aa r X i v : . [ g r- q c ] J a n Bimetric MOND gravity
Mordehai Milgrom
The Weizmann Institute Center for Astrophysics
A new relativistic formulation of MOND is advanced, involving two metrics as independent degreesof freedom: the MOND metric g µν , to which alone matter couples, and an auxiliary metric ˆ g µν .The main idea hinges on the fact that we can form tensors from the difference of the Levi-Civitaconnections of the two metrics, C αβγ = Γ αβγ − ˆΓ αβγ , 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 a − C αβγ . I look at a class of bimetric MONDtheories governed by the action I = − (16 πG ) − R [ βg / R + α ˆ g / ˆ R − g ˆ g ) / f ( κ ) a M ( ˜Υ /a )] d x + I M ( g µν , ψ i ) + ˆ I M (ˆ g µν , χ i ) , with ˜Υ a scalar quadratic in the C αβγ , κ = ( g/ ˆ g ) / , I M the matter action,and allowing for the existence of twin matter that couples to ˆ g µν alone. Thus, gravity is modified notby modifying the ‘elasticity’ of the space-time in which matter lives, but by the interaction betweenthat space-time and the auxiliary one. In particular, I concentrate on the interesting and simplechoice ˜Υ ∝ g µν ( C γµλ C λνγ − C γµν C λλγ ). This theory introduces only one new constant, a ; it tendssimply to general relativity (GR) in the limit a →
0, and to a phenomenologically valid MONDtheory in the nonrelativistic limit. The theory naturally gives MOND and “dark energy” effects fromthe same term in the action, both controlled by the MOND constant a . As regards gravitationallensing by nonrelativistic systems–a holy grail for relativistic MOND theories–the theory predictsthat the same potential that controls massive-particle motion also dictates lensing in the same wayas in GR: Lensing and massive-particle probing of galactic fields will require the same “halo” of darkmatter to explain the departure of the present theory from GR. This last result can be modifiedwith other choices of ˜Υ, but lensing is still enhanced and MOND-like, with an effective logarithmicpotential. PACS numbers:
I. INTRODUCTION
From the inception of MOND [1] 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[2]. This was the first in a chain of theories of increasing force, culminating in the advent of the tensor-vector-scalartheory (TeVeS) of Bekenstein [3]. Some landmarks along this track are described in [3–8]; see, in particular, thereviews in [5, 8]. All these theories involve as independent degrees of freedom an Einstein metric, whose free action isthe 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 whichmatter couples. TeVeS has a version of the nonrelativistic (NR) theory proposed in [2] as a NR limit.Another line of relativistic theories that aim to reproduce MOND phenomenology has been propounded in [9, 10],based on the omnipresence of a gravitationally polarizable medium proposed in [11].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 [12] a new class of nonrelativistic,bi-potential MOND theories, a subclass of which is governed by a Lagrangian density of the form L = − πG { β ( ~ ∇ φ ) + α ( ~ ∇ ˆ φ ) − a M [( ~ ∇ φ − ~ ∇ ˆ φ ) /a ] } + ρ ( 12 v − φ ) , (1)leading to the field equations ~ ∇ · [ µ ∗ ( | ~ ∇ φ ∗ | /a ) ~ ∇ φ ∗ ] = 4 πGρ, µ ∗ ( y ) ≡ β − α + βα M ′ ( y )∆ φ = ~ ∇ · [(1 − α − M ′ ) ~ ∇ φ ∗ ] = 4 πGβ − ρ + β − ~ ∇ · ( M ′ ~ ∇ φ ∗ ) , (2)with φ ∗ = φ − ˆ φ . I also described in detail the requirements from α, β , and M ( z ) that lead to the required MONDand Newtonian limits of these theories. In particular, I discussed at length the interesting case β + α = 0 ( β = 1 thennormalizes G to be the Newton constant), which leads to the field equations∆ φ ∗ = 4 πGρ, ∆ φ = ~ ∇ · [(1 + M ′ ) ~ ∇ φ ∗ ] = 4 πGρ + ~ ∇ · ( M ′ ~ ∇ φ ∗ ) , (3)with M ′ a function of ( ~ ∇ φ ∗ /a ) , such that M ′ ( z ) → z → ∞ ensures the Newtonian limit, and M ′ ( z ) ≈ z − / inthe MOND regime z ≪
1. This is a particularly tractable MOND theory, as it requires solving only linear differentialequations, with the inevitable MOND nonlinearity entering only algebraically. In all the NR theories above, mattercouples only to one of the potentials: the MOND potential φ , while ˆ φ is an auxiliary potential, and in the special caseof Eq.(3) their difference φ ∗ is exactly the Newtonian potential of the problem.These NR MOND theories have inspired the construction of closely analogous relativistic MOND theories withtwo metrics as independent, gravitational degrees of freedom, which I begin to investigate here. This new class ofBIMOND theories involve only a as a new constant. They tend to general relativity (GR) in the limit a →
0, whichis 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 theappearance 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 [13] considered bimetrictheories, where the auxiliary metric is forced to be flat. More recently, it was found [14] that ghosts appear in alarge class of bimetric theories (apparently not including the present BIMOND). More matter-of-principle questionsregarding bimetric gravities are discussed in [15–18], but these authors confined themselves to metric couplings thatinvolve 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 ofthese theories, showing how they lead to NR MOND theories; section IV demonstrates how the theories go to GR inthe limit a →
0; 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 φ felt by matter, and an auxiliaryone ˆ φ . They point to relativistic BIMOND theories involving the MOND metric g µν , to which matter couples, andwhich in the NR limit reduces to φ , and an auxiliary metric ˆ g µν .Working with two metrics enables us to form nontrivial tensors and scalars from the difference in their Levi-Civitaconnections Γ αβγ and ˆΓ αβγ , C αβγ = Γ αβγ − ˆΓ αβγ , (4)involving only first derivatives of the metrics, which is not possible with a single metric. This is particularly pertinentin the context of MOND, since connections act like gravitational accelerations. So, without introducing new constantsin the relativistic formulation we can write Lagrangian functions of dimensionless scalars constructed from a − C αβγ that enable us to interpolate between the GR limit, a →
0, and the MOND limit, a → ∞ .The tensor C αβγ 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): g µν : λ = g αν C αµλ + g αµ C ανλ ˆ g µν ; λ = − ˆ g αν C αµλ − ˆ g αµ C ανλ , (5) C λαβ = 12 g λρ ( g αρ : β + g βρ : α − g αβ : ρ ) = −
12 ˆ g λρ (ˆ g αρ ; β + ˆ g βρ ; α − ˆ g αβ ; ρ ) , (6)where the covariant derivative (; ) is taken with the connection Γ αβγ and (:) with ˆΓ αβγ . We can form various scalars outof C αβγ and the metrics. One scalar that will be of particular use to us is based on the tensorΥ µν = C γµλ C λνγ − C γµν C λλγ , (7)with the same index combination that appears in the expression for the Ricci tensor R µν = Γ αµα,ν − Γ αµν,α + Γ γµλ Γ λνγ − Γ γµν Γ λλγ , (8) In these theories the two metrics are independent degrees of freedom. Theories like Brans-Dicke, TeVeS, etc., are also sometimesdescribed as being bimetric because they involve two metrics, but those two metrics are a priori related conformally or disformally viaother degrees of freedom such as scalars or vectors. and in ˆ R µν constructed similarly from ˆ g µν . One finds R µν − ˆ R µν = C λµλ ; ν − C λµν ; λ − Υ µν . (9)Thus, using well known manipulations, the scalar Υ ≡ g µν Υ µν connects the two Ricci scalars R = g µν R µν and themixed ˆ R m = g µν ˆ R µν by R − ˆ R m = − Υ + g − / ( g / g µν C λµλ ) ,ν − g − / ( g / g µν C λµν ) ,λ . (10)Similarly, interchanging the roles of g µν and ˆ g µν ,ˆ R − R m = − ˆΥ − ˆ g − / (ˆ g / ˆ g µν C λµλ ) ,ν + ˆ g − / (ˆ g / ˆ g µν C λµν ) ,λ , (11)where ˆΥ = ˆ g µν Υ µν , R m = ˆ g µν R µν , ˆ R is the Ricci scalar of ˆ g µν , and g and ˆ g are minus the determinants of g µν and ˆ g µν respectively.We can construct gravitational Lagrangian densities using the scalars R, ˆ R, R m , ˆ R m , and scalars constructed bycontracting powers of C λµν with the two metrics and their inverses (there are also ˆ g/g , ¯ ω ≡ g µν ˆ g µν , etc. that can beused). If we only contract with g µν and g µν , a quadratic scalar is a linear combination (possibly with coefficientsdepending on scalars such as g/ ˆ g or ¯ ω ) of the following scalars g µν C γµλ C λνγ , ¯ C γ C γ , g µν ¯ C µ ¯ C ν , g µν C µ C ν , g αλ g βµ g γν C αβγ C λµν , (12)where ¯ C γ ≡ g µν C γµν , C γ ≡ C αγα . The choice of scalars to be used may be forced on us by various theoretical andphenomenological desiderata (see below). The main point is that C γµν have only first derivatives of the metrics, thatthey reduce to derivatives of the potential difference in the Newtonian limit (in the sense to be discussed below), andthat we can form dimensionless quantities from them with the MOND acceleration a (or a MOND length ℓ = c /a ).As regards the four curvature scalars, is will be advantageous to include them in the action only linearly, and eschewterms such as in the voguish f ( R ) theories. Such nonlinear terms render the theory a higher derivative one, which Iwould like to avoid. Another reason to avoid such terms in the MOND context is that they do not naturally leadin their NR limit to a single constant a controlling the dynamics. Neither obstacle appears if we allow functions ofscalars made of C αβγ . These contain only first derivatives of the metrics, and give NR limits in which only a appears(see below). We see from Eqs.(10) that g / R and g / ˆ R m differ by Υ plus a total derivative so it is enough to includeone of these in the action, as we anyhow permit functions of Υ. The same is true of the pair ˆ g / R m and ˆ g / ˆ R .Because the number of possible combinations it too large to explore here, I limit myself to the subclass of actions ofthe form I = − c πG Z [ βg / R + α ˆ g / ˆ R − g ˆ g ) / f ( κ ) ℓ − M ( ℓ m Υ ( m ) i )] d x + I M ( g µν , ψ i ) + ˆ I M (ˆ g µν , χ i ) , (13)where ℓ ≡ c /a , κ ≡ ( g/ ˆ g ) / , f (1) = 1, and Υ ( m ) i are scalars formed by contracting a product of (even) m C αβγ , whichcan be used in principle. In what follows, I shall confine myself to quadratic scalars. I have included two matteractions: The first, I M , involves the matter degrees of freedom with which we interact directly, designated symbolicallyas ψ i . It contains only the MOND metric g µν to which matter is coupled in the standard way. The other, ˆ I M , involvesother matter degrees of freedom, χ i , and only ˆ g µν , to account for the possibility that ˆ g µν controls a matter world ofits own. There are no direct (electromagnetic, etc.) interactions between the ψ matter and the twin χ matter. For the same reason I avoid scalars that are higher order in the curvature tensors, such as the different possible contractions of R µν with itself or with ˆ R µν . These are even less appealing as explained in [19]. For example, to account for dimensions correctly, a function of R has to be introduced as f ( ℓ R ), with ℓ some length scale. The NRlimit of R includes c − ∆ φ as the dominant term in φ/c , and second order ones such as c − ( ~ ∇ φ ) , and c − φ ∆ φ . Thus, in the argumentof f the second term will give ( ~ ∇ φ/a ) , with a = c /ℓ , which fits well into the MOND frame. But the first, dominant, term wouldinvolve a time scale ℓ/c , not an acceleration. When R appears linearly, the first term becomes immaterial in the action, as a completederivative, and we are left with terms that are welcome in MOND [the φ ∆ φ term is also ( ~ ∇ φ ) up to a derivative]. The MOND constant a is normalized so that the mass-asymptotic-velocity relation is MGa = V . It defines the scale length ℓ that isused in the coefficient and the argument of M . Any dimensionless factors can be absorbed in the definition of M so that its coefficientis ℓ − = c − a and its argument is as prescribed here. To obviate possible confusion, note that the twin matter is not to play the role of the putative dark matter in galactic systems; this isstill 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 meansthe following: given a non relativistic system of slow masses one can express the metrics solution of the relativistictheory in terms of potentials so that the equations of motion for slow particles in the resulting (multi) potential theoryare those required by NR MOND, with the appropriate MOND and Newtonian limits; this is a phenomenologicalrequirement (by itself it does not dictate the effects on massless particles–e.g., gravitational lensing–even in NRsystems). b. Require that the action gives GR in the limit a →
0. This is not a phenomenological necessary (forexample, TeVeS does not satisfy it), but I feel that it is highly desirable for various reasons. This automaticallycauses the theory to agree with the stringent constraints from the solar system and binary pulsars–which are knownto agree with GR–because the accelerations in these systems are many orders of magnitude larger than a . I alsorequire this limit lest we have to introduce additional constant(s) to the theory, which has to give GR in some limitof its parameters.When the two metrics are conformally related, which might be the case in certain circumstances, g µν = e ϑ ( x ) ˆ g µν , wehave C λ = 2 ϑ ,λ , ¯ C λ = − g λρ ϑ ,ρ , Υ µν = (1 / g µν g αβ ϑ ,α ϑ ,β − ϑ ,µ ϑ ,ν ), Υ = (3 / g µν ϑ ,µ ϑ ,ν . If we a priori constrain ourmetrics to be conformally related (i.e. vary the action only over such pairs) we get the Brans-Dicke theory with thechoice M ( z ) ∝ z (and appropriate choice of the constants and f ( κ ), and possibly using ˆΥ in the argument of M ).With a more general form of M ( z ), we then get the relativistic MOND theory sketched in [2].Without the interaction M term, the theory separates into two disjoint copies of GR. It is important to note thatas a combined structure, the theory then enjoys a larger symmetry involving separate coordinate transformations inthe two separate actions. This double symmetry has to be brought to bear when solving the field equations of thetheory, 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 samecoordinate transformation to the two metrics. However, under certain circumstances the interaction is symmetricunder a more extended set of coordinate transformation, and we must be careful then to employ the larger gaugefreedom. The above mentioned complete decoupling is an example that, as we shall see in section IV, applies in theformal limit a → A. Concrete simple example
I shall hereafter concentrate on a simple special case of the class. Some generalizations will be mentioned brieflybelow, in this section, and in section VI.In the first place, I take M to be a function of only one scalar, quadratic in the C αβγ . In particular, I find the scalarΥ defined above a natural choice for this argument, as it has the same structure as the first-derivative part of theRicci curvature scalar (not itself a scalar) Γ (2) ≡ g µν (Γ γµλ Γ λνγ − Γ γµν Γ λλγ ) . (14)It is well known that one can replace R in the Einstein-Hilbert action by Γ (2) and still get GR. Here we can also dothis, replacing also ˆ R by the corresponding ˆΓ (2) , and making M a function of Υ, which is constructed in the same wayfrom C αβγ . 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 α + β = 0. 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 β = 1 for G to be Newton’s constant.Work in units in which c = 1, and use a = ℓ − to highlight the connection with MOND. Also, anticipating theexpression for NR limit of Υ, I take the argument of M to be − Υ / a . The relativistic action I then consider is I = − πG Z [ g / R − ˆ g / ˆ R − g ˆ g ) / f ( κ ) a M ( − Υ / a )] d x + I M ( g µν , ψ i ) − ˆ I M (ˆ g µν , χ i ) . (15)[Using Eq.(11) we can replace the first two terms by ( g / g µν − ˆ g / ˆ g µν ) R µν + ˆ g / ˆΥ.] I take a mixed volume elementfor the interaction term, with f normalized such that f (1) = 1. Note the change of sign in the definition of the twinmatter action to match the negative sign for the Hilbert-Einstein action of ˆ g µν .Varying over g µν and over ˆ g µν we get, respectively G µν + S µν = − πG T µν , (16)ˆ G µν + ˆ S µν = − πG ˆ T µν , (17)where G µν and ˆ G µν are the Einstein tensors of the two metrics, G µν = R µν − Rg µν , ˆ G µν = ˆ R µν −
12 ˆ R ˆ g µν , (18) T µν and ˆ T µν are the matter energy-momentum tensors (EMT); e.g., δI M ≡ − (1 / R g / T µν δg µν , and S µν , ˆ S µν are thefunctional derivatives (one with an opposite sign) of the interaction term with respect to the two metrics: δ Z − g ˆ g ) / f ( κ ) a M ( − Υ / a ) d x ≡ Z ( g / δg µν S µν − ˆ g / δ ˆ g µν ˆ S µν ) d x. (19)For the present choice of the scalar argument of M , we have S µν = κ − M ′ Υ µν + ( κ − M ′ ˜ S λµν ) ; λ − Λ m g µν , (20)ˆ S µν = ( κ + M ′ ˆ S λµν ) : λ − ˆΛ m ˆ g µν , (21)˜ S λµν = C λµν − δ λ ( µ C ν ) + 12 ( C λ − ¯ C λ ) g µν , (22)ˆ S λµν = q α ( µ C λν ) α + g λρ C αρ ( µ ˆ g ν ) α − ˆ g λρ C αρβ q β ( µ ˆ g ν ) α − q λ ( µ C ν ) + 12 g ∗ µν ˆ g λα C α −
12 ¯ C λ ˆ g µν , (23)Λ m = − κ [ κf ( κ )] ′ a M , ˆΛ m = − κ κ − f ( κ )] ′ a M . (24)Here, C λ ≡ g αλ C α , κ ± ≡ κ ± f ( κ ) , q µα = g µν ˆ g να , g ∗ µν = ˆ g αµ g αβ ˆ g βν , (25)and ( µ...ν ) = { µ...ν + ν...µ } / T µν = 18 πG [ κ − M ′ Υ µν + ( κ − M ′ ˜ S λµν ) ; λ ] (26)may be viewed as the EMT of the phantom dark matter (DM); whereas the Λ m term may roughly be viewed as “darkenergy”. Note that the last term in ˜ S λµν , which contributes [ κ − M ′ ( C λ − ¯ C λ )] ; λ g µν may also contribute to the darkenergy due to its form. Define in analogy with ˜ T µν ˆ T µν = 18 πG ( κ + M ′ ˆ S λµν ) : λ . (27)The Einstein tensors satisfy the usual Bianchi identities G νµ ; ν = ˆ G νµ : ν = 0, derivable from the invariance of theEinstein-Hilbert actions to coordinate transformations. In addition, we have here, for the general action (13), a setof four identities following from the fact that the mixed term is a scalar; these read S νµ ; ν − κ − ˆ S νµ : ν = 0 . (28)Given that the matter EMTs are divergence free (for matter degrees of freedom satisfying their own equations ofmotion): T νµ ; ν = ˆ T νµ : ν = 0, the above identities imply four differential identities satisfied by our 20 field equations. Ifwe write these equations as Q µν = 0 and ˆ Q µν = 0, respectively, then the four relations Q νµ ; ν − κ − ˆ Q νµ : ν = 0 (29) For each tensor indices are raise with the corresponding metric; so, e.g., G νµ = g να G µα , ˆ G νµ = ˆ g να ˆ G µα . hold identically, and, as usual, deprive us of four equations to account for the fact that the solution can be determinedonly up to a coordinate transformation. This seems to leave us with a tractable Cauchy problem, although this requiremore careful checking. Of course, for solutions of the field equations we do have separately S νµ ; ν = ˆ S νµ : ν = 0 , (30)which can be used as useful constraints of the solutions (only one set is independent).Note the useful identities C ν = 12 g αβ g αβ : ν = −
12 ˆ g αβ ˆ g µν ; ν = 2 κ ,ν /κ ¯ C λ = − κ − ( κ g λρ ) : ρ C λ = − ¯ C λ − g λν : ν , (31)˜ S λµν ; λ = ˆ R µν −
12 ˆ R m g µν − ( R µν − Rg µν ) − (Υ µν −
12 Υ g µν ) , ( C λ − ¯ C λ ) ; λ = R − ˆ R m + Υ . (32)Identities (32) follow from Eqs.(9)(10). Similar manipulations are possible for ˆ S λµν , and the field equation (17).Contracting Eq.(16) with g µν gives R − [ κ − M ′ ( C λ − ¯ C λ )] ; λ − κ − M ′ Υ + 4Λ m = 8 πG T . (33)Contracting Eq.(17) with ˆ g µν givesˆ R − [ κ + M ′ ( 12 ¯ ω ˆ g λρ C ρ − ¯ C λ − ˆ g λρ q αµ C µρα )] : λ + 4 ˆΛ m = 8 πG ˆ T , (34)where ¯ ω = g µν ˆ g µν . We can thus replace Eq.(16) by R µν + κ − M ′ (Υ µν −
12 Υ g µν ) − [ κ − M ′ ( δ λ ( µ C ν ) − C λµν )] ; λ + Λ m g µν = − πG ( T µν − T g µν ) , (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 withits own volume element.It was deduced in [14] 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 ofPauli-Fierz actions for the different metrics, which are quadratic in the metric departures. This, however, leads to alinear theory in this limit, which is at odds with MOND: MOND phenomenology dictates that at g µν = ˆ g µν = η µν any BIMOND theory (or any relativistic MOND theory for that matter) is not even analytic in the squares of thedepartures g µν − η µν , ˆ g µν − η µν (where the argument of M ′ in the above version of the theory vanishes, and M ′ diverges). It thus remains to be seen if obstacles similar to these are at all relevant to BIMOND, and if they are towhat extent they are deleterious.For conformally related metrics g µν = e ϑ ( x ) ˆ g µν , we have ˜ S λµν = e ϑ ( x ) ˆ S λµν . B. Generalizations
Some generalizations of the above simple theory include the following.1. Instead of using Υ as the argument of M , we can use other scalars, or several scalar variables. A quadraticscalar variable can be written, most generally, as Ξ = Q βγµναλ C αβγ C λµν , (36) As a result of identities (28) and the Bianchi identities, the four expressions G µ + S µ − κ − ( ˆ G µ + ˆ S µ ) contain only up to first timederivatives of the metric, and cannot be used to propagate the problem in time. Instead, the initial conditions have to satisfy the fourequations Q µ − κ − ˆ Q µ = 0, and the remaining sixteen field equations, with the aid of four gauge conditions, propagate us in time, andinsure that these four are always satisfied. where Q βγµναλ it built from g µν , ˆ g µν , their inverses, δ αβ , and scalars such as κ and ¯ ω . In this case the Λ terms take amore general form, and so do terms that are second order in the C αβγ . The only terms in S µν and ˆ S µν that survive inthe NR limit, which we treat below, are those involving ˜ S λµν and ˆ S λµν . For these we now have for example˜ S λµν = 2 U γλ ( µ g ν ) γ − U γσρ g µγ g νσ g λρ , U µνλ = − Q βγµναλ C αβγ , (37)which I shall need in what follows.For example, taking as the argument of M , − C γµλ C λνγ / a instead of − Υ / a , would leave us with only the firstterm in expression (22) for ˜ S λµν , and with the first three terms in expression (23) for ˆ S λµν .2. One can consider more general α, β values.3. We can increase the symmetry with respect to the two metrics by taking interaction terms of the form M (Υ ˆΥ), M (Υ) M ( ˆΥ), etc..4. One can make M a function of scalars such as κ and ¯ ω .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 allquantities with the dimensions of velocities, such as v, √ φ , etc., are much smaller than the speed of light. In thecontext of GR this limit is attained by formally taking c → ∞ everywhere in the relativistic theory. In the context ofMOND one has to be more specific, since system attributes with the dimensions of acceleration, such as v /R, ~ ∇ φ ,etc., cannot be assumed very small in the limiting process, even though they have velocities in the numerator. Wewant to consider systems, such as galaxies, in which these are finite compared with the MOND acceleration, whichis also a relevant parameter. The NR limit in MOND is thus formally attained by taking everywhere c → ∞ , but atthe same time ℓ → ∞ , so that a = c /ℓ 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 T = ρ . I also neglect here the possible effects of thepresence of twin matter. First, I consider the system in a double Minkowski background. This is aesthetically themost 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 symmetricin the two metrics than our simple action (15) is] that have cosmological solutions with ˆ g µν = g µν , either at all times,or as vacuum solutions, which might be appropriate for today (see section VI). In this case we have C αβγ = 0 for thecosmological background, and finite C αβγ values occur only due to local inhomogeneities. We can then take locally, onscales much smaller then cosmological ones, a double Minkowski background. Departures from this assumption willbe discussed below.Write, then, the metrics as slightly perturbed from Minkowski. Because the source system is time-reversal symmetricin the approximation we treat it (neglecting motions in the source), we are looking for a solution for which the mixedspace-time elements of the two metrics vanish. We can then write most generally g µν = η µν − φδ µν + h µν , ˆ g µν = η µν − φδ µν + ˆ h µν , (38)where h µ = h µ = ˆ h µ = ˆ h µ = 0. We denote the differences g ∗ µν = g µν − ˆ g µν = − φ ∗ δ µν + h ∗ µν , (39)with φ ∗ = φ − ˆ φ , h ∗ µν = h µν − ˆ h µν . We wish to solve the field equations to first order in the potentials φ, ˆ φ, h ij , ˆ h ij (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 potentialfor an isolated mass diverges logarithmically at infinity; so, strictly speaking we cannot formulate a first-order theory This is justified if this matter is nonexistent, or of it is smoothly distributed so its local contribution is negligible, or if there does nothappen to exist a twin body in the near vicinity of the ψ body under study. We 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 thesolution satisfies this ansatz; see the end of this subsection. The ansatz simplifies the presentation, and is justified a posteriori by ourshowing below that such a solution exists. for such an isolated mass assuming φ ≪ C αβγ are C i = C i = C i = − g ∗ ,i = φ ∗ ,i ,C ijk = 12 ( g ∗ ij,k + g ∗ ik,j − g ∗ jk,i ) = 12 ( h ∗ ij,k + h ∗ ik,j − h ∗ jk,i ) + φ ∗ ,i δ jk − φ ∗ ,j δ ik − φ ∗ ,k δ ij . (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 R µν ) are R = − ∆ φ, R ij = 12 H ij − ∆ φδ ij , (41)with H ij ≡ ∆ h ij + h ,i,j − h k ( i,j ) ,k , (42)where h is the trace of h ij . The nonvanising components of the Einstein tensor are G = − φ + 14 H, G ij = 12 ( H ij − Hδ ij ) (43)( H is the trace of H ij ). The same expressions exist for the hatted and for the starred quantities. We are nowready to use these expressions in the field equations (16)(17). We neglect the small cosmological-constant terms (inline with our assuming background Minkowski metrics), and note that terms such as Υ µν are of second order inthe potentials, so they can be neglected. Also, ˜ S λµν and ˆ S λµν are linear in components of the tensor C αβγ , which arefirst order in the potentials; so everywhere else in these expressions we can take the metrics as Minkowski, so that f ( κ ) ≈ , κ ± ≈ , q λµ ≈ δ λµ , etc.. Also, for the same reason, the covariant derivatives can be replaced by normalderivatives. All in all we get that the two terms involving ˜ S λµν and ˆ S λµν are equal. Thus, taking the difference of thetwo field equations we get G ∗ = − πG T , G ∗ ij = 0 . (44)Substituting from Eq.(43) we get ∆ φ ∗ − H ∗ = 4 πGρ, H ∗ ij − H ∗ δ ij = 0 . (45)Taking the trace of the second part we get H ∗ = 0, and substituting in the first we get∆ φ ∗ = 4 πGρ. (46)We impose for φ ∗ the boundary condition at infinity φ ∗ →
0, which establishes it as the Newtonian potential of theproblem. But the second equation (45) does not, in itself, determine h ∗ ij , because G ∗ ij , like G ij , satisfy three Bianchiidentities G ∗ ij,j = 0, which are the reductions of identities (29) to our case (the fourth identity is automatically satisfiedfor our choice of vanishing mixed elements of the metrics).We have only used one of the field equations (or rather their difference). Now consider the first field equation alonein the form (35). Again, neglect the second order Υ terms, etc. to get R µν + [ M ′ ( ¯ S iµν −
12 ¯ S i η µν )] ,i = − πGρδ µν , (47) Because the metric derivatives, connections, and curvature components are already first order, all the metrics that are used to contractthem can be taken as η µν . where ¯ S iµν is the NR limit of ˜ S λµν , and ¯ S i its (four) trace. The (0 i ) components of the equations hold identically (tofirst order) since ¯ S i j = 0. The (00) and ( ij ) components give, respectively, − ∆ φ + [ M ′ ( ¯ S k + 12 ¯ S k )] ,k = − πGρ. (48)12 H ij − ∆ φδ ij + [ M ′ ( ¯ S kij −
12 ¯ S k δ ij )] ,k = − πGρδ ij . (49)Multiply Eq.(48) by δ ij and subtract from Eq.(49) to get12 H ij + [ M ′ ( ¯ S kij − ¯ S kmm δ ij )] ,k = 0 , (50)which I use instead of Eq.(49). Equation (50) does not satisfy any more identities and thus gives six independentequations, which together with the above four unused independent equations, and the remaining freedom to choosethree gauge conditions, should determine the remaining 13 potentials φ, h ij , h ∗ ij .It is beneficial to employ three of these six equations encapsuled in Eq.(50) by taking its divergence, taking theBianchi identities for H ij into account, to get ( M ′ ¯ S kij ) ,k,j = 0 . (51)These three equations, together with the three independent equations in the second of (45) now involve only the six h ∗ ij (so we managed to decouple these from φ and h ij ; φ ∗ , which also appears in these equations is already known),and can be solved for these. Once this is done (imposing boundary conditions at infinity) φ is determined fromEq.(48) by solving a Poisson equation, and h ij are likewise determined from Eq.(50) with the aid of gauge conditions.The remaining gauge freedom is associated with coordinate transformations that preserve our assumed form of themetrics; i.e., near-Minkowski and stationary (time independent and lacking mixed elements). These are of the generalform t = t ′ , x i = x ′ i + ǫ i ( x ′ ) , (52)with ǫ i ( x ′ ) first order in the potentials. They do not affect g ∗ µν and so leave h ∗ ij and φ ∗ intact, changing only h ij by ǫ i,j + ǫ j,i .Everything so far is valid for an arbitrary choice of quadratic scalar argument. I now specialize to my preferredchoice of scalar argument − Υ /
2. In this case we have¯ S i = 2 φ ∗ ,i + 12 ( h ∗ ij,j − h ∗ ,i ) , ¯ S ijk = 12 ( h ∗ ij,k + h ∗ ik,j − h ∗ jk,i ) + 14 [2( h ∗ ,i − h ∗ im,m ) δ jk − h ∗ ,k δ ij − h ∗ ,j δ ik ] . (53)What is special about this case is that the space components ¯ S kij depend only on h ∗ ij , not on φ ∗ . This greatly simplifiesthe solution of Eqs.(45)(51), which has to be h ∗ ij = 0 (with boundary conditions h ∗ ij → The fact that h ∗ ij = 0 causes the C αβγ , as given in Eq.(40), to be linear combinations of derivatives of φ ∗ , and theargument of M ′ becomes a function of ( ~ ∇ φ ∗ ) . Now Eq.(48) reads∆ φ = 4 πGρ + ~ ∇ · {M ′ [( ~ ∇ φ ∗ /a ) ] ~ ∇ φ ∗ } . (54)Equation (50) becomes H ij = 0 , (55) To see this note that from Eq.(37) we have ¯ S i j ∝ (2 Q βγjiα − Q βγ iαj ) C αβγ . Now, the NR limit of the Q tensor is constructed only from η µν and η µν . This means that its only nonvanishing components must have three pairs of equal indices. This means, in turn, that theonly contributions to ¯ S i j come from C αβγ with one or three time indices; but these all vanish. These are coupled nonlinear equations, since h ∗ ij appear also in the argument os M ′ . Since g ∗ µν is already first order, in our approximation the transformation affects it only to zeroth order; i.e., not at all. All the above holds if we use both Υ and Υ ∗ as variables, because they degenerate into one in the NR limit. Also, note that ¯ C = 0,and ¯ C i = h ∗ ij,j − (1 / h ∗ ,i ; so, adding the scalar g µν ¯ C µ ¯ C n should not change this conclusion. h ij –as would be done in GR, where this equation is alwayssatisfied. This implies, with the appropriate boundary conditions, that there is a gauge in which h ij = 0, and we workin this gauge.The matter action for a system of slowly moving masses is, to our present approximation, I M ≈ Z ρ ( v − φ ) d xdt ; (56)So the motion of such particles is governed by the potential φ , which is thus identified as the MOND potential. Itis determined from the field equation Eq.(54) (with φ ∗ being the Newtonian potential of the system), which is theQUMOND formulation described by Eq.(3), and discussed at length in [12]. Note that it requires solving only lineardifferential equations. To have the required Newtonian limit we have to have M ′ ( z ) → z → ∞ (i.e., a → z ≪ M ′ ( z ) ≈ z − / to get space-time scale invariance, which is the definingtenet of the NR MOND limit [20] (the normalization is absorbed in the definition of a ).To recapitulate, we end up with the simple result in the chosen gauge: g µν = η µν − φδ µν , ˆ g µν = η µν − φδ µν , (57)with φ ∗ = φ − ˆ φ being the Newtonian potential, and the MOND potential φ determined from the QUMOND Eq.(3).The relation between the first-order MOND metric and the MOND potential is thus exactly the same as that betweenthe first-order GR metric and the Newtonian potential.Suppose we have not assumed a priori that the mixed elements of the first-order metrics vanish. This does notchange expressions (40) for C αβγ , but we now have additional nonvanishing elements C i j = ( h ∗ i,j − h ∗ j,i ) / , C ij = − ( h ∗ i,j + h ∗ j,i ) / . (58)The (0 i ) component of the difference Ricci tensor is R ∗ i = (∆ h ∗ i − h ∗ j,i,j ) /
2. So Eq.(45) is now complemented by(∆ h ∗ i − h ∗ j,i,j ) = 0 . (59)With our boundary conditions h ∗ i → h ∗ i = v ∗ ,i for some v ∗ ( x ) (the lefthand side is identically divergence free, which leaves us with only two independent equations). Note now that the first-order limit of the theory, discussed here, enjoys a less obvious symmetry beside the general invariance to simultaneouscoordinate transformations: it is invariant under a ‘small’ transformation of the form t = t ′ − u ( x ′ ) applied separately to the g µν and the ˆ g µν sectors (with u first order in the potentials). In other words, there is symmetry to transforming g i → g i + u ,i , ˆ g i → ˆ g i + ˆ u ,i (hence g ∗ i → g ∗ i + u ∗ ,i ; u ∗ = u − ˆ u ), with u and ˆ u free for us to choose (the EMTs areunchanged to lowest order). We thus have the freedom to choose a gauge for ˆ g µν alone, in which v ∗ = 0, and hence h ∗ i = 0. This means that C i j = C ij = 0, and so the (0 i ) components of Eq.(47) read(∆ h i − h j,i,j ) = 0 , (60)whose solution is h i = v ,i for some v ( x ). We still have the gauge freedom to chose v ( x ) = 0, and so we do. We thusend up with a gauge in which the field equations themselves dictate h i = ˆ h i = 0, as we assumed a priori.The double gauge symmetry we use can be seen to apply directly to the first-order equations. It can be tracedback to the fact that the NR limit of Υ is invariant to it: If we do not assume a priori that the mixed elements h ∗ i vanish, then the addition to the lowest order expression for Υ is C i k C k i , because C i j is antisymmetric in i, j , while C ij is symmetric. But, under the double gauge transformation h ∗ i changes by u ∗ ,i , so C i k is invariant, and so is Υ.It remains to be checked if this symmetry is a remnant of some symmetry enjoyed by the relativistic theory itself.Anticipating the discussion of the next subsection, note that not all scalar arguments are invariant to this doublegauge in their NR limit. For example, the change induced in the scalar g µν ¯ C µ ¯ C ν is − h ∗ i,i ∆ u ∗ . For the more generalcase, Eq.(59) is still valid, and again gives h ∗ i = v ∗ ,i , while instead of Eq.(60) we have more generally(∆ h i − h j,j,i ) / M ′ ¯ S k i ] ,k = 0 . (61)The first term is identically divergence free so we can write one of these three equations as[ M ′ ¯ S k i ] ,k,i = 0 . (62)Now, ¯ S k i is linear in v ∗ (which may also appear in the argument of M ′ ), so this equation generically dictates v ∗ = 0.For the scalar Υ this does not work because at this stage we already have ¯ S k i = 0, but in return we have the doublegauge freedom to help us remove v ∗ . For the scalar g µν ¯ C µ ¯ C ν we do not have the double gauge freedom but we canwrite ¯ S k i ∝ δ ki ∆ v ∗ , so Eq.(62) gives ∆( M ′ ∆ v ∗ ) = 0, which implies v ∗ = 0. In any event we can always have h ∗ i = 0,and continue from there as before to show that there is always a gauge where the mixed elements of the metricsvanish.1 A. Other choices of the scalar argument of M Here I consider the NR limit of theories with other choices of the quadratic scalar argument of M . The mainpurpose is to see whether these give theories that are different in their NR limits, and so can be distinguished usingobservations of NR systems such as rotation curves and lensing in galaxies.Take then the general quadratic argument as given by Eq.(36). All of our procedures in the present section, up toEq.(52) remain valid. Up to that point we had not made use of the particular expressions for ˜ S λµν and ˆ S λµν , only ofthe fact that they become equal to first order in the potentials, and this is still the case. Departure from the above occurs, however, for more general scalars in the employment of Eq.(51). Now, the spacecomponents ¯ S ijk do, in general, depend on the gradient of φ ∗ , and it is easy to see that h ∗ ij = 0 is no longer a solution,in general: Substituting h ∗ ij = 0 in Eq.(51) would result in three constraints on the Newtonian potential φ ∗ , whichit does not satisfy ( φ ∗ is really arbitrary if we allow arbitrary density distributions, including negative ones). So, ingeneral, after φ ∗ is calculated as the Newtonian potential of the system, we have to solve the nine coupled equations(45) (second part) and (51) (only six of which are independent) for the h ∗ ij . After this is done we determine φ fromEq.(48), and h ij from Eq.(50) with the aid of gauge conditions. Note that in the Newtonian limit, a → φ becomesthe Newtonian potential, and h ij → M ′ does.We can derive some scaling properties of the h ∗ ij , even in the general case: It is easy to see that the second Eq.(45)and Eq.(51) are invariant under m i → λm i , r → λ / r , where m i stand for the masses in the system. This is becausethe only quantities appearing in these equations, beside the variables h ∗ ij , are φ ∗ ,i and a both of the same dimensionsas m i G/r . This means that the h ∗ ij,k are invariant under this scaling.Consider now in more detail a spherically symmetric problem. We are looking for solutions in which the variouspotential tensors h ij , ˆ h ij , h ∗ ij are of the form exemplified by h ∗ ij = h ∗ ( r ) δ ij + h ∗ ( r ) n i n j , (63)where n = r /r . It can be shown that h ∗ ij satisfying the second Eq.(45), namely, annuling H ∗ ij , is tantamount to h ∗ = rh ∗ ′ , which means, in turn, that h ∗ ij = q ,i,j for some q ( r ). It thus remains to determine q from Eq.(51). Becauseof the spherical symmetry, the different i components of Eq.(51) give equivalent equations; so we are left with onlyone equation from which to determine q ( r ).Once q ( r ) is known, we use Eq.(50) to solve for h ij . Write h ij in the form (63). We can use the remaining gaugefreedom to eliminate one of the two functions. In the spherically symmetric case the remaining freedom is to transform r = r ′ [1 + ǫ ( r ′ )] for some ǫ ( r ′ ), treated to first order. This transformation takes h ij → h ij + 2 ǫδ ij + 2 rǫ ′ n i n j ; so, wecan use such a transformation to eliminate either of the functions in the expression for h ij . For example, let us choosethe gauge in which h ij = ϕ ( r ) δ ij . (64)The general NR MOND metric is thus diagonal for this choice of gauge with g = − − φ, g ij = δ ij [1 − φ + ϕ )] . (65)For the form (64) of h ij we have H ij − Hδ ij = − ( ϕ ′′ + r − ϕ ′ ) δ ij + ( ϕ ′′ − r − ϕ ′ ) n i n j ≡ aδ ij + bn i n j . (66)Note that a ′ = − r − ( r b ) ′ , from the fact that the expression is divergence free. We now use H ij − Hδ ij = − M ′ ¯ S kij ) ,k , (67)obtained from Eq.(50), to solve for ϕ . Since the right hand side of this equation is already known to be divergencefree, from Eq.(51), we get only one independent equation of the form r ( r − ϕ ′ ) ′ = p ( r ) , (68) This follows from the asymmetry of C αbγ to interchange of the matrices, and from the fact that to first order we can put everywhere elseˆ g µν ≈ g µν ≈ η µν . This equation has to read in the spherical case P [ q ( r )] r = 0 where P is a differential operator acting on q ( r ), and we get one equation P [ q ( r )] = 0. p ( r ) is a known function. This we finally solve for ϕ , which we permit to behave asymptotically as ln ( r ).Here we note already an interesting difference from the theories that have Υ as scalar argument, which have Eqs.(2)and (3) as their NR limit (even with general α, β –see the next subsection). In such theories, in the spherical case theMOND acceleration is an algebraic function of the Newtonian one, with the relation being unique for the theory. Inthe general case this is not so: To get the MOND acceleration in the spherical case we apply the Gauss theorem toEq.(48). The expression we then get is some functional of q ( r ) that cannot be written as a function of the Newtonianacceleration − dφ ∗ /dr . This can lead to different predictions even for massive-particle motions.The spherical problem can be solved analytically for the case where φ ∗ = Ar θ –for example when we are outsidethe mass where θ = −
1, and we are in a region where M ′ ( z ) ∝ z − σ , for example, in the deep-MOND regime wherewe will have to have σ = 1 /
4. Then the solution can be shown to be of the form q = λr φ ∗ , with λ determined fromEq.(51) depending on θ and σ . With this ansatz M ′ ∝ ( | A | /a ) − σ ( a + bλ + cλ ) − σ r − σ ( θ − , and Eq.(51) then gives( A/a ) − σ ( a + bλ + cλ ) − σ A (¯ a + ¯ bλ ) r ζ n = 0, with ζ = (1 − σ )( θ − − − a, ¯ b depending on θ, σ and the choice of scalar argument of M for the specific theory (¯ a comes from the terms in ¯ S kij linearin the gradient of φ ∗ , and ¯ b from those linear in the gradient of h ∗ ij ). So λ = − ¯ a/ ¯ b gives us the solution (for the choiceof Υ as argument ¯ a = 0). Equation (68) then gives ϕ = ξ ( A/a ) − σ Ar ζ +3 for ζ = −
3, and ϕ = ξ ( A/a ) − σ Aln ( r ) for ζ = −
3, with the dimensionless ξ determined.Take, for instance the interesting case where we are asymptotically outside matter and in the deep-MOND regime,where A = − M G and ζ = −
3. We then get ϕ = ξ ( M Ga ) / ln ( r ), asymptotically. In this case we also have φ = ( M Ga ) / ln ( r ), by definition; so, ϕ = ξφ . For example, for the choice of argument − g µν C γµλ C λνγ / a , I find,following the above procedure for the deep-MOND ( σ = 1 / θ = −
1) case: ξ = 4 / g = − − φ, g ij = δ ij [1 − φ (1 + ξ )] . (69)Remember that while ϕ = ξφ in the MOND regime, in the Newtonian regime φ becomes the Newtonian potentialwhile ϕ vanishes as fast as dictated by the vanishing of M ′ at high values of its argument. B. General α β values
The NR limit of the field equation in a theory governed by the action (13) for general α and β values is β ( R µν − η µν R ) + ( M ′ ¯ S iµν ) ,i = − πGρδ µ δ ν , (70) α ( ˆ R µν − η µν ˆ R ) − ( M ′ ¯ S iµν ) ,i = 0 , (71)with all quantities taken to first order in the potentials φ , ˆ φ , h ij , ˆ h ij . ¯ S iµν is linear in the first-order expression for C αβγ , as given in Eq.(40), and M ′ is a function of a quadratic scalar built from these. Multiply the second equationby − β/α and add to the first to get β ( R ∗ µν − η µν R ∗ ) + α + βα ( M ′ ¯ S iµν ) ,i = − πGρδ µ δ ν . (72)The (0 j ) components of this equation hold identically, as before. Equations (41)-(43) are then used to write the (00)and ( ij ) components of this equation as β (∆ φ ∗ − H ∗ ) − α + β α ( M ′ ¯ S i ) ,i = 4 πGρ, H ∗ ij − H ∗ δ ij + 2( α + β ) αβ ( M ′ ¯ S kij ) ,k = 0 . (73)Taking the trace of the second and substituting for H ∗ in the first we get β ∆ φ ∗ − α + β α [ M ′ ( ¯ S i + ¯ S imm )] ,i = 4 πGρ. (74)Equation (73) comprises seven independent equations that can be solved for φ ∗ and the h ∗ ij . Once these are knownwe can get φ and h ij from Eq.(70), or equivalently from βR µν + [ M ′ ( ¯ S iµν −
12 ¯ S i η µν )] ,i = − πGρδ µν . (75)3Its (00) component gives ∆ φ = 4 πGβ − ρ + β − [ 12 M ′ ( ¯ S i + ¯ S ikk )] ,i (76)from which φ can be gotten (since the right hand side is now known). The ( ij ) component gives [after making use ofEq.(76)] H ij − Hδ ij = − β − ( M ′ ¯ S kij ) ,k . (77)From these h ij can be gotten. Note that the divergence of Eq.(77) is identically the same as that of the secondEq.(73); so we only have here three independent equations for the six h ij . However, again we have the gauge freedomto eliminate this indeterminacy.Specializing to our preferred choice of scalar Ξ = − Υ / S kij does not contain the derivatives of φ ∗ . This implies that the second term in the second Eq.(73) islinear in the derivatives of h ∗ ij , and so the solution of this equation is easy to get: h ∗ ij = 0, (again, with the asymptoticboundary conditions h ∗ ij → M ′ is now a function of ( ~ ∇ φ ∗ /a ) , and that, in fact,¯ S ijk = 0. As a result the first of Eq.(73) becomes identical with the first of Eq.(2), while Eq.(76) becomes identicalwith the second of Eq.(2). In addition, from Eq.(77) we get h ij = 0 as before.We thus end up with a NR limit in which g µν = η µν − φδ µν and ˆ g µν = η µν − φδ µν as in Eq.(57), with the MONDpotential φ determined now from the NR MOND theory described by Eq.(2). This theory has been discussed atlength in [12]. The relation between the first-order MOND metric and the MOND potential is thus, again, the sameas that between the first-order GR metric and the Newtonian potential.Note in this context as well that using a scalar argument that is a combination of Υ and ¯Υ = g µν ¯ C µ ¯ C ν , leads tothe same first-order metric and NR limit. C. Other backgrounds
So far I assumed that the two metrics have the same cosmological background and so, for systems small on thecosmological scale both can be taken as nearly Minkowski.Here I consider some possible departures from this assumption. One possibility, for example, is that for today’scosmology the two metrics are conformally related ˆ g µν = λg µν with constant λ ; this still gives C αβγ = 0 for thecosmological background. In this case we can take locally the background metrics to be g Bµν = η µν , ˆ g Bµν = λη µν andexpand around these: g µν = η µν − φδ µν + h µν , ˆ g µν = λ ( η µν − φδ µν + ˆ h µν ) , (78)instead of Eq.(38). The expressions of ˆΓ αβγ , C αβγ , and ˆ R µν in terms of the potentials remain the same as before. Also,‘small’ coordinate transformations of the type shown in Eq.(52) still do not affect the potential differences only the h ij . Work with the general theory for arbitrary α, β . The NR limit of the field equations is now β ( R µν − η µν R ) + λf ( λ − )( M ′ ¯ S iµν ) ,i = − πG T µν , (79) α ( ˆ R µν − η µν ˆ R ) − f ( λ − )( M ′ ¯ S iµν ) ,i = 0 , (80)with the λ powers coming from factors such as κ ± , etc.. Defining ˜ α = λα , and ˜ M ′ = λf ( λ − ) M ′ , we get backthe λ = 1 case, but with α replaced by ˜ α , and M ′ by ˜ M ′ . As before, with our favorite choice of scalar argumentΞ = − Υ / a we have h ij = 0 in the appropriate gauge; so we get the relation between the MOND metric and theMOND potential as before.For a more general background we can write the background metrics,locally for a small system, g µν = η µν , ˆ g µν = λ ( η µν − uδ µ δ ν ) . (81)This leads to more complex NR limits, which I do not discuss here.We see then that the NR limit of the theory as applied to system that are small on cosmic scale depends on thebackground metrics. If the relation between the background metrics vary with cosmological time (I assume that itdoes not–see section VI) the application of BIMOND to local inhomogeneities also varies with cosmic time. But I willnot discuss this possibility further here.4 IV. GENERAL RELATIVITY LIMIT
I showed in [12] that the NR theories with β = 1, but arbitrary α , have a Newtonian limit if M ′ ( z ) → z → ∞ .This carries over to the relativistic theories. The same property of M causes the relativistic theory to go to GR inthe same limit a →
0, because M ′ → T µν , ˆ T µν →
0, and since in this case M ( z ) /z must also vanish in thelimit we also have Λ m , ˆΛ m →
0. This gives GR, with g µν satisfying the Einstein equation with the standard matterEMT as source. In this limit ˆ g µν satisfies its own Einstein equation. It decouples from g µν anyway, so it affects matterneither directly nor indirectly.We do not have tight phenomenological constraints on how fast MOND approaches Newtonian dynamics for highaccelerations. There are indications from solar system constraints [1, 21, 22] that M ′ ( z ) decreases at least as fastas z − , but it might turn out to do so much more precipitously. If so, any departure from GR might be practicallywiped out in a very-high-acceleration system, such as a laboratory on earth, the inner solar system, or a close binarypulsar. , In the limit a →
0, the vanishing of ˜ T µν and ˆ T µν might occur much faster than that of Λ m and ˆΛ m . For example,if M ( ∞ ) is finite, the latter vanish as a , wile the former might vanish much faster. We can thus, as an intermediateapproximation, keep the Λ m and ˆΛ m terms in the theory, and write the limiting field equations (16)(17) as R µν − Rg µν − Λ m ( ∞ ) g µν = − πG T µν , (82)ˆ R µν −
12 ˆ R ˆ g µν − ˆΛ m ( ∞ )ˆ g µν = − πG ˆ T µν . (83)For general β values it remains to be checked whether the requirement on M ′ from the NR limit, deduced in [12]suffices to guarantee a GR limit for a → V. GRAVITATIONAL LENSING
One of the two main phenomenological duties we expect from a relativistic MOND theory is to predict gravitationallensing correctly. In particular, we know that lensing analysis of galactic systems indicate mass discrepancies that arenot very different to those derived from massive-particle motions (e.g., rotation curves). In other words, lensing in theMOND regime is found observationally to be greatly enhanced over the GR prediction without DM. Reproducing thisfact has been a pressing desideratum in constructing relativistic MOND theories, achieved finally in TeVeS throughthe efforts of Sanders [4] and Bekenstein [3].In the present BIMOND class such enhanced, MOND-like lensing is predicted naturally by all the theories in theclass. For a choice of the scalar argument that is a combinations of Υ, Υ ∗ , and ¯Υ (and with any α, β ), relation(57) between the first-order MOND metric and the MOND potential holds. So, to this order the MOND connectionΓ λµν is expressed in terms of the MOND potential φ in the same way as the GR connection is expressed in terms ofthe Newtonian potential. This means, in turn, that the MOND potential describes the dynamics of both massiveand massless particles in the same way as the Newtonian potential does in GR. In other words, such theories predictthat analyzing lensing and massive-particle dynamics by a NR system assuming GR, should give the same effectivepotential (or the same distribution of “phantom matter”). This is consistent with observations.We also saw that there are other choices of the scalar argument of M for which the NR MOND metric is characterizedby additional potentials h ij . However, these vanish quickly for accelerations much above a , while in the MOND regimethey are of the same order as the MOND potential φ . We then expect these theories to yield somewhat different lensingto that expected with the GR relation between metric and potential, albeit still with the MOND characteristics.For example, we saw that far from a central mass M , and in the deep-MOND regime, the form of the MONDmetric is given by Eq.(69), and this leads to lensing that is multiplied by a factor 1 + ξ/ The presence of the galactic field still induces the departures discussed in [21]. A theory like TeVeS also has a surrogate for a that appears in its NR limit, which is constructed out of the constants characterizingthe theory. However, TeVeS does not become GR in the limit a → VI. COSMOLOGY
I cannot at present offer a specific BIMOND cosmology. There are two obstacles to doing this. In the firstplace we cannot even pinpoint the exact BIMOND theory out of the various versions possible. NR phenomenologycan assist somewhat in pinpointing the NR limit. However, even for a given NR MOND theory there are differentrelativistic versions having this limit, which differ greatly in the relativistic regime, and specifically in their applicationto cosmology.Second, as we well know from a century of experience with GR, having the underlying theory is one thing; pinpointingthe cosmology is another: In dealing with standard GR cosmology, the cosmological evolution and the present state ofthe universe does not emerge uniquely from first principles. There are major observational constraints, assumptionsabout symmetries, initial conditions, and matter content, that are put in by hand into cosmological theory. We donot know, for instance, the initial conditions for our universe from first principles, so an initial singularity (as opposedsay to a steady state universe with continuous matter creation, or to a static universe as Einstein would have itinitially) is imposed by hand. Early inflation (the mechanism for which is still moot) is put in by hand to accountfor various observational facts. Cosmic acceleration, whose cause remains unknown, is imposed by hand by invoking‘dark energy’, modified gravity, or other mechanisms. The material content of the universe (e.g. the very existence ofbaryon asymmetry–mechanism still unknown) is an input in cosmology. A priori, we could have had a cosmos with aspace that is inhomogeneous or anisotropic on cosmological scales; but, the cosmological principle is imposed basedon what our eyes tell us about our universe. All this is even more acute in light of recent developments in quantumcosmology.In the case of BIMOND we are on even shakier ground when coming to construct a cosmology. Here we are dealingwith two space-times, only one of which we can sense directly. We have no direct knowledge of many of the globalproperties of the other space-time. Did it have a Big Bang? Did it undergo inflation? Is it spatially flat? (in Rosen’stheory the auxiliary metric is constrained to be flat)? Is it spatially homogeneous and isotropic? What is the natureof the twin matter? Is it there at all? Is it always homogeneously distributed, or does it clump? Is it characterizedby the same baryon asymmetry, etc.?MOND phenomenology in systems that are small on cosmological scales is particularly simple and clear cut in adouble Minkowski background. Such a background applies if in cosmology ˆ g µν = g µν . BIMOND cosmology that havedisparate metrics, so that the cosmological values of C αβγ are appreciable, do not seem to make phenomenologicalsense in small systems.I thus assume, as an additional cosmological assumption to the many above, that on cosmological scales we haveˆ g µν = g µν . This could emerge, as an external constraint on BIMOND, from the world picture that underlies it. Or,more appealingly, it could, at least, correspond to a solution of BIMOND itself in some version.I thus consider here briefly only cosmologies with ˆ g µν = g µν , or with the somewhat relaxed assumption ˆ g µν = λg µν ,with a constant λ . In either case we have C αβγ = 0 in cosmology; so, finite values of C αβγ are produced only due to localinhomogeneities. This greatly simplifies the equations of motion, since the only contributions of the interaction termthat survive are the Λ m terms. With this ansatz, the equations of motion for the more general action (13) are then βG µν + qa M (0) g µν = − πG T µν ( g µν , ψ i ) ,αG µν + ˆ qa M (0) g µν = − πG ˆ T µν ( λg µν , χ i ) , (84)where q = ( λ/ κf ( κ )] ′ κ = λ − , ˆ q = − ( λ − / κ − f ( κ )] ′ κ = λ − , (85)and I used the fact that with the above ansatz ˆ G µν = G µν . For the two equalities in equation (84) to hold simultane-ously, we need to start with a BIMOND theory with some symmetry with respect to the two metrics. As an exampleconsider a special, more symmetric, case of the gravitational Lagrangian in Eq.(13) written as βg / ( R − a ¯ M ) + α ˆ g / ( ˆ R − a ¯ M ) . (86)This choice corresponds to f ( κ ) = ( βκ + ακ − ) / ( α + β ), and M = ( α + β ) ¯ M . It gives q = β/ ( α + β ), ˆ q = λα/ ( α + β ).So the field equations are now G µν + a ¯ M (0) g µν = − πGβ − T µν ( g µν , ψ i ) ,G µν + λa ¯ M (0) g µν = − πGα − ˆ T µν ( λg µν , χ i ) . (87)6Clearly, ˆ g µν = g µν ( λ = 1) always corresponds to a vacuum solution of this theory, with both space-times being a deSitter or Anti de Sitter, with a cosmological constant Λ = − a ¯ M (0). Furthermore, if we also have from symmetry,for two identical configurations in the two sectors, ˆ T µν = ( α/β ) T µν , then the two equations are the same evenwith matter. The cosmology we then get is, quite interestingly, standard GR cosmology (taking β = 1) with Λ ascosmological constant. This would be reassuring since it would automatically insure that we are not bereft of thesuccesses of standard cosmology regarding inflation, nucleosynthesis, etc. This picture would also force us to consider more seriously the nature of the twin matter, and its possible visibleeffects in our space-time. If it is homogeneously distributed it will be difficult to detect any direct effects of it. Ifit clumps it could have various effects; for example, it may produce some effects that are otherwise attributed tocosmological dark matter.It is also possible that BIMOND can replace cosmological DM by the distribution and fluctuations in ˜ T µν , whichis constructed from the two metrics alone, not directly from matter. This would be similar in vein to what has beendiscussed in connection with such a possible role of auxiliary fields in other theories [8, 18, 23, 26].The above is only one example of a BIMOND theory that has a cosmology with ˆ g µν = g µν . This particular versionshould not be assumed for the special case β + α = 0, since then ¯ M drops from the theory in the NR limit. Howeverthere are other versions of BIMOND that can accommodate our cosmological ansatz for this case. For example, wecan take as the gravitational Lagrangian density (say with β = 1) g / ( R − pa ) − ˆ g / ( ˆ R − pa ) − g ˆ g ) / f ( κ ) a M == g / R − ˆ g / ˆ R − a ( g ˆ g ) / [ pκ − ˆ pκ − + f ( κ ) M ] , (88)This theory also has the NR limit we discussed above, and has a cosmological solution with ˆ g µν = g µν if a certainrelation between p, ˆ p , and M (0) holds. For example, if M (0) = 0 we have the symmetric theory with p = ˆ p , in whichcase − pa is the cosmological constant. More generally, we can take the gravitational Lagrangian density (keepingthe symmetry) g / ( R − M a ) − ˆ g / ( ˆ R − M a ) − g ˆ g ) / f ( κ ) a M == g / R − ˆ g / ˆ R − a ( g ˆ g ) / [( κ − κ − ) ¯ M + f ( κ ) M ] (89)(with ¯ M also a function of the scalar argument Υ /a ). Then, ¯ M does not appear in the NR limit on a doubleMinkowski background, which is governed by M , and a cosmology with our ansatz is a solution, with − a ¯ M (0) beingthe cosmological constant [for M (0) = 0].We saw that in the GR limit a M ∞ plays the role of a cosmological constant [ M ∞ = M ( ∞ )], while in the presentcontext it is a M (0). More generally, the Λ m ∝ a M term, which is variable, and possibly other terms in ˜ T µν , maygive rise to “dark energy” effects. M ( z ) need not change much for the full range of z , since for small values it alsohas to go to some constant, as M ( z ) ≈ M + (4 / z / for z ≪
1. In fact, changes in M ( z ) over the full z range are,generically, of order unity, since M ∞ − M ( z ) = R ∞ z M ′ ( z ) dz is of order unity if M ′ decreases beyond z = 1 fasterthan z − (unlike M , which is known only up to an additive constant, M ′ is determined by MOND phenomenology). M ∞ is a dimensionless constant characterizing the theory. If |M ∞ | ∼
1, then |M| is always of order unity. We thenautomatically get the well known, but otherwise mysterious, proximity between a , as determined from the dynamicsof small systems, and Λ, the density of ‘dark energy’, as deduced from cosmology. Note that irrespective of the relevance to MOND, bimetric theories of the type presented here provide a frame fordiscussing ‘dark energy’ as modified gravity, which could be an alternative to schemes such as f ( R ) theories. This ensures that without the interaction, physics is the same in the two sectors. This line of thinking seems to indicate that the theories with α = β = 1 are preferable. The possible proximity of Λ and a , thus hinges on the dimensionless M being of order unity. Because of the way the normalizationof M is defined, this means that the scale over which M varies as a function of Υ, and the scale that determines the magnitude of the M term in the Lagrangian, which have the same dimensions of length − , are also of the same magnitude. This need not be the case,just as not all mass parameters that appear in the standard model of particle physics are of similar values. So, the apparent proximityΛ ∼ a that we get here is only a plausibility not a corollary. A. Deep-MOND relativistic systems?
In principle, BIMOND theories enable one to study the structure of deep-MOND, relativistic systems such asdeep-MOND black holes. As has been stressed many times in the past, such a deep-MOND system would have tohave its typical curvature radius much larger than the MOND length ℓ = c /a . This length is, however, of theorder of the Hubble radius today, and certainly in the past. In practice then, the universe seems to be the only suchlow-acceleration (rather, intermediate-acceleration) relativistic system, at present. VII. DISCUSSION
I have described a class of bimetric MOND (BIMOND) theories. Matter lives in the space-time described by oneof the metrics, which, in turn, couples to another through the interaction M term. If we heuristically view gravityas reflecting an effective ‘elasticity’ of space-time we can view the double-metric nature of our theory as representingtwo coexisting elastic bodies, each with its own elasticity as encapsuled in the respective R , ˆ R terms in the action.Thus MOND departure from GR is introduced not through a modification of the ‘elasticity’ properties of space-time,but rather through the interaction of the space-time that is the arena for matter with the auxiliary one. The strengthof the interaction between these two space-time ‘membranes’ depends on the gradient difference. The response ofour home space-time to matter is affected by its interaction with the other space-time, which modifies its effectiveelasticity. However, once its shape is determined, this home space-time affects matter in the standard way. Withour assumption that on cosmological scales ˆ g µν = g µν the two membranes are, in a sense, stuck together on thesescales, and ‘separate’ only locally due to inhomogeneities. Such heuristics may help pinpoint the fundamental conceptunderpinning the MOND paradigm. For example, it may give meaning to the length ℓ = c /a that appears in theNR limit as a .The BIMOND theories have the (yet unproven) potential to account for all the components of the dark sector(galactic DM, cosmological DM, and dark energy) from one term in the action, all controlled by a .My main objective has been to point out that there exists such a class of relativistic theories that have MOND-liketheories as their NR limit, and, which produce enhanced, MOND-like gravitational lensing. We are, however, still farfrom pinpointing the exact version of the theory that is the most suitable. This is particularly true in the context ofcosmology, which depends crucially on the choice of version. Hopefully, theoretical and phenomenological constraintswill be brought to bear on this by future studies. It remains to be seen whether a version of BIMOND can be foundthat pass muster given all such requirements. Recent discussions of matter-of-principle questions, such as the causalstructure of bimetric theories of a different type (where the interaction term is a function of the metrics themselves,not their derivatives) can be found, e.g., in [16, 17].Finally, it has to be realized that however useful such theories may turn out to be, they must be only effective,approximate theories, as evinced by the appearance of the a priori unspecified function M and the length ℓ (or theMOND acceleration a ). These will, hopefully, be calculated from a theory at a deeper stratum. Acknowledgements
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