# Energy nonconservation and relativistic trajectories: Unimodular gravity and
beyond

Y. Bonder<EMAIL_ADDRESS>J.E. Herrera  A.M. Rubiol  Instituto de
Ciencias Nucleares, Universidad Nacional Autónoma de México
Apartado Postal 70-543, Cd. Mx., 04510, México

###### Abstract

Energy conservation has the status of a fundamental physical principle.
However, measurements in quantum mechanics do not comply with energy
conservation. Therefore, it is expected that a more fundamental theory of
gravity —one that is less incompatible with quantum mechanics— should admit
energy nonconservations. This paper begins by identifying the conditions for a
theory to have an energy-momentum tensor that is not conserved. Then, the
trajectory equation for pointlike particles that lose energy is derived,
showing that energy nonconservation produces a particular acceleration. As an
example, the unimodular theory of gravity is studied. Interestingly, in
spherical symmetry, given that there is a generalized Birkhoff theorem and
that the energy-momentum tensor divergence is a closed form, the trajectories
of test particles that lose energy can be found using well known methods.
Finally, limits on the energy nonconservation parameters are set using Solar
system observations.

##  I Introduction

Energy conservation is regarded as one of the most fundamental pillars in
physics Lindsay (1971). However, energy can fail to be conserved during a
quantum-mechanical measurement Maudlin  _et al._ (2020). An evident example is
given by a wave packet that is wide in energy. When an energy measurement is
performed, the state collapses into an energy eigenstate, and the hamiltonian
expectation value will, most likely, change. Usually, it is argued that energy
conservation is restored when considering the interaction with the (classical)
measuring apparatus. However, using entangled states, it is easy to devise
examples where this cannot occur, since the energy exchange with the measuring
apparatus can be arbitrarily small when compared to the change in the
hamiltonian expectation value Maudlin  _et al._ (2020). What is more, in
several quantum theories that try to solve the measurement problem, like most
Dynamical Reduction Models Ghirardi  _et al._ (1986); Bassi and Ghirardi
(2003), energy is explicitly not conserved, and similar issues show up in
semiclassical gravity Diez-Tejedor and Sudarsky (2012).

Nowadays, General relativity (GR) is the paradigmatic theory for gravitation.
In GR, there is no generic notion of gravitational energy, and thus, no energy
conservation law can be proposed Bondi (1990) (see Refs. Katz, 2005; Katz  _et
al._ , 2006 for examples of situations where gravitational energy can be
defined). Yet, the energy of matter111Matter refers to all nongravitational
fields. is conserved in the sense that the energy-momentum tensor has
vanishing divergence. In turn, this is a consequence diffeomorphism invariance
and it played a key role in the construction of GR Weinstein (2013).

One of the main challenges of modern physics is to consistently couple gravity
with quantum-mechanical matter Oriti (2009). This, together with the
prediction of spacetime singularities Penrose (1965), and the necessity to add
dark matter Bergström (2000) and dark energy Peebles and Ratra (2003),
strongly suggests that GR may be replaced by a more fundamental theory. This
has motivated the community to propose alternative theories of gravity, some
of which remain in the framework of pseudoriemannian geometry.

The most popular alternative theories of gravity lie within the
$\mathfrak{f}(R)$ framework Sotiriou and Faraoni (2010), which gives rise to
conventional matter energy conservation. However, there are other mechanisms
to construct alternative theories of gravity that lead to energy
nonconservation. In fact, one could argue that theories with unconventional
conservation laws should be “less incompatible” with quantum mechanics. As
such, it is relevant to verify if energy nonconservation can be incorporated
into the relativistic framework: this is the main goal of the paper.

For concreteness, attention is set on the propagation of pointlike particles
that do not comply with energy conservation. Of course, pointlike particles
play a key role in the construction of any geometrical theory Aguilar  _et
al._ (2012); Bonder  _et al._ (2018), and, in GR, it is well understood how
these particles lose energy through gravitational radiation Gralla and Wald
(2008, 2011); Barack (2009). In this paper, the energy-loss mechanism is
assumed to be general and should be thought to arise from more fundamental
physics. Also, notice that gravitational backreaction is not a priori
neglected, even though it is not considered in the examples in the last part
of the paper.

Throughout the text, the notation and conventions of Ref. Wald, 1984 are
followed. Use is made of abstract indexes, which are denoted with lowercase
Latin characters from the beginning of the alphabet: $a,b,c,\ldots$. Component
indexes are represented by Greek characters. As is customary, a pair of
repeated indexes indicates the corresponding contraction. Moreover, the
spacetime metric is denoted by $g_{ab}$, and $g$, without indexes, stands for
the determinant of $g_{\mu\nu}$. When convenient, indexes are raised (lowered)
with the inverse metric $g^{ab}$ ($g_{ab}$). The metric-compatible and
torsion-free derivative operator is $\nabla_{a}$ and ${R_{abc}}^{d}$ is the
Riemann tensor associated with it. The Ricci tensor is
$R_{ab}\equiv{R_{acb}}^{c}$ and $R\equiv g^{ab}R_{ab}$ is the curvature
scalar. Finally, pairs of indexes in between parenthesis represent its
symmetrical part weighted by $1/2$. Units where $G=1=c$ are used, however, $c$
is reintroduced when it helps to make certain approximations.

A description of the structure of the paper may be useful: in the next
section, the conditions for a theory to have a divergence free energy-momentum
tensor are deduced. In Sec. III, the equation for the trajectory of a
pointlike particle that loses energy is derived. To make concrete
calculations, the assumptions of staticity and spherical symmetric are
considered in Sec. IV. Sec. V deals with the application of the formalism to
the unimodular theory of gravity, and, in Sec. VI, some empirical bounds are
set using Solar system data. The concluding remarks are presented in Sec. VII.

##  II When is the energy-momentum tensor divergence free?

This section is devoted to the conditions that lead to a vanishing divergence
of the energy-momentum tensor. It is assumed that gravity is geometrical, that
spacetime is four dimensional, and that matter is described by conventional
tensor fields222An example of unconventional fields are bitensorial fields
Bonder and Morales (2022)., which are standard assumptions in most alternative
theories of gravity. The additional hypotheses under consideration are:

  1. 1.

Gravity is completely described by the metric.

  2. 2.

Minimal coupling: the total action can be naturally separated as

$S[g,\psi]=\frac{1}{2\kappa}S_{G}[g]+S_{M}[g,\psi],$ (1)

where $\kappa$ is the gravitational coupling constant. Also, $S_{G}$ and
$S_{M}$ are respectively known as the gravitational and matter actions.

  3. 3.

The theory is invariant under general diffeomorphisms. In particular, $S_{M}$
is invariant under such transformations.

Hypothesis 1 implies that the relevant derivative operator is $\nabla_{a}$,
which, being metric compatible and torsion free, is determined by $g_{ab}$
(Wald, 1984, chapter 3.1). Moreover, hypothesis 3 is closely related to the
assumption that all the fields in the action are dynamical Corral and Bonder
(2019) and it may be regarded as an application of the principle of general
covariance Wald (1984).

Under these hypotheses, invariance of the matter action under a generic
infinitesimal diffeomorphism produces

$0=\delta S_{M}[g,\psi]=\int d^{4}x\left(\frac{\delta\mathcal{L}_{M}}{\delta
g^{ab}}\delta
g^{ab}+\frac{\delta\mathcal{L}_{M}}{\delta\psi}\delta\psi\right),$ (2)

where $\mathcal{L}_{M}$ is the matter lagrangian density. It is clear that, on
shell, $\delta\mathcal{L}_{M}/\delta\psi=0$. In addition, the energy-momentum
tensor is defined as

$T_{ab}\equiv-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{M}}{\delta
g^{ab}}=T_{(ab)}.$ (3)

Since the action variation is taken with respect to an infinitesimal
diffeomorphism along an arbitrary vector field $\xi^{a}$, the inverse metric
variation is the corresponding Lie derivative, namely, $\delta
g^{ab}=-2\nabla^{(a}\xi^{b)}$. Inserting these results into Eq. (2) leads to

$0=-\int d^{4}x\sqrt{-g}\xi^{b}\nabla^{a}T_{ab},$ (4)

where an integration by parts is performed and the boundary term is ignored;
these terms are disregarded throughout the manuscript. Since Eq. (4) is valid
for all $\xi^{a}$, it follows that $\nabla^{a}T_{ab}=0$. Therefore, any theory
that satisfies the three listed hypotheses gives rise to a divergence free
energy-momentum tensor. In particular, Lovelock’s theorem Lovelock (1972)
implies that GR with a cosmological constant is the only theory that satisfies
that above listed conditions and that leads to second order equations of
motion. On the other hand, $\mathfrak{f}(R)$ theories, where $S_{G}=\int
d^{4}x\sqrt{-g}\ \mathfrak{f}(R)$ for a given function $\mathfrak{f}$, satisfy
the listed hypotheses, and therefore, the corresponding energy-momentum tensor
is conserved.

This work is devoted to theories where at least one of the listed hypotheses
is not met. In such cases, a generalization of the previous argument can be
used to find the modified conservation law; these laws are referred to as
“energy nonconservation” conditions. A particular example for when hypothesis
1 is negated is to consider an independent connection $C_{ab}^{c}$. The matter
action variation, in this case, takes the form

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\delta S_{M}[g,C,\psi]$
$\displaystyle=$ $\displaystyle\int
d^{4}x\left(\frac{\delta\mathcal{L}_{M}}{\delta g^{ab}}\delta
g^{ab}+\frac{\delta\mathcal{L}_{M}}{\delta C_{ab}^{c}}\delta
C_{ab}^{c}+\frac{\delta\mathcal{L}_{M}}{\delta\psi}\delta\psi\right)$
$\displaystyle=$ $\displaystyle-\int
d^{4}x\sqrt{-g}\xi^{d}\left[\nabla^{a}T_{ad}+\Sigma^{ab}_{c}\nabla_{d}C^{c}_{ab}\right.$
$\displaystyle\left.+\nabla_{c}(\Sigma^{ab}_{d}C_{ab}^{c})-\nabla_{a}(\Sigma^{ab}_{c}C_{db}^{c})-\nabla_{b}(\Sigma^{ab}_{c}C_{ad}^{c})\right],$

where $\Sigma^{ab}_{c}\equiv-(1/\sqrt{-g})(\delta\mathcal{L}_{M}/\delta
C_{ab}^{c})$, and $\delta C_{ab}^{c}$ is given by the Lie derivative of
$C_{ab}^{c}$, which is expressed in terms of $\nabla_{a}$. Clearly, for Eq.
(LABEL:actionvariation2) to be valid for all $\xi^{a}$, it is necessary that
what is inside the brackets vanishes, leading to the corresponding energy
nonconservation law. Notice that the key condition for energy nonconservation,
in this case, is that the matter action depends on additional gravitational
degrees of freedom. Other theories with additional gravitational degrees of
freedom include the DHOST theories Langlois and Noui (2016), among others.

When hypothesis 2 is invalid, there is no clear definition of $T_{ab}$, and
hence, no reason for it to have a vanishing divergence. Still, there are
theories, like that of Ref. Bertolami  _et al._ , 2008, where there is a
natural energy-momentum tensor that does not have vanishing divergence.
Finally, when invariance under diffeomorphisms is explicitly broken, $\xi^{a}$
is not general, and it is impossible to conclude, from Eq. (4), that
$\nabla_{a}T^{ab}=0$. Restricted diffeomorphisms appear in theories with
nondynamical fields, such as the unimodular theory of gravity or in the
parametrization of Lorentz violation known as the Standard Model Extension
Colladay and Kostelecký (1997, 1998); Kostelecký (2004) (for concrete examples
see Refs. Bonder, 2013; Bonder and Corral, 2018a; Bluhm  _et al._ , 2019;
Bonder and Peterson, 2020). In fact, to produce a conventional energy
conservation law, many studies in the context of the Standard Model Extension
assume that Lorentz and diffeomorphism invariance are spontaneously broken
Kostelecký (2004); Bluhm and Kostelecký (2005); Bluhm  _et al._ (2008); Bonder
(2015); Bonder and León (2017); Bonder and Peterson (2021).

##  III Pointlike particles

In this section, the equation for the trajectory of a pointlike particle that
loses energy is derived following Papapetrou’s method Papapetrou (1951). It is
assumed that the particle’s energy loss is given by
$\tilde{j}^{b}\equiv\nabla_{a}\tilde{t}^{ab}$, where
$\tilde{t}^{ab}\equiv\sqrt{-g}t^{ab}$ is the tensor density associated with
the particle’s energy-momentum tensor, $t^{ab}=t^{ba}$. Note that, in this
section, the spacetime metric is considered to be known.

The method begins by assuming that the particle is much smaller than the
characteristic radius of the gravitational field, where the “size” of the
particle is extracted from the support of $t^{ab}$. Eventually, the limit
where the size of the particle goes to zero is considered, which eliminates
the dependence on the foliation and the spatial coordinates. As the particle
evolves, it traces a world tube. Within this world tube, an arbitrary smooth
temporal curve $X$ is chosen. Fermi normal coordinates associated with $X$ are
used333These Fermi normal coordinates are defined as follows: The curve $X$ is
affinely parametrized by $t$, and an arbitrary orthonormal basis, whose
timelike vector is parallel to the curve tangent, $u^{a}$, is specified at a
certain point on $X$. Then, this basis is transported along $X$ in such a way
that the basis is kept orthonormal and its timelike vector always coincides
with $u^{a}$ Poisson (2004). Finally, to give coordinates to a point $q$, it
is necessary to find the point $p\in X$ and the vector $v^{a}$ at $p$ such
that $g_{ab}u^{a}v^{b}=0$ and the geodesic that emanates from $p$ with tangent
$v^{a}$ “lands” on $q$ after an affine distance $1$. The coordinates of $q$
are $x^{0}=t$ and the three spatial coordinates, $x^{1}$, $x^{2}$, and
$x^{3}$, are the nonzero components of $v^{a}$ in the transported basis.
Notably, there always exists a neighborhood of $X$ where this coordinates are
well defined Poisson (2004), and here it is assumed that such a region
contains the particle’s world tube.. Inside the support of $t^{ab}$, these
coordinates generate a foliation by constant $t$ hypersurfaces, $\Sigma_{t}$
(see Fig. 1). In what remains of this section, these coordinates are utilized,
as it is emphasized by the use of Greek indexes.

Figure 1: Papapetrou’s method ($t$-$x^{1}$ plane). The gray region represents
the world tube of the particle that is given by the support of its energy-
momentum tensor. The arbitrary curve $X$ and the $t=t_{0}$ hypersurface,
$\Sigma_{t_{0}}$, are depicted.

From its definition, it is possible to see that

$\tilde{j}^{\alpha}=\partial_{\mu}\tilde{t}^{\alpha\mu}+\Gamma^{\alpha}_{\mu\nu}\tilde{t}^{\mu\nu}.$
(6)

Multiplying by $x^{\beta}$, taking the symmetric part, and adding
$\tilde{t}^{\alpha\beta}$, yields

$\tilde{t}^{\alpha\beta}+x^{(\alpha}\tilde{j}^{\beta)}=\tilde{t}^{\alpha\beta}+x^{(\alpha}\partial_{\mu}\tilde{t}^{\beta)\mu}+x^{(\alpha}\Gamma^{\beta)}_{\mu\nu}\tilde{t}^{\mu\nu}.$
(7)

Introducing a Kronecker delta as $\partial_{\mu}x^{\alpha}$, the last equation
takes the form

$\partial_{\mu}(x^{(\alpha}\tilde{t}^{\beta)\mu})=\tilde{t}^{\alpha\beta}+x^{(\alpha}\tilde{j}^{\beta)}-x^{(\alpha}\Gamma^{\beta)}_{\mu\nu}\tilde{t}^{\mu\nu}.$
(8)

Integrating Eqs. (6) and (8) in $\Sigma_{t}$ yields

$\displaystyle\frac{d}{dt}\int d^{3}x\ \tilde{t}^{t\alpha}$ $\displaystyle=$
$\displaystyle\int
d^{3}x\left[\tilde{j}^{\alpha}-\tilde{t}^{\mu\nu}\Gamma^{\alpha}_{\mu\nu}\right],$
(9) $\displaystyle\frac{d}{dt}\int d^{3}x\ x^{(\alpha}\tilde{t}^{\beta)t}$
$\displaystyle=$ $\displaystyle\int d^{3}x$
$\displaystyle\left[\tilde{t}^{\alpha\beta}+x^{(\alpha}\tilde{j}^{\beta)}-x^{(\alpha}\Gamma^{\beta)}_{\mu\nu}\tilde{t}^{\mu\nu}\right],$

where the integrals of the spatial divergences are zero by virtue of the
divergence theorem and the fact that the integrands have compact support, and
the index $t$ refers to the component along the $t$ coordinate. Recall that
the integrands are densities, and thus, the integrals are well defined.

Importantly, in the point particle approximation (also known as the monopole
approximation), which is the relevant regime for this work, it is enough to
work to zeroth order in $\delta x^{\alpha}\equiv x^{\alpha}-X^{\alpha}$. Eqs.
(9)-(LABEL:Pap2), to zeroth order in $\delta x^{\alpha}$, become

$\frac{d}{dt}\int d^{3}x\ \tilde{t}^{t\alpha}=\int d^{3}x\
\tilde{j}^{\alpha}-\Gamma^{\alpha}_{\mu\nu}(X)\int d^{3}x\
\tilde{t}^{\mu\nu},$ (11)

and

$\displaystyle\frac{dX^{(\alpha}}{dt}\int d^{3}x\
\tilde{t}^{\beta)t}+X^{(\alpha}\frac{d}{dt}\int d^{3}x\ \tilde{t}^{\beta)t}$
(12) $\displaystyle=$ $\displaystyle\int d^{3}x\
\tilde{t}^{\alpha\beta}+X^{(\alpha}\int d^{3}x\ \tilde{j}^{\beta)}$
$\displaystyle-X^{(\alpha}\Gamma^{\beta)}_{\mu\nu}(X)\int d^{3}x\
\tilde{t}^{\mu\nu}.$

These equations can be combined as

$\int d^{3}x\ \tilde{t}^{\alpha\beta}=\frac{dX^{(\alpha}}{dt}\int d^{3}x\
\tilde{t}^{\beta)t}.$ (13)

Recall that $t$ is an affine parameter along $X$, and thus, $u^{\alpha}\equiv
dX^{\alpha}/dt$ is such that $u^{\alpha}u_{\alpha}=-1$ and $u^{t}=1$. What is
more, it is convenient to define $M^{\alpha\beta}\equiv\int d^{3}x\
\tilde{t}^{\alpha\beta}$. Then, Eq. (13) takes the form

$M^{\alpha\beta}=u^{(\alpha}M^{\beta)t},$ (14)

and its $\alpha t$ component satisfies

$\displaystyle M^{\alpha t}=u^{\alpha}M^{tt}.$ (15)

By means of Eq. (15), Eq. (14) can be written as

$M^{\alpha\beta}=mu^{\alpha}u^{\beta},$ (16)

with $m\equiv M^{tt}$, which is taken as positive invoking the weak energy
condition Wald (1984).

With these results, Eq. (11) becomes

$\frac{u^{\alpha}}{m}\frac{dm}{dt}+u^{\beta}\nabla_{\beta}u^{\alpha}=J^{\alpha},$
(17)

where $J^{\alpha}\equiv(1/m)\int d^{3}x\ \tilde{j}^{\alpha}$. Multiplying by
$u_{\alpha}$ yields

$\frac{-1}{m}\frac{dm}{dt}=J^{\alpha}u_{\alpha},$ (18)

where use is made of the fact that
$u_{\alpha}u^{\beta}\nabla_{\beta}u^{\alpha}=(1/2)u^{\beta}\nabla_{\beta}(-1)=0$.
With all this, Eq. (17) can be written as

$u^{\beta}\nabla_{\beta}u^{\alpha}=J_{\perp}^{\alpha},$ (19)

with $J_{\perp}^{\alpha}\equiv
J^{\beta}(\delta^{\alpha}_{\beta}+u^{\alpha}u_{\beta})$ being components of
the projection of $J^{a}$ orthogonal to $u^{a}$. Notice that a projection is
expected by the fact that $t$ is an affine parameter. In addition, when
$J_{\perp}^{\alpha}=0$, the geodesic equation is recovered. Yet, in general,
due to energy loss, the particle trajectory is subject to an acceleration
$J_{\perp}^{\alpha}$. Clearly, for Eq. (19) to be coordinate invariant, the
limit where the size of the particle goes to zero, while keeping
$J_{\perp}^{\alpha}$ finite, must be taken. Still, the result has the form
shown in Eq. (19), which is the main result of this section and whose
consequences are studied in what follows.

##  IV Staticity and spherical symmetry

As is well known, the gravitational field produced by the Sun, which is the
relevant environment used here to test the method, is properly modeled within
the assumptions of staticity and spherical symmetry. The former symmetry means
that there exists a timelike Killing vector field that is hypersurface
orthogonal Wald (1984), thus, if $t$ is the time coordinate such that this
Killing field is $\left(\partial/\partial t\right)^{a}$, there is a $t\to-t$
invariance. Spherical symmetry, on the other hand, can be rigorously defined
Wald (1984) and it matches the intuitive notion. Moreover, spherical symmetry
gives rise to adapted coordinates in the constant $t$ hypersurfaces that, in
the regions under consideration, are a generalization of the conventional
spherical coordinates $r$, $\theta$, and $\phi$, with $r>0$.

To proceed, an expression for the spacetime metric is required. Let $f(r)$ and
$h(r)$ be such that the static and spherically symmetric metric, in the
coordinates at hand, can be brought to the form

$ds^{2}=-f(r)dt^{2}+h(r)dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right).$ (20)

Importantly, it is assumed that, in the regions of spacetime that are relevant
for this work, $f>0$ and $h>0$, avoiding possible singularities.

In addition, one can verify that any vector field $v^{a}$ that is invariant
under these symmetries must have the following form

$v^{a}=A(r)\left(\frac{\partial}{\partial
t}\right)^{a}+B(r)\left(\frac{\partial}{\partial r}\right)^{a},$ (21)

where $A(r)$ and $B(r)$ are arbitrary functions. It is assumed that, for
simplicity, the nonconservation current $J^{a}$ is subject to these
symmetries, thus, it has two independent components. What is more, the
orthogonal current $J_{\perp}^{a}$ is completely characterized by a single
function $J_{\perp}^{r}(r)$ since

$J_{\perp}^{t}=\frac{hu^{r}}{fu^{t}}J_{\perp}^{r},$ (22)

where $u^{t}>0$.

Importantly, an equation of the form of Eq. (19), when staticity and
spherically symmetry are imposed, does not imply that there exists a constant
of motion associated with its time component [see for comparison Eq. (23)].
Still, the trajectory is restricted to a plane, say $\theta=\pi/2$, and there
is a constant, along each particle trajectory, associated to the Killing field
$\psi^{a}\equiv(\partial/\partial\phi)^{a}$. This constant is given by
$l\equiv g_{ab}u^{a}\psi^{b}=r^{2}\sin^{2}\theta\dot{\phi}$, where the overdot
represents a derivative with respect to the curve parameter. To see that $l$
is constant, one must calculate its change along the trajectory:

$\displaystyle u^{a}\nabla_{a}l$ $\displaystyle=$ $\displaystyle
u^{a}\nabla_{a}\left(u^{b}\psi_{b}\right)$ (23) $\displaystyle=$
$\displaystyle u^{b}u^{a}\nabla_{(a}\psi_{b)}+J_{\perp}^{\phi}=0,$

where, in the last step, the Killing equation and the fact that
$J_{\perp}^{\phi}=0$ are used.

The facts that the motion is restricted to a plane and that there is one
constant of motion, make the process of solving Eq. (19) simpler. In addition,
it can be seen that the velocity norm, $\kappa\equiv u^{a}u_{a}$, is constant
along the curve:

$u^{a}\nabla_{a}\kappa=2u_{b}u^{a}\nabla_{a}u^{b}=2u_{b}J_{\perp}^{b}=0.$ (24)

Inserting the constants of motion in the velocity normalization condition
produces

$\kappa=g_{ab}u^{a}u^{b}=-f\dot{t}^{2}+h\dot{r}^{2}+\frac{l^{2}}{r^{2}}.$ (25)

In turn, the last expression can be written as

$\displaystyle\dot{t}^{2}$ $\displaystyle=$
$\displaystyle\frac{h}{f}\dot{r}^{2}+\frac{l^{2}}{fr^{2}}-\frac{\kappa}{f}.$
(26)

To proceed, note that the $r$ component of Eq. (19) takes the form

$\ddot{r}+\frac{f^{\prime}}{2h}\dot{t}^{2}+\frac{h^{\prime}}{2h}\dot{r}^{2}-\frac{r}{h}\dot{\phi}^{2}=J_{\perp}^{r},$
(27)

where the prime represents an $r$ derivative. Utilizing Eq. (26) and the
expression for $l$, Eq. (27) can be written as

$\ddot{r}+\left(\frac{f^{\prime}}{f}+\frac{h^{\prime}}{h}\right)\frac{\dot{r}^{2}}{2}+\frac{f^{\prime}}{2fh}\left(\frac{l^{2}}{r^{2}}-\kappa\right)-\frac{l^{2}}{hr^{3}}=J_{\perp}^{r}.$
(28)

Note that $\ddot{r}=(1/2)d\dot{r}^{2}/dr$, which is a total $r$ derivative.
Still, the remaining terms in Eq. (28) cannot be written as a total $r$
derivative. Thus, in general, Eq. (28) cannot be trivially integrated.
Remarkably, in unimodular gravity it is possible perform this integration. For
this reason, the unimodular theory of gravity is studied in what follows.

##  V Unimodular gravity

###  V.1 Basic aspects

The unimodular theory is an alternative theory of gravity with some
interesting features Einstein (1919); Anderson and Finkelstein (1971); van der
Bij  _et al._ (1982); Buchmuller and Dragon (1988); Unruh (1989); Henneaux and
Teitelboim (1989); Ng and van Dam (1991); Finkelstein  _et al._ (2001); Ng and
Van Dam (2001); Ellis  _et al._ (2011); Bonder and Corral (2018b); Carballo-
Rubio  _et al._ (2022). It can be described as a theory in four spacetime
dimensions where, in addition to the conventional Einstein-Hilbert action
term, there is a Lagrange multiplier that constraints the volume form to
coincide with a nondynamical $4$-form. In turn, the presence of this
nondynamical $4$-form partially breaks invariance under diffeomorphisms.
Concretely, the unimodular action, written as the integral of a scalar
density, takes the form

$S=\int d^{4}x\
\frac{1}{2\kappa}\left[\sqrt{-g}R+\lambda\left(\sqrt{-g}-F\right)+\mathcal{L}_{M}(g,\psi)\right],$
(29)

where $\lambda$ is the Lagrange multiplier, $F$ is a nondynamical scalar
density associated to the nondynamical $4$-form, and $\psi$ collectively
represents all the matter fields. The equations of motion are

$\displaystyle G_{ab}-\frac{1}{2}\lambda g_{ab}$ $\displaystyle=$
$\displaystyle\kappa T_{ab},$ (30) $\displaystyle\sqrt{-g}$ $\displaystyle=$
$\displaystyle F,$ (31)
$\displaystyle\frac{\delta\mathcal{L}_{M}}{\delta\psi}$ $\displaystyle=$
$\displaystyle 0,$ (32)

where $G_{ab}\equiv R_{ab}-g_{ab}R/2$ is the Einstein tensor and the energy-
momentum tensor is defined by Eq. (3). Also, Eq. (31) is known as the
unimodular constraint.

The trace of Eq. (30) generates

$\lambda=-\frac{1}{2}\left(\kappa T+R\right)$ (33)

where $T\equiv g^{ab}T_{ab}$. Using this expression, Eq. (30) can be recast as

$R^{\rm TL}_{ab}\equiv
R_{ab}-\frac{1}{4}Rg_{ab}=\kappa\left(T_{ab}-\frac{1}{4}Tg_{ab}\right).$ (34)

It is easy to see that this equation is traceless, and, together with the
unimodular constraint, they comprise a set of ten equations.

The divergence of Eq. (34) produces

$\frac{1}{4}\nabla_{a}R=\kappa\left(\nabla^{b}T_{ab}-\frac{1}{4}\nabla_{a}T\right).$
(35)

Therefore, in cases when $\nabla^{b}T_{ab}=0$, $\tilde{\Lambda}\equiv(R+\kappa
T)/4$ is a constant. Interestingly, inserting $\tilde{\Lambda}$ into Eq. (34)
yields the conventional Einstein equations with $\tilde{\Lambda}$ acting as a
cosmological constant. In other words, unimodular gravity, with the additional
assumption that the energy-momentum tensor is divergence free, is dynamically
equivalent to GR with a cosmological constant that arises as an integration
constant. As such, this constant is completely independent from the vacuum
energy of the matter fields, offering a plausible explanation to the
“cosmological constant problem” Weinberg (1989) (see also Ref. Bengochea  _et
al._ , 2020).

Moreover, $F$, being nondynamical, does not transform under diffeomorphisms.
As a consequence, the action (29) is not invariant under all diffeomorphisms.
To show this, consider the simpler case when $T_{ab}=0$. The action variation
under an infinitesimal diffeomorphism associated with $\xi^{a}$ takes the form

$\delta S=\int d^{4}x\
\sqrt{-g}\lambda\nabla_{a}\left(\frac{F}{\sqrt{-g}}\xi^{a}\right),$ (36)

where the Bianchi identity $\nabla_{a}G^{ab}=0$ is used. On shell,
$F=\sqrt{-g}$, and the action is invariant if and only if
$\nabla_{a}\xi^{a}=0$. Now, to find the conservation law, it is necessary to
express a divergence free vector field $\xi^{a}$ in terms of an arbitrary
tensor. This is achieved through an antisymmetric tensor $\alpha_{ab}$. Let

$\xi^{a}=\epsilon^{abcd}\nabla_{b}\alpha_{cd},$ (37)

where $\epsilon_{abcd}$ is the metric volume form, i.e., the $4$-form such
that $\epsilon^{abcd}\epsilon_{abcd}=-4!$ and $\nabla_{a}\epsilon_{bcde}=0$.
Then,

$\displaystyle\nabla_{a}\xi^{a}$ $\displaystyle=$
$\displaystyle\nabla_{a}\left(\epsilon^{abcd}\nabla_{b}\alpha_{cd}\right)$
(38) $\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon^{abcd}\left(\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a}\right)\alpha_{cd}$
$\displaystyle=$ $\displaystyle\epsilon^{abcd}{R_{[abc]}}^{e}\alpha_{ed}=0,$

where the squared brackets denote the totally antisymmetric part and, in the
last step, the identity ${R_{[abc]}}^{d}=0$ is used.

The matter action variation is still given by Eq. (2) but now the vector field
$\xi^{a}$ is restricted to be divergence free. Expressing $\xi^{a}$ in terms
of $\alpha_{ab}$ yields

$0=\delta S_{M}=\int d^{4}x\
\sqrt{-g}\alpha_{de}\epsilon^{bcde}\nabla_{c}\nabla^{a}T_{ab},$ (39)

where a double integration by parts is performed. Given that $\alpha_{ab}$ is
arbitrary, the corresponding energy nonconservation law can be readily found;
it reads

$0=\epsilon^{bcde}\nabla_{c}\nabla^{a}T_{ab}.$ (40)

Clearly, this conservation law is satisfied when $\nabla^{a}T_{ab}=0$, in
which case the theory reduces to GR, as is argued above. Yet, Eq. (40) has
more general solutions. In fact, denoting $j^{a}\equiv\nabla_{b}T^{ab}$, Eq.
(40) implies that the exterior derivative of $j_{a}$ has to vanish, namely
${\rm d}j=0$. What is more, using Poincare’s lemma Nakahara (2016), it is
possible to show that, if spacetime is contractible, as it is assumed hereon,
there exists a scalar function $\Phi$, called the nonconservation potential,
such that

$j_{a}=-\nabla_{a}\Phi.$ (41)

This fact makes unimodular gravity tractable with the standard particle
trajectory methods. The other result is a generalization a well-known theorem,
which is presented in the next subsection.

###  V.2 Generalized Birkhoff theorem

The Birkhoff theorem states that the only spherically symmetric solution of
the vacuum Einstein equation, $R_{ab}=0$, is the Schwarzschild solution, which
is static. In this subsection, an analogous result is derived in the context
of vacuum unimodular gravity, following the method presented in Ref. (Hartle,
2003, page 468). The goal is to solve Eq. (34) in vacuum, namely, $R^{\rm
TL}_{ab}=0$, in spherical symmetry. In adapted coordinates, the most general
spherically symmetric line element, which is not necessarily static, is

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-A\left(t,r\right)dt^{2}+2B\left(t,r\right)dtdr+C\left(t,r\right)dr^{2}$
(42)
$\displaystyle+r^{2}\left(d\theta^{2}+r^{2}\sin^{2}\theta\mathrm{d}\phi^{2}\right),$

where $A$, $B$, and $C$ are arbitrary functions. With a coordinate
transformation, $\tilde{t}=\tilde{t}(t,r)$, it is possible to absorb the $B$
function, obtaining

$ds^{2}=-e^{\nu\left(\tilde{t},r\right)}d\tilde{t}^{2}+e^{\rho\left(\tilde{t},r\right)}dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}\theta\
d\phi^{2}\right).$ (43)

With this metric, the $\tilde{t}r$ component of $R^{\rm TL}_{ab}=0$ is simply

$\frac{1}{r}\frac{\partial\rho}{\partial\tilde{t}}=0,$ (44)

which is solved by $\rho=\rho(r)$. In addition,

$0=re^{\rho-\nu}R^{\rm TL}_{\tilde{t}\tilde{t}}+rR^{\rm
TL}_{rr}=\nu^{\prime}+\rho^{\prime},$ (45)

which implies

$\nu\left(\tilde{t},r\right)=\beta\left(\tilde{t}\right)-\rho\left(r\right).$
(46)

What is more, the $\theta\theta$ component of $R^{\rm TL}_{ab}=0$ becomes

$0=\frac{4}{r^{2}}e^{\rho}R^{\rm
TL}_{\theta\theta}=\frac{2}{r^{2}}(e^{\rho}-1)+\rho^{\prime
2}-\rho^{\prime\prime},$ (47)

whose solution is

$\rho\left(r\right)=-\ln\left(1-\frac{2M}{r}+\Lambda r^{2}\right),$ (48)

with $M$ and $\Lambda$ constants known as the mass and the cosmological
constant, respectively.

Using Eqs. (46) and (48), the metric (43) takes the form

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{\beta\left(\tilde{t}\right)}\left(1-\frac{2M}{r}+\Lambda
r^{2}\right)d\tilde{t}^{2}+\frac{dr^{2}}{1-\frac{2M}{r}+\Lambda r^{2}}$ (49)
$\displaystyle+r^{2}\left(d\theta^{2}+\sin^{2}\theta\ d\phi^{2}\right).$

Finally, it is possible to change coordinates once again and introduce
$\hat{t}=\hat{t}(\tilde{t})$ to absorb $\beta$, producing

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}+\Lambda
r^{2}\right)d\hat{t}^{2}+\frac{dr^{2}}{1-\frac{2M}{r}+\Lambda r^{2}}$ (50)
$\displaystyle+r^{2}\left(d\theta^{2}+\sin^{2}\theta\ d\phi^{2}\right),$

which is the Schwarzschild, Schwarzschild-de Sitter, or Schwarzschild-antide
Sitter metric, depending on the sign of $\Lambda$, which are static. In
addition, it can be verified that the metric (50) solves all the components of
$R^{\rm TL}_{ab}=0$. This result is clearly compatible with the fact,
discussed above, that, in vacuum, unimodular gravity reduces to GR with a
cosmological constant. In this sense, this result is a generalization,
relevant for unimodular gravity, of Birkhoff’s celebrated theorem.

##  VI Empirical constraints

In this section, Solar system observations are used to test particular models
for energy nonconservation. This is done as a proof of concept; the goal is to
show that energy nonconservation produces physical effects. For simplicity, a
situation where all the calculations can be performed analytically is chosen.
The proposal is to consider the vacuum propagation of pointlike _test_
particles that lose energy according to unimodular gravity, in a background
that is a solution of this theory. Also, spherical symmetry is assumed, for
the background metric and for the energy nonconservation potential, since it
is a good approximation to account for Solar system observations. Moreover,
$c$ is restored to be able to expand in $c^{-2}$.

Interestingly, vacuum unimodular gravity, under the symmetries at hand, is a
theory where Eq. (28) is a total derivative, which greatly simplifies the
calculations. This follows from Eq. (41) and from the generalized Birkhoff
theorem, which implies $f=h^{-1}$. As a consequence, the following identities
can be verified:

$\displaystyle\frac{f^{\prime}}{f}+\frac{h^{\prime}}{h}$ $\displaystyle=$
$\displaystyle 0,$ (51)
$\displaystyle\frac{f^{\prime}}{2fh}\left(\frac{l^{2}}{r^{2}}-\kappa\right)-\frac{l^{2}}{hr^{3}}$
$\displaystyle=$
$\displaystyle\frac{d}{dr}\left[\frac{f}{2}\left(\frac{l^{2}}{r^{2}}-\kappa\right)\right].$
(52)

Now, Eq. (41) states that $j_{a}=-\nabla_{a}\Phi/c^{2}$, where the factor
$c^{-2}$ is introduced so that the loss of energy is at the same order than
the gravitational effects. Clearly, in spherical symmetry, $\Phi=\Phi(r)$, and
thus, the right-hand side of Eq. (28) is also a total $r$ derivative. Notice
that the right-hand side of Eq. (28), when given by this derivative, should be
regarded as an additional, albeit well motivated, assumption, since what
appears on the right-hand side of Eq. (19) is proportional to the integral of
$j_{a}$ in the limit when the particle’s size goes to zero.

Eq. (28), with these considerations, becomes

$\frac{1}{2}\frac{d\dot{r}^{2}}{dr}+\frac{d}{dr}\left[\frac{f}{2}\left(\frac{l^{2}}{r^{2}}-\kappa\right)\right]=-\frac{1}{c^{2}}\frac{d\Phi}{dr},$
(53)

which can be integrated to produce

$E=\frac{1}{2}\dot{r}^{2}+V_{\rm eff}(r),$ (54)

where $E$ is an integration constant and

$V_{\rm
eff}(r)\equiv\frac{f}{2}\left(\frac{l^{2}}{r^{2}}-\kappa\right)+\frac{1}{c^{2}}\Phi.$
(55)

Clearly, Eq. (54) has the form of an energy equation in conventional Newtonian
dynamics for a system of one degree of freedom.

Before proceeding, note that the observed Adam  _et al._ (2016) cosmological
constant, $\Lambda$, is such that, for all the relevant values of $r$,
$\Lambda r^{2}\ll 1$. Thus, such contributions are neglected and
$f=1-2M/(c^{2}r)$. In consequence, the utilized metric is the Schwarzschild
metric and the only unconventional effects that are considered here are due to
$\Phi$. In what follows, comparisons with Solar system data are performed.

###  VI.1 Null trajectories

Given that light follows null trajectories contained in a plane, $\kappa=0$
and $\theta=\pi/2$. What is more, it is useful to define $b\equiv
l/\sqrt{2E}$, which, when $\Phi=0$, is the impact parameter. Assuming444The
case where $l<0$ is related to what is done here by changing the coordinates’
orientation; radial trajectories, where $l=0$, must be analyzed independently.
that $l>0$, it is possible to express Eq. (54) as

$\frac{1}{b^{2}}=\frac{1}{l^{2}}\dot{r}^{2}+V_{\rm eff}^{\rm n},$ (56)

where

$V_{\rm eff}^{\rm n}\equiv\frac{1}{r^{2}}f+\frac{2}{c^{2}l^{2}}\Phi_{\rm n},$
(57)

and $\Phi_{\rm n}$ is the nonconservation potential for null trajectories.

To proceed, a concrete expression for $\Phi_{\rm n}$ is necessary. Such a
potential is assumed to vanish as $r\to\infty$ to avoid sourcing energy
nonconservation by objects located arbitrarily far away. Moreover, $\Phi_{\rm
n}$ is taken to be such that it maintains the structure of the maxima and
minima of the effective potential as compared with the $\Phi_{\rm n}=0$ case.
This is because a change in this structure would produce dramatic effects that
have not been observed. Given that $V_{\rm eff}^{\rm n}$, when $\Phi_{\rm
n}=0$, has $r^{-2}$ and $r^{-3}$ terms, the following form for the
nonconservation potential is considered:

$\Phi_{\rm n}=\frac{b^{\rm n}_{2}}{r^{2}}+\frac{b^{\rm n}_{3}}{r^{3}},$ (58)

where $b^{\rm n}_{2}$ and $b^{\rm n}_{3}$ are free parameters. From
inspection, it can be verified that, for these parameters to keep the form of
the effective potential, it is necessary that

$\displaystyle l^{2}c^{2}+2b^{\rm n}_{2}>0,$ (59) $\displaystyle Ml^{2}-b^{\rm
n}_{3}>0.$ (60)

These are restrictions on $l$ and on the nonconservation parameters, and are
assumed hereon. The qualitative form of $V_{\rm eff}^{\rm n}$ is plotted in
Fig. 2. Importantly, in general, the parameters of this nonconservation
potential cannot be absorbed into $M$ and $l$.

Figure 2: Effective potential for null trajectories as a function of the
radial coordinate $r$. For a given $b$, there is a radius of maximal approach,
$r_{0}$.

####  VI.1.1 Deflection of light

The deflection of light was the first prediction of GR to be empirically
confirmed, and refinements of this effect have been proposed, arising, for
example, from QED radiative corrections Drummond and Hathrell (1980) or from
higher-order geometrical theories Accioly  _et al._ (2017). Here, this effect
is computed for unimodular gravity and with the above presented
nonconservation potential.

Light trajectories are described by the angle $\phi$ as a function of $r$. In
turn, $\phi(r)$ can be found after integrating $d\phi/dr=\dot{\phi}/\dot{r}$;
the numerator is $\dot{\phi}=l/r^{2}$, and the denominator is obtained from
Eq. (54), producing

$\frac{d\phi}{dr}=\frac{1}{r^{2}\sqrt{\frac{1}{b^{2}}-\frac{f}{r^{2}}-\frac{2}{l^{2}c^{2}}\Phi_{\rm
n}}}.$ (61)

To calculate the total deflection angle, $\Delta\phi_{\rm def}$, it is
customary to consider that the light beam emerges and returns to $r\to\infty$.
Given the symmetries of the problem, the net effect is twice the $r$-integral
of Eq. (61) from infinity to the radius of maximal approach, $r_{0}$, which
corresponds to the radius where $\dot{r}=0$. From Eq. (56) at $r_{0}$, it can
be seen that

$\frac{1}{b^{2}}=\frac{1}{r_{0}^{2}}f(r_{0})+\frac{2}{c^{2}l^{2}}\Phi_{\rm
n}(r_{0}).$ (62)

With this, it can be verified that the total deflection angle satisfies

$\Delta\phi_{\rm def}=\int_{r_{0}}^{\infty}\frac{2\
dr}{r^{2}\sqrt{\frac{f(r_{0})}{r_{0}^{2}}-\frac{f(r)}{r^{2}}+\frac{2}{c^{2}l^{2}}\left[\Phi_{\rm
n}(r_{0})-\Phi_{\rm n}(r)\right]}}.$ (63)

To integrate, it is helpful to expand the integrand in powers of $c^{-2}$; the
leading contribution is

$\Delta\phi^{(0)}_{\rm def}=\int_{r_{0}}^{\infty}\frac{2\
dr}{r^{2}\sqrt{\frac{1}{r_{0}^{2}}-\frac{1}{r^{2}}}}=\pi.$ (64)

Moreover, the first correction takes the form

$\displaystyle\Delta\phi^{(1)}_{\rm def}$ $\displaystyle=$
$\displaystyle\frac{1}{c^{2}}\int_{r_{0}}^{\infty}\frac{dr\left\\{\frac{2M}{r_{0}^{3}}-\frac{2M}{r^{3}}-\frac{2}{l^{2}}\left[\Phi_{\rm
n}(r_{0})-\Phi_{\rm
n}(r)\right]\right\\}}{r^{2}\left(\frac{1}{r_{0}^{2}}-\frac{1}{r^{2}}\right)^{3/2}}$
(65) $\displaystyle=$ $\displaystyle\frac{4(Ml^{2}-b^{\rm
n}_{3})}{c^{2}l^{2}r_{0}}-\frac{\pi b^{\rm n}_{2}}{c^{2}l^{2}}.$

Considering that $\Delta\phi_{\rm def}=\pi$ corresponds to a light beam that
follows a straight trajectory, the relevant deflection angle is

$\delta\phi_{\rm def}\equiv\Delta\phi_{\rm def}-\pi=\frac{4(Ml^{2}-b^{\rm
n}_{3})}{c^{2}l^{2}r_{0}}-\frac{\pi b^{\rm n}_{2}}{c^{2}l^{2}}.$ (66)

At this point, it is convenient to use $r_{0}=b+O\left(c^{-2}\right)$, which
is derived from Eq. (62). Also, Eq. (56) in the region $r\to\infty$ implies
$l=b$, where the fact that $\dot{r}\to 1$ is utilized555Here, $b$ is the
geometrical impact parameter up to $O(c^{-2})$ corrections, which can be
neglected.. The final expression reads

$\delta\phi_{\rm def}=\frac{4M}{c^{2}b}-\frac{\pi b^{\rm
n}_{2}}{c^{2}b^{2}}-\frac{4b^{\rm n}_{3}}{c^{2}b^{3}}.$ (67)

As expected, $\delta\phi_{\rm def}$ has the GR term plus modifications due to
$b^{\rm n}_{2}$ and $b^{\rm n}_{3}$. The former produces an effect that goes
like $b^{-2}$, while the latter goes like $b^{-3}$, in agreement with naive
dimensional considerations.

Bounds on $b^{\rm n}_{2}$ and $b^{\rm n}_{3}$ are set by using the well-known
Parameterized Post-Newtonian formalism (PPN) Will (2018). In turn, the PPN
bounds are set using light rays that pass extremely close to the surface of
the Sun, therefore, the impact parameter is approximately the Sun’s radius,
$R_{\odot}$. Moreover, the corresponding PPN expression reads Will (2014)

$\delta\phi^{\rm PPN}_{\rm def}=(1+\gamma)\frac{2M_{\odot}}{c^{2}R_{\odot}},$
(68)

where $\gamma$ is the PPN parameter and $M_{\odot}$ is the mass of the Sun.
Comparison with Eq. (67) yields

$\frac{\pi b^{\rm n}_{2}}{2M_{\odot}R_{\odot}}+\frac{2b^{\rm
n}_{3}}{M_{\odot}R_{\odot}^{2}}=1-\gamma.$ (69)

The corresponding PPN bounds are Lambert and Le Poncin-Lafitte (2011)

$-2.0\times 10^{-4}<\gamma-1<0.4\times 10^{-4},$ (70)

which, using $R_{\odot}\approx 5\times 10^{5}M_{\odot}$, translate to

$-51<4.0\frac{b^{\rm n}_{2}}{M_{\odot}^{2}}+(1.1\times 10^{-5})\frac{b^{\rm
n}_{3}}{M_{\odot}^{3}}<255.$ (71)

Note that a change in the reference scale, form $M_{\odot}$ to, say, ${\rm
GeV}\sim 10^{-57}M_{\odot}$, would drastically change the numbers in Eq. (71).
Finally, it is important to emphasize that these limits are compatible with
conditions (59)-(60) provided that

$\frac{b^{\rm n}_{3}}{M_{\odot}^{3}}<2.5\times 10^{11}.$ (72)

An interesting proposal666Due to an anonymous referee. is to find an effective
index of refraction, $n_{\rm eff}$, that describes the effects of gravity and
energy nonconservation. In Ref. Fischbach and Freeman (1980) a gravitational
effective index of refraction is found by comparing the effects of a media and
those of spacetime curvature in Maxwell’s equations. Thus, to utilize the
methods of Ref. Fischbach and Freeman (1980), it is necessary to incorporate
energy nonconservation into an effective metric. For the case at hand, this
can be done by inspecting Eq. (57) and noticing that there is an effective
function, $f_{\rm eff}$, such that $V_{\rm eff}^{\rm n}$ has the same
functional form as in GR but with $f_{\rm eff}$ in place of $f$. This function
has the form

$f_{\rm eff}\equiv f+\frac{2r^{2}}{c^{2}l^{2}}\Phi_{\rm n}=1+\frac{2b^{\rm
n}_{2}}{c^{2}l^{2}}-\frac{2M}{c^{2}r}\left(1-\frac{b^{\rm
n}_{3}}{Ml^{2}}\right),$ (73)

and the corresponding “effective metric” has the form of Eq. (50) but with
$f_{\rm eff}$ instead of $f$. Note, however, that energy nonconservation is
not “geometrizable” in the sense that $f_{\rm eff}$ depends on the light
trajectory via $l$. With this observation, finding $n_{\rm eff}$ reduces to
writing the metric in isotropic coordinates and reading off the time and
spatial components Fischbach and Freeman (1980); the result is

$n_{\rm eff}=1-\frac{b^{\rm n}_{2}}{c^{2}l^{2}}+\frac{M-b^{\rm
n}_{3}/l^{2}}{c^{2}r},$ (74)

where, for simplicity, an expansion on $c^{-2}$ is performed. Eq. (74) reduces
to the correct results when the parameters for energy nonconservation are set
to zero. What is more, light trajectories could be found from $n_{\rm eff}$
following the iterative procedure described in Ref. Fischbach and Freeman
(1980). More interestingly, Eq. (74) could lead to new experiments to search
for static and spherically symmetric forms of energy nonconservation,
particularly in the realm of tabletop optics experiments.

###  VI.2 Time delay

The time delay, as originally devised by Shapiro Shapiro (1964), is calculated
in this subsection for the case where there is energy nonconservation. This
delay concerns the time spend by a light signal that travels from the Earth to
a reflecting satellite that orbits the Sun, and back, when the light beam
passes close to the Sun. In addition, this effect is calculated neglecting the
change in the position of the Earth and the satellite. The relevant expression
is $t=t(r)$, which is obtained by integrating $dt/dr=\dot{t}/\dot{r}$, where
the fact that $\dot{r}\neq 0$ is used. From Eq. (26), it is possible to write

$\frac{dt}{dr}=\pm\frac{1}{cf}\sqrt{1+\frac{l^{2}f}{r^{2}\dot{r}^{2}}}\ ,$
(75)

where the sign has to be appropriately chosen depending on whether $r$ grows
or decreases. Eq. (56) evaluated at an arbitrary radius $r$ minus the same
equation at $r_{0}$ is

$\frac{1}{l^{2}}\dot{r}^{2}=\frac{1}{r_{0}^{2}}f(r_{0})-\frac{1}{r^{2}}f(r)+\frac{2}{c^{2}l^{2}}\left[\Phi_{\rm
n}(r_{0})-\Phi_{\rm n}(r)\right].$ (76)

In turn, this can be used to bring Eq. (75) to the form

$\frac{dt}{dr}=\pm\frac{1}{cf}\sqrt{1+\frac{f/r^{2}}{\frac{1}{r_{0}^{2}}f(r_{0})-\frac{1}{r^{2}}f(r)+\frac{2}{c^{2}l^{2}}\left[\Phi_{\rm
n}(r_{0})-\Phi_{\rm n}(r)\right]}}\ .$ (77)

The time it takes the light to travel from $r_{1}$ to $r_{2}$ is obtained
through integration:

$\Delta
t(r_{1},r_{2})=\int_{r_{1}}^{r_{2}}dr\frac{dt}{dr}=\pm\int_{r_{1}}^{r_{2}}\frac{dr}{cf}\sqrt{1+\frac{f/r^{2}}{\frac{1}{r_{0}^{2}}f(r_{0})-\frac{1}{r^{2}}f(r)+\frac{2}{c^{2}l^{2}}\left[\Phi_{\rm
n}(r_{0})-\Phi_{\rm n}(r)\right]}}.$ (78)

Note that $\Delta t(r_{1},r_{2})=\Delta t(r_{2},r_{1})$ as the sign that
arises when changing the limits of the integral compensates the sign
associated with $\dot{r}$. This is also clear on physical grounds (recall that
the change of position of the Earth and the satellite is neglected). The total
time traveled by the light signal is given by $\Delta t_{\rm tot}\equiv
2\Delta t(r_{0},R_{E})+2\Delta t(r_{0},R_{S})$, where $R_{E}$ and $R_{S}$ are,
the radius of the orbit of the Earth and the satellite, respectively. Thus, it
is necessary to find expressions for $\Delta t(r_{0},R)$ for an arbitrary
$R>r_{0}$.

Again, the integral can be solved analytically when expanding in $c^{-2}$. To
lowest order, the integral becomes

$\Delta
t(r_{0},R)^{(0)}=\frac{1}{c}\int_{r_{0}}^{R}dr\frac{r}{\sqrt{r^{2}-r_{0}^{2}}}=\frac{1}{c}\sqrt{R^{2}-r_{0}^{2}}.$
(79)

The next order effect is

$\displaystyle\Delta t(r_{0},R)^{(1)}$ $\displaystyle=$
$\displaystyle\int_{r_{0}}^{R}\frac{l^{2}M(r-r_{0})(2r+3r_{0})+rr_{0}^{4}\left[\Phi_{\rm
n}(r)-\Phi_{\rm
n}(r_{0})\right]}{c^{3}l^{2}\left(r^{2}-r_{0}^{2}\right)^{3/2}}$ (80)
$\displaystyle=$
$\displaystyle\frac{M}{c^{3}}\left[\sqrt{\frac{R-r_{0}}{R+r_{0}}}+4\operatorname{arcsinh}\left(\sqrt{\frac{R-r_{0}}{2r_{0}}}\right)\right]$
$\displaystyle+\frac{-\pi b^{\rm n}_{2}Rr_{0}+4b^{\rm
n}_{2}Rr_{0}\arctan\left(\frac{R-\sqrt{R^{2}-r_{0}^{2}}}{r_{0}}\right)-2b^{\rm
n}_{3}(2R+r_{0})\sqrt{\frac{2R}{R+r_{0}}-1}}{2c^{3}l^{2}R}.$

The “excess time,” $\delta t$, is defined as $\Delta t_{\rm tot}$ minus the
flight time in flat spacetime and in a theory with energy conservation (i.e.,
when $M$, $b^{\rm n}_{2}$, and $b^{\rm n}_{3}$ vanish). It is given by

$\displaystyle\delta t$ $\displaystyle=$ $\displaystyle 2\Delta
t(r_{0},R_{E})+2\Delta
t(r_{0},R_{S})-\frac{2}{c}\sqrt{R_{E}^{2}-r_{0}^{2}}-\frac{2}{c}\sqrt{R_{S}^{2}-r_{0}^{2}}$
(81) $\displaystyle=$ $\displaystyle\frac{-\pi b^{\rm n}_{2}R_{E}r_{0}+4b^{\rm
n}_{2}R_{E}r_{0}\arctan\left(\frac{R_{E}-\sqrt{R_{E}^{2}-r_{0}^{2}}}{r_{0}}\right)-2b^{\rm
n}_{3}(2R_{E}+r_{0})\sqrt{\frac{2R_{E}}{R_{E}+r_{0}}-1}}{c^{3}l^{2}R_{E}}$
$\displaystyle+\frac{-\pi b^{\rm n}_{2}R_{S}r_{0}+4b^{\rm
n}_{2}R_{S}r_{0}\arctan\left(\frac{R_{S}-\sqrt{R_{S}^{2}-r_{0}^{2}}}{r_{0}}\right)-2b^{\rm
n}_{3}(2R_{S}+r_{0})\sqrt{\frac{2R_{S}}{R_{S}+r_{0}}-1}}{c^{3}l^{2}R_{S}}$
$\displaystyle\approx$
$\displaystyle\frac{4M}{c^{3}}\left[1+\ln\left(\frac{4R_{E}R_{S}}{r_{0}^{2}}\right)\right]-\frac{4b^{\rm
n}_{3}}{c^{3}r_{0}^{2}},$

where, in the last step, an expansion in $r_{0}/R_{E}$ and $r_{0}/R_{S}$ is
performed. Also, from Eq. (76) in the $r\to\infty$ limit, it is possible to
show that $l=r_{0}+O\left(c^{-2}\right)$, which is used in the last term of
Eq. (81).

Importantly, it can be verified that the first part of Eq. (81) coincides with
the corresponding GR expression (see, e.g., (Hartle, 2003, page 214)).
Moreover, $b^{\rm n}_{3}$ produces a shift on $\delta t$ that is independent
on $R_{E}$ and $R_{S}$, and, at this level, there are no effects due to
$b^{\rm n}_{2}$. Unfortunately, when analyzing this type of empirical data, a
constant shift in $\delta t$ is usually discarded. In fact, the PPN limits
Will (2014) are set using the logarithmic part of $\delta t$. Therefore, PPN
limits set with time delay do not lead to limits on $b^{\rm n}_{3}$, which was
impossible to foresee before making the calculations.

###  VI.3 Timelike trajectories

Under the symmetries at hand, timelike trajectories, which describe the
propagation of massive pointlike particles, are also contained in the
$\theta=\pi/2$ plane, but now $\kappa=-c^{2}$. Therefore,

$E-\frac{c^{2}}{2}=\frac{1}{2}\dot{r}^{2}+V^{\rm t}_{\rm eff},$ (82)

where the superindex ${\rm t}$ refers to timelike curves. The effective
potential is

$\displaystyle V^{\rm t}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{f}{2}\left(\frac{l^{2}}{r^{2}}+c^{2}\right)-\frac{c^{2}}{2}+\frac{1}{c^{2}}\Phi_{\rm
t}$ (83) $\displaystyle=$
$\displaystyle-\frac{l^{2}M}{c^{2}r^{3}}+\frac{l^{2}}{2r^{2}}-\frac{M}{r}+\frac{1}{c^{2}}\Phi_{\rm
t}.$

Here, $\Phi_{\rm t}=\Phi_{\rm t}(r)$ is the nonconservation potential for
timelike trajectories. Also, note that a term $-c^{2}/2$ is introduced in both
sides of Eq. (82) so that the energy conserving part of $V^{\rm t}_{\rm eff}$
tends to zero as $r\to\infty$.

The effective potential $V^{\rm t}_{\rm eff}$, without energy nonconservation,
has terms that are proportional to $r^{-1}$, $r^{-2}$, and $r^{-3}$.
Therefore, for $\Phi_{\rm t}$ to maintain the form of $V^{\rm t}_{\rm eff}$,
it is necessary that

$\Phi_{\rm t}=\frac{b^{\rm t}_{1}}{r}+\frac{b^{\rm t}_{2}}{r^{2}}+\frac{b^{\rm
t}_{3}}{r^{3}},$ (84)

where the free parameters, in this case, are given by $b^{\rm t}_{1}$, $b^{\rm
t}_{2}$, and $b^{\rm t}_{3}$. Moreover, it is necessary that the coefficients
of $r^{-3}$, $r^{-2}$, and $r^{-1}$, are, respectively, negative, positive,
and negative, and, for the effective potential to have two extrema, it is also
needed that

$\displaystyle Mc^{2}-b^{\rm t}_{1}$ $\displaystyle>$ $\displaystyle 0,$ (85)
$\displaystyle c^{2}l^{2}+2b^{\rm t}_{2}$ $\displaystyle>$ $\displaystyle 0,$
(86) $\displaystyle Ml^{2}-b^{\rm t}_{3}$ $\displaystyle>$ $\displaystyle 0,$
(87)

and, in addition,

$12\left(Mc^{2}-b^{\rm t}_{1}\right)\left(Ml^{2}-b^{\rm
t}_{3}\right)\leq\left(l^{2}c^{2}+2b^{\rm t}_{2}\right)^{2}.$ (88)

The behavior of $V^{\rm t}_{\rm eff}$ is plotted in Fig. 3. Note that $V^{\rm
t}_{\rm eff}$ allows the particles to follow orbits, that is, timelike
trajectories that are contained in between two radii $r_{1}$ and $r_{2}\geq
r_{1}$; this happens when $E$ is such that $V_{\rm eff}(r_{\rm min})\leq
E-c^{2}/2<0$, where $r_{\rm min}$ is the radius that minimizes the effective
potential. These additional conditions, together with Eqs. (85)-(88), are
assumed in what follows. Again, it can be verified that the free parameters
cannot be all absorbed into effective values for $M$ and $l$.

Figure 3: $V^{\rm t}_{\rm eff}$ as a function of $r$. When $E<c^{2}/2$, the
potential admits orbits that are confined between $r_{1}$ and $r_{2}$.

####  VI.3.1 Perihelion shift

Orbits can be described by the angular coordinate as a function of the radial
coordinate: $\phi=\phi(r)$. Once again, this function can be obtained by
integrating $d\phi/dr=\dot{\phi}/\dot{r}$. From the definition of $l$ and Eq.
(82), it is possible to write

$\frac{d\phi}{dr}=\pm\frac{l/r^{2}}{\sqrt{2E-c^{2}-2V^{\rm t}_{\rm eff}(r)}},$
(89)

where the sign is positive (negative) whenever $\dot{r}$ is positive
(negative). Let $r_{1}>0$ and $r_{2}>r_{1}$ be the radii where $\dot{r}=0$,
which are known as the return points of the orbit777There are three solutions
to $\dot{r}=0$, however, one solution does not correspond to orbits. In
addition, circular orbits ($r_{1}=r_{2}$) can be considered by taking, at the
end of the calculation, the $\epsilon\to 0$ limit, where $\epsilon$ is the
orbit’s eccentricity.. From Eq. (82), it follows that

$E-\frac{c^{2}}{2}=V^{\rm t}_{\rm eff}(r_{1}),$ (90)

and

$l=\sqrt{\frac{c^{2}[f(r_{2})-f(r_{1})]+\frac{2}{c^{2}}[\Phi_{\rm
t}(r_{2})-\Phi_{\rm
t}(r_{1})]}{\frac{f(r_{1})}{r_{1}^{2}}-\frac{f(r_{2})}{r_{2}^{2}}}},$ (91)

where $l$ is taken as positive.

An orbit is defined as the trajectory from the minimal radius $r_{1}$, to
$r_{2}>r_{1}$, and back. It turns out that the angle spanned when going from
$r_{1}$ to $r_{2}$ is equal to that from $r_{2}$ to $r_{1}$, since the change
in the integration limits compensates the global sign. Thus, the angle spanned
during a full orbit is

$\displaystyle\Delta\phi$ $\displaystyle=$ $\displaystyle
2l\int_{r_{1}}^{r_{2}}\frac{dr}{r^{2}\sqrt{2V^{\rm t}_{\rm eff}(r_{1})-2V^{\rm
t}_{\rm eff}(r)}},$ (92)

where Eq. (90) is used. Moreover, Eq. (91) can be utilized to replace $l$,
which appears inside $V^{\rm t}_{\rm eff}$, by a function of $r_{1}$ and
$r_{2}$.

To integrate, an expansion in powers of $c^{-2}$ is performed. The dominant
term is

$\displaystyle\Delta\phi^{(0)}$ $\displaystyle=$ $\displaystyle
2\int_{r_{1}}^{r_{2}}\frac{dr\
\sqrt{r_{1}r_{2}}}{r\sqrt{(r-r_{1})(r_{2}-r)}}=2\pi,$ (93)

and the next order effect is given by

$\displaystyle\Delta\phi^{(1)}$ $\displaystyle=$
$\displaystyle\int_{r_{1}}^{r_{2}}\left\\{\frac{2M[r(r_{1}+r_{2})+r_{1}r_{2}]}{c^{2}r^{2}\sqrt{r_{1}r_{2}}(r-r_{1})^{1/2}(r_{2}-r)^{1/2}}\right.$
(94)
$\displaystyle-\frac{\sqrt{r_{1}r_{2}}(r_{1}+r_{2})}{c^{2}M}\frac{r\Phi_{\rm
t}(r)}{(r-r_{1})^{3/2}(r_{2}-r)^{3/2}}$
$\displaystyle+\frac{\sqrt{r_{1}r_{2}}(r_{1}+r_{2})}{c^{2}M(r_{1}-r_{2})}\frac{r_{1}\Phi_{\rm
t}(r_{1})}{(r-r_{1})^{3/2}(r_{2}-r)^{1/2}}$
$\displaystyle\left.-\frac{\sqrt{r_{1}r_{2}}(r_{1}+r_{2})}{c^{2}M(r_{1}-r_{2})}\frac{r_{2}\Phi_{\rm
t}(r_{2})}{(r-r_{1})^{1/2}(r_{2}-r)^{3/2}}\right\\}$ $\displaystyle=$
$\displaystyle\left(\frac{3\pi M}{c^{2}}-\frac{\pi b^{\rm
t}_{2}}{c^{2}M}\right)\frac{r_{1}+r_{2}}{r_{1}r_{2}}$
$\displaystyle-\frac{3\pi b^{\rm
t}_{3}}{2c^{2}M}\left(\frac{r_{1}+r_{2}}{r_{1}r_{2}}\right)^{2}.$

Observe that this last expression is independent of $b^{\rm t}_{1}$.

The deflection angle, $\delta\phi$, is the shift in the orbit’s perihelion and
it is defined as $\Delta\phi$ minus $2\pi$, which corresponds to a closed
orbit. Moreover, it is customary to write the result in terms of the length of
the semimajor axis, $a$, and the orbit’s eccentricity, $\epsilon$, which are
such that $r_{1}=a(1-\epsilon)$ and $r_{2}=a(1+\epsilon)$. Also, units where
$c=1$ are reintroduced, for simplicity. Taking all this into the account, it
is possible to write

$\delta\phi=\frac{6\pi M}{a(1-\epsilon^{2})}\left(1-\frac{b^{\rm
t}_{2}}{3M^{2}}\right)-\frac{6\pi b^{\rm t}_{3}}{Ma^{2}(1-\epsilon^{2})^{2}}.$
(95)

Relevantly, when $b^{\rm t}_{2}=0=b^{\rm t}_{3}$, the general relativistic
effect is recovered. Note that $b^{\rm t}_{2}$ depends on $a$ and $\epsilon$
in the same way than the GR effect. On the other hand, $b^{\rm t}_{3}$ goes
like $a^{-2}(1-\epsilon^{2})^{-2}$. Thus, it is possible to set separate
bounds on these nonconservation parameters using Solar system data,
particularly, if such bounds were set using data of several planets. In fact,
there are reported bounds for Mercury Verma  _et al._ (2014) and Mars Konopliv
_et al._ (2011), however, such an analysis lies outside the scope of this
paper.

The most stringent PPN bounds arise from studies of Mercury’s orbit by the
MESSENGER spacecraft Verma  _et al._ (2014). The corresponding PPN expression
is Will (2014)

$\displaystyle\delta\phi^{\rm PPN}$ $\displaystyle=$ $\displaystyle\frac{2\pi
M_{\odot}}{a_{\mercury}(1-\epsilon^{2}_{\mercury})}(2+2\gamma-\beta)$ (96)
$\displaystyle\approx$ $\displaystyle
43^{\prime\prime}\left(1+\frac{2(\gamma-1)-(\beta-1)}{3}\right)$

where $\gamma$ and $\beta$ are the PPN parameters and the subindex $\mercury$
stands for Mercury. In the last step, the GR value of nearly 43 arcseconds per
century for the deflection angle is used, which corresponds to $2.1\times
10^{-4}$ radians per century. In fact, these limits are actually set on
$\beta$ after assuming $\gamma-1=(2.1\pm 2.3)\times 10^{-5}$ Bertotti  _et
al._ (2003). The resulting limits are $\beta-1=(-4.1\pm 7.8)\times 10^{-5}$.
With these data

$\displaystyle\delta\phi^{\rm PPN}$ $\displaystyle\approx$ $\displaystyle
43^{\prime\prime}\left[1+(2.7\pm 4.1)\times 10^{-5}\right],$ (97)

which, when compared to Eq. (95), yields

$-6.8<(3.3\times 10^{4})\frac{b^{\rm t}_{2}}{M_{\odot}^{2}}+1.1\frac{b^{\rm
t}_{3}}{M_{\odot}^{3}}<1.4.$ (98)

These are the bounds on the nonconservation parameters, and they are
compatible with the inequalities (85)-(88), as required for consistency.

##  VII Conclusions

In this paper, it is argued that matter energy conservation could be abandoned
in theories that try to reconcile gravity and quantum mechanics, as in the
latter theory there are processes, associated with measurements, where energy
is explicitly not conserved. It is shown that, to produce a geometrical theory
of gravity that is compatible with energy nonconservation, it is necessary to
have either additional gravitational degrees of freedom, or nonminimal
couplings, or to consider nondynamical fields.

The core of this papers is devoted to study the trajectories of pointlike
particles subject to energy nonconservation. A trajectory equation is found
using Papapetrou’s method and the result shows that energy nonconservation
generates a particular acceleration, as expected. This trajectory equation is
further studied under the assumptions of staticity and spherical symmetry.

Interestingly, the unimodular theory of gravity, which is a proposal to tackle
the cosmological constant problem through a nondynamical structure, satisfies
two conditions that greatly simplify the trajectories’ study. The first of
these conditions is the existence of a generalized Birkhoff theorem that can
be applied in unimodular gravity. The second condition has to do with the fact
that, in unimodular gravity, the energy-momentum divergence, when viewed as a
differential $1$-form, has to be closed. This, in turn, implies that the
effects of energy nonconservation are encoded in a scalar function, dubbed
nonconservation potential. By virtue of these results, the trajectories of
light and massive particles, in the test particle approximation, can be
studied using equations that resemble a system of one degree of freedom.

To probe the method, concrete nonconservation potentials are proposed for null
and timelike trajectories. The deflection of light, time delay, and the shift
in the perihelion are explicitly calculated, and the results are compared with
Solar system data. These comparisons produce bounds on the parameters of the
nonconservation potentials, implying that energy nonconservation can generate
physical effects.

The ideas presented here can be generalized. Perhaps the most intriguing of
such generalizations is to study the particles’ trajectory equation to the
next order in Papapetrou’s approximation, where the (classical) spin is
considered. This would allow one to test spin-dependent models of energy
nonconservation, particularly those that produce an effective cosmological
constant that is compatible with the observations Perez  _et al._ (2018);
Perez and Sudarsky (2019). With the methods presented here, spin-dependent
models could be compared with Solar system data, producing independent tests.
Note, however, that the models of Refs. Perez  _et al._ , 2018; Perez and
Sudarsky, 2019 use the curvature scalar, $R$, as a measure for energy
diffusion, and $R$ vanishes for the relevant symmetries. Still, similar models
could be analyzed, where, for example, the Kretschmann scalar plays the role
of $R$.

###### Acknowledgements.

We acknowledge getting valuable feedback from H.A. Morales-Técotl, U.
Nucamendi, E. Okon, J.A. García-Zenteno, and D. Sudarsky. This research was
funded by UNAM-DGAPA-PAPIIT Grant IG100120 and CONACyT FORDECYT-PRONACES Grant
140630, and also by CONACyT through the graduate school scholarships.

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