Einstein double field equations

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 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.

Stringy Newton gravity with H -flux

“…Developments over the last decade in an area of string theory, now called Double Field Theory (DFT), have gradually unveiled a new form of pure gravity [5][6][7][8][9][10][11][12][13][14][15]. It is based on the O(D, D) Symmetry Principle, with D denoting the spacetime dimension.…”

“…[19]), R-R sector [20][21][22][23], full-order supersymmetrizations [24,25], point particles [26,27], fundamental strings [28][29][30][31][32][33], and the Standard Model itself [34]. Naturally, the Einstein Field Equations are augmented to an O(D, D)-symmetric form [15] (c.f. [35] for a short summary): [14], while T AB is the on-shell conserved energy-momentum tensor, defined through the variation of the matter Lagrangian with respect to {H AB , d} [15].…”

“…Naturally, the Einstein Field Equations are augmented to an O(D, D)-symmetric form [15] (c.f. [35] for a short summary): [14], while T AB is the on-shell conserved energy-momentum tensor, defined through the variation of the matter Lagrangian with respect to {H AB , d} [15]. Remarkably, the perfectly O(D, D)-symmetric vacua, satisfying G AB = 0, turned out to be a topological phase which allows no moduli and no interpretation within Riemannian geometry, thus escaping beyond the realm of GR [30,36,37].…”

“…We stress that for each matter content, the O(D, D) symmetry determines its coupling to the closed-string massless NS-NS sector of {B µν , g µν , φ} and hence fixes K µν , T (0) completely [15]. For example, a point particle should couple to the string frame metric g µν onlyminimally in the standard way-resulting in K [µν] = 0, T (0) = 0, and…”

“…Further, in contrast, gauge bosons couple to g and φ, but do not interact with the B-field (44) [34]. While the B-field does not interact with the electromagnetic force nor point particles, its electric H-flux nevertheless contributes to the mass formula [15],…”