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Non-local Gravity

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Infrared Non-local Modifications of General Relativity

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

We are now ready to consider generally-covariant extensions of the non-local field theories introduced in the second chapter.

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Notes

  1. 1.

    Indeed, as shown in Appendix A.3.2, the property \([\square , \square _\mathrm{r}^{-1}] = 0\) for fields with finite past still holds for globally hyperbolic space-times.

  2. 2.

    Simply put, unlike in the case of differential forms, the presence of symmetric pairs of indices when \(s \ge 2\) forces the use of \(\nabla _{\mu }\) in the action. This in turn implies that the gauge symmetry also depends on \(\nabla \) and can therefore not be achieved on arbitrary space-times.

  3. 3.

    In the Proca case the equation of motion in this form is simply modified by \(\square \rightarrow \square - m^2\).

  4. 4.

    Indeed, as discussed in Sect. 3.2.1, one cannot quantize a non-local theory without either enlarging the set of solutions in the classical limit, or losing unitarity.

  5. 5.

    These overall factors are not seen in the linearized limit because they multiply second-order terms in the action, or first-order terms in the equations of motion, and thus reduce to \(e^{c \theta } \rightarrow 1\).

  6. 6.

    The above problems could be resolved if we replace the fixed masses by a scalar field \(\phi \) sitting in a non-trivial minimum of its potential and transforming homogeneously under (4.1.34)

    This allows us to use all the curvature invariants, since we can compensate their inhomogeneous transformation with the one of the kinetic term of \(\phi \), while at the same time there are no fixed masses and thus no leftover exponential factors under (4.1.34). The problem now however is that we have one more dynamical field \(\phi \) and the gauge symmetry either neutralizes the latter or the scalar mode in \(g_{\mu \nu }\), not both. Thus, we still have one more dynamical field than what we started with.

  7. 7.

    Indeed, the direct use of Lagrange multipliers to enforce relations among fields is rather the usual procedure [13, 18–27].

  8. 8.

    In the Weyl model the tensor structure will correspond to the one of a 4-tensor, but since the source is a 2-tensor, the saturated propagator will reveal the same type of structure as the one of \(h_{\mu \nu }\).

  9. 9.

    This is similar to what happens in Barvinsky’s non-local theory (3.3.2) [12]. Indeed, the linearized action is the one of GR, and has thus a healthy propagator, but the non-linear localized action contains an auxiliary tensor \(\varphi ^{\mu \nu }\) on top of the metric, and the latter has obviously ghost modes. We thus have that the propagator of the diagonalized/localized theory has ghost poles that are compensated by healthy ones, as is clear from Eq. (28) of [12]. Thus, the ghost propagator simply appears to shift the graviton propagator, canceling it exactly for \(\alpha = 1\).

  10. 10.

    With the correct normalization for the source.

  11. 11.

    Of course here too we could use the Weyl tensor but only if we accept derivatives acting on curvature, i.e. terms like \(_a {(} \tilde{\square }^{-2}\nabla ^{\rho } \nabla ^{\sigma } W_{\mu \rho \nu \sigma } {)}^\mathrm{T}\).

  12. 12.

    To see this, suppose such an action exists. Then, term \(\sim \) \( \nabla _{(\mu } \phi _{\nu )}\) in the first equation, which would correspond to the equation of motion of \(g_{\mu \nu }\), would be a total derivative \(\nabla _{\mu } \phi ^{\mu }\) in the hypothetical action. Thus, such “friction” terms cannot derive from an action.

  13. 13.

    So that we can neglect non-linearities for h.

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  1. Department of Physics, Institute for Strings, Cosmology and Astroparticle Physics, Columbia University, New York, USA

    Ermis Mitsou

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Mitsou, E. (2016). Non-local Gravity. In: Infrared Non-local Modifications of General Relativity . Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-31729-8_4

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