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We consider a scalar field action for which the Lagrangian density is a powerof the massless Klein-Gordon Lagrangian. The coupling of gravity to this matteraction is considered. In this case, we show the existence of nontrivial scalarfield configurations with vanishing energy-momentum tensor on any static,spherically symmetric vacuum solutions of the Einstein equations. Theseconfigurations in spite of being coupled to gravity do not affect the curvatureof spacetime. The properties of this particular matter action are alsoanalyzed. For a particular value of the exponent, the extended Klein-Gordonaction is shown to exhibit a conformal invariance without requiring theintroduction of a nonminimal coupling. We also establish a correspondencebetween this action and a non-relativistic isentropic fluid in one fewerdimension. This fluid can be identified with the (generalized) Chaplygin gasfor a particular value of the power. It is also shown that the non-relativisticfluid admits, apart from the Galileo symmetry, an additional symmetry whoseaction is a rescaling of the time.Comment: 7 pages, two columns. Minor change

Extended Klein-Gordon action, gravity and nonrelativistic fluid