and a scalar field. This scalar mode exerts an extra attraction in the massive case compared to the massless case. Hence, if one wants measurements of the force exerted between nonrelativistic masses to agree, the coupling constant of the massive theory should be smaller than that of the massless theory. But light bending is blind to the scalar sector, because the stress-energy tensor of light is traceless. Hence, provided the two theories agree on the force between nonrelativistic probes, the massive theory would predict a smaller light bending than the massless one. Vainshtein screening It was argued by Vainshtein two years later that the vDVZ discontinuity is an artifact of the linear theory, and that the predictions of general relativity are in fact recovered at small scales when one takes into account nonlinear effects, i.e., higher than quadratic terms in . Heuristically speaking, within a region known as the Vainshtein radius, fluctuations of the scalar mode become nonlinear, and its higher-order derivative terms become larger than the canonical kinetic term. Canonically normalizing the scalar around this background therefore leads to a heavily suppressed kinetic term, which damps fluctuations of the scalar within the Vainshtein radius. Because the extra force mediated by the scalar is proportional to (minus) its gradient, this leads to a much smaller extra force than we would have calculated just using the linear Fierz–Pauli theory. This phenomenon, known as Vainshtein screening, is at play not just in massive gravity, but also in related theories of modified gravity such as DGP and certain scalar-tensor theories, where it is crucial for hiding the effects of modified gravity in the solar system. This allows these theories to match terrestrial and solar-system tests of gravity as well as general relativity does, while maintaining large deviations at larger distances. In this way these theories can lead to cosmic acceleration and have observable imprints on the large-scale structure of the Universe without running afoul of other, much more stringent constraints from observations closer to home. The Boulware–Deser ghost As a response to Freund–Maheshwari–Schonberg finite-range gravity model, and around the same time as the vDVZ discontinuity and Vainshtein mechanism were discovered, David Boulware and Stanley Deser found in 1972 that generic nonlinear extensions of the Fierz–Pauli theory reintroduced the dangerous ghost mode; the tuning which ensured this mode's absence at quadratic order was, they found, generally broken at cubic and higher orders, reintroducing the