Problems with Propagation and Time Evolution in Gravity
Teleparallel theories of gravity have a long history. They include a special case referred to as the Teleparallel Equivalent of General Relativity (TEGR, aka GR). Recently this theory has been generalized to gravity. Tight constraints from observations suggest that gravity is not as robust as initially hoped. This might hint at hitherto undiscovered problems at the theoretical level. In this work, we point out that a generic theory can be expected to have certain problems including superluminal propagating modes, the presence of which can be revealed by using the characteristic equations that govern the dynamics in gravity and/or the Hamiltonian structure of the theory via Dirac constraint analysis. We use several examples from simpler gauge field theories to explain how such superluminal modes could arise. We also point out problems with the Cauchy development of a constant time hypersurface in FLRW spacetime in gravity. The time evolution from a FLRW (and as a special case, Minkowski spacetime) initial condition is not unique.
Einstein proposed the idea of teleparallelism, or Fernparallelismus (aka absolute parallelism, distant parallelism), in order to unify gravity and electromagnetism Einstein (). Unlike general relativity, spacetime in teleparallelism is equipped with a connection with zero curvature, but non-vanishing torsion. Since the spacetime is flat, the parallel transport of a vector is independent of path, and hence the name teleparallel, meaning “parallel at a distance”. Einstein’s quest for unification via teleparallel theory can be found in the interesting account of Sauer (). Despite Einstein’s failure to formulate a unified theory, teleparallelism was later revived and studied as a pure theory of gravity. The main motivations were (1) that the teleparallel formulation was considered to have advantages with regard to the identification of the energy-momentum of gravitating systems, and (2) teleparallel theory can be regarded as a gauge theory of local translations (see, e.g., Moller:1961jj (); Cho:1975dh (); Nester1989 (); Hehl:1994ue (); Itin:1999wi (); deAndrade:2000kr (); Blagojevic:2000pi (); Blagojevic:2002du (); Hehl:2012pi ()).111For recent study of teleparallelism as a higher gauge theory, see Baez (). In fact Einstein’s general theory of relativity (GR) can be rewritten in teleparallel language (see, e.g., Hayashi (); Pereira (); Kleinert (); Sonester ()). The result is a theory equivalent to GR known variously as the Teleparallel Equivalent of General Relativity (TEGR) or GR. Recently TEGR has been generalized to theory, a theory of modified gravity formed in the same spirit as generalizing general relativity to gravity 1 ().
It seems however that observational data puts rather tight constraints on gravity, effectively reducing the theory (at least for some choices of ) back to TEGR (However, see also CRC () and ABCL ()). This includes constraints from considering a varying fine structure constant haowei (), the formation of large-scale structures baojiu () and a varying gravitational constant haowei2 (), as well as solar system constraints lorenzo (). This could hint at problems with gravity at the theoretical level.
In this paper we will briefly review TEGR and more general theories of teleparallel gravity, including gravity, in section II. In section III, we discuss the degrees of freedom for theory. We know that there are extra degrees of freedom in the theory beyond those of GR; that the number of extra degrees of freedom is generically 3 was clearly revealed by the Dirac constraint analysis carried out by Li et al. Li (). Despite this, in a careful linear perturbative analysis up to second order on flat Friendmann-Lemaître-Robertson-Walker (FLRW) background, no extra mode appears IzumiOng (). In the case of Minkowski background, this had already been pointed out in Li (). Disappearance of extra degrees of freedom is known to happen in some gravity theories (e.g., in massive gravity, see Gumrukcuoglu:2011ew ()); this can be caused by the accidental disappearance of kinetic terms at the linear level. If so, as we shall argue, it is likely to be a signal of superluminal propagation. We will also discuss the related danger of the nonlinear constraint effect, which plagued some other theories of gravity in the past. Our conclusion is that theory is very likely to give rise to unwanted problems due to the effects of nonlinear constraints, in addition to propagation outside of the null cone. As a consequence, the number and type of degrees of freedom could vary as a physical system evolves. This is further explored via the method of characteristics in section IV.
The method of characteristics is properly introduced in section IV, where we also provide an illuminating example in an application to the nonlinear Proca field. The same example is given a Hamiltonian analysis that clearly shows the close connection of superluminal propagating modes and nonlinear constraints. Although the characteristic equations allow one to obtain superluminal propagating degrees of freedom, one must be careful to distinguish between physical degrees of freedom and gauge degrees of freedom, as the example of the Maxwell field illustrates.
In section V, we investigate the dynamical equations in theory and derive a system of partial differential equations that describe the characteristics of the theory. This calculation reveals nonlinear features that are very prone to allow superluminal propagation outside of the metric null cone.222We should remark here that superluminal propagation may or may not be achronal, so it is not certain that such propagation really does lead to a violation of causality. We make further comments on this in section IV. Furthermore, we also show that the constant-time hypersurface in FLRW spacetime, and as a special case Minkowski spacetime, can be a Cauchy horizon, which means that even the infinitesimal future time evolution from it is not unique. We conclude with some discussion in section VI.
Ii Teleparallel Theories of Gravitation
Let be a semi-Riemannian manifold of signature with metric . The teleparallel theories we wish to discuss333There are more general teleparallel theories which also have nonmetricity Hehl:1994ue (); in fact there are symmetric teleparallel theories—with vanishing torsion but non-vanishing nonmetricity—including one that is equivalent to GR Nester:1998mp (); Adak:2005cd (). use the vierbein field or tetrad as the fundamental dynamical variables. We use Greek letters for spacetime coordinate indices and Latin letters to label the tetrad vectors.
Suppose that the manifold is parallelizable, i.e., there exist vector fields such that at any point the tangent vectors ’s provide a basis of the tangent space at . Note that this particular set of vector fields must be able to span the tangent spaces at all points of the manifold, i.e., it is a set of everywhere linearly independent globally non-vanishing vector fields. Then we can view the mapping between the bases of the tangent space in a coordinate frame to that of a non-coordinate frame as an isomorphism . Orthonormality of the tetrad is thus imposed by the Minkowski metric on the tangent space , related to the metric on the manifold by .
where . Equivalently, the connection coefficients are
In local coordinates, the components of the torsion tensor are
The Weitzenböck connection can be shown to be curvature-free.444There is an alternative formulation for teleparallel theories that has some advantages; it uses the teleparallel connection as an a priori independent dynamical variable that is constrained to have vanishing curvature via a Lagrange multiplier Kopczynski (); Hehl:1994ue (); Blagojevic:2000pi (); Blagojevic:2000qs (); Blagojevic:2002du (); Obukhov:2002tm (). Here, to more simply communicate our ideas to the audience we most want to reach, we will use the formulation familiar in the works.
The Weitzenböck connection differs from the Levi-Civita connection by the contortion tensor555We remark that the proper term is contortion instead of the often used “contorsion”. See e.g., Hehl ().
The Einstein-Hilbert action of general relativity is (in units where )
Up to a total divergence (which does not affect the field equations) this action is equivalent to
and is a certain quadratic-in-torsion scalar:
the latter is the so-called torsion scalar.
In this teleparallel language, (6) is called the TEGR action. gravity simply promotes the “torsion scalar” in the TEGR Lagrangian to a function, i.e. the action of gravity is666For simplicity here we follow the somewhat non-standard convention used in IzumiOng (), in which the factor is absorbed into , i.e., TEGR corresponds to .
One can then obtain the field equation for theory of modified gravity by varying with respect to the frame , this leads to
We remark that the (quadratic-in-)torsion scalar defined above is a very special one. We note that mathematically we can define a more general torsion “scalar” by relaxing the coefficients:777In fact one could also include two more quadratic-in-torsion combinations which have odd parity; they can be found in Baekler and Hehl Baekler:2011jt ().
In particular one can construct the one-parameter combination
where . Years ago a teleparallel theory using this as the Lagrangian was proposed as an alternative to GR. This theory passed all observational (as of, perhaps up to, 1990) SS (); NH () and some theoretical tests KN (). But then certain theoretical problems were found, which we will discuss in the next section.
We summarize how teleparallel theories work: We start with choosing a good local orthonormal frame which is declared to be covariantly constant under parallel transport by imposing the Weitzenböck connection. This defines a global frame field that sometimes goes by the name orthoteleparallel frame or OT frame. Recall that there are infinitely many possible choices of frames that span the same tangent space at , related to each other by elements of the proper orthochronous Lorentz group. The torsion tensor and consequently the torsion scalar, can thus be expressed in terms of any one of these infinitely many bases. Once we get a parallelization, it is defined up to global Lorentz transformation. This defines a Weitzenböck geometry. The set of all possible parallelizations of is then partitioned into equivalence classes consisting of parallelizations that relate to each other in the same class by a global Lorentz transformation (thus, one cannot pass from one parallelizations to another by a local Lorentz transformation). Note that at this stage, nonzero torsion does not mean that we have some “distinguished frames” or preferred frames; any equivalence class of frames are on the same footing, in the sense that they parallelized the spacetime equally well. However, once we introduce the physics via the action, we get field equations. Now we can only admit certain classes of parallelizations, just as in general relativity where we can only admit metrics that satisfy the Einstein Field Equations.
We remark that due to the lack of local Lorentz invariance (See also the discussions in Barrow1 () and Barrow2 ()), unlike in general relativity where we can change coordinate systems and use any frame fields, this is not so in generic teleparallel theories. As a consequence, this implies that we cannot directly extract the tetrad from the metric in the straightforward way. TEGR is special since its action does not determine the admissible OT frame but only the metric. From now onwards, we will suppress all the explicit overscript ’s on the torsion scalar and connection coefficients etc.
Iii Extra Degrees of Freedom and the Danger of Nonlinear Constraints
We are reminded of the problem faced by the one-parameter teleparallel theory in which the teleparallel Lagrangian has the form (12). This theory has 4 extra degrees of freedom Nester3 (). Kopczyński Kopczynski () first pointed out that this theory has predictability problems; he argued that one is unable to determine uniquely the evolution of the teleparallel geometry. However, further analysis Nester2 () showed that the problem is not generic, instead it only occurs for a special class of solutions. Cheng et al. verified this behavior using the Hamiltonian approach and identified the troubles as a certain effect of nonlinear constraints, there referred to as constraint bifurcation Nester3 (), in which the chain of constraints could “bifurcate” depending on the values of the fields. That is to say, the number or type of constraints would depend on the values of the phase space variables. One could imagine that the evolution of some system as depicted by a curve in phase space could pass through regions with a different number or type of constraints. This would be rather strange (since this means that the number of gauges and physical degrees of freedom also change), and one will find it hard, if at all possible, to predict the evolution of the system. To put it in another way, the matrix with the Poisson brackets of the constraints as entries may not have constant rank, due to the nonlinearity of the constraints. The number or type of constraints may change as one approaches a point in the phase space where the rank changes. The more general Poincaré gauge theory also has similar problems Nester8 (); Nester9 (); Nester10 (). We suspect that this might also be the case for theory (and probably for many teleparallel theories). It is of course conceivable that with a specific choice of , we could prevent the solutions from ever approaching such problematic regions in phase space, e.g., if they would act as a dynamical repeller.
The number of degrees of freedom of a generic theory of gravity (in -dimensions) has been found to be five by Miao Li et al. Li () using a Hamiltonian approach.888His analysis was based on the Hamiltonian formulation of Maluf Maluf:2000ag (); Maluf:2001rg (). Additional insight could probably be found using the other more general Hamiltonian approaches to gravity theories which specialize to teleparallel theory Blagojevic:2000pi (); Blagojevic:2000qs (); Nester:1991yd (); Chen:1998aw (). That is, there are three extra degrees of freedom, which the authors suggest could correspond to one massive vector field or one massless vector field with one scalar field. To understand the properties of the degrees of freedom in a theory, one should first try to analyze them by a perturbative approach. However, it could happen that not all degrees of freedom show up in the perturbation level.
For the case of gravity, as remarked by Li et al. at the end of their paper Li (), for some special backgrounds (including Minkowski space with ), some of the extra degrees of freedom do not appear in the linear perturbation. It is also found that IzumiOng (), on the flat FLRW background with a scalar field, linear perturbation up to second order does not reveal any extra degree of freedom. In other words, theory is highly nonlinear: the number and type of constraints in the linear theory is different from that of the full, nonlinear theory. This is also one of the problematic features discussed in the context of Poincaré Gauge Theory Nester8 (); Nester9 (); Nester10 (). The worry of such behavior is that linearized modes which are “good” may cease to be so in the full nonlinear theory, perhaps accompanied by anomalous characteristics, as we will further comment on in the later sections.
The revealing Hamiltonian discussion of Li et al. for gravity in (3+1) dimensional spacetime is rather complicated and some of the results are only inferred implicitly. However they also worked out explicitly the analogous but somewhat simpler Hamiltonian formulation for gravity in -dimensions; for this case the problem we wish to address is clearly evident. In this case there are six first class constraints999We do not explain notations of Li et al. Li () here since we will not need them in detail. , where and , as well as four second class constraints , which leads to two degrees of freedom.
They define the quantities
The self-consistency equation in matrix form is shown to be , where and
satisfies . This matrix generically is of rank 4. However, as previously mentioned, some of the Poisson brackets could vanish in some cases, which would result in rank changes of the matrix as the system evolves. Miao Li et al. found the explicit formulas for the Lagrange multipliers:
While they were interested in the most general case and thus only considered the case with , this type of special case is precisely what we are interested in in this work. Indeed one sees that in principle can vanish, yet the numerators of these expressions are not generally constrained to vanish at the same time, which allows one or more of the Lagrange multipliers to become unbounded. As we will explain below, this is signaling a superluminal propagation mode.
Iv The Method of Characteristics
A viable theory of gravity should satisfy certain theoretical criteria, including the lack of tachyonic modes, or modes that carry negative kinetic energy (i.e., ghosts).101010However, one may still argue that theories with such seemingly pathological features may still be acceptable. We will make further comments in section VI. The theory should also support a well-posed initial value problem, that is, satisfy the Cauchy-Kowalevski theorem Cauchy (); Kowalevski (); Nakhushev (). Moreover any propagation mode in the theory should also be described by hyperbolic quasi-linear partial differential equations with well-behaved characteristics, i.e., the characteristic surfaces should be non-spacelike. One should also consider a theory with a “good” Minkowski limit as preferable to a theory that does not have such behavior.
The method for studying characteristics is well-known; see for example the detailed classical work of Courant and Hilbert Courant (), or Chapter 8 of the more recent text of Strichartz (). A discussion of the method of characteristics in the case of general relativity can be found in Lecture 14 of Buchdahl (). For more rigorous treatment see Friedlander (), as well as in Christodoulou (). See also Zwanziger () for some useful discussions in the context of external fields in gauge theories. Here we only give a brief summary. Recall that if is a linear differential operator of order , then we can consider as a polynomial in the derivative . In multi-index notation we can write this as
We may ask in what directions it is really of order . For an ODE for example, it is obvious that is the condition required for the equation to be of order everywhere. For PDEs with multiple variables, we need to be careful. Consider for example the Laplacian in . This is clearly of order 2 in both the - and -direction, however for an operator which is mixed, e.g., , it is not of order 2 in either the - or -direction. Nevertheless, it is a second-order operator, which can be revealed by introducing new variables, , which renders . That is, this operator is of order 2 in the and directions. In general then, given any operator at , and a direction , we can make an orthogonal change of variable so that points along one of the new coordinate axes, say the -axis. Now, if the coefficient of the partial derivative (in the new coordinate system) is nonzero at , then we can say that is of order at in the direction . We refer to such situation as noncharacteristic. That is, characteristic refers to the case in which the coefficient of vanishes at .
The total symbol or top-order symbol of is simply a polynomial obtained by replacing the derivative with a variable, say . The principal symbol, denoted , is the highest degree component of the total symbol. That is,
The principal symbol almost completely determines the qualitative behavior of the solutions of the system. Furthermore, it is well known that for hyperbolic (as well as parabolic) partial differential equations, the zeros of the principal symbol describe the characteristics of the system. That is to say, the characteristic directions are exactly those for which the principal symbol vanishes. Indeed the “standard procedure” is to begin with the equation of motion or the field equation, keep only the highest derivative terms, and then replace the said derivatives with components of the normal vector to the characteristics and set the equation to zero. This is called the characteristic equation. We can study how the characteristics propagate by looking for what sorts of vectors are allowed as solutions to the characteristic equation.
Note that for a characteristic equation that is a matrix equation, called the characteristic matrix, it suffices to consider the vanishing of the determinant of the characteristic equation (known as the characteristic determinant) instead of the vanishing of the equation itself (the latter is of course a stronger statement).
In this paper, we are interested in one (or a few) special characteristic direction that gives the signal of a superluminal mode. By analyzing the characteristic determinant, we can examine all directions at once.
The characteristic surface which is orthogonal to the characteristic direction coincides with the edge of Cauchy development, i.e., the Cauchy horizon. This is because the higher order derivative term with respect to the characteristic direction disappears. The disappearance of the higher order derivative term causes the evolution to be singular. Thus, the Cauchy development of the characteristic surface is only the characteristic surface itself and the evolution from it is meaningless even if it is spacelike.
Note that for application in theories of gravity, we would like our characteristic directions to be null or spacelike, as a timelike characteristic direction is the signal of superluminal propagation (and could violate causality, although this is not necessarily so even in theories with local Lorentz invariance Bruneton (); Afshordi (); Geroch1 ()). Correspondingly, there should not be any non-trivial solution that corresponds to any timelike vector.
iv.1 Example: Nonlinear Proca Field
The characteristic method is best illustrated via an example or two.
Consider the Lagrangian of the Proca field with a nonlinear term Velo:1970ur ():
where and The signature is . The field equations are
Upon taking the divergence of this equation, because the first term vanishes by the antisymmetry of , we obtain an implicit constraint satisfied by the system:
For , this gives , which upon substituting back into the field equation (23) gives
which is the Klein-Gordon equation. The characteristic equation for this case is just , i.e., the characteristic is null.
To find the characteristics in the general case, replace the highest derivative terms by . Here means it is not the value of the vector , but represents the change of the vector in a certain direction.
We then get the relations
A linear combination of these gives
Now from eq. (26) one can see that modes with propagate with null characteristics, but from eq. (28) one can see that modes with generally have non-null characteristics. The normal to the characteristic surface could be timelike, indicating a superluminal characteristic. In that case there is a Lorentz frame in which . In such a frame the condition is
We want to emphasize that this tachyonic mode of the nonlinear Proca field can be detected using the Hamiltonian formulation, as we now explain. From the Lagrangian, the canonical momenta are
That is, explicitly, and ; the latter is a primary constraint.
where the primary constraint 111111Here “” denotes Dirac’s weak equality, i.e., it only holds on the constraint surface within the phase space. has been included with an unknown Lagrange multiplier . From the Hamiltonian evolution equation
one finds the meaning of the multiplier, it is the missing “velocity”. Preserving the primary constraint leads to the secondary constraint:
The Poisson bracket of the two constraints is
Generically this is non-vanishing, so they make up a 2nd class pair.
However, there is an important exception when the RHS vanishes, this is exactly the same as the anomalous characteristic condition (29). The dynamical consequence shows up when we require preservation of the constraint, which is the relation that determines the “unknown multiplier” :
where denotes a collection of field dependent terms which, generically, are non-vanishing. Consequently, the field “velocity” , given by the Lagrange multiplier, , becomes unbounded at any point(s) where approaches 0. That is,
This is an indication that, with respect to this constant time spacelike hypersurface, there is instantaneous propagation of the mode. The nonlinear constraint has led to a field-dependent constraint Poisson bracket value, a signal for the occurrence of superluminal propagation, in complete agreement with the previous analysis obtained from the characteristic equation.
iv.2 Example: Scalar Field and Maxwell Field
One issue that we have to be careful with is the possible presence of gauge degrees of freedom in a theory (which was not present in the previous nonlinear Proca field example).
In order to appreciate the issue, first we shall look at the simple example of scalar fields. We consider the Lagrangian for the single scalar field defined by
We know that the scalar field propagates in the null direction. In terms of characteristics, we can understand it as follows. The equation of motion is
So the characteristic equation for is , which means the characteristic is null.
Let us introduce another scalar field which does not appear in the Lagrangian. Trivially, the Lagrangian does not change under the transformation . This property is a certain kind of gauge transformation. The equation of motion which comes from the variation of the Lagrangian (37) with respect to is trivial, i.e., . Thus the characteristic equation for and becomes , which is trivially satisfied for any . The triviality comes from the gauge mode . Since does not appear in the Lagrangian, we should not include it in the discussion of characteristics.
The situation for any gauge field theory is similar. For simplicity, we consider the Maxwell theory of the electromagnetic field; the Lagrangian is
where . The equation of motion is obtained to be
The principal symbol yields the characteristic equation
where we want to seek some nontrivial . If is null then gives a propagating mode on the null cone. However, if is not null, then the characteristic equation can be satisfied by any for arbitrary constant . In particular can be chosen to be timelike, which means that the characteristic direction will become timelike. This superluminal mode however, as we shall see, actually corresponds to gauge mode and so it is not physical.
Alternatively we can consider the determinant method. If we write down the characteristic equation without removing the gauge degree of freedom, we will obtain characteristic matrix that satisfies
The left-hand side of this equation is for any , and so this equation becomes trivial. The origin of the triviality is the gauge degree of freedom. If we operate to the left-hand of eq. (40), it becomes algebraically . Therefore, eq. (40) has the information of only three independent equations. We know the Maxwell field has -gauge, i.e., the Lagrangian is invariant under the -gauge transformation . Since the gauge degree of freedom does not appear in the Lagrangian, the variation of the Lagrangian with respect to the gauge degree of freedom must be trivial, which is the origin of the above trivial equation. According to the above simple example of scalar fields, we should eliminate the contribution from gauge degrees of freedom in the discussion of characteristics. Subtracting the contribution from the gauge degree of freedom, i.e., considering the matrix whose basis can be anything independentof , we can obtain the exact characteristic equation, which is satisfied only when is null. A more rigorous treatment of the removal of gauge degrees of freedom and the characteristics of Maxwell equations can be found in Christodoulou ().
V Characteristics Equations and Problematic Time Evolution in Gravity
In this section, we discuss the Cauchy problem for gravity by an analysis of the characteristics. First, we shall determine the characteristic equations of gravity. After that, we will show that a constant-time hypersurface of the FLRW metric does not provide a good initial condition and that the time evolution of the FLRW metric is not well behaved. Finally, we concretely show the problematic solution where the time evolution from a FLRW initial condition is not unique.
v.1 Characteristic Equation of Gravity
From eq. (10), we see that the vacuum field equation in gravity is explicitly given by
This can be rewritten as
The highest order derivative terms are just
We recall that is linear in , so we may write
for some which is algebraic in the tetrad . The explicit form of is
where the bracket means antisymmetrization, i.e.,
Consequently we have
Thus, we can re-write eq. (45) in a more simplified form as
Following the standard procedure we can replace in eq. (50) to obtain the characteristic equations for theory of gravity
Here the notation indicates that these are not the values of the frame, but rather represent the change of the frame in a certain direction. We note that eq. (51) thus provide quadratic polynomial conditions on the four ’s.
That is, the characteristic matrix becomes
Since the tetrad always has the inverse , the property of the matrix is the same as that of
Therefore for convenience, we shall analyze the matrix in eq. (54) instead.
Before discussing the property of the characteristic matrix of eq. (54), we must first find out the gauge degrees of freedom. We have 16 equations of motion (10), but not all of them are independent due to gauge degrees of freedom. We actually have four nontrivial identities.
Indeed, let us consider the invariance of the action under an infinitesimal coordinate transformation:121212Geometrically this represents an infinitesimal diffeomporhism.
This leads to the transformation of the tetrad by
and we know this is related to a gauge mode, i.e., , where are four scalar fields with an internal index. Varying the action with respect to the component
one obtains the nontrivial Noether differential identities Noether ()
Since the infinitesimal coordinate transformation also leads to a transformation of metric, the metric has the information about all the gauge degrees of freedom. Therefore, all the degrees of freedom of the gauge modes can be eliminated by gauge fixing of the metric components.
Now, we move on to the discussion of the characteristic determinant which is constructed by the part of the characteristic matrix in eq. (54) having removed the gauge degrees of freedom. Here, we analyze the characteristic matrix by the decomposition into symmetric and antisymmetric parts with respect to and and also with respect to and . Without loss of generality we can choose an appropriate basis to write down the matrix such that the left half submatrix represents the symmetric part and the right half submatrix represents the antisymmetric part.
First, we consider the first term of the characteristic matrix in eq. (54). The first term of the symmetrized characteristic matrix is well behaved because it is the only contribution in TEGR. The symmetric part of the characteristic equation is related to the metric component, which a priori has 10 degrees of freedom. However we know that 4 of the degrees of freedom are gauge modes and we have only 6 physical modes. We claim that its rank is generically 6, and indeed after removing the gauge degrees of freedom the second order derivatives are associated with all components of the symmetrized tetrad. We consider next the antisymmetrized component with respect to and . The first term then gives zero contribution, which means that the rank of the first term is indeed for general .
To summarize, the characteristic matrix of gravity, after removing the gauge degrees of freedom, can be written as a matrix, with four block submatrices. The generic rank of the upper left block is 6, corresponding to the symmetrized tetrads.
Next, we consider the antisymmetrized components of the second term in eq. (54). The form of the second term is the product of two and its rank is one (since one of the eigenvalue vanishes). One can thus form a matrix at the very center of the matrix, in which each element belongs to one of the four blocks, respectively. Together with the block corresponding to the symmetrized tetrad, we thus get a matrix.
Therefore we have found that the ranks of the first term and the second term of are 6 and 1, respectively. By the fact that is generically nonzero, we can prove that the rank of matrix is generically 7. This rests on the fact that the determinant of the matrix is generically nonzero, which follows from elementary linear algebra (See Appendix.A).
It is important to note that the matrix at the center, being of rank 1, corresponds to only one of the extra degrees of freedom in gravity. However we know that the theory contains 3 extra degrees of freedom – so where are the remaining two?
In fact the five antisymmetrized components corresponding to the tetrad (), which satisfy
become zero. This means that in the equations for the five tetrads the second order derivative does not appear and that the characteristic equation should be at most a first order differential equation. In other words, the remaining two simply do not show up in the characteristic matrix!
Thus, in order to obtain the complete characteristic we must derive the characteristic equation for separately. This is what we will do next for the sake of completeness, although it is not a crucial part for our subsequent analysis (since it is sufficient to show that the one degree of freedom that does show up in the characteristic matrix, leads to problematic behavior).
In order to derive the characteristic equation for , we must go back to the equation of motion eq. (10), which can be written as
If we consider equation of motion from the variation of the action with respect to , we can easily show that the first term in eq. (60) does not appear. The equation of motion (60) is obtained from variation of the action
and for the first term of the right-hand side becomes zero due to eq. (59). The second term in eq. (60) is also zero because of antisymmetrization. Eventually, only the last term in eq. (60) gives a non-zero contribution.
Although the last term in eq. (60) seems to be a higher order equation of the first order derivative of (which, if true, will render the characteristic method unworkable), we can show that it is actually linear. Indeed, we can multiply it by for convenience. We then have the antisymmetrized equation with respect to and
This gives the two branches of the equation, and
If we choose the first branch, i.e., , the five equations become degenerate, and thus, time evolution cannot be fixed uniquely. This causes trouble regarding the well-posedness of the Cauchy problem. Therefore, we choose the second branch.131313 There is however a priori neither a physical principle nor a mathematical requirement that requires us to choose the second branch. This might be another issue of gravity. The second order terms of the first derivative in eq. (64) can be written as
If both derivative factors are variations of , eq. (65) becomes trivial. Thus at least one of the derivative term is not such a variation. This means that eq. (64) is not second order but first order with respect to and the characteristic equation for can be shown to be
where the notation is similar to , i.e., it also represents the change of the frame in a certain direction. Eq. (LABEL:charaanti) gives the characteristic matrix for :
Then, we have the exact characteristic matrix which is a combination of the matrix and the matrix .141414 The cross terms of the 5 components and the other 7 do not contribute to the characteristic matrix, since their components have only one derivative. Since the nonzero components of the matrix have two derivatives, the contribution of the cross terms with any smaller number of derivatives should not appear in the characteristic equation.
To summarize, the remaining two extra degrees of freedom do not show up in the characteristic matrix, since their corresponding characteristic equation only contains one derivative term. They can nevertheless be described by another characteristic matrix . We remark that despite the single derivative term here, this does not necessary mean that the dynamics of the two extra degrees of freedom is governed by first order equations. This may be analogous to the Dirac equation, which is first order and linear. However, together with its conjugate equation, one can derive a second order Klein-Gordon equation governing the dynamics of the system. Likewise, Maxwell’s equations are first-order but combining them gives the second order wave equation.
In the next section we will give concrete examples to demonstrate problematic behaviors that could arise in gravity due to bad characteristics. To appreciate what is likely to go wrong with the characteristic, let us look at the characteristic equation eq. (51) again. We see that the first term is good since it is the only term present in TEGR, and we know that characteristics are null in general relativity. In fact the first term gives dynamics to the 6 metrical components of . The second term quadratic in the field strength governing one of the extra degrees of freedom however is almost certainly going to give a disaster if for timelike . It suggests that with some field values the characteristics could leak outside the metric null cone, which in turn means that there might be problems with tachyonic modes. This nonlinear behavior is reminiscent of the nonlinearity we encountered in the generalized Proca field example.
Note that the case for TEGR is very different since the theory is locally Lorentz invariant, which allows us to choose a gauge that simplifies a lot of the calculation. In any case, TEGR corresponds to for any value of and there is no problem with the characteristics.
v.2 Absence of Cauchy Development from a Constant-Time Hypersurface in FLRW Geometry
In this section, we will show that in gravity the time direction, i.e., , in the FLRW metric is the characteristic direction, and thus the time evolution of the FLRW metric is not unique even in the infinitesimal future. We analyze the characteristic equation in gravity which has been derived in the previous subsection. The characteristic determinant can be decomposed into two parts, the matrix and the matrix . In our analysis we look into the matrix , which has the information about all of the six symmetric components and one of the antisymmetric components. The antisymmetric component appears only in the second term of the characteristic matrix in eq. (54), and thus for some , the determinant becomes zero if the value of its antisymmetric component is zero. This means that is one of the characteristic directions. To see this, it is enough to solve for . In the case of the FLRW metric, is a solution.
Let us see this explicitly. We consider the cotetrads
These cotetrads give the FLRW geometry with flat spatial section:
Given the cotetrads, we can calculate the torsion tensor and . They are
while the other components all vanish. Here is the Hubble parameter, i.e., . Then, it is trivial that is a solution of .
v.3 Concrete Example of Non-Uniqueness of Time Evolution in FLRW Spacetime
We have shown that the FLRW solution has a spacelike characteristic hypersurface, and thus we cannot properly discuss the time evolution from a constant time hypersurface even into the infinitesimal future. Here, we show a concrete solution where the time evolution from the FLRW initial condition without torsion is not unique in gravity with or without matter.
We introduce a matter action
Then, equation of motion (10) is modified as
Assume that the matter action is locally Lorentz invariant, i.e., it depends on the tetrad only through the metric. Then Eq. (73) can be rewritten as
Let us now consider the ansatz