Dynamical Gravitational Coupling

Dynamical Gravitational Coupling as a
Modified Theory of General Relativity

Felix Finster  and  Christian Röken

April 2016
Fakultät für Mathematik
Universität Regensburg
D-93040 Regensburg
Germany
finster@ur.de, Christian.Roeken@mathematik.ur.de
Abstract.

A modified theory of general relativity is proposed, where the gravitational constant is replaced by a dynamical variable in space-time. The dynamics of the gravitational coupling is described by a family of parametrized null geodesics, implying that the gravitational coupling at a space-time point is determined by solving transport equations along all null geodesics through this point.

General relativity with dynamical gravitational coupling (DGC) is introduced. We motivate DGC from general considerations and explain how it arises in the context of causal fermion systems. The underlying physical idea is that the gravitational coupling is determined by microscopic structures on the Planck scale which propagate with the speed of light.

In order to clarify the mathematical structure, we analyze the conformal behavior and prove local existence and uniqueness of the time evolution. The differences to Einstein’s theory are worked out in the examples of the Friedmann-Robertson-Walker model and the spherically symmetric collapse of a shell of matter. Potential implications for the problem of dark matter and for inflation are discussed. It is shown that the effects in the solar system are too small for being observable in present-day experiments.

1. Introduction

Einstein’s theory of general relativity is the basis of modern astrophysics and cosmology. It was confirmed in various experiments (like the perihelion shift, the bending of light and the red-shift effect for atomic clocks; see for example [29]), and it led to spectacular predictions like black holes, the big bang and gravitational waves (see [19, 28, 25]). However, according to present cosmological models there are major discrepancies between observations and Einstein’s theory, usually associated to dark matter and dark energy (see for example [27, 21]). Instead of trying to fix this problem by introducing new and yet unobserved matter fields, one can also think about resolving it by modifying the Einstein equations. One idea, which goes back to Paul Dirac (see [7] and [8, Section 5]) is that the gravitational constant should be replaced by a dynamical variable in space-time. This idea has been implemented in various modified theories of gravity, notably in the scalar-tensor theories [3, 2] and vector-tensor theories [30, 17]. In these theories, the dynamics of the gravitational coupling is described by additional fields. The disadvantage of this procedure is that these additional fields are introduced ad hoc, and that the description of the dynamics of these fields involves additional free parameters.

Here we propose a different type of modification of the Einstein equations in which the gravitational coupling becomes a dynamical variable. However, in contrast to the above-mentioned scalar-tensor and vector-tensor theories, our model of dynamical gravitational coupling (DGC) does not involve any free parameters. In particular, DGC is not a “deformation” of general relativity, but it is a different theory. It does not involve a parameter such that setting this parameter to zero gives back Einstein’s theory. Moreover, it is not formulated ad hoc, but instead it follows naturally from concepts behind the theory of causal fermion systems, being a recent proposal for a unified physical theory [10].

Generally speaking, the DGC model is based on the physical concept that on a microscopic length scale, space-time can no longer be described by a Lorentzian manifold, but has a different non-trivial structure. This microstructure should break the Lorentz invariance. Considering macroscopic systems, however, the microstructure should not be visible. Instead, it should be possible to describe the effective macroscopic behavior by physical equations (like the Einstein equations, the Dirac equation or the Maxwell equations) which are Lorentz invariant and are formulated geometrically on a Lorentzian manifold.

This physical picture is analogous to that of a crystal in solid state physics: On the atomic scale, the crystal is composed of atoms, which are in a regular configuration which distinguishes certain spatial directions. On scales much larger than the atomic distance, however, the crystal can often be modelled by a homogeneous and isotropic material, and it can be described effectively by macroscopic quantities like temperature, pressure, densities, etc.

The effective macroscopic physical equations should be thought of as arising by a suitable “averaging process” or “homogenization process” from the physical equations which hold on the microscopic scale (there is no consensus on how these microscopic physical equations should look like, and at this stage we do not need to be specific). In this averaging process, most information on the microscopic structure of space-time gets lost. However, for what follows, it is important to take the point of view that one parameter of the macroscopic theory should be a remnant of the microscopic theory: the gravitational coupling constant. The simplest way to take this point of view is to identify the Planck length with the length scale on which the microstructure becomes effective. More generally, one can think of the Planck length as a quantity encoded in the microstructure, but the length scale of the microstructure could for example be even much smaller than the Planck length.

The considerations so far were very general and are widely accepted in the physics community. A conclusion of these considerations is that, if the microscopic structure of space-time has a dynamical behavior, then the gravitational constant should also be dynamical. The crucial question is what the dynamics of the microstructure is and how the effective behavior of the gravitational coupling can be described. Here we make the assumption that the microstructure should have the same dynamics as an ultraviolet regularization to a distributional solution of a hyperbolic differential equation (like the wave or Dirac equation). Since the ultraviolet regularization modifies the solution only on a microscopic scale, its dynamics is governed by the high-frequency behavior of the hyperbolic equation and can thus be described by analyzing the propagation along characteristics. In simple and more physical terms, we assume that the microstructure propagates with the speed of light. As will be explained in more detail in Section 3.1, making this concept precise leads quite naturally to DGC.

We remark that the above assumption that the microstructure should have the dynamics of an ultraviolet regularization has a deeper justification in the theory of causal fermion systems. In this theory, all space-time structures are encoded in an ensemble of fermionic wave functions (the “physical wave functions”). The physical equations are formulated directly in terms of these wave functions via the so-called causal action principle. In a certain limiting case (the “continuum limit”), the Euler-Lagrange equations of the causal action give rise to the Einstein equations (see [10, Section 4.9]), where the gravitational coupling is related to ultraviolet properties of the physical wave functions. This will be explained in more detail in Section 3.2.

Since DGC should be of interest independent of the connection to causal fermion systems, we here argue physically and derive the structure of DGC from general considerations. This also has the benefit that the paper should be easily accessible to a broader readership. Only the specific form of our DGC model as derived in Section 3.2 relies on results from the theory of causal fermion systems. Consequently, since causal fermion systems are used only in Section 3.2, we decided not to give a general introduction to causal fermion systems and the causal action principle, but refer the interested reader to [10] or the survey papers [12, 14].

At present, the physical consequences and implications of DGC are difficult to estimate because the analysis of physically realistic scenarios will take considerable time and effort. Nevertheless, it is a main purpose of this paper to derive and discuss a few physical effects. Our findings are summarized as follows: In a Friedmann-Robertson-Walker space-time, the DGC function is proportional to the square of the scale function (see Section 5). Therefore, DGC suggests that in the early universe, the gravitational attraction should have been weaker than measured at present. This bears some resemblance to inflationary models which also reduce the strength of gravity in the early universe. Moreover, our analysis of the spherically symmetric gravitational collapse (as studied in Section 6 for a shell of matter) suggests that for a heavy collapsing star, the gravitational coupling could in certain situations be larger than at present. In such situations, the star would generate a stronger gravitational field than in Einstein’s theory. If this effect were sufficiently large, it could resolve the problem of dark matter, because already the known matter would generate a sufficiently strong gravitational field to give agreement with physical observations. In the solar system, the effects of DGC seem too weak for being detectable in present-day experiments.

The paper is organized as follows. In Section 2, the DGC model is introduced, and it is explained how to describe a constant gravitational coupling in Minkowski space. Section 3 motivates the model from general considerations and then gives a derivation in the context of causal fermion systems. Section 4 is devoted to the mathematical structure of DGC. Namely, the connection between conservation laws and conformal Killing symmetries as well as the behavior under conformal transformations is worked out (Sections 4.1 and 4.2). Moreover, we prove that the Einstein equations with DGC coupled to usual matter fields (like a perfect fluid, Maxwell or Dirac field, etc.) have a well-defined local time evolution (Section 4.3). In Section 5, the effect of DGC is analyzed in a Friedmann-Robertson-Walker space-time. Section 6 is devoted to the analysis of DGC in the gravitational collapse of a star in the simplified model of a spherically symmetric shell of matter. In Section 7, linearized gravity and the Newtonian limit are worked out, and potential effects in the solar system are discussed. Finally, the appendices provide some additional background material: Appendix A derives formulas for the volume and curvature of a two-surface contained in the light cone. Appendix B introduces the so-called regularized Hadamard expansion which puts the physical arguments of Section 3 on a concise mathematical basis. Finally, in Appendix C, the form of the so-called DGC tensor is derived.

2. General Relativity with Dynamical Gravitational Coupling

For conceptual clarity, we now define general relativity with dynamical coupling. The physical motivation, derivation and discussion will be given later in Section 3.

2.1. The Einstein Equations with Dynamical Gravitational Coupling


Let  be a Lorentzian manifold. We also denote the Lorentzian metric at a point  by  (where ). A parametrized geodesic is a smooth mapping  from an open interval  to  which satisfies the geodesic equation

(2.1)

A parametrized geodesic is null if it satisfies the relations

(2.2)

(it suffices to assume that these relations are satisfied for some , because then the geodesic equation implies that they hold for all ). A null geodesic can be reparametrized in several ways. One obvious freedom is to change the parameter by an additive constant,

(2.3)

Moreover, one can change the orientation of the geodesic, i.e.

(2.4)

If  is time-orientable, one could distinguish an orientation and restrict attention for example to future-directed null geodesics. In order to keep our setting as general as possible, we prefer not to assume a time orientation at this point. Finally, one may reparametrize by a non-negative multiplicative constant , i.e.

(2.5)

This freedom scales the velocity vector by , preserving both (2.1) and (2.2).

In order to describe the dynamical behavior of the gravitational constant, we want to distinguish the multiplicative reparametrization of all null geodesics. Clearly, it suffices to consider maximal geodesics (i.e. geodesics which are inextendable), because all other null geodesics can be obtained by restriction to a smaller parameter domain.

Definition 2.1.

We introduce a set of parametrized null geodesics

(2.6)

with the following properties:

  • For every , reparametrizing by an additive constant (2.3) or changing the orientation (2.4) again gives a geodesic in .

  • For every maximal null geodesic  in , there is exactly one  such that the multiplicative reparametrization (2.5) gives a geodesic .

Next, for any space-time point  we introduce the set

(2.7)

Clearly, this set is a subset of the light cone centered at , i.e.

Moreover, for every non-zero null vector , the ray  intersects  in exactly two points  with . Assuming that  depends smoothly on the null vector , the set  is a smooth two-dimensional surface consisting of two connected components, each of which is topologically a sphere (see Figure 1).

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Figure 1. The two-surface .

The Lorentzian metric on  induces a Riemannian metric on  (for details see Appendix A). We denote the corresponding volume measure on  by . Furthermore, the Gaussian curvature (being one half scalar curvature) of  is denoted by . We introduce the DGC function  at the space-time point  by

(2.8)

The Einstein equations with dynamical gravitational coupling and cosmological constant  are defined by

(2.9)

where  is the Ricci tensor of , is scalar curvature, is the cosmological constant, and  is the usual energy-momentum tensor of matter. The tensor , referred to as the DGC tensor, is a correction term needed in order to get a mathematically consistent system of partial differential equations. In order not to distract from the main ideas, the detailed form of  will be stated and derived in Appendix C (see (C.4)). At this point, we prefer to make a few remarks. First, the DGC tensor is trace-free, so that taking the trace of (2.9) gives the equation

(2.10)

Second, the DGC tensor is formed of integrals of expressions involving  and derivatives of  along null geodesics. In particular, if the energy-momentum tensor vanishes, then  is also zero. Therefore, in the vacuum, the above equations reduce to the usual vacuum Einstein equations . However, if matter is present, the space-time dependence of the DGC function  as described by the dynamics of the null geodesics in (2.8) comes into play and modifies how matter generates curvature.

In this paper, we will restrict attention to the situation that  varies only on a large scale. Then the DGC tensor should be very small, implying that gravity with DGC should have all the properties of usual gravity, except that the strength of the gravitational coupling depends on the space-time point . With this in mind, our main objective is to understand how  depends on .

2.2. A Simplified Model

Since the form of the dynamical coupling function (2.8) is rather involved from the computational point of view, we now propose a simpler model which is easier to analyze but still seems to capture the main features of (2.8) and (2.9).

In order to derive the simplified model, let us assume that the Gaussian curvature is a non-zero constant up to an error which only needs to be taken into account linearly, i.e.

(2.11)

Then, writing (2.8) as

we may insert any power of the Gaussian curvature into numerator and denominator without changing the values of the integral, i.e. for any ,

(2.12)

Namely, substituting (2.11) and expanding, we obtain

which is obviously independent of . Choosing , we may carry out the integral in the denominator in (2.12) using the Gauß-Bonnet theorem. Since  is topologically the disjoint union of two spheres, we obtain

We conclude that

This expression is much simpler than (2.8) because it becomes unnecessary to compute the Gaussian curvature. Thus for the simplified model we replace  in (2.9) by the DGC function  defined by

(2.13)

(clearly, we also replace  by  in the formula for the DGC tensor (C.4)). Another advantage of this ansatz is that this expression is well-defined even if  is not smooth; in fact it suffices that  is locally an -graph over the sphere (see Appendix A).

Using the Hölder inequality, one immediately sees that under general assumptions, the DGC function of the simplified model (2.13) is stronger than that of the full model (2.8):

Lemma 2.2.

Assume that the Gaussian curvature  is everywhere non-negative on . Then

Proof.

Assume that . Then the Hölder inequality implies that

It follows that

Applying (2.8) and (2.13) gives the result. ∎

2.3. Example: Minkowski Space

Let us illustrate the above definitions in a simple example. We let  be Minkowski space. Then the null geodesics are straight lines of the form

Clearly, there are many ways to choose . A simple method is to impose that  for a given parameter . Thus we set

(2.14)

By direct computation, one finds that

(2.15)

This set consists of two -spheres of radius . Hence the scalar curvature and the the volume of  are the constants

(2.16)

Thus the dynamical coupling constant (2.8) is also constant,

(2.17)

This shows that  should be identified with a multiple of the Planck length. The simplified model (2.13) also gives , giving agreement with (2.17). As a consequence, the Einstein equations with DGC (2.9) reduce to the usual Einstein equations with . Hence the parameter , which can be thought of as the Planck length, introduces a length scale which determines the gravitational coupling constant. We point out that the choice of  clearly breaks Lorentz invariance and distinguishes a specific reference frame. Nevertheless, the Einstein equations (2.9) are Lorentz invariant (i.e. are tensor equations). In other words, the violation of the Lorentz invariance by  is not visible in the equations of gravity.

By (2.15), we realized the situation that the reference frame distinguished by the regularization coincides with the rest frame of the considered physical system. Later, we will also consider the situation that the rest frame of the physical system is moving with constant velocity relative to the reference frame distinguished by the regularization. In order to describe this situation, we let  be a future-directed timelike unit vector,

(2.18)

We modify (2.14) to

(2.19)

Consequently, (2.15) becomes

(2.20)

Choosing , we clearly get back (2.14) and (2.15). Moreover, by performing a Lorentz boost, it is obvious that the right equation in (2.16) as well as (2.17) remain valid for general .

3. Derivation of Dynamical Gravitational Coupling

We now explain how to come up with the Einstein equations with DGC. To this end, we begin with general considerations (Section 3.1) and then give a derivation in the context of causal fermion systems (Section 3.2).

3.1. General Considerations Leading to Dynamical Gravitational Coupling

We always work in natural units . Then the masses of the elementary particles give a length scale, the Compton scale (dividing by , we also have a corresponding time scale). As long as gravity does not come into play, the Compton scale determines the length scale in all physical processes. Since time measurements also involve physical processes (for example in an atomic clock), the Compton scale also enters the Lorentzian metric. With this in mind, it is obvious and inevitable that the masses of the elementary particles are constant in space-time (only mass ratios could change, but this will not be considered here). Also, when we talk of length or time scales, these are always to be understood relative to the Compton scale.

In the above units, the gravitational coupling has the dimension of length squared. The resulting length scale is the Planck length ,

It is generally believed that on length scales as tiny as the Planck length, the conventional laws of physics should no longer hold, and yet unknown physical effects should come into play. Many physicists believe that on the Planck scale, the concept of the usual space-time continuum breaks down, giving rise to yet unknown structures of a “quantum space-time” or “quantum geometry.” This idea is also commonly used in quantum field theory, where the Planck length gives a natural length scale for the ultraviolet regularization, thereby preventing the ultraviolet divergences of quantum field theory. In what follows, we use these general concepts, but without specifying in detail what the microscopic structure of space-time on the Planck scale should be.

A general idea behind DGC is that the microscopic space-time structure should have a dynamical behavior, implying that the Planck length and consequently also the gravitational constant should depend on the space-time point. In order to describe the dynamics, we impose that the microstructure propagates with the speed of light. This assumption can be motivated from the above observation that the Planck length gives rise to an ultraviolet regularization and thus affects the high-frequency behavior of the quantum fields. The high-frequency component of a relativistic field, however, behaves typically like an ultrarelativistic wave and thus propagates with the speed of light. A more detailed justification for working with light speed propagation will be given in Section 3.2 using concepts which arise in the context of causal fermion systems.

The behavior of length scales propagating with the speed of light can be described most conveniently by families of null geodesics. In order to explain the method in the simplest possible setting, we consider a two-dimensional oriented and time-oriented space-time and let  and  be two space-time points which can be joined by a null geodesic  (see Figure 2).

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Figure 2. Frequency shift as described by families of null geodesics.

Assume that at the points  and , we have chosen local reference frames (i.e. Gaussian coordinate systems) denoted by  and . Next, we assume that at the point , we have a distinguished time scale . This time scale can be described by choosing a curve , , moving along the -axis with velocity  given by

(3.1)

In order to obtain a corresponding scale at the point , we construct a family of null geodesics  starting at , i.e.

Assume that the null geodesic  intersects the -axis at a point which we denote by . Then the resulting curve  determines a time scale  at , as in analogy to (3.1) it satisfies the relations

(3.2)

In this way, the dynamics of the time scale is described by analyzing the behavior of “neighboring null geodesics.” In the example of Figure 2, where neighboring null geodesics are moving further apart, the resulting time scale  at  will be larger than the time scale  at .

Before generalizing this concept to higher space-time dimensions, it is convenient to formulate the connection between  and  in terms of the geodesic distance function. To this end, we assume for simplicity that  and  are contained in a geodesically convex subset  of space-time, meaning that any pair of points  can be joined by a unique geodesic in  (see for example [1, Definition 1.3.2]). We let  be the length of this geodesic squared, with the sign convention that  is positive in timelike directions and negative in spacelike directions. The fact that the points  and  both lie on the null geodesic  implies that

Differentiating at , we obtain

In order to simplify this formula, it is convenient to parametrize the null geodesic  such that

Then the gradients of  are multiples of the velocity vector of , namely

(for the derivation see Lemma B.1 in Appendix B). We thus obtain

(3.3)

Using that the vector  points in the time direction of the reference frame , and similarly that  points in the time direction of the reference frame , we can compute the inner products in (3.3) with the help of (3.1) and (3.2). This gives

(where the index zero denotes the time component in the respective reference frame). In a general affine parametrization of , this identity can be written as

(where the index zero now refers to the reference frame at ). For convenience, we parametrize the null geodesic such that the constant equals one, i.e.

(3.4)

We conclude that the regularization length changes like one over the time component of the velocity vector of the null geodesic .

In Figure 2, we considered a null geodesic moving to the left. Clearly, one should also consider null geodesics moving to the right. Therefore, one should not think of  as a scalar parameter, but as a function depending on the choice of the spatial direction. Likewise, adapting the above consideration to a time-oriented four-dimensional space-time , the time scale  depends on the choice of a spatial direction, and its dynamics is described similar to (3.4) in terms of the velocity vector of a null geodesic. Taking into account all spatial directions systematically leads us to introducing a set of parametrized null geodesics  according to Definition 2.1. Then the time scale  in a reference frame  in a direction  at the space-time point  is obtained as follows: We choose a future-directed null geodesic in  with  such that the spatial component of  is a positive multiple of . Then, following (3.4), the time scale  is given by

(3.5)

So far, we considered time-oriented space-times. In order to get rid of this assumption, it is preferable to slightly modify the point of view. Namely, instead of associating  to a spatial direction , we shall work with a non-zero null vector . Then, given such a null vector, we choose a null geodesic in  with  such that  is a positive multiple of . In the time-oriented case, this procedure is equivalent to that explained before (3.5) because for every  there is a unique time component  such that . However, working with the null vector  has the advantage that it is no longer necessary to distinguish a time direction. Indeed, the invariance of  under changes of orientation (2.4) ensures that  remains unchanged if the sign of  is flipped.

Since the gravitational coupling constant  is a scalar quantity of dimension length squared, it must have the scaling . Let us think how such a quantity can be obtained in our setting. Clearly, in order not to distinguish a preferred reference frame, should be formed geometrically out of the two-surface  in (2.7). On this surface, one has an induced Riemannian metric  (for details see Appendix A). Moreover, it is shown in Appendix A that the only curvature quantity on  is the Gaussian curvature  (see Lemma A.1). Rescaling the velocity of the null geodesics according to , the volume measure and the Gaussian curvature scales like  and . In view of (3.5), this means that the volume of  has the scaling dimension , whereas the Gaussian curvature has the scaling dimension . This shows that in particular that our formulas for the DGC functions (2.8) and (2.13) indeed have the correct scaling dimension (i.e. the correct dimension of length squared). But there are many other expressions with the correct scaling dimension. For example, one can form powers of integrals of powers of scalar curvature

Using covariant derivatives of Gaussian curvature, one can also form expressions like

The most general expression with the correct length dimension is obtained by combining the above expressions by iteratively applying the operation  with . In order to determine the form of dynamical coupling beyond such scaling arguments, we will now use the detailed structure of the causal action principle used in the theory of causal fermion systems.

3.2. Derivation in the Context of Causal Fermion Systems

The theory of causal fermion systems (see [10] and the references therein) provides concise mathematical structures which should describe physical space-time on all length scales. On the macroscopic scale, these space-time structures reduce to the structures of a Lorentzian manifold. However, on the Planck scale, causal fermion systems allow for non-trivial microstructures which may be discrete or continuous. The physical equations on a causal fermion systems are formulated via a novel variational principle, referred to as the causal action principle. For what follows, we only need a few key features of causal fermion systems, which we now introduce. One feature is that all space-time structures (causal relations, geometric structures, bosonic potentials, etc.) are encoded in an ensemble of quantum mechanical wave functions defined the space-time points. On the macroscopic scale, these wave functions are solutions of the Dirac equation. The non-trivial microstructure is described by modifying the wave functions on the microscopic scale, such that the Dirac equation no longer needs to hold. In more technical terms, this is achieved by introducing an ultraviolet (UV) regularization into the so-called kernel of the fermionic projector . Denoting the length scale of the regularization by a parameter , the dynamical behavior of the resulting regularized kernel  is described by the so-called regularized Hadamard expansion as worked out in Appendix B. This dynamics is consistent with the general considerations in the previous sections and makes these considerations precise in the context of the Dirac equation.

Another feature of causal fermion systems of importance here is that the Euler-Lagrange equations of the causal action give rise to the Einstein equations in the so-called continuum limit, an approximation based on the assumption that the length scale  of the regularization should be much smaller than the length scale  of macroscopic physics (where by “macroscopic physics” we mean the length scales accessible to experiments in high-energy physics). The gravitational coupling is described by a length scale , which must be in the range

We next make the following crucial assumption:

The dynamics on the scale  is the same as that on the scale . (3.6)

This assumption is of course trivially satisfied if the length scales  and  coincide. However, if  defines another length scale which is much larger than , this assumption is questionable, as will be discussed in Remark 3.1 below. Making this assumption, the EL equations in the continuum limit give the equations (see [10, Section 4.9])

(3.7)

where  is a dimensionless constant (which in the case  is much smaller than one), and  is the time scale in (3.5). Since this time scale depends on the null direction, the equation (3.7) is to be evaluated weakly, meaning that we must take “averages” over the null directions. The natural way of doing so is to integrate both sides over . Moreover, rewriting the factor  geometrically gives a constant times . We thus obtain the equations

(3.8)

with a new constant . This procedure of bringing the equations into a geometrical form suffers from the shortcoming that the resulting equations are no longer consistent (this is obvious because dividing (3.8) by  and taking the divergence, the left side of the equation is non-zero, but the right side vanishes). One method for obtaining consistent equations is to bring the equations in variational form. Thus one would have to find a Lagrangian which is diffeomorphism invariant such that the dominant terms of the resulting Euler-Lagrange equations coincide with (3.8). This procedure would give a systematic method of deriving all the correction terms needed to obtain consistent equations.

This variational approach also seems most convincing conceptually in view of the fact that the equations (3.8) are derived in [10] from a variational principle (the causal actinon principle), suggesting that the effective equations should again be of variational form (for more details see [10, Section 4.7]). Also, a variational form would be desirable in order to make Noether’s theorem applicable (the connection between symmetries and conservation laws was established for the causal action principle in [13]). However, bringing the Einstein equations with DGC in variational form seems to make it necessary to also consider the family of geodesics  as variable quantities by formulating an action for the set . This goes beyond the scope of the present paper. For this reason, we simply add correction terms to (3.8) which vanish if  is constant and ensure that the resulting system of partial differential equations is mathematically consistent (for details see Appendix C).

Clearly, the above derivation involved a few assumptions, which we now discuss in detail.

Remark 3.1.

(critical discussion of assumption (3.6)) The above derivation depends crucially on the assumption (3.6). Therefore, we now discuss this assumption in detail. The consideration of Figure 2 (and its higher-dimensional analog) shows that the propagation along null geodesics describes the dynamics of a length scale. Since this consideration is purely geometric, this dynamics is universal and applies to all quantities which move along null geodesics. However, the consideration no longer applies to quantities which propagate slower than the speed of light. For example, mass parameters are constant in space-time, which is no contradiction to our consideration because the mass changes the propagation speed. The parameter  arises in [10] in order to describe two different regularization effects: general surface states and shear states. While the analogy between the curvature of the hypersurface describing the general surface states and the curvature of the mass hyperbola suggests that  should be regarded as a very large mass parameter, it is not clear whether it is really constant in space-time. On the contrary, it seems natural to assume that  is of the same order of magnitude as , in which case they should also have the same dynamical behavior. For the shear states, there is no similarity to a mass, again suggesting a massless dynamics and propagation with the speed of light.

To summarize, the theory of causal fermion systems strongly suggests DGC. However, we would like to point out that DGC is not a compelling consequence of the causal action principle. Assuming a massless dynamics for all regularization effects is an interpolation which could only be justified by a more detailed knowledge or a better understanding of the microscopic structure of space-time.  

Remark 3.2.

(alternative forms of the Einstein equations with DGC) Rewriting the EL equations of the causal action in the continuum limit (3.8) involves some arbitrariness, as we now discuss:

  • In order to keep the setting as simple as possible, we introduced  as a set of null geodesics. This description might be too simple, as we now explain. In the context of causal fermion systems, the dynamics of the microstructure arises by considering the high-frequency behavior of solutions of the Dirac equation. Considering Dirac particles with an electric charge, this high-frequency limit is described by the motion in the presence of the electromagnetic field,

    (3.9)

    This suggests that, if electromagnetic fields are present, the geodesic equation (2.1) should be replaced by (3.9). This might have a relevant effect for example for a star in a strong magnetic field. Similarly, the geodesic equation should also be modified if other fields (like strong or weak gauge fields) are present.

    The question if and how precisely the geodesic equation should be modified depends crucially on the question which elementary particles determine the gravitational coupling function. In [10, Chapter 4] the gravitational constant is encoded in the neutrino sector, giving an indication that the high-frequency limit of the neutrino equation should be considered. Consequently, only the weak gauge fields should be taken into account.

  • Another potential modification is that also the cosmological constant  might have a dynamical behavior. In the derivation in [10], the parameter  in (3.8) can be chosen arbitrarily at every space-time point. In the Einstein equations  without DGC, taking the divergence gives the equation , implying that  must be constant in space-time. But this argument no longer applies with DGC. Therefore, it is conceivable that one should replace  in (3.8) by a cosmological function  (clearly, this would also make it necessary to modify the DGC tensor).

    We remark that in principle, the dynamics of  could be derived from the causal action by analyzing the corresponding EL equations to degree three on the light cone. But this analysis would involve extensive computations which have not yet been carried out.

  • The critical reader may wonder why we evaluated (3.7) weakly by integrating over the null directions. Should (3.7) not be satisfied for all null directions? We took the point of view that (3.7) should be satisfied in the strongest possible sense. Therefore, if a pointwise evaluation in every null direction is impossible, we must take suitable “averages” over the null directions. However, it is conceivable that the EL equations (3.7) can indeed be satisfied pointwise in every null direction if additional perturbations of the fermionic projector are taken into account (this question could only be answered by a detailed and extensive analysis which has not yet been carried out). If these additional perturbations were not dynamical (similar to the microlocal chiral transformation in [10]), then this would have no effect on our results, except that possibly one might have to modify the “averaging procedure” in (3.8). However, if these additional perturbations were dynamical (i.e. if they came with additional hyperbolic field equations), this could change the picture completely. In this case, one could hope that gravity could be described by a modification or extension of the Einstein equations with DGC (2.9).

Despite these potential extensions and modifications, it is fair to say that the Einstein equations (2.9) with a DGC function according to (2.8) or (2.13) are a suitable starting point for exploring the effects of DGC.  

4. The Mathematical Structure of Dynamical Gravitational Coupling

4.1. Conformal Killing Symmetries and Conservation Laws

In preparation, we recall the notion of a conformal Killing field and explain why this notion is useful for studying the dynamics of gravitational coupling. A vector field  is a Killing field if it satisfies the Killing equation

(4.1)

where the brackets denote symmetrization, i.e. . A Killing field describes a continuous symmetry of space-time. According to Noether’s theorem, continuous symmetries give rise to conservation laws. For a geodesic , the corresponding conserved quantity is simply the inner product , as is verified by the computation

(4.2)

If  is a null geodesic, this conservation law can be generalized to a so-called conformal Killing field, where (4.1) is weakened to the conformal Killing equation

(4.3)

Namely,

Taking the trace of (4.3), one finds that , implying that the conformal Killing equation can be written equivalently as

4.2. Behavior under Conformal Transformations

The DGC function has a simple behavior under conformal transformations, as we now explain. We consider a conformal transformation of the metric

(4.4)

with a strictly positive smooth function . We assume that the conformal transformation is trivial outside a space-time region , i.e.

Null geodesics are invariant under conformal transformations, but the affine parameter changes. In order to see this in detail, let  be a null geodesic corresponding to the metric , i.e.

Under the conformal change, the Christoffel symbols transform as follows,

Thus the equation for a null geodesic  becomes

(4.5)

(where in the last step we used that ). Reparametrizing the original null geodesic by setting

(4.6)

we find that

Using these formulas, one sees that  defined by (4.6) indeed satisfies (4.5) and is thus a null geodesic.

Using this transformation law for null geodesics under conformal transformations, one finds that

(4.7)
(4.8)
(4.9)
(4.10)

The last formula is what one would have expected from a naive scaling argument, keeping in mind that  has the dimension of length squared.

4.3. Local Existence and Uniqueness of the Time Evolution

In this section, we prove local existence and uniqueness of the time evolution for the Einstein equations with DGC. Before beginning, we recall that without matter, the Einstein equations with DGC coincide with the usual vacuum Einstein equations  (see the paragraph after (2.10)). This shows in particular that, starting with smooth initial data, singularities may form in finite time (see [5, 18]). This is why we restrict attention to local solutions to the Cauchy problem. In preparation for the Cauchy problem for the whole system, let us consider the Cauchy problem for our DGC models in a given space-time background. In order to pose the Cauchy problem, one must assume that the space-time is globally hyperbolic (for basic definitions see for example [1, 4, 22]). Let  be a Cauchy surface. Then, by definition of global hyperbolicity, every maximal null geodesic intersects  exactly once. Therefore, the set , (2.6), is characterized uniquely by looking at the vectors  on ,

Conversely, initial-data of this form gives rise to a unique family of null geodesics , as is made precise in the following proposition.

Proposition 4.1.

Let  be a set of null vectors

with the following properties:

  • If  is in , so is .

  • For every  and for every non-zero null vectors , there is a unique scalar  such that .

Then there is a unique choice of parametrized null geodesics  (compatible with Definition 2.1) such that .

If the sets  are smooth surfaces for each  and depend smoothly on the base point , then the sets  are smooth surfaces for all .

Proof.

In order to construct , one solves the geodesic equation with initial values on ,

Taking all the resulting maximal geodesics for all choices of  and all  gives a family . The uniqueness of  follows immediately from the fact that every maximal null geodesic intersects , where the parametrization is determined by .

The smoothness is an immediate consequence of the fact that the solutions of ordinary differential equations depend smoothly on parameters and on the initial data. ∎

We next consider the time evolution of the DGC tensor, in a given globally hyperbolic space-time with given . According to (C.4), it can be computed at a space-time point  if for every future-directed  we know the tangent vector

(4.11)

In words, this tangent vector is obtained by integrating along a null geodesic to the past. Since every null geodesic intersects the Cauchy surface , one can describe  uniquely by prescribing it on on the Cauchy surface as a mapping

(4.12)

where the  denotes the set of all future-directed null geodesics in . Then the time evolution of  is obtained by solving the transport equation

(4.13)

giving a mapping  (where  with the obvious fiber bundle structure). In principle, the mapping  should be determined according to (4.11) and (C.4) by integrating along and over all null geodesics to the past. In situations in which this procedure is not feasible (for example if the past development of the Cauchy surface is not known), one should regard  as part of the initial data, which should be chosen depending on the physical situation in mind. Whenever the effect of the DGC tensor is small, one may simply choose .

We now turn our attention to the Cauchy problem for our DGC models. Assume that we are given a Cauchy surface . On  we are given a smooth Riemannian metric  and a smooth second fundamental form . Due to finite propagation speed, it suffices to solve the Cauchy problem in a small open subset of  (then the solution on  is obtained by “glueing together” the local solutions). With this in mind, we may work in a local chart and identify  with a subset of . Working in the wave gauge, the Ricci tensor is obtained from the metric by applying a quasilinear hyperbolic operator (for details see [15, 4]),

Moreover, the Riemannian metric and second fundamental form on  give rise to the Lorentzian metric in the wave gauge and its first time derivative (for details see [4]). Therefore, we may describe the initial data of the gravitational field by

The matter is described by additional fields on . We assume that the equations of matter are such that coupling the matter fields to the usual Einstein equations, one obtains a system of partial differential equations for which the Cauchy problem is well-posed (like for example a perfect fluid, Vlasov matter, the Maxwell field, the Yang-Mills field, a scalar field, the Dirac field or the Klein-Gordon field). We also assume that our initial data satisfies the Einstein constraint equations (see for example [22, Section III.13] or [4, Section VII]). This problem can be treated exactly as for the Einstein’s gravity. Particular solutions of the constraint equations are obtained by taking initial data of the usual Einstein equations and choosing DGC initial data with  and . But clearly, choices with non-trivial DGC on  are also possible.

Under these assumptions we prove the following result:

Theorem 4.2.

The Cauchy problem for the Einstein-matter equations with DGC is well-posed.

The remainder of this section is devoted to the proof of this theorem. In more technical terms, our above assumptions on the matter fields can be restated by saying that the coupled Einstein-matter system can be rewritten as a quasilinear symmetric hyperbolic system (see [22]). In order to show that the Cauchy problem remains well-posed if DGC is taken into account, we need to settle the following issues:

  • Given the gravitational coupling function , the modified Einstein-matter equations can again be written as a quasilinear symmetric hyperbolic system.

  • The dynamics of the DGC function  (as defined by (2.8) or (2.13)) and of the DGC tensor  (as defined by (C.4) in Appendix C) can be described by a quasilinear symmetric hyperbolic system.

Once these points are proven, by combining the symmetric hyperbolic systems of the Einstein-matter system and of the dynamical equations for  and , one obtains a symmetric hyperbolic system for the Einstein-matter equations with DGC. Then Theorem 4.2 follows from the general local existence result for solutions of the initial value problem for quasilinear symmetric hyperbolic systems (see [22, Section 9] or [26, Section 16]).

The next consideration settles the above problem (a): The highest orders of the derivatives of the metric do not involve derivatives of . Thus the resulting equations are of the form

Since the lower order terms only affect the zero order terms in the corresponding first order system, the system is again symmetric hyperbolic.

We next explain how to settle the above problem (b). Our task is to write the dynamics of  and  in terms of a quasilinear symmetric hyperbolic system. To this end, it is convenient to describe the initial data  in Proposition 4.1 by a smooth mapping

(4.14)

Given , we construct the set  for any  as follows. To any  we associate a null vector  by

where the time component  is determined uniquely by imposing that . Now we set

In order to describe the dynamics of the function , we follow the flow of the null geodesics. Thus given , we let  be the null geodesic with initial conditions

(and the dot again denotes the derivative with respect to ). Thus in components,

(4.15)

Using the geodesic equation

the position and the tangent vector of the geodesic have evolved to