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Gürses Metin, Şentürk Çetin. A Modified Gravity Theory: Null Aether. Communications in Theoretical Physics, 2019, 71(3): 312  
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A Modified Gravity Theory: Null Aether
Gürses Metin1, 2, †, Şentürk Çetin2, 3, ‡
1Department of Mathematics, Faculty of Sciences, Bilkent University, Ankara 06800, Turkey
2Department of Physics, Faculty of Sciences,Bilkent University, Ankara 06800, Turkey
3Department of Aeronautical Engineering,University of Turkish Aeronautical Association, Ankara 06790, Turkey

 

† Corresponding author. E-mail: gurses@fen.bilkent.edu.tr csenturk@thk.edu.tr

Abstract
Abstract

General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the “aether”. In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory (NAT)—a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordström-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT: We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions—AdS-plane waves and pp-waves—in this theory in any dimension . Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a “mass” determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.

Keyword:Aether theory;Newtonian limit;black holes;cosmological solutions;Kerr-Schild-Kundt class of metrics;Ads-plane waves;pp-waves
1. Introduction

Lorentz violating theories of gravity have attracted much attention recently. This is mainly due to the fact that some quantum gravity theories, such as string theory and loop quantum gravity, predict that the spacetime structure at very high energies—typically at the Planck scale—may not be smooth and continuous, as assumed by relativity. This means that the rules of relativity do not apply and Lorentz symmetry must break down at or below the Planck distance (see e.g., Ref. [1]).

The simplest way to study Lorentz violation in the context of gravity is to assume that there is a vector field with fixed norm coupling to gravity at each point of spacetime. In other words, the spacetime is locally endowed with a metric tensor and a dynamical vector field with constant norm. The vector field defined in this way is referred to as the “aether” because it establishes a preferred direction at each point in spacetime and thereby explicitly breaks local Lorentz symmetry. The existence of such a vector field would affect the propagation of particles—such as electrons and photons—through spacetime, which manifests itself at very high energies and can be observed by studying the spectrum of high energy cosmic rays. For example, the interactions of these particles with the field would restrict the electron’s maximum speed or cause polarized photons to rotate as they travel through space over long distances. Any observational evidence in these directions would be a direct indication of Lorentz violation, and therefore new physics, at or beyond the Planck scale.

So vector-tensor theories of gravity are of physical importance today because they may shed some light on the internal structure of quantum gravity theories. One such theory is Einstein-Aether theory[2,3] in which the aether field is assumed to be timelike and therefore breaks the boost sector of the Lorentz symmetry. This theory has been investigated over the years from various respects.[4–22] There also appeared some related works,[23–26] which discuss the possibility of a spacelike aether field breaking the rotational invariance of space. The internal structure and dynamics of such theories are still under examination; for example, the stability problem of the aether field has been considered in Refs. [27–28].§ Of course, to gain more understanding in these respects, one also needs explicit analytic solutions to the fairly complicated equations of motion that these theories possess.

In this paper, we propose yet another possibility, namely, the possibility of a null aether field, which dynamically couples to the metric tensor of spacetime. From now on, we shall refer to the theory constructed in this way as Null Aether Theory (NAT). This construction enables us to naturally introduce a scalar degree of freedom, i.e. the spin-0 part of the aether field, which is a scalar field that has a mass in general. By using this freedom, we show that it is possible to construct exact black hole solutions and nonlinear wave solutions in the theory. Indeed, assuming the null aether vector field (vμ) is parallel to the one null leg (lμ) of the viel-bein at each spacetime point, i.e. , where ϕ(x) is the spin-0 aether field, we first discuss the Newtonian limit of NAT and then proceed to construct exact spherically symmetric black hole solutions to the full nonlinear theory in four dimensions. In the Newtonian limit, we considered three different forms of the aether field: (a) where aμ is a constant vector representing the background aether field and kμ is the perturbed aether field. (b) and where is a nonzero constant and ϕ1 is the perturbed scalar aether field. (c) The case where ϕ0=0.

Among the black hole solutions, there are Vaidya-type nonstationary solutions, which do not need the existence of any extra matter field: the null aether field present in the very foundation of the theory behaves, in a sense, as a null matter to produce such solutions. For special values of the parameters of the theory, there are also stationary Schwarzschild-(A)dS type solutions that exist even when there is no explicit cosmological constant in the theory, Reissner-Nordström-(A)dS type solutions with some “charge” sourced by the aether, and solutions of conformal gravity that contain a term growing linearly with radial distance and so associated with the flatness of the galaxy rotation curves. Our exact solutions perfectly match the solutions in the Newtonian limit when the aether field is on the order of the Newtonian potential.

We investigated the cosmological solutions of NAT. Taking the matter distribution as the perfect fluid energy momentum tensor, with cosmological constant, the metric as the spatially flat (k = 0) Friedmann-Lemaître-Robertson-Walker (FLRW) metric and the null aether propagating along the x-axis, we find some exact solutions where the equation of state is of polytropic type. If the parameters of the theory satisfy some special inequalities, then acceleration of the expansion of the universe is possible. This is also supported by some special exact solutions of the field equations. There are two different types of solutions: power law and exponential. In the case of the power law type, there are four different solutions in all of which the pressure and the matter density blow up at t = 0. In the other exponential type solutions case, the metric is of the de Sitter type and there are three different solutions. In all these cases the pressure and the matter density are constants.

On the other hand, the same construction, , also permits us to obtain exact solutions describing gravitational waves in NAT. In searching for such solutions, the Kerr-Schild-Kundt (KSK) class of metrics[33–38] was shown to be a valuable tool to start with: Indeed, recently, it has been proved that these metrics are universal in the sense that they constitute solutions to the field equations of any theory constructed by the contractions of the curvature tensor and its covariant derivatives at any order.[38] In starting this work, one of our motivations was to examine whether such universal metrics are solutions to vector-tensor theories of gravity as well. Later on, we perceived that this is only possible when the vector field in the theory is null and aligned with the propagation direction of the waves. Taking the metric in the KSK class with maximally symmetric backgrounds and assuming further , we show that the AdS-plane waves and pp-waves form a special class of exact solutions to NAT. The whole set of field equations of the theory are reduced to two coupled differential equations, in general, one for a scalar function related to the profile function of the wave and one for the “massive” spin-0 aether field . When the background spacetime is AdS, it is possible to solve these coupled differential equations exactly in three dimensions and explicitly construct plane waves propagating in the AdS spacetime. Such constructions are possible also in dimensions higher than three but with the simplifying assumption that the profile function describing the AdS-plane wave does not depend on the transverse D-3 coordinates. The main conclusion of these computations is that the mass corresponding to the spin-0 aether field acquires an upper bound (the Breitenlohner-Freedman bound[39]) determined by the value of the cosmological constant of the background spacetime. In the case of pp-waves, where the background is flat, the scalar field equations decouple and form one Laplace equation for a scalar function related to the profile function of the wave and one massive Klein-Gordon equation for the spin-0 aether field in (D-2)-dimensional Euclidean flat space. Because of this decoupling, plane wave solutions, which are the subset of pp-waves, can always be constructed in NAT.

The paper is structured as follows. In Sec. 2, we introduce NAT and present the field equations. In Sec. 3, we study the Newtonian limit of the theory to see the effect of the null vector field on the solar system observations. In Sec. 4, we construct exact spherically symmetric black hole solutions in their full generality in four dimensions. In Sec. 5, we study the FLRW cosmology with spatially flat metric and null aether propagating along the x direction. We find mainly two different exact solutions in the power and exponential forms. We also investigate the possible choices of the parameters of the theory where the expansion of the universe is accelerating. In Sec. 6, we study the nonlinear wave solutions of NAT propagating in nonflat backgrounds, which are assumed to be maximally symmetric, by taking the metric in the KSK class. In Sec. 7, we specifically consider AdS-plane waves describing plane waves moving in the AdS spacetime in dimensions. In Sec. 8, we briefly describe the pp-wave spacetimes and show that they provide exact solutions to NAT. We also discuss the availability of the subclass plane waves under certain conditions. Finally, in Sec. 9, we summarize our results.

We shall use the metric signature throughout the paper.

2. Null Aether Theory

The theory we shall consider is defined in D dimensions and described by, in the absence of matter fields, the action

where

Here Λ is the cosmological constant and is the so-called aether field, which dynamically couples to the metric tensor and has the fixed-norm constraint

which is introduced into the theory by the Lagrange multiplier λ in Eq. (1). Accordingly, the aether field is a timelike (spacelike) vector field when ε=+1 (ε=−1), and it is a null vector field when ε=0.∥ The constant coefficients c1, c2, c3 and c4 appearing in Eq. (2) are the dimensionless parameters of the theory.**

The equations of motion can be obtained by varying the action (1) with respect to the independent variables: Variation with respect to λ produces the constraint equation (3) and variation with respect to and vμ produces the respective, dynamical field equations

where and

In writing Eq. (4), we made use of the constraint (3). From now on, we will assume that the aether field is null (i.e., ε=0) and refer to the above theory as Null Aether Theory, which we have dubbed NAT. This fact enables us to obtain from the aether equation (5) by contracting it by the vector ; that is,

Here we assume that to exclude the trivial zero vector; i.e., . It is obvious that flat Minkowski metric () and a constant null vector ( = ), together with λ=0, constitute a solution to NAT. The trivial case where and Ricci flat metrics constitute another solution of NAT. As an example, at each point of a 4-dimensional spacetime it is possible to define a null tetrad where and are real null vectors with , and is a complex null vector orthogonal to and . The spacetime metric can then be expressed as

This form of the metric is invariant under the local SL(2,C) transformation. For asymptotically flat spacetimes, the metric is assumed to reduce asymptotically to the Minkowski metric ,

where is the null tetrad of the flat Minkowski spacetime and is the asymptotic limit of the null tetrad . Our first assumption in this work is that the null aether is proportional to the null vector ; i.e., , where is a scalar function. In Petrov-Pirani-Penrose classification of spacetime geometries, the null vectors and play essential roles. In special types, such as type-D and type-N, the vector is the principal null direction of the Weyl tensor. Hence, with our assumption, the null aether vector gains a geometrical meaning. Physical implications of the aether field comes from the scalar field which carries a nonzero charge. Certainly the zero aether, , or the trivial solution satisfies field equations (4) and (5). To distinguish the nontrivial solution from the trivial one, in addition to the field equations (4) and (5), we impose certain nontrivial initial and boundary conditions for . This is an important point in initial and boundary value problems in mathematics. In any initial and boundary value problem, when the partial differential equation is homogenous, such as the massless Klein-Gordon equation, the trivial solution is excluded by either the boundary or initial conditions. Trivial solution exists only when both boundary and initial values are zero. Therefore, our second assumption in this work is that in stationary problems the scalar field ϕ carries a nonzero scalar charge and in non-stationary problems it satisfies a non-trivial initial condition.

In the case of black hole solutions and Newtonian approximation, the vector field is taken as where lμ asymptotically approaches a constant vector and ϕ(x) behaves like a scalar field carrying some null aether charge. In the case of the wave solutions, ϕ(x) becomes a massive scalar field.

Null Aether Theory, to our knowledge, is introduced for the first time in this paper. There are some number of open problems to be attacked such as Newtonian limit, black holes, exact solutions, stability, etc. In this work, we investigate the Newtonian limit, the spherically symmetric black hole solutions (in D = 4), cosmological solutions, and the AdS wave and pp-wave solutions of NAT. In all these cases, we assume that , where is a null leg of the viel-bein at each spacetime point and is a scalar field defined as the spin-0 aether field that has a mass in general. The covariant derivative of the null vector can always be decomposed in terms of the optical scalars: expansion, twist, and shear.[47]

3. Newtonian Limit of Null Aether Theory

Now we shall examine the Newtonian limit of NAT to see whether there are any contributions to the Poisson equation coming from the null aether field. For this purpose, as usual, we shall assume that the gravitational field is weak and static and produced by a nonrelativistic matter field. Also, we know that the cosmological constant—playing a significant role in cosmology—is totally negligible in this context.

Let us take the metric in the Newtonian limit as

where . We assume that the matter energy-momentum distribution takes the form

where , and pm are the mass density and pressure of matter, and is the stress tensor with . We obtain the following cases.

Let the null vector be

where is a constant null vector representing the background aether and represents the perturbed null aether. Nullity of the aether field vμ implies

at the perturbation order. Since the metric is symmetric under rotations, we can take, without loosing any generality, and for simplicity we will assume that . Then we obtain , , , and

It turns out that the gravitational potential Φ satisfies the equation

where

which implies that Newton’s gravitation constant G is scaled as in Refs. [16,42]. The constraint can be removed by taking the stress part into account in the energy momentum tensor, then there remains only the constraint

In this spacetime, a null vector can also be defined, up to a multiplicative function of , as

where with i=1,2,3. Now we write the null aether field as (since we are studying with a null vector, we always have this freedom) and assume that where is an arbitrary constant not equal to zero and ϕ1 is some arbitrary function at the same order as Φ and/or Ψ. Next, in the Eistein-Aether equations (4) and (5), we consider only the zeroth and first order (linear) terms in , Φ, and Ψ. The zeroth order aether scalar field is different from zero, . In this case the zeroth order field equations give and , and consistency conditions in the linear equations give and . Then we get and

which implies that

This is a very restricted aether theory because there exist only one independent parameter c1 left in the theory.

The zeroth order scalar aether field in case 2 is zero, . This means that is at the same order as Φ and/or Ψ. In the Eistein-Aether equations (4) and (5), we consider only the linear terms in , Φ, and Ψ. Then the zeroth component of the aether equation (5) gives, at the linear order,

where , and the i-th component gives, at the linear order,

after eliminating λ using Eq. (21). Since the aether contribution to the equation (4) is zero at the linear order, the only contribution comes from the nonrelativistic matter for which we have (12). Here we are assuming that the matter fields do not couple to the aether field at the linear order. Therefore, the only nonzero components of Eq. (4) are the 00 and the ij component (the 0i component is satisfied identically). Taking the trace of the ij component produces

which enforces

for the spacetime to be asymptotically flat. Using this fact, we can write, from the 00 component of Eq. (4),

Thus we see that the Poisson equation is unaffected by the null aether field at the linear order in G.

The Poisson equation (25) determines the Newtonian potential. To see the effect of the Newtonian potential on a test particle, one should consider the geodesic equation in the Newtonian limit in which the particle is assumed to be moving nonrelativistically (i.e., ) in a static (i.e., ) and weak (i.e., with ) gravitational field. In fact, by taking the metric in the form (11), one can easily show that the geodesic equation reduces to the Newtonian equation of motion for a nonrelativistic particle.

Outside of a spherically symmetric mass distribution, the Poisson equation (25) reduces to the Laplace equation which gives

On the other hand, for spherical symmetry, the condition (22) can be solved and yields

where a1 and a2 are arbitrary constants and

This solution immediately puts the following condition on the parameters of the theory

Specifically, when , we have

when , we have

or when , we have

In this last case, asymptotically, letting , , where Q is the NAT charge.

4. Black Hole Solutions in Null Aether Theory

In this section, we shall construct spherically symmetric black hole solutions to NAT in D = 4. Let us start with the generic spherically symmetric metric in the following form with :

where is the cosmological constant. For f(u,r)=0, this becomes the metric of the usual (A)dS spacetime. Since the aether field is null, we take it to be with being the null vector of the geometry.

With the metric ansatz (33), from the u component of the aether equation (5), we obtain

and from the r component, we have

where the prime denotes differentiation with respect to r and the dot denotes differentiation with respect to u. The equation (35) can easily be solved and the generic solution is

for some arbitrary functions and , where

When and , then where . Here , where Q is the NAT charge.

Note that when , the square root in Eq. (37) vanishes and the roots coincide to give . Inserting this solution into the Einstein equations (4) yields, for the ur component,

with the identifications

Thus we obtain

where and are arbitrary functions. Notice that the last case occurs only when . If we plug Eq. (40) into the other components, we identically satisfy all the equations except for the uu component which, together with λ from Eq. (34), produces

for and , and

for . The last case immediately leads to

where m is the integration constant. Thus we see that Vaidya-type solutions can be obtained in NAT without introducing any extra matter fields, which is unlike the case in general relativity. Observe also that when f(u,r)=0, we should obtain the (A)dS metric as a solution to NAT (see Eq. (33)). Then it is obvious from Eq. (38) that this is the case, for example, if corresponding to

where d is an arbitrary constant and a(u) is an arbitrary function.

Defining a new time coordinate t by the transformation

one can bring the metric (33) into the Schwarzschild coordinates

where the function g(t,r) should satisfy

When and are constants, since then, the condition (47) says that g=g(t) and so it can be absorbed into the time coordinate t, meaning that g(t,r) can be set equal to unity in Eqs. (45) and (46). In this case, the solution (46) will describe a spherically symmetric stationary black hole spacetime. The horizons of this solution should then be determined by solving the equation

where , m=const., and

When , we let , and the first case () in Eq. (48) becomes

This is a black hole solution with event horizons located at the zeros of the function h(r) which depend also on the constant Q. This clearly shows that the corresponding black hole carries an NAT charge Q. The second case (q = 0) in Eq. (48) is the usual Schwarzschild-(A)dS spacetime. At this point, it is important to note that when a1 and a2 are in the order of the Newton’s constant G, i.e. and , since h(r) depends on the squares of a1 and a2, we recover the Newtonian limit discussed in Sec. 3 for , and D = 4. For special values of the parameters of the theory, the first case () of Eq. (48) becomes a polynomial of r; for example,

When (q = 3), : This is a Schwarzschild-(A)dS type solution if A = 0. Solutions involving terms like can be found in, e.g., Ref. [9,48].

When (q = 1), : This is a Reissner-Nordström-(A)dS type solution if A = 0.

When (q = 2), : This solution with A = 0 has been obtained by Mannheim and Kazanas[49] in conformal gravity who also argue that the linear term Br can explain the flatness of the galaxy rotation curves.

Here A and B are the appropriate combinations of the constants appearing in Eq. (48). For such cases, the equation h(r)=0 may have at least one real root corresponding to the event horizon of the black hole. For generic values of the parameters, however, the existence of the real roots of h(r)=0 depends on the signs and values of the constants , b1, b2, and in Eq. (48). When q is an integer, the roots can be found by solving the polynomial equation h(r)=0, as in the examples given above. When q is not an integer, finding the roots of h(r) is not so easy, but when the signs of and are opposite, we can say that there must be at least one real root of this function. Since the signs of these limits depends on the signs of the constants , b1, b2, and , we have the following cases in which h(r) has at least one real root:

If , , and

If , , and ,

If , , and ,

If , , and .

Of course, these are not the only possibilities, but we give these examples to show the existence of black hole solutions of NAT in the general case.

5. Cosmological Solutions in Null Aether Theory

The aim of this section is to construct cosmological solutions to the NAT field equations (4) and (5). We expect to see the gravitational effects of the null aether in the context of cosmology. We will look for spatially flat cosmological solutions, especially the ones which have power law and exponential behavior for the scale factor.

Taking the metric in the standard FLRW form and studying in Cartesian coordinates for spatially flat models, we have

The homogeneity and isotropy of the space dictates that the “matter” energy-momentum tensor is of a perfect fluid; i.e.,

where and we made the redefinitions

for, respectively, the density and pressure of the fluid which are functions only of t. Therefore, with the inclusion of the matter energy-momentum tensor (52), the Einstein equation (4) take the form

where denotes the null aether contribution on the right hand side of Eq. (4). Since first two terms in this equation have zero covariant divergences by construction, the energy conservation equation for the fluid turns out as usual; i.e., from , we have

where the dot denotes differentiation with respect to t.

Now we shall take the aether field as

which is obviously null, i.e. , with respect to the metric (51). Then there are only two aether equations: one coming from the time component of Eq. (5) and the other coming from the x component. Solving the time component for the lagrange multiplier field, we obtain

where , and inserting this into the x component, we obtain

Also, eliminating λ from the Einstein equations (54) by using Eq. (57), we obtain

from and , respectively, and

from (or from ). The equation is identically satisfied thanks to Eq. (58).

To get an idea how the null aether contributes to the acceleration of the expansion of the universe, we define (the Hubble function) and . Then Eqs. (58) and (61) respectively become

Eliminating between these equations, we obtain

It is now possible to make the sign of positive by assuming that

which means that the universe’s expansion is accelerating.

In the following sub-sections we give exact solutions of the above field equations in some special forms.

5.1. Power Law Solution

Let us assume the scale factor has the behavior

where R0 and ω are constants. Then the equation (58) can easily be solved for ϕ to obtain

where ϕ1 and ϕ2 are arbitrary constants and

Now plugging Eqs. (66) and (67) into Eq. (61), one can obtain the following condition on the parameters:

where

The interesting cases are

Using the definition of β in Eq. (68), we can now put some constraints on the parameters of the theory.

(β=0)

In this case, it turns out that

where we define

which must satisfy . Then we have

Here is a new constant defined by . The last two equations say that

where

Thus, for dust () to be a solution, it is obvious that

(, ) and (, )

In these two cases, we have

where the subscript i represents “1” for Case 2 and “2” for Case 3. Adding Eqs. (81) and (82), we also have

where

It is interesting to note that the null aether is linearly increasing with time and, together with the parameters of the theory, determines the cosmological constant in the theory. For example, for dust () to be a solution, it can be shown that

Since by definition (see Eq. (68)), in Case 2 and in Case 3. So the dust solution (85) can be realized only in Case 2.

,

In these cases, using the definition of β given in Eq. (68), we immediately obtain

We should also have by definition. Then we find

Here i represents “2” for Case 4 and “1” for Case 5. So as in Case 1,

where

In all the cases above, the Hubble function and hence . Then corresponds to the acceleration of the expansion of the universe, and in all our solutions above, there are indeed cases in which

5.2. Exponential Solution

Now assume that the scale factor has the exponential behavior

where R0 and ω are constants. Following the same steps performed in the power law case, we obtain

where ϕ1 and ϕ2 are new constants and

and the condition

where

Then the interesting cases are

where we defined . It should be noted again that by definition (See Eq. (94)). In all these three cases, we find that

where ω is arbitrary. When , this is the usual de Sitter solution which, describes a radiation dominated expanding universe.

6. Wave Solutions in Null Aether Theory: Kerr-Schild-Kundt Class of Metrics

Now we shall construct exact wave solutions to NAT by studying in generic dimensions. For this purpose, we start with the general KSK metrics[33–38] of the form

with the properties

where is an arbitrary vector field for the time being. It should be noted that is not a Killing vector. From these relations it follows that

In Eq. (99), is the background metric assumed to be maximally symmetric; i.e. its curvature tensor has the form

with

It is therefore either Minkowski, de Sitter (dS), or anti-de Sitter (AdS) spacetime, depending on whether K = 0, , or . All the properties in Eq. (100), together with the inverse metric

imply that (see, e.g., Ref. [34])

and the Einstein tensor is calculated as

with

where and is the covariant derivative with respect to the background metric .

To solve the NAT field equations we now let and assume . By these assumptions we find that Eqs. (6) and (7) are worked out to be

Then one can compute the field equations (4) and (5) as

where and use has been made of the identity . For the KSK metric (99), these equations become

From these, we deduce that

where we eliminated the term that appears in Eq. (115) by using the aether equation Eq. (119) and assuming .

Now let us make the ansatz

for some arbitrary constant . With this, we can write Eq. (118) as

Here there are two possible choices for . The first one is for which Eq. (121) becomes

and reduces to

when K = 0 and , which is the pp-wave case to be discussed in Sec. 8. The other choice, , drops the second term in Eq. (121) and produces

Here it should be stressed that this last case is present only when the background metric is nonflat (i.e. ) and/or .

On the other hand, the aether equation (119) can be written as

where, assuming λ is constant, we defined

since . The equation (125) can be considered as the equation of the spin-0 aether field with m being the “mass” of the field. The definition (126) requires that

the same constraint as in Eq. (186) when K = 0. Obviously, the field becomes “massless” if

Thus we have shown that, for any solution of Eq. (125), there corresponds a solution V0 of Eq. (122) for or of Eq. (124) for , and we can construct an exact wave solution with nonflat background given by Eq. (99) with the profile function Eq. (120) in NAT.

7. AdS-Plane Waves in Null Aether Theory

In this section, we shall specifically consider AdS-plane waves for which the background metric is the usual D-dimensional AdS spacetime with the curvature constant

where is the radius of curvature of the spacetime. We shall represent the spacetime by the conformally flat coordinates for simplicity; i.e. with and

where u and v are the double null coordinates. In these coordinates, the boundary of the AdS spacetime lies at z = 0.

Now if we take the null vector in the full spacetime of the Kerr-Schild form Eq. (99) as , then using Eq. (104) along with ,

so the functions V and are independent of the coordinate v; that is, and . Therefore the full spacetime metric defined by Eq. (99) will be

with the background metric (130). It is now straightforward to show that (see also Ref. [34])

where we used the second property in Eq. (105) to convert the full covariant derivative to the background one , and with . Comparing Eq. (133) with the defining relation in Eq. (100), we see that

where we again used Eq. (104) together with .

Thus, for the AdS-plane wave ansatz Eq. (132) with the profile function

to be an exact solution of NAT, the equations that must be solved are the aether equation (125), which takes the form

where and

and the equation (122) for , which becomes

or the equation (124) for , which becomes

where .

7.1. AdS-Plane Waves in Three Dimensions

It is remarkable that the equations (136), (138), and (139) can be solved exactly in D = 3. In that case , and so, and . Then Eq. (136) becomes

with

and has the general solution, when ,

where and are arbitrary functions. With this solution, Eqs. (138) and (139) can be written compactly as

where

The general solution of Eq. (143) is

with the arbitrary functions and . Note that the second term can always be absorbed into the AdS part of the metric (132) by a redefinition of the null coordinate v, which means that one can always set here and in the following solutions without loosing any generality. In obtaining Eq. (146), we assumed that . If, on the other hand, , then the above solution becomes

and if , it becomes

At this point, a physical discussion must be made about the forms of the solutions Eq. (142) and Eq. (146): As we pointed out earlier, the point z = 0 represents the boundary of the background AdS spacetime; so, in order to have an asymptotically AdS behavior as we approach z = 0, we should have (the Breitenlohner-Freedman bound[39])

Since in three dimensions, this restricts the mass to the range

which, in terms of λ through Eq. (141), becomes

Thus we have shown that the metric

with the profile function

describes an exact plane wave solution, propagating in the three-dimensional AdS background, in NAT.

Up to now, we consider the case . The case m = 0, which corresponds to the choice in Eq. (141), needs special handling. The solution of Eq. (140) when m = 0 is

with the arbitrary functions and . Inserting this into Eqs. (138) and (139) for D = 3 produces

where

The general solution of Eq. (156) can be obtained as

7.2. AdS-Plane Waves in D Dimensions: A Special Solution

Let us now study AdS-Plane Waves in D Dimensions: A Special SolutionLet us now study the problem in D dimensions. Of course, in this case, it is not possible to find the most general solutions of the coupled differential equations (136), (138), and (139). However, it is possible to give a special solution, which may be thought of as the higher-dimensional generalization of the previous three-dimensional solution (154).

The D-dimensional spacetime has the coordinates with . Now assume that the functions V0 and are homogeneous along the transverse coordinates xi; i.e., take

In that case, the differential equation (136) becomes

where m is given by Eq. (137), whose general solution is, for ,

where and are two arbitrary functions and

Inserting Eq. (162) into Eqs. (138) and (139) yields

where

The general solution of Eq. (164) can be obtained as

where and are arbitrary functions. This solution is valid only if

When , we have

and, when , we have

For , all these expressions reduce to the corresponding ones in the previous section when D = 3.

As we discuss in the previous subsection, these solutions should behave like asymptotically AdS as we approach z = 0. This means that

With Eqs. (163) and (129), this condition gives

where . For D = 4 and taking the present value of the cosmological constant, (GeV)2, we obtain the upper bound for the mass of the spin-0 aether field .

Therefore the metric

with the profile function

describes an exact plane wave, propagating in the D-dimensional AdS background, in NAT.

8. pp-Waves in Null Aether Theory

As a last example of KSK metrics, we shall consider pp-waves, These are defined to be spacetimes that admit a covariantly constant null vector field i.e.,

These spacetimes are of great importance in general relativity in that they constitute exact solutions to the full nonlinear field equations of the theory, which may represent gravitational, electromagnetic, or some other forms of matter waves.[47]

In the coordinate system with adapted to the null Killing vector , the pp-wave metrics take the Kerr-Schild form[50,51]

where u and v are the double null coordinates and is the profile function of the wave. For such metrics, the Ricci tensor and the Ricci scalar become

where . A particular subclass of pp-waves are plane waves for which the profile function is quadratic in the transverse coordinates xi, that is,

where the symmetric tensor contains the information about the polarization and amplitude of the wave. In this case the Ricci tensor takes the form

where Tr(h) denotes the trace of the matrix .

Now we will show that pp-wave spacetimes described above constitute exact solutions to NAT. As before, we define the null aether field as , but this time we let the scalar function and the vector field satisfy the following conditions

Note that this is a special case of the previous analysis achieved by taking the background is flat (i.e. K = 0) and there. Then it immediately follows from Eqs. (112), (113), and (114) that

and the field equations are

where we have eliminated the term that should appear in Eq. (183) by using the aether equation (184) assuming . The right-hand side of the equation (183) is in the form of the energy-momentum tensor of a null dust, i.e. with

The condition requires that ††

On the other hand, the equation (184) gives Klein-Gordon equation for the field :

where we defined the “mass” by

which is consistent with the constraint (186).

With the pp-wave ansatz (177), the field equations (183) and (184) become

Therefore, the profile function of pp-waves should satisfy

since it must be that . At this point, we can make the following ansatz

where is an arbitrary constant. Now plugging this into Eq. (191), we obtain

and since we are free to choose any value for , we get

Thus, any solution of Eq. (190) together with the solution of the Laplace equation (194) constitutes a pp-wave metric (177) with the profile function given by Eq. (192).

Let us now consider the plane wave solutions described by the profile function (179). In that case, we can investigate the following two special cases.

When (or, through Eq. (194)), it is obvious from Eq. (192) that the function , satisfying Eq. (190), detaches from the function V and we should have . This means that the profile function satisfies the Laplace equation, i.e.,

which is solved by only if . Thus we have shown that plane waves are solutions in NAT provided the equation (190) is satisfied independently. For example, in four dimensions with the coordinates , the metric

describes a plane wave propagating along the null coordinate v (related to the aether field through with satisfying Eq. (190)) in flat spacetime. Here the function is related to the polarization of the wave and, for a wave with constant linear polarization, it can always be set equal to zero by performing a rotation in the transverse plane coordinates x and y.

In this case, the Laplace equation (194) says that , and from Eq. (192) we have

Inserting this into Eq. (190), we obtain

This condition is trivially satisfied if , but this is just the previous case in which . Nontrivially, however, the condition Eq. (198) can be satisfied by setting the coefficient of the first term and the mass m (or, equivalently, the Lagrange multiplier λ) equal to zero. Then again plane waves occur in NAT.

9. Conclusions

In this work, we introduced the Null Aether Theory (NAT), which is a vector-tensor theory of gravity in which the vector field defining the aether is assumed to be null at each point of spacetime. This construction allows us to take the aether field () to be proportional to one null leg () of the viel-bein defined at each point of spacetime, i.e. with being the spin-0 part of the aether field. We first investigated the Newtonian limit of this theory and then constructed exact spherically symmetric black hole solutions in D = 4 and nonlinear wave solutions in in the theory. Among the black hole solutions, we have Vaidya-type nonstationary solutions, which do not need any extra matter fields for their very existence: the aether behaves in a sense as a null matter field to produce such solutions. Besides these, there are also (i) Schwarzschild-(A)dS type solutions with for that exist even when there is no explicit cosmological constant in the theory, (ii) Reissner-Nordström-(A)dS type solutions with for , (iii) solutions with for , which were also obtained and used to explain the flatness of the galaxy rotation curves in conformal gravity, and so on. All these solutions have at least one event horizon and describe stationary black holes in NAT. We also discussed the existence of black hole solutions for arbitrary values of the parameters {}.

We studied the cosmological implications of NAT in FLRW spacetimes. We assumed the null aether is propagating along the x direction and found mainly two different types of solutions. In the first type, the null aether scalar field and radius function R(t) are given as (power law) where σ is expressed in terms of the parameters of the theory. The pressure and the matter density functions blow up when t = 0 (Big-bang singularity). The second type is the de Sitter universe with exponentially decaying aether filed. In this case the pressure and the matter density functions are constants. We showed that the accelerated expansion of the universe is possible in NAT if the parameters of the theory satisfy some special inequalities.

As for the wave solutions, we specifically studied the Kerr-Schild-Kundt class of metrics in this context and showed that the full field equations of NAT reduce to just two, in general coupled, partial differential equations when the background spacetime takes the maximally symmetric form. One of these equations describes the massive spin-0 aether field . When the background is AdS, we solved these equations explicitly and thereby constructed exact AdS-plane wave solutions of NAT in three dimensions and in higher dimensions than three if the profile function describing the wave is independent of the transverse coordinates. When the background is flat, on the other hand, the pp-wave spacetimes constitute exact solutions, for generic values of the coupling constants, to the theory by reducing the whole set of field equations to two decoupled differential equations: one Laplace equation for a scalar function related to the profile function of the wave and one massive Klein-Gordon equation for the spin-0 aether field in -dimensional Euclidean flat space. We also showed that the plane waves, subset of pp-waves, are solutions to the field equations of NAT provided that the parameter c3 vanishes. When c3 is nonvanishing, however, the solution of the Laplace equation should satisfy certain conditions and the spin-0 aether field must be massless, i.e., λ=0. The main conclusion of these computations is that the spin-0 part of the aether field has a mass in general determined by the cosmological constant and the Lagrange multiplier given in the theory and in the case of AdS background this mass acquires an upper bound (the Breitenlohner-Freedman bound) determined by the value of the background cosmological constant.

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