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 O(D, D) 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, G AB = 8π GT AB , which we dub the Einstein Double Field Equations. As an example, we study the most general D = 4 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.

Taking O(D, D) covariant field variables as its truly fundamental constituents, Double Field Theory can accommodate not only conventional supergravity but also non-Riemannian gravities that may be classified by two non-negative integers, (n,n). Such non-Riemannian backgrounds render a propagating string chiral and anti-chiral over n andn dimensions respectively. Examples include, but are not limited to, Newton-Cartan, Carroll, or Gomis-Ooguri. Here we analyze the variational principle with care for a generic (n,n) non-Riemannian sector. We recognize a nontrivial subtlety for nn = 0, which seems to suggest that the various non-Riemannian gravities should better be identified as different solution sectors of Double Field Theory rather than viewed as independent theories. Separate verification of our results as string worldsheet beta-functions may enlarge the scope of the string landscape far beyond Riemann. arXiv:1909.10711v1 [hep-th] 24 Sep 2019 1.1 Double Field Theory as the O(D, D) completion of General Relativity While the initial motivation of Double Field Theory was to reformulate supergravity in an O(D, D) manifest manner [2-7] ([58-60] for reviews), through subsequent further developments [61-64], DFT has evolved and can be now identified as Stringy Gravity, i.e. pure gravitational theory that string theory seems to predict foremost. 1 More specifically, DFT is the string theory based, O(D, D) completion of General Relativity: taking the O(D, D) symmetry of string theory as the first principle, this Stringy Gravity assumes the whole massless NS-NS sector of closed string as the fundamental gravitational multiplet and interacts with other superstring sectors (R-R [67-69], R-NS [70], and heterotic Yang-Mills [71-73]). Having said that, regardless of supersymmetry, it can also couple to various matter fields which may appear in lower dimensional effective field theories [70, 74, 75], just as General Relativity (GR) does so. In particular, supersymmetric extensions have been completed to the full (i.e. quartic) order in fermions for D = 10 case 1 At least formally let alone its phenomenological validity, c.f. [65, 66].

A Symmetry Principle has been shown to augment unambiguously the Einstein Field Equations, promoting the whole closed-string massless NS-NS sector to stringy graviton fields. Here we consider its weak field approximation, take a non-relativistic limit, and derive the stringy augmentation of Newton Gravity:Not only the mass density ρ but also the current density K is intrinsic to matter. Sourcing H which is of NS-NS H-flux origin, K is nontrivial if the matter is 'stringy'. H contributes quadratically to the Newton potential, but otherwise is decoupled from the point particle dynamics, i.e.ẍ = −∇Φ. We define 'stringization' analogous to magnetization and discuss regular as well as monopole-like singular solutions.

We propose a novel Kaluza-Klein scheme which assumes the internal space to be maximally non-Riemannian, meaning that no Riemannian metric can be defined for any subspace. Its description is only possible through Double Field Theory but not within supergravity. We spell out the corresponding Scherk-Schwarz twistable Kaluza-Klein ansatz, and point out that the internal space prevents rigidly any graviscalar moduli. Plugging the same ansatz into higher-dimensional pure Double Field Theory and also to a known doubled-yet-gauged string action, we recover heterotic supergravity as well as heterotic worldsheet action. In this way, we show that 1) supergravity and Yang-Mills theory can be unified into higher-dimensional pure Double Field Theory, free of moduli, and 2) heterotic string theory may have a higher-dimensional non-Riemannian origin.

In string theory the closed-string massless NS-NS sector forms a multiplet of $$\mathbf {O}(D,D)$$ O ( D , D ) symmetry. This suggests a specific modification to General Relativity in which the entire NS-NS sector is promoted to stringy graviton fields. Imposing off-shell $$\mathbf {O}(D,D)$$ O ( D , D ) symmetry fixes the correct couplings to other matter fields and the Einstein field equations are enriched to comprise $$D^{2}+1$$ D 2 + 1 components, dubbed recently as the Einstein Double Field Equations. Here we explore the cosmological implications of this framework. We derive the most general homogeneous and isotropic ansatzes for both stringy graviton fields and the $$\mathbf {O}(D,D)$$ O ( D , D ) -covariant energy-momentum tensor. Crucially, the former admits space-filling magnetic H-flux. Substituting them into the Einstein Double Field Equations, we obtain the $$\mathbf {O}(D,D)$$ O ( D , D ) completion of the Friedmann equations along with a generalized continuity equation. We discuss how solutions in this framework may be characterized by two equation-of-state parameters, w and $$\lambda $$ λ , where the latter characterizes the relative intensities of scalar and tensor forces. When $$\lambda +3w=1$$ λ + 3 w = 1 , the dilaton remains constant throughout the cosmological evolution, and one recovers the standard Friedmann equations for generic matter content (i.e. for any w). We further point out that, in contrast to General Relativity, neither an $$\mathbf {O}(D,D)$$ O ( D , D ) -symmetric cosmological constant nor a scalar field with positive energy density gives rise to a de Sitter solution.