Post-Newtonian parameters and of scalar-tensor gravity for a homogeneous gravitating sphere
We calculate the parameters and in the parametrized post-Newtonian (PPN) formalism for scalar-tensor gravity (STG) with an arbitrary potential, under the assumption that the source matter is given by a non-rotating sphere of constant density, pressure and internal energy. For our calculation we write the STG field equations in a form which is manifestly invariant under conformal transformations of the metric and redefinitions of the scalar field. This easily shows that also the obtained PPN parameters are invariant under such transformations. Our result is consistent with the expectation that STG is a fully conservative theory, i.e., only and differ from their general relativity values , which indicates the absence of preferred frame and preferred location effects. We find that the values of the PPN parameters depend on both the radius of the gravitating mass source and the distance between the source and the observer. Most interestingly, we find that also at large distances from the source does not approach , but receives corrections due to a modified gravitational self-energy of the source. Finally, we compare our result to a number of measurements of and in the Solar System. We find that in particular measurements of improve the previously obtained bounds on the theory parameters, due to the aforementioned long-distance corrections.
General relativity (GR) is currently the most established theory of gravitation. It correctly describes a number of observations, such as planetary orbits in the Solar System, the motion of masses in the Earth’s gravitational field BETA.Will:2014kxa, the recently discovered gravitational waves BETA.Ligo or the CDM model in cosmology BETA.Planck. However successful on these scales, GR itself does not provide sufficient answers to fundamental open questions such as the reason for the accelerated expansion of the universe, the phase of inflation or the nature of dark matter. Further tension arises from the fact that so far no attempt to extend GR to a full quantum theory has succeeded.
GR is expected to be challenged by different upcoming experiments on ground and in space, such as high precision clocks BETA.cacciapuoti2011atomic and atom interferometers in Earth orbits, pulsar timing experiments BETA.Pulsar and direct observations of black hole shadows BETA.Goddi; BETA.Broderick. This plethora of existing and expected experimental data, together with the tension with cosmological observations, motivates studying alternative theories of gravitation BETA.Nojiri. In particular, the upcoming experiments are expected to give more stringent constraints on the parameter spaces of such theories or even find violations of GR’s predictions.
One class of alternative theories are scalar-tensor theories of gravity (STG) - an extension to GR that contains a scalar degree of freedom in addition to the metric tensor. The detection of the Higgs proved that scalar particles exist in nature BETA.higgs and scalar fields are a popular explanation for inflation BETA.inflation.guth and dark energy BETA.Quintessence.and.the.Rest.of.the.World. Further, effective scalar fields can arise, e.g., from compactified extra dimensions BETA.compactified.extra.dimensions or string theory BETA.Damour.
While the motivation for such alternative theories of gravitation is often related to cosmology, of course any such theory must also pass Solar System tests. The most prominent class of such tests is based on the post-Newtonian limit of the theory under consideration, which is usually discussed in the parametrized post-Newtonian (PPN) framework BETA.will.book; BETA.Will:2014kxa. It allows characterizing theories of gravitation in the weak field limit in terms of a number of parameters, that can be calculated from the field equations of the theory, and will, in general, deviate from the parameters predicted by general relativity. These parameters can be constrained using observational data and experiments BETA.Fomalont:2009zg; BETA.Bertotti:2003rm; BETA.Hofmann:2010; BETA.Verma:2013ata; BETA.Devi:2011zz. In this work we are interested in the parameters and only, as these are the only parameters that may differ in fully conservative gravity theories, to which also STG belongs BETA.will.book.
The most thoroughly studied standard example of a scalar-tensor theory is Brans-Dicke theory BETA.Brans-Dicke.1961, which contains a massless scalar field, whose non-minimal coupling to gravity is determined by a single parameter . This theory predicts the PPN Parameter , in contrast to in GR. Both theories predict . Adding a scalar potential gives the scalar field a mass, which means that its linearized field equation assumes the form of a Klein-Gordon equation, which is solved by a Yukawa potential in the case of a point-like source. In this massive scalar field case, the PPN parameter becomes a function of the radial coordinate BETA.Olmo1; *BETA.Olmo2; BETA.Perivolaropoulos.
Scalar-tensor theories can be expressed in different but equivalent conformal frames. This means that the form of the general scalar-tensor action is invariant under conformal transformations of the metric, which depend on the value of the scalar field. There are two such frames that are most often considered: In the Jordan frame, test particles move along geodesics of the frame metric while in the Einstein frame, the scalar field is minimally coupled to curvature. The PPN parameters and for scalar-tensor theories with a non-constant coupling have been calculated in the Jordan BETA.HohmannPPN2013; *BETA.HohmannPPN2013E and in the Einstein frame BETA.SchaererPPN2014. These works consider a spacetime consisting of a point source surrounded by vacuum. As will be elucidated below, this assumption leads to problems when it comes to the definition and calculation of the PPN parameter .
Applying conformal transformations and scalar field redefinitions allows to transform STG actions, field equations and observable quantities between different frames. It is important to note that these different frames are physically equivalent, as they yield the same observable quantities BETA.Postma:2014vaa; BETA.Flanagan. Hence, STG actions which differ only by conformal transformations and field redefinitions should be regarded not as different theories, but as different descriptions of the same underlying theory. This observation motivates the definition of quantities which are invariant under the aforementioned transformations, and to express observable quantities such as the PPN parameters or the slow roll parameters characteristic for models of inflation fully in terms of these invariants BETA.JarvInvariants2015; BETA.KuuskInvariantsMSTG2016; BETA.Jarv:2016sow; BETA.Karam:2017zno.
The PPN parameters and were calculated for a point source BETA.HohmannPPN2013; *BETA.HohmannPPN2013E; BETA.SchaererPPN2014, and later expressed in terms of invariants BETA.JarvInvariants2015; BETA.KuuskInvariantsMSTG2016. However, the assumption of a point source leads to a number of conceptual problems. The most important of these problems is the fact that, in terms of post-Newtonian potentials, the Newtonian gravitational potential becomes infinite at the location of the source, so that its gravitational self-energy diverges. It is therefore impossible to account for possible observable effects caused by a modified gravitational self-energy of the source in a theory that differs from GR. We therefore conclude that the assumption of a point source is not appropriate for a full application of the PPN formalism to STG. This has been realized earlier in the particular case of STG with screening mechanisms BETA.SchaererPPN2014; BETA.Zhang:2016njn.
The goal of this article is to improve on the previously obtained results for the PPN parameters and for a general class of scalar-tensor theories, in which the divergent gravitational self-energy has been neglected. Instead of a point mass source, the gravitating mass source we consider in this article is given by a sphere with homogeneous density, pressure and internal energy that is surrounded by vacuum. In this case the gravitational self-energy remains finite, and can therefore be taken into account. During our calculation we do not fix a particular frame, but instead make use of the formalism of invariants mentioned above already from the beginning in order to calculate the effective gravitational constant as well as the PPN parameters and .
The article is structured as follows. In Sec. II we discuss the scalar-tensor theory action, the field equations and the invariants. The perturbative expansion of relevant terms is outlined in Sec. III and the expanded field equations are provided in Sec. IV. Next, in Sec. V, these are solved explicitly for a non-rotating homogeneous sphere and the PPN parameters are derived. Sec. VI applies our results to observations. Finally, we conclude with a discussion and outlook in Sec. VII. The main part of our article is supplemented by Appendix A, in which we list the coefficients appearing in the post-Newtonian field equations and their solutions.
We start our discussion with a brief review of the class of scalar-tensor tensor theories we consider. The most general form of the action, in a general frame, is displayed in section II.1. We then show the metric and scalar field equations derived from this action in section II.2. Finally, we provide the definition of the relevant invariant quantities, and express the field equations in terms of these, in section II.3.
We consider the class of scalar-tensor gravity theories with a single scalar field besides the metric tensor , and no derivative couplings. Its action in a general conformal frame is given by BETA.Flanagan
Any particular theory in this class is determined by a choice of the four free functions and , each of which depends on the scalar field . The function determines the kinetic energy part of the action. The scalar potential is given by ; a non-vanishing potential may be used to model inflation, a cosmological constant or give a mass to the scalar field. The last part is the matter part of the action. The matter fields, which we collectively denote by , couple to the so-called Jordan frame metric . It is conformally related to the general frame metric . The latter is used to raise and lower indices and determines the spacetime geometry in terms of its Christoffel symbols, Riemann tensor and further derived quantities. In general, the scalar field is non-minimally coupled to curvature. This coupling is determined by the function .
There are different common choices of the conformal frame; see BETA.JarvInvariants2015 for an overview. In the Jordan frame, one has and the matter fields couple directly to the metric . By a redefinition of the scalar field one may further set . Typically, one considers the coupling function . This particular choice of the parametrization is also known as Brans-Dicke-Bergmann-Wagoner parametrization.
Another possible choice for the conformal frame is the Einstein frame, in which the field couples minimally to curvature, . However, in this case the matter fields in general do not couple to the frame metric directly, but through a non-vanishing coupling function . In this case one may also choose the canonical parametrization .
We call the scalar field minimally coupled if the Jordan and Einstein frames coincide, i.e., if one can achieve and simultaneously through a conformal transformation of the metric.
ii.2 Field equations
The metric field equations are obtained by varying the action (1) with respect to the metric. Written in the trace-reversed form they are
where we use the d’Alembertian and the notation . Taking the variation with respect to the scalar field gives the scalar field equation
The function introduced on the left hand side is defined by
Note that these equations simplify significantly in the Einstein frame and . We will make use of this fact in the following, when we express the field equations in terms of invariant quantities.
Further, note that the functions and should be chosen such that . A negative would lead to a positive kinetic term in the Einstein frame, causing a ghost scalar field that should be avoided.
Given a scalar-tensor theory in a particular frame, it can equivalently be expressed in a different frame by applying a Weyl transformation of the metric tensor and a reparametrization of the scalar field
We defined in (4) since it transforms as a tensor under scalar field redefinition and is invariant under Weyl transformation,
In order to have a frame independent description, we want to express everything in terms of invariants, i.e., quantities that are invariant under the transformations given above. The matter coupling and the scalar potential can be written in an invariant form by introducing the two invariants BETA.JarvInvariants2015
Given the action in a general frame, we can define the invariant Einstein and Jordan frame metrics by
which are related by
Note that if the action is already given in the Einstein frame, the metric coincides with the Einstein frame metric defined above, , and the same holds for the Jordan frame. We define the Einstein frame metric (8a) as it significantly simplifies the field equations. The metric field equations reduce to
is the trace-reversed energy-momentum tensor in the Einstein frame. It is invariant under conformal transformations and field redefinitions, since also the left hand side of the field equations (10) is invariant. Note that we use the invariant Einstein metric for taking the trace and moving indices here, in order to retain the invariance of this tensor. For later use, we also define the invariant Jordan frame energy-momentum tensor
Similarly to the metric field equations, we obtain the scalar field equation (3)
These are the field equations we will be working with. In order to solve them in a post-Newtonian approximation, we will perform a perturbative expansion of the dynamical fields around a flat background solution. This will be done in the following section.
Iii PPN formalism and expansion of terms
In the preceding section we have expressed the field equations of scalar-tensor gravity completely in terms of invariant quantities. In order to solve these field equations in a post-Newtonian approximation, we make use of the well known PPN formalism. Since we are dealing with different invariant metrics and their corresponding conformal frames, we briefly review the relevant parts of the PPN formalism for this situation. We start by introducing velocity orders in section III.1. These are used to define the PPN expansions of the scalar field in section III.2, the invariant metrics in section III.3, the energy-momentum tensor in section III.4 and the Ricci tensor in section III.5.
iii.1 Slow-moving source matter and velocity orders
Starting point of the PPN formalism is the assumption of perfect fluid matter, for which the (Jordan frame) energy-stress tensor is given by
Since test particles fall on geodesics of the Jordan frame metric, we consider this as the ‘physical metric’ and we define mass density , pressure and specific internal energy in this frame. By we denote the four-velocity, normalized such that , where indices are raised and lowered using the Jordan frame metric .
We now consider the PPN framework to expand and solve the field equations up to the first post-Newtonian order. For this purpose we assume that the source matter is slow-moving, . We use this assumption to expand all dynamical quantities in velocity orders . Note that and each contribute at order , while contributes at . The velocity terms are, obviously, of order . We finally assume a quasi-static solution, where any time evolution is caused by the motion of the source matter. Hence, each time derivative increases the velocity order of a term by one.
iii.2 PPN expansion of the scalar field
We now expand the scalar field around its cosmological background value in terms of velocity orders,
where is of order and is of order . Other velocity orders either vanish due to conservation laws or are not relevant for the PPN calculation. Any function of the scalar field can then be expanded in a Taylor series as
For convenience, we denote the Taylor expansion coefficients, which are given by the values of the functions and their derivatives evaluated at the background value, in the form , , , , , and similarly for all functions of the scalar field.
iii.3 PPN expansion of the metric tensors
In the next step, we assume that the Jordan frame metric, which governs the geodesic motion of test masses, is asymptotically flat, and can be expanded around a Minkowski vacuum solution in suitably chosen Cartesian coordinates. The expansion of the Jordan frame metric components up to the first post-Newtonian order is then given by
It can be shown that these are all relevant and non-vanishing components. A similar expansion of the Einstein frame metric can be defined as
The ’s on the left sides are required in order to satisfy (9). The expansion coefficients in the two frames are then related by
as one easily checks. Conversely, one finds the inverse relations
iii.4 PPN expansion of the energy-momentum tensors
We now come to the PPN expansion of the energy-momentum tensors. Here we restrict ourselves to displaying the expansion of the invariant energy-momentum tensor in the Einstein frame, since this is the frame we will be using for solving the field equations. It is related to the invariant Jordan frame energy-momentum tensor by . Its PPN expansion follows from the standard PPN expansion of the energy-momentum tensor in the Jordan frame BETA.will.book and is given by
Its trace, taken using the Einstein frame metric, has the PPN expansion
Consequently, the trace-reversed energy-momentum tensor is given by
iii.5 Invariant Ricci tensor
Finally, we come to the PPN expansion of the Ricci tensor of the invariant Einstein metric. We will do this in a particular gauge, which is determined by the gauge conditions
which will simplify the calculation. In this gauge, the components of the Ricci tensor to the orders that will be required are given by
We now have expanded all dynamical quantities which appear in the field equations into velocity orders. By inserting these expansions into the field equations, we can perform a similar expansion of the field equations, and decompose them into different velocity orders. This will be done in the next section.
Iv Expanded field equations
We will now make use of the PPN expansions displayed in the previous section and insert them into the field equations. This will yield us a system of equations, which are expressed in terms of the metric and scalar field perturbations that we aim to solve for. We start with the zeroth order field equations in section IV.1, which are the equations for the Minkowski background, and will give us conditions on the invariant potential . We then proceed with the second order metric equation in section IV.2, the second order scalar equation in section IV.3, the third order metric equation in section IV.4, the fourth order metric equation in section IV.5 and finally the fourth order scalar equation in section IV.6.
iv.1 Zeroth order metric and scalar equations
At the zeroth velocity order, the metric equations (10) are given by
which is satisfied only for , and hence restricts the choice of the invariant potential . At the same velocity order, the scalar equation reads
and is solved only by , so that we obtain another restriction on the allowed potential . In the following, we will only consider theories in which these conditions on are satisfied.
iv.2 Second order metric and
At the second velocity order we find the -metric field equation
Inserting the expansion of the Ricci tensor shown in section III.5 and using and we solve for and find the Poisson equation
where we introduced the Newtonian gravitational constant
The -equations at the same order are given by
which similarly reduces to
Note that the diagonal components satisfy the same equation (29) as .
iv.3 Second order scalar field
The second order scalar field equation is given by
It is convenient to introduce the scalar field mass by
We assume that , since otherwise the scalar field would be a tachyon. Then, the second order scalar field equation takes the form of a screened Poisson equation,
We will see that can be interpreted as the mass of the scalar field, while is a measure for the non-minimal coupling of the scalar field at the linear level. We finally remark that is an invariant, while transforms as a tangent vector to the real line of scalar field values BETA.JarvInvariants2015.
iv.4 Third order metric
The third order metric equation reads
Thus we can solve for the third order metric perturbation and obtain another Poisson equation,
Note that the source terms on the right hand side of this equation are given by time derivatives of other metric components and moving source matter, and hence vanish for static solutions and non-moving sources.
iv.5 Fourth order metric
The fourth order metric field equation reads
Solving for the fourth order metric perturbation then yields
Also this equation has the form of a Poisson equation.
iv.6 Fourth order scalar field
Finally, for the scalar field we have the fourth order equation
Solving for the fourth order scalar perturbation then yields
This is again a screened Poisson equation, which contains the same mass parameter as the second order scalar field equation (36).
These are all necessary equations in order to determine the relevant perturbations of the invariant Einstein frame metric and the scalar field. We will solve them in the next section, under the assumption of a massive scalar field, , and a static, homogeneous, spherically symmetric source mass.
V Massive field and spherical source
In the previous section we derived the gravitational field equations up to the required post-Newtonian order. We will now solve these field equations for the special case of a homogeneous, non-rotating spherical mass distribution. This mass distribution, as well as the corresponding ansatz for the PPN metric perturbation and the PPN parameters, are defined in section V.1. We then solve the field equations by increasing order. The second order equations for the invariant Einstein frame metric and the scalar field are solved in sections V.2 and V.3, while the corresponding fourth order equations are solved in sections V.4 and V.5. From these solutions we read off the effective gravitational constant as well as the PPN parameters and in section V.6. A few limiting cases of this result are discussed in section V.7.
v.1 Ansatz for homogeneous, spherical mass source
In the following we consider a static sphere of radius with homogeneous rest mass density, pressure and specific internal energy, surrounded by vacuum. Its density , pressure and specific internal energy are then given by
where is the radial coordinate and we use isotropic spherical coordinates. We further assume that the mass source is non-rotating and at rest with respect to our chosen coordinate system, so that the velocity vanishes.
For the metric perturbation corresponding to this matter distribution, which is likewise spherically symmetric, we now use the ansatz
Here denote the standard PPN potentials, which satisfy the Poisson equations BETA.will.book
For the homogeneous, spherically symmetric mass source we consider they are given by