Einstein   Double  Field  Equations

Department of Physics, Sogang University, 35 Baekbeom-ro, Mapo-gu, Seoul 04107, KOREA

stephenangus4@gmail.com  khcho23@sogang.ac.kr   park@sogang.ac.kr

Upon treating the whole closed string massless sector as stringy graviton fields, Double Field Theory may evolve into Stringy Gravity, i.e. the stringy augmentation of General Relativity. Equipped with an covariant differential geometry beyond Riemann, we spell out the definition of the Energy-Momentum tensor in Stringy Gravity and derive its on-shell conservation law from doubled general covariance. Equating it with the recently identified stringy Einstein curvature tensor, all the equations of motion of the closed string massless sector are unified into a single expression, , which we dub the Einstein Double Field Equations. As an example, we study the most general static, asymptotically flat, spherically symmetric, ‘regular’ solution, sourced by the stringy Energy-Momentum tensor which is nontrivial only up to a finite radius from the center. Outside this radius, the solution matches the known vacuum geometry which has four constant parameters. We express these as volume integrals of the interior stringy Energy-Momentum tensor and discuss relevant energy conditions.

One must be prepared to follow up the consequence of theory, and feel that

one just has to accept the consequences no matter where they lead.

Paul Dirac


Our mistake is not that we take our theories too seriously, but that we do not take them seriously enough.

Steven Weinberg


1 Introduction

The Einstein-Hilbert action is often referred to as ‘pure’ gravity, as it is formed by the unique two-derivative scalar curvature of the Riemannian metric. Minimal coupling to matter follows unambiguously through the usual covariant derivatives,

(1.1)

which ensures covariance under both diffeomorphisms and local Lorentz symmetry. In the words of Cheng-Ning Yang, symmetry dictates interaction. The torsionless Christoffel symbols of the connection and the spin connection are fixed by the requirement of compatibility with the metric and the vielbein. The existence of Riemann normal coordinates supports the Equivalence Principle, as the Christoffel symbols vanish pointwise. Needless to say, in General Relativity (GR), the metric is privileged to be the only geometric and thus gravitational field, on account of the adopted differential geometry a la Riemann, while all other fields are automatically categorized as additional matter.

String theory may put some twist on this Riemannian paradigm. First of all, the metric is merely one segment of closed string massless sector which consists of a two-form gauge potential, , and a scalar dilaton, , in addition to the metric, . A genuine stringy symmetry called T-duality then converts one to another [1, 2]. Namely, the closed string massless sector forms multiplets of T-duality. This may well hint at the existence of Stringy Gravity as an alternative to GR, which takes the entire massless sector as geometric and therefore gravitational. In recent years this idea has been realized concretely through the developments of so-called Double Field Theory (DFT) [3, 4, 5, 6, 7, 8]. The relevant covariant derivative has been identified [9, 10] and reads schematically,

(1.2)

where is the DFT version of the Christoffel symbols for generalized diffeomorphisms, while and are the two spin connections for the twofold local Lorentz symmetries, . They are compatible with, and thus formed by, the closed string massless sector, containing in particular the -flux (). The doubling of the spin group implies the existence of two separate locally inertial frames for left and right closed string modes, respectively [11]. In a sense, it is a prediction of DFT (and also Generalized Geometry [12]) that there must in principle exist two distinct kinds of fermions [13]. The DFT-Christoffel symbols constitute DFT curvatures: scalar and ‘Ricci’. The scalar curvature naturally defines the pure DFT Lagrangian in analogy with GR. However, in Stringy Gravity the Equivalence Principle is generically broken [13, 14]: there exist no normal coordinates in which the DFT-Christoffel symbols would vanish pointwise. This should not be a surprise since, strictly speaking, the principle holds only for a point particle and does not apply to an extended object like a string, which is subject to ‘tidal forces’ via coupling to the -flux.

Beyond the original goal of reformulating supergravities in a duality-manifest framework, DFT turns out to have quite a rich spectrum. It describes not only the Riemannian supergravities but also various non-Riemannian theories in which the Riemannian metric cannot be defined [15], such as non-relativistic Newton–Cartan or ultra-relativistic Carroll gravities [16], the Gomis–Ooguri non-relativistic string [17, 18], and various chiral theories including the one by Siegel [19]. Without resorting to Riemannian variables, supersymmetrizations have been also completed to the full order in fermions, both on target spacetime [20, 21] and on worldsheet [22].

Combining the scalar and ‘Ricci’ curvatures, the DFT version of the Einstein curvature, , which is identically conserved, , and generically asymmetric, , has been identified [23]. Given this identification, it is natural to anticipate the ‘Energy-Momentum’ tensor in DFT, say , which should counterbalance the stringy Einstein curvature through the Einstein Double Field Equations, i.e. the equations of motion of the entire closed string massless sector as the stringy graviton fields,

(1.3)

where (without any subscript index) denotes Newton’s constant. For consistency, the stringy Energy-Momentum tensor should be asymmetric, , and conserved, , especially on-shell, i.e. up to the equations of motion of the additional matter fields.

In order to compare the ‘gravitational’ aspects of DFT and GR, circular geodesic motions around the most general spherically symmetric solution to ‘’ have been studied in [24] for the case of . While the solution was a re-derivation of a previously known result in the supergravity literature [25], the new interpretation was that it is the ‘vacuum’ solution to DFT, with the right-hand side of (1.3) vanishing: it is analogous to the Schwarzschild solution in GR. The DFT spherical vacuum solution turns out to have four (or three, up to a radial coordinate shift) free parameters, in contrast to the Schwarzschild geometry which possesses only one free parameter, i.e. mass. With these extra free parameters, DFT modifies GR at ‘short’ scales in terms of the dimensionless parameter , i.e. the radial distance normalized by the mass times Newton’s constant. For large , DFT converges to GR, but for finite they differ generically. It is an intriguing fact that the dark matter and dark energy problems all arise from astronomical observations at smaller , corresponding to long distance divided by far heavier mass [24, 14]. Such a ‘uroboros’ spectrum of is listed below in natural units.

The purpose of the present paper is twofold: i) to propose the definition of the stringy Energy-Momentum tensor which completes the Einstein Double Field Equations spelled out in (1.3), and ii) to analyze the most general spherically symmetric ‘regular’ solution which will teach us the physical meanings of the free parameters appearing in the vacuum solution of [24, 25]. The rest of the paper is organized as follows.

  • We start section 2 by reviewing DFT as Stringy Gravity. We then consider coupling to generic matter fields, propose the definition of the stringy Energy-Momentum tensor, and discuss its properties including the conservation law. Some examples will follow.

  • In section 3 we devise a method to address isometries in the vielbein formulation of Stringy Gravity. We generalize the known generalized Lie derivative one step further, to a ‘further-generalized Lie derivative’, which acts not only on vector indices but also on all the local Lorentz indices.

  • Section 4 is devoted to the study of the most general, asymptotically flat, spherically symmetric, static ‘regular’ solution to the Einstein Double Field Equations. We postulate that the stringy Energy-Momentum tensor is nontrivial only up to a finite cutoff radius, . While we recover the vacuum solution of [24] for , we derive integral expressions for its constant parameters in terms of the stringy Energy-Momentum tensor for , and discuss relevant energy conditions.

  • We conclude with our summary and comments in section 5.

  • In Appendix A we collect some known features of GR, such as the general properties of the Energy-Momentum tensor and the most general spherically symmetric (Schwarzschild type) regular solution to the undoubled Einstein Field Equations, which we double-field-theorize in the present paper.

2 Einstein Double Field Equations

In this section we first give for completeness a self-contained review of DFT as Stringy Gravity, following which we propose the DFT, or stringy, extensions of the Energy-Momentum tensor and the Einstein Field Equations.

2.1 Review of DFT as Stringy Gravity

We review DFT following the geometrically logical—rather than historical—order: i) conventions, ii) the doubled-yet-gauged coordinate system with associated diffeomorphisms, iii) the field content of stringy gravitons, iv) DFT extensions of the Christoffel symbols and spin connection, and v) covariant derivatives and curvatures. For complementary aspects, we refer readers to [33, 34, 35] as well as [36, 37].

Symmetries and conventions
The built-in symmetries of Stringy Gravity are as follows.

  • T-duality

  • DFT diffeomorphisms

  • Twofold local Lorentz symmetries,111In the most general case, the two spin groups can have different dimensions [15]. .

We shall use capital Latin letters, for the vector indices, while unbarred small Latin letters, or Greek letters, will be used for the vectorial or spinorial indices of , respectively. Similarly, barred letters denote the other representations: (vectorial) and (spinorial). In particular, each vectorial index can be freely lowered or raised by the relevant invariant metric,

(2.1)

Doubled-yet-gauged coordinates and diffeomorphisms
By construction, functions admitted to Stringy Gravity are of special type. Let us denote the set of all the functions in Stringy Gravity by , which should include not only physical fields but also local symmetry parameters. First of all, each function, , has doubled coordinates, , , as its arguments. Not surprisingly, the set is closed under addition, product and differentiation such that, if and , then

(2.2)

and hence is . The truly nontrivial property of is that every function therein is invariant under a special class of translations: for arbitrary ,

(2.3)

where is said to be derivative-index-valued. We emphasize that this very notion is only possible thanks to the built-in group structure, whereby the invariant metric can raise the vector index of the partial derivative, . It is straightforward to show222Consider the power series expansion of around , where we have introduced a real parameter, . The linear-order term gives , which in turn, after replacing and by and , implies that . Consequently, is a nilpotent matrix and thus must be traceless,  [39]. that the above translational invariance is equivalent to the so-called ‘section condition’,

(2.4)

which is of practical utility. From (2.3), we infer that ‘physics’ should be invariant under such a shift of . This observation further suggests that the doubled coordinates may be gauged by an equivalence relation [38],

(2.5)

Diffeomorphisms in the doubled-yet-gauged spacetime are then generated (actively) by the generalized Lie derivative, , which was introduced initially by Siegel [4], and also later by Hull and Zwiebach [6]. Acting on an arbitrary tensor density, , with weight , it reads

(2.6)

Thanks to the section condition, the generalized Lie derivative forms a closed algebra,

(2.7)

where the so-called C-bracket is given by

(2.8)

Along with this expression, it is worthwhile to note the ‘sum’,

(2.9)

Further, if the parameter of the generalized Lie derivative, , is ‘derivative-index-valued’, the first two terms on the right-hand side of (2.6) are trivial. Moreover, if this parameter is ‘exact’ as , the generalized Lie derivative itself vanishes identically. Now, the closure (2.7) implies that the generalized Lie derivative is itself diffeomorphism-covariant:

(2.10)

where in the last step, from (2.8), (2.9), we have used the fact that , which is exact and hence null as a diffeomorphism parameter. However, if the tensor density carries additional indices, e.g. , its generalized Lie derivative is not local-Lorentz-covariant. Hence the generalized Lie derivative is covariant for doubled-yet-gauged diffeomorphisms but not for local Lorentz symmetries. We shall fix this limitation in section 3 by further generalizing the generalized Lie derivative.

In contrast to ordinary Riemannian geometry, the infinitesimal one-form, , is not (passively) diffeomorphism covariant in doubled-yet-gauged spacetime,

(2.11)

Furthermore, it is not invariant under the coordinate gauge symmetry shift, . However, if we gauge explicitly by introducing a derivative-index-valued gauge potential, ,

(2.12)

we can ensure both the diffeomorphism covariance and the coordinate gauge symmetry invariance,

(2.13)

Utilizing the gauged infinitesimal one-form, , it is then possible to define the duality-covariant ‘proper length’ in doubled-yet-gauge spacetime [15, 14], and construct associated sigma models such as for the point particle [24, 41], bosonic string [40, 39], Green-Schwarz superstring [22] (and its coupling to the R–R sector [42]), exceptional string [43, 44], etc.

With the decomposition of the doubled coordinates, , in accordance with the form of the invariant metric,  (2.1), the section condition reads . Thus up to rotations, the section condition is generically solved by setting , removing the dependence on coordinates. It follows that and hence the coordinates are indeed gauged, .

Stringy graviton fields from the closed string massless sector
The T-duality group is a fundamental structure in Stringy Gravity. All the fields therein must assume one representation of it, such that the covariance is manifest.

The stringy graviton fields consist of the DFT dilaton, , and DFT metric, . The former gives the integral measure in Stringy Gravity after exponentiation, , which is a scalar density of unit weight. The latter is then, by definition, a symmetric element:

(2.14)

Combining and , we acquire a pair of symmetric projection matrices,

(2.15)

which are orthogonal and complete,

(2.16)

It follows that the infinitesimal variations of the projection matrices satisfy

(2.17)

Further, taking the “square roots” of the projectors,

(2.18)

we acquire a pair of DFT vielbeins, which satisfy four defining properties:

(2.19)

such that (2.15) and (2.16) hold. Essentially, , when viewed as a matrix, diagonalizes and simultaneously into ‘’ and ‘’, respectively. The presence of twofold vielbeins as well as spin groups are a truly stringy feature, as it indicates two distinct locally inertial frames existing separately for the left-moving and right-moving closed string sectors [11], and may be a testable prediction of Stringy Gravity in itself [13].

It is absolutely crucial to note that DFT [3, 4, 8] and its supersymmetric extensions [20, 21, 22] are formulatable in terms of nothing but the very fields satisfying precisely the defining relations (2.14), (2.19). The most general solutions to the defining equations turn out to be classified by two non-negative integers, . With and , the DFT metric is of the most general form [15],

(2.20)

where  i) and are symmetric, but is skew-symmetric, i.e. , ,  ;
ii) and admit kernels, ,  ;
iii) a completeness relation must be met,  .
It follows from the linear independence of the kernel eigenvectors that

With the section choice and the parameter decomposition , the generalized Lie derivative, , reduces to the ordinary (i.e. undoubled) Lie derivative, , plus -field gauge symmetry,

(2.21)

Only in the case of can and be identified with the (invertible) Riemannian metric and its inverse. The Riemannian DFT metric then takes the rather well-known form,

(2.22)

and the corresponding DFT vielbeins read

(2.23)

where and are a pair of Riemannian vielbeins for the common Riemannian metric,

(2.24)

With the non-vanishing determinant, , the DFT dilaton can be further parametrized by

(2.25)

In this way, the stringy gravitons may represent the conventional closed string massless sector, .

Other cases of are then generically non-Riemannian, as the Riemannian metric cannot be defined. They include or for non-relativistic Newton–Cartan or ultra-relativistic Carroll gravities [16], for the Gomis–Ooguri non-relativistic string [17, 18], and various chiral theories, e.g. [19].

For later use, it is worth noting that the two-indexed projectors generate in turn a pair of multi-indexed projectors,

(2.26)

satisfying

(2.27)

They are symmetric and traceless in the following sense:

(2.28)

Covariant derivatives with stringy Christoffel symbols and spin connections
The ‘master’ covariant derivative in Stringy Gravity,

(2.29)

is equipped with the stringy Christoffel symbols of the diffeomorphism connection [9],

(2.30)

and the spin connections for the twofold local Lorentz symmetries [10],

(2.31)

In the above, we set

(2.32)

which, ignoring any local Lorentz indices, acts explicitly on a tensor density with weight as

(2.33)

The stringy Christoffel symbols (2.30) can be uniquely determined by requiring three properties:

  • full compatibility with all the stringy graviton fields,

    (2.34)

    which implies, in particular,

    (2.35)
  • a cyclic property (traceless condition),

    (2.36)

    which makes compatible with the generalized Lie derivative (2.6) as well as the C-bracket (2.8), such that we may freely replace the ordinary derivatives therein by ,

    (2.37)
  • projection constraints,

    (2.38)

    which ensure the uniqueness.

Unlike the Christoffel symbols in GR, there exist no normal coordinates where the stringy Christoffel symbols would vanish pointwise. The Equivalence Principle holds for the point particle but not for the string [13, 14].

Once the stringy Christoffel symbols are fixed, the spin connections (2.31) follow immediately from the compatibility with the DFT vielbeins,

(2.39)

The master derivative is also compatible with the two sets of local Lorentz metrics and gamma matrices,

(2.40)

such that, as in GR,

(2.41)

The master derivative (2.29) acts explicitly as

(2.42)

Unsurprisingly the master derivative is completely covariant for the twofold local Lorentz symmetries. The characteristic of the master derivative, , as well as is that they are actually ‘semi-covariant’ under doubled-yet-gauged diffeomorphisms: the stringy Christoffel symbols transform as

(2.43)

such that and are not automatically diffeomorphism-covariant, e.g.