Wyman's "other" scalar field solution, Sultana's generalization, and their Brans-Dicke and $R^2$ relatives

TL;DR

This paper explores Wyman's lesser-known scalar field solutions, their generalizations, and their relations to Brans-Dicke and quadratic gravity theories, revealing their properties, singularities, and physical interpretations.

Contribution

It provides a detailed analysis of Wyman's solutions, their extensions, and their connections to alternative gravity theories, including explicit mappings and physical insights.

Findings

Wyman's solutions contain central naked singularities.

Conformal transformations relate these solutions to Brans-Dicke geometries.

The geometries also solve purely quadratic gravity equations.

Abstract

Wyman's less known static and spherically symmetric solution of the Einstein-Klein-Gordon equations and its recent generalization for positive cosmological constant are discussed, showing that they contain central naked singularities. By mapping back to the Jordan frame, we obtain the conformal cousins of these geometries that solve the vacuum Brans-Dicke field equations and are time-dependent and spherical. Their physical nature is discussed and the geometry is shown to be also a solution of purely quadratic gravity.

Equations

$$d{\tilde{s}}^2=-\kappa r^{2}d{t}^2+2d{r}^2+r^{2}d{\Omega }_{(2)}^2,$$

$$\tilde{\varphi }(t)={\tilde{\varphi }}_{0}t,$$

$$\displaystyle\tilde{\mathcal{R}}_{ab}-\frac{1}{2}\,\tilde{g}_{ab}\tilde{\mathcal{R}}=\kappa\Big{(}\tilde{\nabla}_{a}\tilde{\phi}\tilde{\nabla}_{b}\tilde{\phi}-\frac{1}{2}\tilde{g}_{ab}\tilde{\nabla}^{c}\tilde{\phi}\tilde{\nabla}_{c}\tilde{\phi}\Big{)}\,,$$

$$\displaystyle\tilde{\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}}\tilde{\phi}=0\,,$$

$$d{\tilde{s}}^2=-\kappa r^{2}d{t}^2+\frac{2d{r}^2}{1-\frac{2\Lambda r^2}{3}}+r^{2}d{\Omega }_{(2)}^2,$$

$$d{\tilde{s}}^2=-A^2(t,r)d{t}^2+B^2(t,r)d{r}^2+r^{2}d{\Omega }_{(2)}^2.$$

$$\displaystyle\tilde{\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}}\tilde{\phi}=0\,,$$

$$\frac{\dot{B}}{B}=4\pi r\dot{\tilde{\varphi }}{\tilde{\varphi }}^{\prime },$$

$$\displaystyle\frac{2B^{\prime}}{B^{3}r}-\frac{1}{B^{2}r^{2}}+\frac{1}{r^{2}}=\frac{\kappa}{A^{2}}\Big{[}\dot{\tilde{\phi}}^{2}+\frac{A^{2}}{2}\Big{(}-\frac{\dot{\tilde{\phi}}^{2}}{A^{2}}+\frac{\tilde{\phi}^{\prime 2}}{B^{2}}\Big{)}+\frac{\Lambda A^{2}}{\kappa}\Big{]}\,,$$

$$\displaystyle\frac{2A^{\prime}}{Ar}-\frac{B^{2}}{r^{2}}+\frac{1}{r^{2}}=\kappa\Big{[}\tilde{\phi}^{\prime 2}-\frac{B^{2}}{2}\Big{(}-\frac{\dot{\tilde{\phi}}^{2}}{A^{2}}+\frac{\tilde{\phi}^{\prime 2}}{B^{2}}\Big{)}-\frac{\Lambda B^{2}}{\kappa}\Big{]}\,,$$

$$\displaystyle\frac{r}{B^{3}}\Big{(}\frac{A^{\prime}B}{A}-B^{\prime}-\frac{rB^{2}\ddot{B}}{A^{2}}+\frac{r\dot{A}B^{2}\dot{B}}{A^{2}}-\frac{rA^{\prime}B^{\prime}}{A}+\frac{rA^{\prime\prime}B}{A}\Big{)}=\kappa\Big{[}-\frac{r^{2}}{2}\Big{(}-\frac{\dot{\tilde{\phi}}^{2}}{A^{2}}+\frac{\tilde{\phi}^{\prime 2}}{B^{2}}\Big{)}-\frac{\Lambda r^{2}}{\kappa}\Big{]}\,,$$

$$-\infty <t<+\infty ,0\le r<\sqrt{\frac{3}{2\Lambda }}.$$

$$\tilde{\mathcal{R}}=\kappa {\tilde{g}}^{ab}{\tilde{\nabla }}_a\tilde{\varphi }{\tilde{\nabla }}_b\tilde{\varphi }=4\Lambda -\frac{{\widetilde{{\varphi }_0}}^2}{r^2},$$

$$\displaystyle\tilde{g}^{ab}\,\tilde{\nabla}_{a}r\tilde{\nabla}_{b}r=\tilde{g}^{rr}=\Big{(}1-\frac{2\Lambda r^{2}}{3}\Big{)}\frac{1}{2}=0\,,$$

$$r_H=\sqrt{\frac{3}{2\Lambda }}$$

$${\tilde{\mathcal{R}}}_{ab}=\kappa {\nabla }_a\tilde{\varphi }{\nabla }_b\tilde{\varphi }=\kappa {\tilde{\varphi }}_0^2{\delta }_{a0}{\delta }_{b0}$$

$$M_{MSH}(r)=\frac{r}{4}\Big{(}1+\frac{2\Lambda r^{2}}{3}\Big{)}\,.$$

$$\displaystyle S_{BD}=\int d^{4}x\,\frac{\sqrt{-g}}{8\pi}\Big{[}\phi\mathcal{R}-\frac{\omega}{\phi}\,\nabla^{c}\phi\nabla_{c}\phi-V(\phi)\Big{]}$$

$$g_{ab}\to {\tilde{g}}_{ab}={\Omega }^2{g}_{ab}=\varphi g_{ab},$$

$$\displaystyle\phi\rightarrow\tilde{\phi}=\sqrt{\frac{|2\omega+3|}{16\pi}}\ln\Big{(}\frac{\phi}{\phi_{\ast}}\Big{)}$$

$$S_{BD}$$

$$\displaystyle U(\tilde{\phi})=\frac{V(\phi)}{\phi^{2}}\Big{|}_{\phi=\phi(\tilde{\phi})}$$

$$V(\varphi )=\frac{m^2{\varphi }^2}{2}$$

$$m^2=\frac{2\Lambda }{\kappa }>0.$$

$$d{s}^2$$

$$\varphi (t)$$

$$d\tau =e^{-\sqrt{\frac{4\pi }{|2\omega +3|}}\widetilde{{\varphi }_0}t}dt,$$

$$\displaystyle\tau=\sqrt{\frac{|2\omega+3|}{4\pi\tilde{\phi_{0}}^{2}}}\bigg{[}1-\mbox{e}^{-\sqrt{\frac{4\pi}{|2\omega+3|}}\,\tilde{\phi_{0}}t}\bigg{]}\,,$$

$${\tau }_{\ast }\equiv \sqrt{\frac{|2\omega +3|}{4\pi {\widetilde{{\varphi }_0}}^2}},$$

$$d{s}^2$$

$$\varphi (\tau )$$

Full text

I Introduction

Scalar fields are ubiquitous in theoretical physics, beginning with particle physics, where one encounters the Higgs field of the Standard Model Higgs1 ; Higgs2 and other scalar fields in its supersymmetric extensions; in gravitation, beginning with the gravitational scalar of Brans-Dicke theory BD and its scalar-tensor extensions ST , the dilaton of string theories GreenSchwarzWitten ; Polchinski , and the moduli fields of multi-dimensional physics. The low-energy limit of the bosonic string theory produces an $\omega=-1$ Brans-Dicke theory Callanetal85 ; FradkinTseytlin85 .

The general solution of the Einstein equations sourced by a minimally coupled, free and massless scalar field111The reason for using a tilde over the quantities appearing in this section will be clear in the following, where the same variables are regarded as Einstein frame variables of a scalar-tensor theory, which are usually denoted with a tilde. $\tilde{\phi}$ that is static, spherically symmetric, and asymptotically flat is well known and has been rediscovered several times. Under the assumption that the matter field $\phi$ depends only on the radial coordinate, the unique solution was found by Fisher Fisher48 and, in other coordinates or in other forms, by Bergmann and Leipnik BL57 , Janis, Newman and Winicour JNW68 , Buchdahl Buchdahl72 , and Wyman Wyman81 (“FBLJNWBW”). Wyman gave the most general form of the solution. In doing so, he showed that another family of solutions can be obtained by assuming the geometry to be spherically symmetric and static, while the scalar depends only on time, $\tilde{\phi}=\tilde{\phi}(t)$. Unfortunately, this class of solutions can, in general, be expressed only in the form of power series, which is not useful from the calculational point of view. However, one of the solutions has the almost trivial form

[TABLE]

[TABLE]

where $\kappa=8\pi G$, $G$ is Newton’s constant and $d\Omega^{2}_{(2)}=d\theta^{2}+\sin^{2}\theta\,d\varphi^{2}$ is the line element on the unit two-sphere. We will refer to this solution of the coupled Einstein-massless Klein-Gordon equations

[TABLE]

[TABLE]

(where \tilde{\hbox{\hbox to0.0pt{$\sqcup$\hss}\sqcap}}\equiv\tilde{g}^{ab}\tilde{\nabla}_{a}\tilde{\nabla}_{b} is the d’Alembertian) as Wyman’s “other” solution. Recently, this solution has been generalized by Sultana Sultana15 to include a positive cosmological constant $\Lambda$. The geometry changes according to

[TABLE]

while the scalar field is still linear in time and given by Eq. (1.2). Wyman’s ”other” solution is recovered, of course, in the limit $\Lambda\rightarrow 0$.

In this article we clarify the physical nature of the Sultana and the Wyman solutions, which was left unclear in the literature thus far, and we regard Sultana’s solution as the Einstein frame version of a solution of Brans-Dicke theory. In Sec. III, by mapping back this solution222In Ref. Sultana15 , Sultana generated new solutions of conformally coupled scalar field theory with a Higgs potential using a technique developed by Bekenstein Bekenstein . to the Jordan frame, we generate a new two-parameter class of solutions of the Brans-Dicke field equations.

In general, the mapping from the Einstein to the Jordan frame generates solutions with very contrived Jordan frame potentials, even though the Einstein frame potential is physically well motivated. However, in our case the constant Einstein frame potential $U=\Lambda/\kappa$ of Sultana’s solution, when mapped to the Jordan conformal frame, generates a physically relevant mass term $V(\phi)=m^{2}\phi^{2}/2$ for the Jordan frame scalar $\phi$. The new Brans-Dicke solution is derived in Sec. III, while Sec. IV shows that this geometry solves also the field equations of quadratic $f({\cal R})$ gravity. Sec. V contains the conclusions. We follow the notation of Ref. Waldbook ; the metric signature is $-+++$, and we use units in which Newton’s constant $G$ and the speed of light $c$ are unity.

II Sultana solution and its Wyman limit

The most general spherically symmetric line element can be written in the form

[TABLE]

Moreover, if only the cosmological constant $\Lambda$ is present, the scalar field satisfies

[TABLE]

since the constant potential $U(\tilde{\phi})=\Lambda/\kappa$ gives zero contribution to the full Klein-Gordon equation \tilde{\hbox{\hbox to0.0pt{$\sqcup$\hss}\sqcap}}\tilde{\phi}-dU/d\tilde{\phi}=0. The coupled Einstein-Klein-Gordon equations assume the form

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where an overdot and a prime denote differentiation with respect to $t$ and $r$, respectively. It is straightforward to verify that (1.5), (1.2) satisfies Eqs. (2.3)-(2.6) and (2.2) provided that $\tilde{\phi}_{0}=1$ (in the notation adopted, $t$ is a dimensionless coordinate).

Let us discuss the Sultana solution (1.5), (1.2) with regard to its physical properties. The coordinates $t$ and $r$ span the ranges

[TABLE]

As will be clear shortly, the presence of $\Lambda>0$ causes the presence of a cosmological-type horizon at $r_{H}=\sqrt{\frac{3}{2\Lambda}}$ and the locally static coordinates $\left(t,r,\theta,\varphi\right)$ cover only the patch below this horizon.

In the limit $\Lambda\rightarrow 0$, Wyman’s “other” solution (1.1), (1.2) is recovered, the horizon is pushed to infinity, and $0\leq r\leq+\infty$. Interestingly, the Wyman geometry (1.1) corresponds to a special case of a geometry found by Carloni and Dunsby CarloniDunsby in scalar-tensor gravity, but with a scalar $\tilde{\phi}=\tilde{\phi}(r)$ that depends only on the radial (instead of the temporal) coordinate and for a power-law potential.

The Ricci scalar of the Sultana solution is

[TABLE]

and it diverges as $r\rightarrow 0^{+}$, where the coordinate $r$ coincides with the physical (areal) radius. If present, horizons are located by the roots of the equation

[TABLE]

which provides the radius of the cosmological horizon

[TABLE]

created by $\Lambda$. This horizon is smaller than the de Sitter horizon of radius $\sqrt{3/\Lambda}$. There are no other horizons, therefore the Sultana solution is interpreted as a naked central singularity embedded in a “background” created by the cosmological constant. (We use quotation marks because, due to the non-linearity of the field equations, it is impossible to split a metric into a “background” and a “deviation” from it in a covariant way (barring special algebraic structures as, e.g., in generalized Kerr-Schild metrics, but this is not the case here.)

In the $\Lambda\rightarrow 0$ limit corresponding to Wyman’s “other” solution, there is no horizon and the geometry can be interpreted physically as a naked singularity embedded in a space which is not asymptotically flat since, in this case, the Ricci tensor

[TABLE]

does not depend on $r$ and does not go to zero as $r\rightarrow+\infty$.

The Misner-Sharp-Hernandez mass MSH contained in a ball of radius $r$ is given by $1-2M_{MSH}/r\equiv\nabla^{c}r\nabla_{c}r$, which yields

[TABLE]

Comparing with the Misner-Sharp-Hernandez mass of de Sitter space, which is $M_{dS}=\Lambda r^{3}/6$, we see that the scalar field gives an additional positive contribution $r/4$.

III The Brans-Dicke counterpart of Sultana’s solution

Let us consider vacuum (Jordan frame) Brans-Dicke theory, described by the action

[TABLE]

where $\phi>0$ is the Brans-Dicke field, the constant $\omega$ is the “Brans-Dicke coupling”, and $g$ is the determinant of the metric. The conformal transformation of the metric

[TABLE]

and the scalar field redefinition

[TABLE]

(where $\phi_{\ast}$ is a positive constant) recast the Brans-Dicke action in its Einstein frame form

[TABLE]

where

[TABLE]

and a tilde denotes Einstein frame quantities. In this form, the theory has all the appearances of general relativity with a minimally coupled scalar field as a source. Since the Sultana solution solves the corresponding field equations with constant potential $U(\tilde{\phi})=\Lambda/\kappa$, we can map this solution to the Jordan frame to obtain a new family of solutions of the vacuum Brans-Dicke field equations. The potential ruling the dynamics of the gravitational Brans-Dicke field $\phi$ in the Jordan frame is given by Eq. (3.5) as the simple mass term

[TABLE]

where

[TABLE]

The Jordan frame potential $V(\phi)$ no longer contains a cosmological constant and the original parameter $\Lambda$ now parametrizes the scalar field mass $m$. The solution-generating technique used here, originally introduced by Bekenstein for non-minimally coupled scalar fields (a different scalar-tensor theory), has been employed a number of times in the literature on scalar-tensor gravity. By applying Eqs. (3.2) and (3.3) to the Sultana solution (1.5), (1.2) one finds

[TABLE]

Since $\phi_{\ast}>0$, the Jordan frame scalar is positive. It is convenient to introduce the new time coordinate $\tau$ defined by

[TABLE]

or

[TABLE]

by imposing that $\tau(t=0)=0$. Introducing

[TABLE]

the new Brans-Dicke solution reads

[TABLE]

In this solution, $\omega$ and $\Lambda$ are parameters of the theory, while $\phi_{\ast}$ plays the role of an initial condition.

III.1 Nature of the solution

Let us examine the nature of this new solution. The Ricci scalar is

[TABLE]

For any value of $\omega$, the Ricci scalar diverges as $\tau\rightarrow\tau_{\ast}^{-}$; this value of the time coordinate corresponds to a Big Crunch-type singularity, where the scalar $\phi$ also diverges.

If $\omega\neq 0$, $\mathcal{R}$ diverges also as $r\rightarrow 0^{+}$ (the case $\omega=0$ will be analyzed separately). The areal radius is

[TABLE]

and $R\rightarrow 0$ as $r\rightarrow 0$. Hence, there is a central singularity if $\omega\neq 0$. Next, one would like to answer the question of whether this singularity is covered by horizons. If present, horizons are located by the roots of

[TABLE]

As in the case of the Sultana spacetime, this is a horizon created by the cosmological constant at

[TABLE]

We have again a naked central singularity embedded in a “background” created by $\Lambda$ and $\phi$, which ends its history at a finite future $\tau_{\ast}$.

III.2 The case $\omega=0$

For completeness, we now discuss the case $\omega=0$ in which the Ricci scalar (3.14) is regular as $r\rightarrow 0^{+}$. In this case, the square of the Ricci tensor \mathcal{R}_{ab}=\phi^{-1}\Big{(}\nabla_{a}\nabla_{b}\phi+Vg_{ab}/2\Big{)} is

[TABLE]

Since

[TABLE]

[TABLE]

and

[TABLE]

one obtains after straightforward manipulations

[TABLE]

This quantity diverges as $r\rightarrow 0^{+}$ and $R\rightarrow 0^{+}$, therefore the naked singularity at $R=0$ persists for $\omega=0$.

In the special case $\Lambda=0$, we have the Brans-Dicke counterpart of Wyman’s “other” solution for which $\mathcal{R}=0$ but $\mathcal{R}_{ab}\mathcal{R}^{ab}\rightarrow\infty$ as $R\rightarrow 0^{+}$.

IV A new solution of purely quadratic gravity

It is well known that metric $f({\cal R})$ gravity is equivalent to an $\omega=0$ Brans-Dicke theory with a complicated potential review1 ; review2 ; review3 . Specifically, the action

[TABLE]

where $f$ is a non-linear function, can be shown review1 ; review2 ; review3 to be dynamically equivalent to the other action

[TABLE]

if $f^{\prime\prime}\neq 0$, where $\phi=f^{\prime}({\cal R})$ and

[TABLE]

is a potential which, in general, is only defined implicitly. The action (4.2) is clearly an $\omega=0$ Brans-Dicke action with this complicated potential for the scalar $\phi$. The question arises of whether the geometry and scalar field obtained by setting $\omega=0$ in Eqs. (LABEL:new1) and (3.13) provides a new solution of $f({\cal R})$ gravity. If this is the case, then what is the corresponding form of the function $f({\cal R})$?

To answer, one notes that Eqs. (3.14) and (3.13) give, for $\omega=0$,

[TABLE]

Then it must be

[TABLE]

which integrates to

[TABLE]

where $f_{0}$ is an integration constant corresponding to a new (i.e., distinct from $\Lambda$) cosmological constant in the theory. Now, by imposing that the scalar field potential be given by Eq. (4.3), one must necessarily have

[TABLE]

However, the condition $V(\phi)=m^{2}\phi^{2}/2$ must be satisfied simultaneously, with the extra information that $m^{2}=2\Lambda/\kappa$, which yields

[TABLE]

The comparison of Eqs. (4.7) and (4.8) yields $f_{0}=0$. Therefore, the time-dependent geometry (LABEL:new1) (with $\omega=0$ in the expression of $\tau_{*}$) is a solution of the fourth order theory

[TABLE]

This theory, which has scale-invariance properties, does not admit a Newtonian limit PechlanerSexl66 and therefore is not adequate to describe the present universe, but is instead a good approximation of the Starobinski model $f({\cal R})={\cal R}+\alpha{\cal R}^{2}$ describing a viable (and even preferred by observations Starob-obs ) scenario of inflation Starobinsky . It is also the subject of several studies in the context of early universe and of black hole physics rsquared .

As already established following Eq. (3.22), this spacetime hosts a naked central singularity where, however, the Brans-Dicke scalar $\phi=f^{\prime}({\cal R})\propto{\cal R}$ is regular. In addition, a spacelike singularity is reached simultaneously at all spatial points as $\tau\rightarrow\tau_{*}$.

V Conclusions

Scalar-tensor theories of gravity are the prototypical alternatives to Einstein’s general relativity and, in various forms (but especially $f({\cal R})$ gravity, which is an $\omega=0$ Brans-Dicke theory with a complicated potential review1 ; review2 ; review3 ), they are studied intensively in order to explain the present acceleration of the universe without an ad hoc dark energy FujiiMaeda ; mymonograph ; cosmography . However, the catalog of exact solutions of the scalar-tensor field equations, which would allow a better understanding of these theories, is still very limited. With this perspective, we have clarified in Sec. II the physical nature of Wyman’s “other” solution Wyman81 and of its generalization obtained by Sultana for a positive cosmological constant $\Lambda$ Sultana15 . These solutions describe naked central singularities embedded in curved “backgrounds” created by $\phi$, or by $\Lambda$ and $\phi$, respectively. For comparison, the more well known FBLJNWBW solution Fisher48 ; BL57 ; JNW68 ; Buchdahl72 ; Wyman81 and its Brans-Dicke counterpart always describe naked singularities or wormhole throats VFAT . The solutions of the Einstein equations discussed here do not violate the Birkhoff theorem because: a) they are not vacuum solutions, and b) they are not asymptotically flat.

While the FBLJNWBW solution and its Brans-Dicke counterpart are static, our new Brans-Dicke solution derived in Sec. III is time-dependent and inhomogeneous. Only a handful of such time-dependent solutions, which are harder to obtain, are presently known in Brans-Dicke theory climobarr ; myclimobarr ; confonarev or $f({\cal R})$ gravity Clifton ; myclifton ; confonarev . Further, the new solution corresponds to a mass term potential, which is physically very reasonable and even desired, while the solution-generating technique that we employed (i.e., mapping from the Einstein to the Jordan frame) usually produces contrived and unphysical scalar field potentials $V(\phi)$ in the Jordan frame. Realistic scalar field configurations seem to collapse to black holes without hair nohair , while other scalar field configurations seem to be unrealistic, including Wyman’s “other” solution Wyman81 and its generalization Sultana15 (the nature of which, although not difficult to study, is shown here for the first time). Unfortunately, the solutions discussed are probably unstable and, therefore, not very relevant physically (their stability will be studied in a future publication). However, any new analytical solution gives some insight into a complicated non-linear theory and has some value. In any case, since the more well known FBLJNWBW geometry has already been mapped into the general (spherical, static, asymptotically flat) solution of Brans-Dicke theory Bronnikov ; VFAT , it is natural to do the same for its lesser known sister, Wyman’s “other” solution, as done in Sec. III. The new Brans-Dicke solution (LABEL:new1), (3.13) fills this gap in the literature and the special case $\omega=0$ adds (as shown in Sec. IV) to the catalogue of solutions of scale-invariant $f({\cal R})={\cal R}^{2}$ gravity.

Acknowledgements.

This work is supported by Bishop’s University and by the Natural Sciences & Engineering Research Council of Canada (Grant No. 2016-03803 to V.F.). A. Banijamali and B. Fazlpour would like to thank Bishop’s University for hospitality during their visit.