Scalar field as a null dust

TL;DR

This paper demonstrates that a massless, minimally coupled scalar field with a null gradient behaves exactly like a null dust fluid, whereas nonminimally coupled scalar fields generally do not, unless they possess a Killing vector symmetry.

Contribution

It establishes the equivalence of a minimally coupled scalar field to null dust and clarifies conditions under which scalar-tensor gravity scalars can be similarly represented.

Findings

Minimally coupled scalar fields with null gradients are equivalent to null dust.

Nonminimally coupled scalar fields generally cannot be represented as null dust.

Scalar-tensor gravity scalars require a Killing vector for null dust representation.

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.

Equations

$$S_{(\varphi )}=\int d^{4}x\sqrt{-g}\left[-\frac{1}{2}{\nabla }^c\varphi {\nabla }_c\varphi -V(\varphi )\right],$$

$$T_{ab}^{(\varphi )}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{(\varphi )}}{\delta g^{ab}}={\nabla }_a\varphi {\nabla }_b\varphi -\frac{1}{2}{g}_{ab}{\nabla }^c\varphi {\nabla }_c\varphi -V(\varphi )g_{ab}$$

$$\square \varphi -\frac{dV}{d\varphi }=0,$$

$$u_a=\frac{{\nabla }_a\varphi }{\sqrt{-{\nabla }^c\varphi {\nabla }_c\varphi }}.$$

$$T_{ab}^{(nd)}=\rho k_a{k}_b,k_c{k}^c=0,$$

$$S_{(ST)}$$

$$R_{ab}-\frac{1}{2}{g}_{ab}R=\frac{8\pi }{\Phi }{T}_{ab}^{(m)}$$

$$T_{ab}^{(\Phi )}=\left(P_{(\Phi )}+{\rho }_{(\Phi )}\right)u_a{u}_b+P_{(\Phi )}{g}_{ab}+q_a^{(\Phi )}{u}_b+q_b^{(\Phi )}{u}_a.$$

$$u_a=\frac{{\nabla }_a\Phi }{\sqrt{-{\nabla }^c\Phi {\nabla }_c\Phi }},u_c{u}^c=-1,$$

$$q_c^{(\Phi )}{q}_{(\Phi )}^c>0,q_c^{(\Phi )}{u}^c=0.$$

$$T_{ab}^{(\varphi )}={\nabla }_a\varphi {\nabla }_b\varphi \dot{=}{k}_a{k}_b,k_a\dot{=}{\nabla }_a\varphi ,\rho \dot{=}1.$$

$$k^b{\nabla }_b{k}_c=-\left({\nabla }^b{k}_b\right)k_c,$$

$$T_{ab}^{(\varphi )}={\nabla }_a\varphi {\nabla }_b\varphi =\rho k_a{k}_b,k_a=\frac{{\nabla }_a\varphi }{\sqrt{\rho }}.$$

$${\nabla }^b{T}_{ab}^{(\varphi )}=\left(k^b{\nabla }_b\rho +\rho {\nabla }^b{k}_b\right)k_a+\rho k^b{\nabla }_b{k}_a=0.$$

$${\nabla }^a{k}_a=-\frac{{\nabla }^a\rho {\nabla }_a\varphi }{2{\rho }^{3/2}}$$

$$k^b{\nabla }_b{k}_c=\frac{1}{\rho }\left(-k^b{\nabla }_b\rho +\frac{k^b{\nabla }_b\rho }{2\sqrt{\rho }}\right)k_c.$$

$$k^a{\nabla }_b{k}_a=0$$

$${\nabla }^c{k}_c=0.$$

$$\square k_a-R_{ab}{k}^b=0,$$

$$\square k_a={\nabla }^b{\nabla }_b{k}_a={\nabla }^b\left({\nabla }_b{\nabla }_a\varphi \right)={\nabla }^b\left({\nabla }_a{\nabla }_b\varphi \right)={\nabla }^b{\nabla }_a{k}_b$$

$$\Big{[}\nabla_{a},\nabla_{b}\Big{]}k_{c}={R_{abc}}^{d}k_{d}=R_{abcd}k^{d}$$

$${\nabla }_b{\nabla }_a{k}_c={\nabla }_a{\nabla }_b{k}_c-R_{abcd}{k}^d,$$

$${\nabla }_b{\nabla }_a{k}^b={\nabla }_a{\nabla }_b{k}^b-{{R_{ab}}^b}_d{k}^d\equiv R_{ad}{k}^d$$

$$\square k^c=0.$$

$$T_{ab}^{(\Phi )}=\frac{\omega }{{\Phi }^2}{\nabla }_a\Phi {\nabla }_b\Phi +\frac{{\nabla }_a{\nabla }_b\Phi }{\Phi },$$

$$T_{ab}^{(\Phi )}=\omega k_a{k}_b+\frac{{\nabla }_a{\nabla }_b\Phi }{\Phi }.$$

$${\nabla }^c{k}_c=\frac{\square \Phi }{\Phi }-\frac{{\nabla }^a\Phi {\nabla }_a\Phi }{{\Phi }^2}=0,$$

$$\frac{{\nabla }_a{\nabla }_b\Phi }{\Phi }={\nabla }_{(a}\left(\frac{{\nabla }_{b)}\Phi }{\Phi }\right)+\frac{{\nabla }_a\Phi {\nabla }_b\Phi }{{\Phi }^2},$$

$$T_{ab}^{(\Phi )}=\left(\omega +1\right)k_a{k}_b+\frac{{\nabla }_a{k}_b+{\nabla }_b{k}_a}{2},$$

$$\square k_a+R_{ab}{k}^b=0$$

Full text

I Introduction

Scalar fields give rise to the simplest field theory and are ubiquitous in cosmology, particle physics, and theories and models of classical and quantum gravity Waldbook ; Carroll ; Bojowald ; Linde ; AmendolaTsujikawabook ; reviews . A minimally coupled scalar field $\phi$ with canonical kinetic energy is described by the action

[TABLE]

where $g$ 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. Waldbook ). The scalar field stress-energy tensor is

[TABLE]

and $\phi$ satisfies the Klein-Gordon equation obtained from the covariant conservation $\nabla^{b}T_{ab}^{(\phi)}=0$,

[TABLE]

where $\Box\equiv g^{ab}\nabla_{a}\nabla_{b}$ is the curved space d’Alembertian. The question of whether this scalar field can be represented as an effective fluid has been posed, and answered, long ago Ellis71 ; Madsen ; myphifluid . If the gradient $\nabla^{c}\phi$ is timelike in a spacetime region, the scalar field stress-energy tensor (1.2) is equivalent to a perfect fluid stress-energy tensor Ellis71 ; Madsen ; myphifluid with fluid four-velocity

[TABLE]

Spacelike scalar field gradients have also been considered myphifluid ; Semiz , although they are not physically very relevant. Here we focus on the remaining case, apparently not covered in the literature, in which $\nabla^{c}\phi$ is null on a spacetime region, $\nabla^{c}\phi\nabla_{c}\phi=0$. A 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 Waldbook . A null dust KSMcH ; KucharBicak corresponds to the limit in which the four-velocity becomes null, and is described by

[TABLE]

where $\rho>0$ and the trace $T\equiv{T^{c}}_{c}$ vanishes. The covariant conservation of $T_{ab}^{(nd)}$ implies that $k^{c}$ is also geodesic KSMcH ; KucharBicak . The null dust is interpreted as a coherent zero rest mass field propagating at the speed of light in the null direction $k^{a}$. Naturally, a null dust can be realized by propagating electromagnetic Robinson or gravitational Griffiths ; KSMcH waves. Here we consider the analogous problem for a massless scalar field. The null dust plays a non-negligible role in the literature on classical and quantum gravity, especially in the study of Vaidya Vaidya , $pp$-wave KSMcH ; Krasinski ; Griffiths , Robinson-Trautman RobinsonTrautman , and twisting twisting solutions of the Einstein or Einstein-Maxwell equations, classical and quantum gravitational collapse, horizon formation, mass inflation exactcollapse , black hole evaporation evaporation , and canonical Lagrangians and Hamiltonians KucharBicak ; Horvath ; Myers . More recently, null dust has been studied in relation with the fluid-gravity correspondence and holography various ; Minwalla ; Myers . The collision of special scalar field-null dust solutions was studied long ago in ndsf and scalar-Vaidya solutions are of interest in the AdS/CFT correspondence Minwalla ; Myers . 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 (1.5) 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 KucharBicak .

A natural question arises: is a scalar field $\phi$ with null gradient in a region of spacetime equivalent to an effective null dust? Special solutions with this property are known ndsf ; Griffiths . As expected, the answer is affirmative but only if $\phi$ is massless and there is no potential $V(\phi)$. Moreover, being generated by a scalar, this effective null dust is irrotational, as expected. A second question also arises naturally. Given that scalar fields play an important role in theories of gravity alternative to Einstein’s general relativity (GR) searched for in tests of gravity tests , the simplest alternative being Brans-Dicke theory BD and its scalar-tensor generalizations ST which contain a gravitational scalar field $\Phi$ (approximately equivalent to the inverse of the effective gravitational coupling strength), is it possible that this scalar is equivalent to an effective null dust if $\nabla^{c}\Phi\nabla_{c}\Phi=0$? The scalar-tensor action is BD ; ST

[TABLE]

where the function $\omega(\Phi)$ (which was a constant parameter in the original Brans-Dicke theory BD ) is the “Brans-Dicke coupling”, $V(\Phi)$ is a scalar field potential (absent in the original Brans-Dicke theory), and $S_{(m)}=\int d^{4}x\sqrt{-g}\,{\cal L}_{(m)}$ describes the matter sector of the theory. The scalar-tensor field equations obtained from the variation of this action take the form of effective Einstein equations,

[TABLE]

where $T^{(m)}\equiv g^{ab}T_{ab}^{(m)}$ is the trace of the matter stress-energy tensor $T_{ab}^{(m)}$. It is well known that, if the gradient $\nabla^{c}\Phi$ is timelike, the Brans-Dicke-like scalar $\Phi$ acts as an effective fluid source which, however, is not a perfect fluid as in the minimally coupled case, but is instead an (effective) imperfect fluid with a heat flux density, described by an effective stress-energy tensor of the form Madsen ; Pimentel89 ; Kolassisetal88 ; Pimenteletal6 ; FaraoniCote

[TABLE]

When $\nabla^{c}\Phi\nabla_{c}\Phi<0$, the effective fluid four-velocity is

[TABLE]

and

[TABLE]

Explicit expressions of the effective fluid quantities $\rho_{\Phi},P_{\Phi}$, and $q_{a}^{(\Phi)}$ are given in Refs. Pimentel89 ; FaraoniCote .

An important conceptual step has been taken in moving the question from the minimally coupled scalar $\phi$ to the Brans-Dicke-like field $\Phi$: 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 equations (LABEL:BDfe1). 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 $\Phi$ cannot be regarded as a null dust, which is a perfect fluid. In fact, in the stress-energy $T_{ab}^{(\Phi)}$ given by Eq. (1.9), the terms $\nabla_{a}\nabla_{b}\Phi,\Box\Phi$ linear in the second covariant derivatives of $\Phi$ always introduce an (effective) imperfect fluid component, i.e., a heat flux. By contrast, the canonical terms $\nabla_{a}\Phi\nabla_{b}\Phi,\nabla^{c}\Phi\nabla_{c}\Phi$ 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 ourBDlimit . 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.

II Massless canonical scalar field as an effective null dust in GR

In this section we restrict to Einstein’s theory. In GR, a canonical scalar field (whether it is a test field or a gravitating one) has stress-energy tensor (1.2) and satisfies the Klein-Gordon equation (1.3). We assume that $\nabla^{c}\phi\nabla_{c}\phi=0$ in a certain spacetime region, to which we will implicitely limit ourselves in the rest of this work. Then, the requirement that $T_{ab}^{(\phi)}$ assumes the null dust form $T_{ab}=\rho\,k_{a}k_{b}$ with $k^{c}$ null, necessarily implies $T^{(\phi)}=0$ and $V(\phi)=0$, as expected. Then, the Klein-Gordon equation reduces to $\Box\phi=0$ and (1.2) reduces to $T_{ab}^{(\phi)}=\nabla_{a}\phi\nabla_{b}\phi$. 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

[TABLE]

For a general null dust, this representation excludes the affine parametrization for the null geodesics tangent to $k^{a}$. However, here these two representations become compatible because of the irrotationality of this scalar field-dust. In fact, the covariant conservation equation $\nabla^{b}T_{ab}^{(\phi)}=0$ yields

[TABLE]

but the divergence of $k^{c}$ vanishes because $k_{c}$ is a pure gradient. Indeed, $\nabla^{a}k_{a}=\Box\phi=0$ by virtue of the Klein-Gordon equation and Eq. (2.2) reduces to the affinely parametrized geodesic equation $k^{b}\nabla_{b}k_{c}=0$. For a scalar field-null dust, therefore, the representation with unit energy density $\rho=1$ coincides with the affine parametrization along the null geodesics tangent to $k^{a}$, due to irrotationality.

The second a priori possibility consists of keeping $\rho$ general and choosing

[TABLE]

Covariant conservation then gives

[TABLE]

Using the Klein-Gordon equation $\Box\phi=0$, we now have

[TABLE]

so that Eq. (2.4) becomes

[TABLE]

Also in this representation it could seem that affine parametrization is incompatible with the choice $\rho\equiv 1$ but, again, this is not the case. Let us adopt the representation $\rho\,\dot{=}\,1$; then we have $k^{c}k_{c}=0$,

[TABLE]

(which follows from taking the covariant derivative of the normalization $k^{c}k_{c}=0$), and

[TABLE]

At this point is is easy to derive also a wave equation for the null vector field $k^{c}$,

[TABLE]

in which $k^{c}=\nabla^{c}\phi$ couples explicitly to the curvature (to the Ricci tensor $R_{ab}$) even though $\phi$ does not couple (to the Ricci scalar $R$) in the Klein-Gordon equation. To obtain Eq. (2.9), first note that

[TABLE]

and that the identity Waldbook

[TABLE]

yields

[TABLE]

so that

[TABLE]

and $\Box k_{a}=\nabla_{b}\nabla_{a}k^{b}=R_{cd}k^{d}$.

Thus far, the equations of this section apply whether $\phi$ is a test or a gravitating field. In the case in which $\phi$ is the only matter source of the Einstein equations, one can go further and note that in this special situation $T^{(\phi)}=0$ yields $R=0$ and the Einstein equations reduce to $R_{ab}=8\pi\nabla_{a}\phi\nabla_{b}\phi$, from which it follows that $R_{ab}k^{b}=8\pi k_{a}k_{b}k^{b}=0$ and

[TABLE]

III Scalar-tensor gravity

Let us consider now scalar-tensor gravity. In this case, the Brans-Dicke-like field $\Phi$ cannot be a test field and always gravitates. Assuming the gradient $\nabla^{c}\Phi$ to be null, the effective stress-energy tensor $T_{ab}^{(\Phi)}$ cannot be a null dust. In fact, assume $V(\Phi)\equiv 0$ and $T^{(m)}=0$, then Eq. (LABEL:BDfe2) gives $\Box\Phi=0$ and (1.9) reduces to

[TABLE]

which satisfies $T^{(\Phi)}=0$. One can set $k_{a}=\nabla_{a}\ln\Phi$ and

[TABLE]

Then $k^{c}$ is null and is also divergence-free since

[TABLE]

where the first term vanishes because of the field equation for $\Phi$. Using the identity

[TABLE]

one obtains

[TABLE]

which is not the stress-energy tensor of a null dust due to the second term on the right hand side. Unless $k^{a}$ is a (null) Killing vector field Waldbook satisfying the Killing equation $\nabla_{a}k_{b}+\nabla_{b}k_{a}=2\nabla_{a}\nabla_{b}\Phi=0$, this second term does not vanish. It is clear that this restriction can be obeyed only by very special geometries and, therefore, we will not pursue it further.

By proceeding in analogy with Sec. II, one derives the wave equation for $k^{c}$

[TABLE]

(note the opposite sign to Eq. (2.9) in the coupling to the Ricci tensor). In fact, the covariant conservation $\nabla^{b}T_{ab}^{(\Phi)}=0$ gives

[TABLE]

where the first bracket vanishes because $k^{a}$ is divergence-free and because of the affinely parameterized geodesic equation. The use of Eq. (2.13) then yields Eq. (3.6). If, in addition, $T^{(m)}=0$, Eq. (3.6) simplifies to $\Box k^{c}=0$.

IV 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 $\Phi$ in scalar-tensor gravity BD ; ST , we have found that the equivalence does not carry over, unless the null gradient of $\Phi$ is also a Killing vector. This is a very strong restriction, which makes this situation rather uninteresting from the physical point of view.

Acknowledgements.

This work is supported, in part, by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2016-03803 to V.F.).