready to use these expressions in the field equations (16)(17). We neglect the small cosmological-constant terms (in line with our assuming background Minkowski metrics), and note that terms such as are of second order in the potentials, so they can be neglected. Also, and are linear in components of the tensor , which are first order in the potentials; so everywhere else in these expressions we can take the metrics as Minkowski, so that , etc.. Also, for the same reason, the covariant derivatives can be replaced by normal derivatives. All in all we get that the two terms involving and are equal. Thus, taking the difference of the two field equations we get

(44) |

Substituting from Eq.(43) we get

(45) |

Taking the trace of the second part we get , and substituting in the first we get

(46) |

We impose for the boundary condition at infinity , which establishes it as the Newtonian potential of the problem. But the second equation (45) does not, in itself, determine , because , like , satisfy three Bianchi identities