Abstract
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 mathrm {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}=8pi G T_{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.
Highlights
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
It is a prediction of Double Field Theory (DFT) that there must in principle exist two distinct kinds of fermions [13]
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
Summary
We review DFT following the geometrically logical – rather than historical – order: (i) conventions, (ii) the doubled-yetgauged coordinate system with associated diffeomorphisms, (iii) the field content of stringy gravitons, (iv) DFT extensions of the Christoffel symbols and spin connection, and where M is said to be derivative-index-valued. The generalized Lie derivative is covariant for doubled-yet-gauged diffeomorphisms but not for local Lorentz symmetries. We shall fix this limitation in Sect. The characteristic of the master derivative, DA, as well as ∇A is that they are ‘semi-covariant’ under doubled-yet-gauged diffeomorphisms: the stringy Christoffel symbols transform as δξ C AB = Lξ C AB + 2[(P + P )C AB F DE. In a similar fashion to (A.12), the (localLorentz-symmetric) matter Lagrangian transforms, up to total derivatives ( ), as δ Lmatter δ
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