# Scalar field as a null dust

## Abstract

We show that a canonical, minimally coupled scalar field which is non-self interacting and massless is equivalent to a null dust fluid (whether it is a test or a gravitating field), in a spacetime region in which its gradient is null. Under similar conditions, the gravitating and nonminimally coupled Brans-Dicke-like scalar of scalar-tensor gravity, instead, cannot be represented as a null dust unless its gradient is also a Killing vector field.

## 1 Introduction

*g*is the determinant of the spacetime metric \(g_{ab}\), \(\nabla _a\) denotes the covariant derivative operator of \(g_{ab}\), and \(V(\phi )\) is the scalar field potential (we use units in which the speed of light and Newton’s constant are unity and we follow the notation of Ref. [1]). The scalar field stress-energy tensor is

*dust*corresponds to a fluid with timelike four-velocity \(u^a\), energy density \(\rho \), and zero pressure described by the energy-momentum tensor \(T_{ab}^{(dust)}=\rho \, u_a u_b \). Because there is no pressure gradient, the fluid elements of the dust follow timelike geodesics, as can be deduced from covariant conservation [1]. A

*null dust*[15, 16] corresponds to the limit in which the four-velocity becomes null, and is described by

*pp*-wave [15, 18, 21], Robinson-Trautman [22], and twisting [23, 24] solutions of the Einstein or Einstein-Maxwell equations, classical and quantum gravitational collapse, horizon formation, mass inflation [25, 26, 27, 28, 29, 30, 31, 32, 33, 34], black hole evaporation [35, 36, 37], and canonical Lagrangians and Hamiltonians [16, 38, 39]. More recently, null dust has been studied in relation with the fluid-gravity correspondence and holography [39, 40, 41, 42, 43, 44, 45, 46, 47, 48]. The collision of special scalar field-null dust solutions was studied long ago in [49, 50] and scalar-Vaidya solutions are of interest in the AdS/CFT correspondence [39, 48]. Since the null vector \(k^a\) can be rescaled by a positive function without changing its causal character, it is possible to find a representation of the stress-energy tensor (5) in which \(\rho \, \dot{=} \, 1\) and \(T_{ab}^{(nd)} \, \dot{=} \, k_a k_b\) (a dot over an equal sign denotes the fact that the equality is only valid in that representation). However, in general, in this representation, the null geodesics tangent to \(k^a\) are not affinely parametrized unless \(k^a\) is divergence-free [16].

An important conceptual step has been taken in moving the question from the minimally coupled scalar \(\phi \) to the Brans-Dicke-like field \(\varPhi \): the former can be a test field or a matter source of the Einstein equations, while the latter always contributes to sourcing the metric in the scalar-tensor field Eq. (7). This distinction will have to be kept in mind in the following sections.

It turns out that, contrary to its minimally coupled counterpart \(\phi \), a Brans–Dicke-like field \(\varPhi \) cannot be regarded as a null dust, which is a perfect fluid. In fact, in the stress-energy \(T_{ab}^{(\varPhi )}\) given by Eq. (9), the terms \(\nabla _a \nabla _b \varPhi , \Box \varPhi \) linear in the second covariant derivatives of \(\varPhi \) always introduce an (effective) imperfect fluid component, *i.e.*, a heat flux. By contrast, the canonical terms \(\nabla _a \varPhi \nabla _b \varPhi , \nabla ^c \varPhi \nabla _c \varPhi \) quadratic in the first order covariant derivatives correspond to (effective) perfect fluid terms.

The effective imperfect fluid description of scalar-tensor gravity has been applied recently to elucidate anomalies in the limit to GR of electrovacuum Brans-Dicke theory [63]. However, it is not possible to do so for the corresponding scalar-tensor solutions describing null fields because a null dust description of scalar-tensor gravity is missing in this case.

## 2 Massless canonical scalar field as an effective null dust in GR

*A priori*, there are two possibilities to identify this stress-energy tensor with that of a null dust. One could choose the representation in which

*a priori*possibility consists of keeping \(\rho \) general and choosing

*R*) in the Klein-Gordon equation. To obtain Eq. (20), first note that

## 3 Scalar-tensor gravity

## 4 Conclusions

We have filled a gap in the literature regarding the equivalence between a scalar field \(\phi \) and an effective null dust when the gradient \(\nabla ^c \phi \) of this scalar is a null vector field over a region of spacetime. A canonical, minimally coupled, free and massless scalar field with null gradient is equivalent to an irrotational null dust. When attempting to generalize this property to a gravitating Brans-Dicke-like scalar field \(\varPhi \) in scalar-tensor gravity [55, 56, 57, 58], we have found that the equivalence does not carry over, unless the null gradient of \(\varPhi \) is also a Killing vector. This is a very strong restriction, which makes this situation rather uninteresting from the physical point of view. There does not seem to be much room to apply or extend the null fluid picture. However, it is true that non-trivial topological null fluid solutions exist which can be mapped into knotted solutions of source-free electromagnetism [64]. Perhaps analogous topological configurations exist in the scalar-tensor gravity equivalent of a null dust. This issue will be explored in the future.

## Notes

### Acknowledgements

We are grateful to Carlos Hoyos for pointing out Ref. [64]. This work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2016-03803 to V.F.).

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