# Canonical potential in bootstrapped Newtonian gravity: approximating compact
objects

Roberto Casadioab, Iberê Kuntzc, and Octavian Micud

aDipartimento di Fisica e Astronomia, Università di Bologna
via Irnerio 46, 40126 Bologna, Italy

bI.N.F.N., Sezione di Bologna, I.S. FLAG
viale B. Pichat 6/2, 40127 Bologna, Italy

cDepartamento de Física, Universidade Federal do Paraná
PO Box 19044, Curitiba – PR, 81531-980, Brazil

dInstitute of Space Science
Bucharest-Magurele, Romania E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider compact objects in a classical and non-relativistic generalisation
of Newtonian gravity, dubbed bootstrapped Newtonian theory, which includes
higher-order derivative interaction terms of the kind generically present in
the strong-field regime of gravity. By means of a field redefinition, the
original bootstrapped Newtonian action is written in a canonical Newtonian
form with non-linear source terms. Exact analytic solutions remain
unattainable, but we show that perturbative solutions of the canonical theory
can be used to derive approximate descriptions of compact objects. In
particular, the ADM (Arnowitt-Deser-Misner) mass always differs from the
proper mass due to the non-linear couplings in the theory. A few examples of
sources with different density profiles are explicitly analysed.

##  1 Introduction

Despite gravity being the oldest of the forces known to science, many
gravitational phenomena and their quantum foundations remain, to a large
degree, fully open questions. Roughly speaking, this can be mainly attributed
to its weakness (as measured by the gravitational coupling $G_{\rm N}\sim
10^{-11}\,$m${}^{3}\,$kg${}^{-1}\,$s-1). When Newton’s constant is combined
with other “small” parameters (like $\hbar$ in a quantum regime),
gravitational effects seem to become utterly negligible at laboratory scales.
Testing aspects of strong gravity thus requires objects with huge masses (or
large compactness) in order to produce sizeable effects. On the other hand,
gravity is a non-linear phenomenon at its core and the theoretical study of
compact objects is a difficult endeavour due to the lack of general techniques
to solve non-linear differential equations.

Most of the known results are obtained perturbatively in the weak-field regime
of general relativity, far away from the compact source. Perturbation theory
indeed fails in the strong-field regime around compact objects, where an
infinite tower of couplings are generated in the non-relativistic
approximation given by the expansion of the Einstein-Hilbert action. In such a
regime, all terms contribute equally and the infinite series cannot be
truncated. In order to avoid this kind of issue, one could take a bottom-up
approach and construct the Newtonian theory by also including, from the onset,
the leading order terms that would show up in the aforementioned expansion.
Rather than truncating the series obtained from general relativity, the
resulting action is viewed as a new theory, where finitely many terms are
treated on the same foot. From this perspective, the model functions as an
alternative, rather than an extension, of Newtonian gravity, in very much the
same way that Stelle’s higher-derivative gravity [1] differs from general
relativity. The theory so obtained is called bootstrapped Newtonian gravity
[2].

In this work, we extend our previous investigations of compact sources in
bootstrapped Newtonian gravity [2, 3, 4, 5, 6] by applying an idea introduced
in Ref. [7], whereby the kinetic term of the bootstrapped Newtonian Lagrangian
is put in canonical form by performing a field redefinition. The role of the
field redefinition is to replace derivative couplings, which are hard to deal
with, by standard (albeit non-linear) couplings. In particular, we shall
employ the same Taylor expansion in powers of the radial coordinate $r$ from
Refs. [2, 3] and show that the error one makes when truncating the transformed
solution (to second order in $r$) is negligible with respect to the truncated
solution before the field redefinition. We stress that, because of the non-
linear nature of the problem and of the required field redefinition, this is a
non-trivial result. It is also practically relevant because working with the
redefined canonical field makes it easier to find approximate solutions for
describing compact objects in the bootstrapped Newtonian theory.

This paper is organised as follows: in Section 2, we review the bootstrapped
Newtonian gravity and detail the aforementioned field redefinition that brings
the kinetic term to the canonical form; in Section 3, we discuss approximate
solutions that can be obtained for generic source terms and some particular
cases. The main results we will obtain are that the bootstrapped Newtonian
solutions are identical (at least) to second order in $r$ and coincide with
the Newtonian potential inside a homogeneous source, regardless of the actual
density and pressure profiles; terms with odd powers of $r$ must vanish and
the difference with respect to the Newtonian potential appears at order
$r^{4}$ (or higher); the ADM-like mass [8] $M$ and the proper mass $M_{0}$ are
always different. Finally, we will draw some more conclusions in Section 4.

##  2 Bootstrapped Newtonian gravity

We start by recalling that, in its most general form, the Lagrangian for the
bootstrapped Newtonian potential $V=V(r)$ for static and spherically symmetric
systems is given by [2] 111We shall use units with $c=1$.

$\displaystyle L[V]$ $\displaystyle\\!\\!=\\!\\!$ $\displaystyle L_{\rm
N}[V]-4\,\pi\int_{0}^{\infty}r^{2}\,\mbox{${\rm
d}$}r\left[q_{V}\,\mathcal{J}_{V}\,V+q_{p}\,\mathcal{J}_{p}\,V+q_{\rho}\,\mathcal{J}_{\rho}\left(\rho+3\,q_{p}\,p\right)\right]$
(2.1) $\displaystyle\\!\\!=\\!\\!$
$\displaystyle-4\,\pi\int_{0}^{\infty}r^{2}\,\mbox{${\rm
d}$}r\left[\frac{\left(V^{\prime}\right)^{2}}{8\,\pi\,G_{\rm
N}}\left(1-4\,q_{V}\,V\right)+\left(\rho+3\,q_{p}\,p\right)V\left(1-2\,q_{\rho}\,V\right)\right]\
,$

where $f^{\prime}\equiv\mbox{${\rm d}$}f/\mbox{${\rm d}$}r$. The motivation
for each term was described extensively in previous publications [2, 3, 4, 5,
6], so in the next paragraphs we will briefly recall the meaning of them.

The standard Newtonian Lagrangian,

$\displaystyle L_{\rm N}[V]=-4\,\pi\int_{0}^{\infty}r^{2}\,\mbox{${\rm
d}$}r\left[\frac{\left(V^{\prime}\right)^{2}}{8\,\pi\,G_{\rm
N}}+\rho\,V\right]\ ,$ (2.2)

yields the Poisson equation

$\displaystyle r^{-2}\left(r^{2}\,V^{\prime}\right)^{\prime}\equiv\triangle
V=4\,\pi\,G_{\rm N}\,\rho$ (2.3)

for the Newtonian potential $V=V_{\rm N}$ generated by the matter energy
density $\rho=\rho(r)$. As it was detailed in Refs. [2, 9], the gravitational
self-coupling contribution is then sourced by the gravitational energy $U_{\rm
N}$ per unit volume, to wit

$\displaystyle\mathcal{J}_{V}\simeq\frac{\mbox{${\rm d}$}U_{\rm
N}}{\mbox{${\rm
d}$}\mathcal{V}}=-\frac{\left[V^{\prime}(r)\right]^{2}}{2\,\pi\,G_{\rm N}}\ .$
(2.4)

The static pressure $p=p(r)$ becomes very large for compact sources with a
compactness [2]

$\displaystyle X\equiv\frac{G_{\rm N}\,M}{R}\gtrsim 1\ ,$ (2.5)

where $M$ is the ADM-like mass that one would measure when studying orbits
[10] and $R$ is the radius of the source [6]. For this reason, a corresponding
potential energy $U_{p}$ was added such that

$\displaystyle\mathcal{J}_{p}\simeq-\frac{\mbox{${\rm d}$}U_{p}}{\mbox{${\rm
d}$}\mathcal{V}}=3\,p\ .$ (2.6)

Since the above just adds to $\rho$, it can be easily included by simply
shifting $\rho\to\rho+3\,q_{p}\,p$, where $q_{p}$ is a positive constant which
formally allows us to implement the non-relativistic limit as $q_{p}\to 0$.
Upon including these new source terms, and the analogous higher-order term
$\mathcal{J}_{\rho}=-2\,V^{2}$ which couples with the matter source, we obtain
the total Lagrangian (2.1).

The Euler-Lagrange equation for $V$ is then given by

$\displaystyle\triangle V=4\,\pi\,G_{\rm
N}\left(\rho+3\,q_{p}\,p\right)\frac{1-4\,q_{\rho}\,V}{1-4\,q_{V}\,V}+\frac{2\,q_{V}\left(V^{\prime}\right)^{2}}{1-4\,q_{V}\,V}\
.$ (2.7)

We remark that the (dimensionless) coupling constants $q_{V}$, $q_{p}$ and
$q_{\rho}$ track the effects of each additional contribution and could be
related to different specific theories of the interaction between gravity and
matter (for similar considerations, see, e.g. Ref. [11]).

###  2.1 Field redefinition

The Lagrangian (2.2) can be generalised to non-static configurations
$V=V(x^{\mu})$ in flat spacetime, thus yielding the kinetic term [7]

$\displaystyle
K=-\left(1-4\,q_{V}\,V\right)\frac{\partial_{\mu}V\,\partial^{\mu}V}{8\,\pi\,G_{\rm
N}}\ ,$ (2.8)

which is not in canonical form, and neither is $V$ of the canonical dimension
for a scalar field. In fact, we can change $K$ into the precise canonical form

$\displaystyle K=-\frac{1}{2}\,\partial_{\mu}\psi\,\partial^{\mu}\psi$ (2.9)

by means of the transformation [7]

$\displaystyle\psi\equiv\psi(V)=\frac{1}{6\,\alpha}\left[1-\left(1-4\,q_{V}V\right)^{3/2}\right]\
,$ (2.10)

where

$\displaystyle\alpha=q_{V}\,\sqrt{G_{\rm N}}\ .$ (2.11)

The inverse relationship is given by

$V\equiv
V(\psi)=\frac{1}{4\,q_{V}}\left[1-\left(1-6\,\alpha\,\psi\right)^{2/3}\right]\
,$ (2.12)

and, after some algebra, the total Lagrangian (2.2) for static and isotropic
configurations $\psi=\psi(r)$ reads

$\displaystyle L[\psi]=-4\,\pi\int_{0}^{\infty}r^{2}\,\mbox{${\rm
d}$}r\left[\frac{\left(\psi^{\prime}\right)^{2}}{8\,\pi}+\left(J_{\rho}+3\,q_{p}\,J_{p}\right)\xi(\psi)\right]\
,$ (2.13)

in which the matter density was rescaled as

$\displaystyle J_{\rho}=\sqrt{G_{\rm N}}\,\rho\ ,$ (2.14)

like the pressure contribution

$\displaystyle J_{p}=\sqrt{G_{\rm N}}\,p\ .$ (2.15)

The interaction terms, which do not contain any derivatives of the new field
$\psi$, are all included in the non-linear coupling to the sources $J_{\rho}$
and $J_{p}$, that is

$\displaystyle\xi(\psi)=\frac{1}{4\,\alpha}\left[1-(1-6\,\alpha\,\psi)^{2/3}\right]\left\\{1-\frac{\beta}{2\,\alpha}\left[1-\left(1-6\,\alpha\,\psi\right)^{2/3}\right]\right\\}\
,$ (2.16)

where

$\displaystyle\beta=q_{\rho}\,\sqrt{G_{\rm N}}\ .$ (2.17)

The most general form of the Euler-Lagrange equation for the canonically
normalised $\psi=\psi(r)$ is finally given by

$\displaystyle\triangle\psi=4\,\pi\,J\,\frac{\alpha-\beta\left[1-\left(1-6\,\alpha\,\psi\right)^{2/3}\right]}{\alpha\left(1-6\,\alpha\,\psi\right)^{1/3}}\
,$ (2.18)

where $J\equiv J_{\rho}+3\,q_{p}\,J_{p}$ is the total effective density. In
the following, we shall show that (approximate) solutions of Eq. (2.18) can
indeed be efficiently employed in order to determine (approximate) solutions
of the original Eq. (2.7).

###  2.2 Vacuum solution

In the vacuum outside a source of mass $M$ and radius $r=R$, we have $J=0$ and
the above Eq. (2.18) reduces to

$\displaystyle\triangle\psi=0\ .$ (2.19)

Of course, the exact solution for $r>R$ satisfying the proper asymptotic
behaviour is the (canonically normalised) Newtonian potential

$\displaystyle\psi_{\rm out}=-\frac{\sqrt{G_{\rm N}}\,M}{r}\ ,$ (2.20)

which transforms back to [7]

$\displaystyle V_{\rm out}=V(\psi_{\rm out})\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\frac{1}{4\,q_{V}}\left[1-\left(1+6\,q_{V}\,\frac{G_{\rm
N}\,M}{r}\right)^{2/3}\right]\ .$ (2.21)

This is the exact solution of the bootstrapped Newtonian Eq. (2.7) where
$\rho=p=0$ with the expected asymptotic behaviour [2].

One important aspect that is not apparent from the above derivation is that
the mass $M$ in Eq. (2.21) is not equal to the proper mass $M_{0}$ of the
source [2, 6]. This follows precisely because of the non-linearity of Eq.
(2.7) and the equivalent Eq. (2.18). We shall further investigate this aspect
for the canonical field $\psi$ and various density profiles in the next
sections.

##  3 Quadratic approximation for the inner canonical potential

To work with simpler equations, in the remainder of the paper we set
$q_{V}=q_{\rho}$ (equivalent to $\alpha=\beta$), which simplifies the field
equation (2.18) to

$\displaystyle\triangle\psi=4\,\pi\,J\left(1-6\,\alpha\,\psi\right)^{1/3}\ ,$
(3.1)

where the effective density $J=J(r)$ vanishes outside the source of radius
$r=R$.

Any solution of Eq. (2.7) for the bootstrapped Newtonian potential $V_{\rm
in}=V(0\leq r<R)$ needs to match smoothly with the outer vacuum solution
$V_{\rm out}$ in Eq. (2.21) across the boundary $r=R$ of the source. It is
very easy to show that identical constraints must then hold for the field
$\psi=\psi(V)$ satisfying Eq. (3.1). More precisely

$\displaystyle V_{\rm in}(R)=V_{\rm out}(R)\equiv
V_{R}\quad\Leftrightarrow\quad\psi_{\rm in}(R)=\psi_{\rm
out}(R)\equiv\psi_{R}\ ,$ (3.2)

and

$\displaystyle V^{\prime}_{\rm in}(R)=V^{\prime}_{\rm out}(R)\equiv
V^{\prime}_{R}\quad\Leftrightarrow\quad\psi^{\prime}_{\rm
in}(R)=\psi^{\prime}_{\rm out}(R)\equiv\psi^{\prime}_{R}\ ,$ (3.3)

where we defined $\psi_{\rm in}=\psi(0\leq r\leq R)$ and $\psi_{\rm
out}=\psi(R\leq r)$. Furthermore, we are looking for potentials generated by
density profiles that are finite in the centre. Therefore, the inner solution
also needs to satisfy the regularity condition $V^{\prime}_{\rm in}(0)=0$,
which in turn means that

$\displaystyle\psi_{\rm in}^{\prime}(0)=0\ .$ (3.4)

Obtaining exact analytic solutions $V_{\rm in}$ for Eq. (2.7) or $\psi_{\rm
in}$ for Eq. (3.1) is not feasible, even for an object with constant density
and negligible pressure. Hence, we will here focus on finding a good
approximation by Taylor expanding around $r=0$. Odd powers can be shown to
vanish when imposing the constraint in Eq. (3.4) [2], so that the bootstrapped
Newtonian potential up to second order reads

$\displaystyle V_{\rm in}\simeq V_{0}+V_{2}\,r^{2}\ .$ (3.5)

This approximation was shown to work well for sources of small and
intermediate compactness $X$ by comparing with numerical solutions in Refs.
[2, 3]. Therefore, we limit this case study to $X\lesssim 1$, which excludes
objects hidden behind a horizon.

The same Taylor expansion for the canonical potential reads

$\displaystyle\psi_{\rm in}\simeq\psi_{0}+\psi_{2}\,r^{2}\ ,$ (3.6)

and the mapping in Eq. (2.12) will then yield $\tilde{V}_{\rm
in}=V(\psi_{0}+\psi_{2}\,r^{2})$, which we can compare with $V_{\rm in}$ in
Eq. (3.5). Note that the two results will contain different powers of $r$ and
their comparison is therefore not straightforward. However, one can estimate
quantitatively how close the two approximations are to one another by
calculating their relative difference.

###  3.1 General density and pressure profiles

Let us first consider a generic $J=J(r)$, with the only constraint that
$J^{\prime}(0)=0$. Upon inserting the Taylor expansion (3.6) in the equation
of motion (3.1) and employing the boundary condition (3.3), we find

$\displaystyle\psi_{0}=\frac{1}{3\,\alpha}\left[1-\frac{27\,G_{\rm
N}^{3/2}\,M^{3}}{R^{9}\,J^{3}_{0}}\right]$ (3.7)

where $J_{0}\equiv J(0)$, and

$\displaystyle\psi_{2}=\frac{\sqrt{G_{\rm N}}\,M}{2\,R^{3}}\ .$ (3.8)

From the matching condition in Eq. (3.2), one obtains

$\displaystyle\psi_{0}=\frac{3\,\sqrt{G_{\rm N}}\,M}{2\,R}\ .$ (3.9)

The approximate expression for the inner canonical field thus becomes

$\displaystyle\psi_{\rm in}\simeq-\frac{\sqrt{G_{\rm
N}}\,M}{2\,R}\left(3-\frac{r^{2}}{R^{2}}\right)\ ,$ (3.10)

which is in fact the (canonically normalised) Newtonian solution for the inner
potential inside a homogeneous source satisfying

$\displaystyle\triangle\psi=4\,\pi\,J_{0}\ .$ (3.11)

This result shows that, as long as the quadratic approximations (3.5) and
(3.6) hold, the canonical $\psi_{\rm in}$ depends only on the mass $M$ and the
size $R$ of the source. From the inverse map (2.12), we conclude that the
bootstrapped Newtonian potential $V_{\rm in}$ is the same regardless of the
density (and pressure) profile, up to terms of order $(r^{4})^{2/3}$.

We emphasise that, in the Newtonian case, the ADM mass $M$ equals the proper
mass

$\displaystyle M_{0}=4\,\pi\int_{0}^{R}r^{2}\,\mbox{${\rm d}$}r\,\rho(r)\ .$
(3.12)

In bootstrapped Newtonian gravity, $M$ and $M_{0}$ instead differ [2, 6],
since demanding that $M=M_{0}$ would over constrain the problem and yield no
solution, as will become apparent in the following subsections. Calculating
$M=M(M_{0})$ is then very instructive, but it can only be done for given
$J=J(r)$ and the corresponding expression can be very cumbersome.

###  3.2 Homogeneous ball with negligible pressure

As a simple application, we directly consider a ball with homogeneous density

$\displaystyle\rho=\rho_{0}\,\Theta(R-r)=\frac{3\,M_{0}}{4\,\pi\,R^{3}}\,\Theta(R-r)\
,$ (3.13)

where $\Theta$ is the Heaviside step function enforcing the density to vanish
for $r>R$ and $M_{0}$ is the proper mass defined in Eq. (3.12). We also assume
that the pressure be negligible, so that Eq. (3.1) further simplifies to

$\displaystyle\triangle\psi=4\,\pi\,J_{\rho}\,(1-6\,\alpha\,\psi)^{1/3}=4\,\pi\,J_{0}\,(1-6\,\alpha\,\psi)^{1/3}\
,$ (3.14)

where $J_{0}=\sqrt{G_{\rm N}}\,\rho_{0}$, and we limit the investigation to
values of the compactness $X\lesssim 1$, as stated earlier.

####  3.2.1 Inner potential via field redefinition

By plugging the Taylor expansion for the canonical $\psi_{\rm in}$ in Eq.
(3.6) into the equation of motion (3.14), we find

$\displaystyle\psi_{2}\simeq\frac{\sqrt{G_{\rm
N}}\,M_{0}}{2\,R^{3}}\left(1-6\,\alpha\,\psi_{0}\right)^{1/3}\ .$ (3.15)

Eq. (3.3) can next be used to determine

$\displaystyle\psi_{0}\simeq\frac{1}{6\,\alpha}\left(1-\frac{M^{3}}{M_{0}^{3}}\right)\
.$ (3.16)

The matching condition (3.2) can be used to express the proper mass in terms
of $M$, yielding

$\displaystyle\tilde{M}_{0}=\frac{M}{\left(1+9\,q_{V}\,G_{\rm
N}\,M/R\right)^{1/3}}=\frac{M}{\left(1+9\,q_{V}\,X\right)^{1/3}}\ .$ (3.17)

Finally, we find the same approximate expression (3.10) corresponding to the
Newtonian solution of Eq. (3.11). This shows that, at least in the quadratic
approximation, the change in the coupling

$\displaystyle\xi(V)=1\to\xi(\psi)=(1-6\,\alpha\,\psi)^{1/3}$ (3.18)

is equivalent to rescaling the Newtonian mass $M_{0}$ into the ADM mass $M$
according to Eq. (3.17). One could therefore solve the simpler Newtonian
problem (3.11) and just write $M$ instead of $M_{0}$ in the solution.

The bootstrapped Newtonian potential is obtained by substituting Eq. (3.10) in
Eq. (2.12) and reads

$\displaystyle\tilde{V}_{\rm in}\\!\\!\\!\\!$ $\displaystyle\simeq$
$\displaystyle\\!\\!\\!\\!\frac{1}{4\,q_{V}}\left\\{1-\left[1+3\,q_{V}\,X\left(3-\frac{r^{2}}{R^{2}}\right)\right]^{2/3}\right\\}$
(3.19) $\displaystyle\simeq$
$\displaystyle\\!\\!\\!\\!\frac{1}{4\,q_{V}}\left\\{1-\left(1+9\,q_{V}\,X\right)^{2/3}\left[1-\frac{2\,q_{V}\,r^{2}/R^{2}}{\left(1+9\,q_{V}\,X\right)}\right]\right\\}\
.$

We can also estimate the relative error for this approximation by replacing
the expression (3.10) into Eq. (3.14), from which we obtain

$\displaystyle\mathcal{E}\equiv\frac{q_{V}\,X}{1+9\,X}\left(\frac{r}{R}\right)^{2}\ll
1\ ,$ (3.20)

which is displayed in Fig. 1. Of course, this error vanishes everywhere inside
the source in the Newtonian limit $q_{V}\to 0$ and is proportional to the
compactness $X$ otherwise.

Figure 1: Relative error from Eq. (3.20) for $q_{V}=1$.

####  3.2.2 Inner bootstrapped Newtonian potential

The approximate solution (3.5) for the homogeneous ball with vanishing
pressure was found in Ref. [2] and is given by

$\displaystyle V_{\rm in}\\!\\!\\!\\!$ $\displaystyle\simeq$
$\displaystyle\\!\\!\\!\\!\frac{1}{4\,q_{V}}\left[1-\frac{1+2\,q_{V}\,X\left(4-r^{2}/R^{2}\right)}{\left(1+6\,q_{V}\,X\right)^{1/3}}\right]$
(3.21) $\displaystyle\simeq$
$\displaystyle\\!\\!\\!\\!\frac{1}{4\,q_{V}}\left\\{1-\left(1+9\,q_{V}\,X\right)^{2/3}\left[1-\frac{2\,q_{V}\,r^{2}/R^{2}}{\left(1+9\,q_{V}\,X\right)}\right]\right\\}\
.$

The matching conditions across the surface also yield

$\displaystyle M_{0}=\frac{M}{(1+6\,q_{V}\,X)^{1/3}}\ ,$ (3.22)

and one notices a different numerical factor multiplying $q_{V}$ in comparison
with Eq. (3.17).

####  3.2.3 Comparing the approximations

The main reason for this analysis is to understand if the field redefinition
that brings the kinetic term in a canonical form, thus simplifying the
equation of motion, leads to results that are in good agreement with those
obtained without this transformation. The relative difference between the
approximations in Eqs. (3.21) and (3.19) for small compactness is given by

$\displaystyle\Delta\equiv\left|\frac{\tilde{V}_{\rm in}-V_{\rm in}}{V_{\rm
in}}\right|\simeq\frac{q_{V}\,X}{6}\left|1-\frac{5\,r^{2}}{3\,R^{2}}\right|\
.$ (3.23)

which is roughly of the same order as the error (3.20) shown in Fig. 1.

Figure 2: Upper panels: inner potentials $\tilde{V}_{\rm in}$ (dotted line),
$V_{\rm in}$ (dashed line), $\hat{V}_{\rm in}$ (solid line) and outer
potential $\tilde{V}_{\rm out}$ (solid line) for $X=1/100$ (left panel),
$X=1/20$ (center panel) and $X=1/5$ (right panel). Lower panels: absolute
value of the relative difference $\Delta=|(\tilde{V}_{\rm in}-V_{\rm
in})/V_{\rm in}|$ for the same compactness as above.

Figure 3: Upper panels: ratios $\tilde{M}_{0}/M$ (dotted line) and $M_{0}/M$
(dashed line) as functions of $X$ for $q_{V}=0.1$ (left panel), $q_{V}=0.5$
(center panel) and $q_{V}=1$ (right panel). Bottom panels: ratios
$\tilde{M}_{0}/M$ (dotted line) and $M_{0}/M$ (dashed line) as functions of
$q_{V}$ for $X=1/100$ (left panel), $X=1/20$ (center panel) and $X=1/5$ (right
panel).

Because of the non-linearity of the field equations, however, the above
estimate remains of questionable relevance. In order to asses how reliable the
analytical approximations are, we solve Eq. (2.7) numerically with the same
boundary conditions (3.2)-(3.4) and denote the numerical solution in the
interior as $\hat{V}_{\rm in}$. From the plot in Fig. 2, we see that
$\tilde{V}_{\rm in}$, $V_{\rm in}$ and $\hat{V}_{\rm in}$ follow each other
very closely, both for small and intermediate values of the compactness. In
the lower panels of the same figure one can also see plots of the relative
difference $\Delta$ for the same values of the compactness. Therefore, even
though the approximate analytical expressions obtained for the potential are
different, both $\tilde{V}_{\rm in}$ and $V_{\rm in}$ appear to be in very
good agreement with the numerical results.

The main difference between the two approximate solutions $\tilde{V}_{\rm in}$
and $V_{\rm in}$ is how $\tilde{M}_{0}$ and $M_{0}$ depend on $M$. Fig. 3
shows the ratios $\tilde{M}_{0}/M$ and $M_{0}/M$ as functions of the
compactness $X$ for the same values of $q_{V}$, respectively as functions of
$q_{V}$ for the same values of $X$.

The value of the coupling $q_{V}$ defines different regimes of the theory. In
the limit $q_{V}\to 0$, one recovers Newtonian physics, as it is obvious from
Eq. (2.1), while the model is expected to approach General Relativity for
$q_{V}\to 1$. This is why in Fig. 4 we display $|(\tilde{M}_{0}-M_{0})/M|$ for
three different values of $q_{V}$. The relative difference increases with the
compactness and with $q_{V}$. Therefore, it is largest for $q_{V}=1$ for the
largest compactness considered of $X=0.2$. One notices that even in this case,
the agreement remains fairly good.

Figure 4: Relative difference $|(\tilde{M}_{0}-M_{0})/M|$ as a function of $X$
for $q_{V}=0.1$ (left panel), $q_{V}=0.5$ (center panel) and $q_{V}=1$ (right
panel).

###  3.3 Gaussian polytropic source

As a much less trivial example, we consider a self-gravitating object
described by the Gaussian density profile

$\displaystyle\rho=\rho_{0}\,e^{-\frac{r^{2}}{b^{2}\,R^{2}}}\,\Theta(R-r)\ .$
(3.24)

This source becomes homogeneous for $b\gg 1$, while it is mostly concentrated
around the centre for $b\ll 1$. We also assume a polytropic equation of state
[12, 4]

$\displaystyle
p=\gamma\,\rho(r)\left[\frac{\rho(r)}{\rho_{0}}\right]^{n-1}\equiv\gamma\,\frac{\rho^{n}(r)}{\rho_{0}^{n-1}}\
,$ (3.25)

where $\gamma$ and $n$ are the polytropic parameters. In this case, the proper
mass (3.12) is given by

$\displaystyle M_{0}=\pi\,b^{3}\,R^{3}\,\rho_{0}\left[\sqrt{\pi}\,{\rm
Erf}\left(\frac{1}{b}\right)-\frac{2}{b}\,e^{-1/b^{2}}\right]\ .$ (3.26)

In the quadratic approximation of Eq. (3.6), from the canonical field equation
(3.1) and the matching condition (3.3), we find

$\displaystyle\psi_{\rm
in}\simeq\frac{64\,\pi^{3}\,R^{9}\,\rho_{0}^{3}\left(1+3\,\gamma\right)^{3}-27\,M^{3}}{384\,\pi^{3}\,\alpha\,R^{9}\,\rho_{0}^{3}\left(1+3\,\gamma\right)^{3}}+\frac{\sqrt{G_{\rm
N}}\,M}{2\,R^{3}}\,r^{2}\ ,$ (3.27)

where, from Eq. (3.26), the central density can be written in terms of $M_{0}$
as

$\displaystyle\rho_{0}=\frac{M_{0}}{\pi\,b^{2}\,R^{3}\left[\sqrt{\pi}\,b\,{\rm
Erf}\left(1/b\right)-2\,e^{-1/b^{2}}\right]}\ .$ (3.28)

The remaining matching condition (3.2) reads

$\displaystyle\frac{64\,\pi^{3}\,R^{9}\,\rho_{0}^{3}\left(1+3\,\gamma\right)^{3}-27\,M^{3}}{384\,\pi^{3}\,\alpha\,R^{9}\,\rho_{0}^{3}\left(1+3\,\gamma\right)^{3}}=-\frac{3\,\sqrt{G_{\rm
N}}\,M}{2\,R}\ ,$ (3.29)

which shows that the solution (3.27) is indeed the same as the one in Eq.
(3.10). Moreover, in this approximation, neither $\psi_{\rm in}$ nor the
boundary condition (3.29) depend on the polytropic index $n$.

The expression (3.29) can be written in terms of the compactness $X$ in Eq.
(2.5) and the analogue proper compactness $X_{0}=G_{\rm N}\,M_{0}/R$ as

$\displaystyle\frac{3}{2}\,X=\frac{27\,b^{6}\,X^{3}\left[\sqrt{\pi}\,b\,{\rm
Erf}\left(1/b\right)-2\,e^{-1/b^{2}}\right]^{3}-64\,X_{0}^{3}\left(1+3\,\gamma\right)^{3}}{384\,q_{V}\,X_{0}^{3}\left(1+3\,\gamma\right)^{3}}\
,$ (3.30)

which shows the dependence of $X$ on $X_{0}$, equivalent to $M=M(M_{0})$ for
fixed $R$. The plot of $X_{0}$ as a function of $X$ for several particular
cases is shown in Fig. 5. It is clear that $X_{0}<X$ in all displayed cases,
which emphasises once more that $M$ and $M_{0}$ are generally different and
setting them equal would only leave the trivial solution $M=M_{0}=0$. Fig. 5
also shows that the ratio $X_{0}/X$ increases with the parameter $b$ for
constant $\gamma$, respectively decreases with $\gamma$ when $b$ is kept the
same.

Figure 5: Plots of $X_{0}$ as a function of $X$ for $q_{V}=1$. Left panel:
$\gamma=0.5$, respectivelly $b=0.4$ (dotted), $b=0.7$ (dashed) and $b=1$
(dash-dotted). Right panel: $b=0.5$, respectively $\gamma=0.1$ (dotted),
$\gamma=0.5$ (dashed) and $\gamma=1$ (dash-dotted).

##  4 Conclusions

After having explored extensively the bootstrapped Newtonian gravity model in
a series of previous papers, we have now tested an alternative approach to
finding solutions for various cases within the same model. In its standard
form, besides the Laplacian, the Euler-Lagrange equation for the bootstrapped
potential contains extra derivatives of the potential. This makes it very
cumbersome (when at all possible) to obtain solutions, in addition to
hindering the true degrees of freedom of the theory. After performing a field
redefinition, one can write the theory in the canonical form, which is easier
to solve and has a more transparent interpretation.

In a quadratic approximation around the centre, we solved the canonical
equations of motion for a general density profile and compared to the results
obtained by solving the non-canonical equations. One striking and seemingly
general property observed in all cases is that the proper mass and the ADM
mass of the source turn out to be different, $M_{0}\neq M$, regardless of the
values for the other parameters. This is indeed expected due to the non-linear
nature of the theory. The discrepancy between $M$ and $M_{0}$ could lead to
important effects in cosmology and astrophysics. Some of these
phenomenological applications are considered in a separate paper [13].

We should also stress that, in principle, field redefinitions cannot change
the physical content of a (classical) theory. Nevertheless, when one considers
non-linear redefinitions, such as the one employed in this paper, approximate
solutions of the canonical equations are not in one-to-one correspondence with
the approximate solutions of the non-canonical ones. In fact, truncating the
solution of either side of the equivalence would require infinitely many terms
from the other side. One should thus expect large errors when comparing
truncated solutions of each side of the correspondence. We, however, showed
that such an error is quite small for the regimes we are interested in, thus
opening up the possibility of solving the equations of motion in a much
simpler way.

## Acknowledgments

R.C. is partially supported by the INFN grant FLAG and his work has also been
carried out in the framework of activities of the National Group of
Mathematical Physics (GNFM, INdAM). O.M. is supported by the Romanian Ministry
of Research, Innovation, and Digitisation, grant no. 16N/2019 within the
National Nucleus Program.

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