To conserve, or not to conserve: A review of nonconservative theories of gravity

The conservation principles are one of the most interesting aspects of physics. They help us to make predictions about the evolution of a given physical system, ensuring that, despite the change, it undergoes certain aspects present in it that shall remain the same. Already in our high school physics and early undergraduate classes, we made contact with such an important property and learned that, under certain circumstances, the energy, linear, and angular momentum of physical systems are preserved (see Maudlin:2019bje for a recent discussion).

It is also usual to associate the notion of conservation with ideal physical situations in which dissipative mechanisms do not take place. Indeed, one of the main pillars of physics is the Hamilton’s principle of stationary action used to derive equations of motion for many conservative systems of varying degrees of complexity. It is worth noting that only recently an extension of the Hamilton’s principle to nonconservative classical systems has been developed Galley:2012hx .

From a strictly, but crude, mathematical standpoint, the notion of conservation is intrinsically related to the way one performs derivatives of physical quantities. Our usual concept of conservation has its foundations in simple laboratory experiments in which classical systems e.g., hydrodynamics experiments, are tested. Starting with a flat spacetime metric with signature $\eta_{\mu\nu}=(-1,+1,+1,+1)$, one defines the derivative of a scalar quantity $\varphi$ simply as $\partial\varphi/\partial x\equiv\varphi_{,x}$ or, in a multi-dimensional spacetime with coordinates denoted by index $\mu$, i.e., $\partial\varphi/\partial x^{\mu}\equiv\varphi_{,\mu}$. Here, the symbol comma $``,``$ refers to an ordinary derivative. However, the formulation of currents and the energy conservation has revealed much more intricacies than that Forger:2003ut .

The tensorial formalism is a more generic structure to represent fluid quantities. In any spacetime, the energy–momentum tensor $T^{\mu\nu}$ can be decomposed in its rest frame components such that $T^{00}=\rho=$ energy density; $T^{0i}$ represents the internal heat-conduction; $T^{i0}$ corresponds to the momentum transferred in the internal energy flux process and $T^{ij}$ is the momentum flux. This tensor is symmetric so that $T^{\mu\nu}=T^{\nu\mu}$. For a fluid element occupying a non expanding volume subjected to energy/particle flowing across its surface, the conservation of energy is stated by $T^{0\mu}_{\quad,\mu}=0$ while momentum conservation by $T^{i\mu}_{\quad,\mu}=0$ where $i$ refers to the momentum component under analysis. This implies in the general conservation law

$$T^{\mu\nu}_{\quad,\mu}=0.$$

(1)

Along the fluid flow, one also defines the particle quadriflux with components $N^{0}=c\times$ particle number density and $N^{i}=$ particle flux. The macroscopic description of relativistic fluids demands the introduction of the four-velocity $u^{\mu}$ for which by convention one has $u_{\mu}=\eta_{\mu\nu}u^{\nu}$ with $u_{0}=-1$ and $u_{i}=0$. Therefore, $N^{\mu}=nu^{\mu}$. The particle flux conservation is then expressed by the law

$$N^{\mu}_{\quad,\mu}=(nu^{\mu})_{,\mu}=0.$$

(2)

Apart from vacuum solutions e.g., black holes, in which the intrinsic gravitational aspects are studied, it is obvious that the universe is not empty. It is therefore mandatory to set up an energy–momentum tensor for relativistic fluids. The simplest possible configuration, and widely used as the standard starting point in the study of relativistic fluids, is to consider the so-called perfect fluids. They are basically non-viscous fluid configurations obeying the structure

$$T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu}+p\eta^{\mu\nu}=\rho u^{\mu}u^{\nu}+ph^{\mu\nu},$$

(3)

where $\rho$ is the energy density and $p$ the fluid pressure. The second equality of (3) states that the pure pressure contribution is associated with the symmetric projection tensor $h^{\mu\nu}=u^{\mu}u^{\nu}+\eta^{\mu\nu}$.

In the context of a covariant gravitational theory as, e.g., GR, the manifestation of the gravitational interaction is seen as an effect of the curvature of the spacetime. Trajectories, flows, and the variation rates of physical quantities should obey new rules that take into account curvature. The mathematical mechanism used to address this issue is the replacement of the flat spacetime metric $\eta^{\mu\nu}$ by the curved one $g^{\mu\nu}$. The metric $g^{\mu\nu}$ is adapted to the physical problem one wants to study and is written in such a way that it describes the geometry of the curved spacetime. Now, the equivalence principle implies that the conservation law (1) is replaced by its version in a curved spacetime

$$T^{\mu\nu}_{\quad;\mu}=T^{\mu\nu}_{\quad,\mu}+T^{\alpha\nu}\Gamma^{\mu}_{\alpha\mu}+T^{\mu\alpha}\Gamma^{\nu}_{\alpha\mu}=0,$$

(4)

where the symbol “;” means covariant derivative. The additional contributions on the right-hand side brings the so-called affine connection, which, for a Riemannian manifold, coincide with the Christoffel symbols, defined as follows:

$$\Gamma^{\alpha}_{\mu\nu}=\frac{g^{\alpha\beta}}{2}\left(g_{\beta\mu,\nu}+g_{\beta\nu,\mu}-g_{\mu\nu,\beta}\right).$$

(5)

There are four different equations within (4) since $\nu=0,1,2,3$. The $\nu=0$ equation denotes conservation of energy while, for $\nu=i=1,2,3$, one has the conservation of the $i^{th}$ component of the momentum.

According to our discussion so far, Equation (4) has been introduced as an extension of the flat spacetime conservation to curved geometries. The gravitational interaction is not implicitly stated at this stage. However, we know that matter curves spacetime via gravity. Then, prior to the appropriate introduction of the gravitational interaction in our discussion, let us continue to describe generic curved manifolds via the definition of the Riemann curvature tensor

$$R^{\alpha}_{\;\;\beta\mu\nu}=\Gamma^{\alpha}_{\beta\nu,\mu}-\Gamma^{\alpha}_{\beta\mu,\nu}+\Gamma^{\alpha}_{\sigma\mu}\Gamma^{\sigma}_{\beta\nu}-\Gamma^{\alpha}_{\sigma\nu}\Gamma^{\sigma}_{\beta\mu}.$$

(6)

Using the fact that second derivatives of the metric tensor are non-vanishing quantities and that partial derivatives commute, the following identity takes place:

$$R_{\alpha\beta\mu\nu}+R_{\alpha\nu\beta\mu}+R_{\alpha\mu\nu\beta}=0.$$

(7)

Similarly, one can find symmetry properties of the Riemann tensor e.g., $R_{\alpha\beta\mu\nu}=-R_{\beta\alpha\mu\nu};\,\,R_{\alpha\beta\mu\nu}=-R_{\alpha\beta\nu\mu};\,\,R_{\alpha\beta\mu\nu}=R_{\mu\nu\alpha\beta}$. Finally, with such results, one can find the desired results for our discussion, the so-called Bianchi identities

$$R_{\alpha\beta\mu\nu;\lambda}+R_{\beta\lambda\mu\nu;\alpha}+R_{\lambda\alpha\mu\nu;\beta}=0.$$

(8)

Now, we can show the consequence of this identity to the conservation of $T^{\mu\nu}$. By contracting Ref. (8) twice, firstly with $g^{\alpha\mu}$, then, with $g^{\beta\nu}$ and using the symmetry properties of the metric tensor, the Bianchi identity becomes

$$(2R^{\mu}_{\lambda}-\delta^{\mu}_{\lambda}R)_{;\mu}=0,$$

(9)

where the Ricci tensor $R_{\alpha\beta}=R^{\mu}_{\;\;\alpha\mu\beta}=R_{\beta\alpha}$ and the Ricci scalar $R=g^{\mu\nu}R_{\mu\nu}$ have been defined. In principle, there is nothing special with Equation (9). Let us analyze an important consequence of (9). As it is well known, the GR field equations may be derived from a variational principle. The starting point of this procedure is the total action below

$$S=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}R+\int d^{4}x\sqrt{-g}{\cal L}_{m}.$$

(10)

The first term on the right-hand side is the so-called Einstein–Hilbert action, defined via the Lagrangian $\mathcal{L}_{EH}=R$, whilst the second one is the matter action defined as usual, in terms of the Lagrangian density associated with the matter fields, ${\cal L}_{m}$. The energy–momentum tensor of arbitrary matter configurations is defined in terms of ${\cal L}_{m}$ in the following way:

$$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}{\cal L}_{m}\right)}{\delta g^{\mu\nu}}.$$

(11)

Taking (11) into account, the variation of the action (10) gives

$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R={\kappa}T_{\mu\nu},$$

(12)

where one immediately recognizes the left-hand side of this equation, also known as the Einstein tensor

$$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R,$$

(13)

with the quantity appearing in (9). Therefore, the covariant derivative in (9) should also apply to the right-hand side of (12) implying conservation of $T_{\mu\nu}$. In the equation above, $\kappa\equiv 8\pi G$ is the gravitational coupling constant by assuming that we are working with $c=1$ units.

Alternatively, in the presence of a cosmological constant, the Einstein–Hilbert Lagrangian would be redefined as ${\cal L}_{EH}\rightarrow{\cal L}_{EH}-2\Lambda$. The resulting field equations in this case are given by

$$G_{\mu\nu}+\Lambda g_{\mu\nu}={\kappa}T_{\mu\nu}.$$

(14)

The above equation also has vanishing covariant divergence, and it is the only field equation of the generic type $F_{\mu\nu}(g_{\alpha\beta},g_{\alpha\beta,\delta},g_{\alpha\beta,\delta\gamma})=T_{\mu\nu}$, where $F_{\mu\nu}$ is a tensor functional, derivable from an action principle in which the gravitational Lagrangian density is a scalar invariant of the metric Lovelock:1971yv .

A remarkable aspect of GR arises when we consider the invariance of the theory under diffeomorphism. Consider an infinitesimal active transformation generated by a given vector field $V^{\mu}$. This corresponds to the following mapping

$$x^{\mu}\rightarrow x^{\mu}+V^{\mu}.$$

(15)

It is well known that such a transformation allows us to introduce a derivative operator which provides the rate of change of a given tensor along the integral curves of $V^{\mu}$. This operator is the so-called Lie derivative, denoted as ${\cal L}_{V}$ (where the index refers to $V^{\mu}$). In any GR textbook, the reader can find a detailed discussion on how such an operator acts on arbitrary-rank tensors carrol ; wald ; dinverno . An interesting relation shows up when one applies such operator on the metric tensor. In this case, we have

$${\cal L}_{V}g_{\mu\nu}=2\nabla_{(\mu}V_{\nu)}.$$

(16)

In order to proceed with our purpose, let us rewrite the total action (10) as follows:

$$S=\frac{1}{2\kappa}S_{EH}[g_{\mu\nu}]+S_{m}[g_{\mu\nu},\psi^{i}].$$

(17)

The theory is diffeomorphism-invariant if such a transformation implies in a variation of the action so that $\delta S=0$. Thus, let us assume this property a priori and examine its consequences. By varying (17) and making it equal to zero, we have

$$\frac{1}{2\kappa}\int d^{4}x\frac{\delta\left(\sqrt{-g}{\cal L}_{EH}\right)}{\delta g_{\mu\nu}}\delta g_{\mu\nu}+\int d^{4}x\frac{\delta\left(\sqrt{-g}{\cal L}_{m}\right)}{\delta g_{\mu\nu}}\delta g_{\mu\nu}+\int d^{4}x\sqrt{-g}\frac{\delta S_{m}}{\delta\psi^{i}}\delta\psi^{i}=0$$

(18)

For a diffeomorphism generated by $V^{\mu}$, the variation of the metric is simply its Lie derivative along $V^{\mu}$, which is given by (16). We can check that the Einstein–Hilbert action is itself invariant under diffeomorphism. To verify this, notice that the first term of (18) corresponds to

	$\displaystyle\delta S_{EH}=\int d^{4}x\frac{\delta\left(\sqrt{-g}{\cal L}_{EH}\right)}{\delta g_{\mu\nu}}\delta g_{\mu\nu}$	$\displaystyle=$	$\displaystyle\int d^{4}x\sqrt{-g}G^{\mu\nu}\delta g_{\mu\nu}$		(19)
		$\displaystyle=$	$\displaystyle\int d^{4}x\sqrt{-g}G^{\mu\nu}{\cal L}_{V}g_{\mu\nu}$
		$\displaystyle=$	$\displaystyle 2\int d^{4}x\sqrt{-g}G^{\mu\nu}\nabla_{\mu}V_{\nu}.$

If we now integrate (19) by parts and use the covariant form of the Gauss law, we find

$\displaystyle\delta S_{EH}$

$\displaystyle=$

$\displaystyle\int d^{4}x\sqrt{-g}\left(\nabla_{\mu}G^{\mu\nu}\right)V_{\nu},$

(20)

where we get rid of the surface integral, as it is assumed that the vector $V^{\mu}$ vanishes on the boundary, although $V^{\mu}$ is arbitrary within the volume enclosed by such a boundary. Thus, for a generic $V^{\mu}$, (20) tells us that the invariance of diffeomorphism of the Einstein–Hilbert action, $\delta S_{EH}=0$, is ensured by $\nabla_{\mu}G^{\mu\nu}=0$, which is the contracted Bianchi identity, previously presented in (9).

It is implicit that the fields $\psi^{i}$ satisfy the matter equations of motion, which leads to the third term vanishing. Thus, the remaining term has necessarily to obey

$$\int d^{4}x\frac{\delta\left(\sqrt{-g}{\cal L}_{m}\right)}{\delta g_{\mu\nu}}\delta g_{\mu\nu}=0.$$

(21)

Using the definition (11) and (16) in (21), one finds

$$\int d^{4}x\frac{\sqrt{-g}}{2}T^{\mu\nu}{\cal L}_{V}g_{\mu\nu}=0.$$

(22)

Analogous to the previous case, we can repeat here the procedure described by (19) and (20). This shall lead (23) to

$$\int d^{4}x\sqrt{-g}\left(\nabla_{\mu}T^{\mu\nu}\right)V_{\nu}=0.$$

(23)

Given an arbitrary $V^{\mu}$, (23) implies in the covariant conservation of $T_{\mu\nu}$, namely (4)

$$\nabla_{\mu}T^{\mu\nu}=0.$$

(24)

Now, we understand how the standard energy–momentum conservation emerges within GR as a product of an essential property of the theory, i.e., the diffeomorphism invariance. In the first few years after General Relativity was formulated, an intense debate existed on whether or not energy conservation was a true mathematical identity of the theory Brading:2005ina . Finally, in noether1 ; noether2 , E. Noether showed that the conservation of energy, linear, and angular momentum of physical systems, as well as other physical quantities, are justified by first principles. The Noether’s claim is that the conservation of a given quantity follows from a specific symmetry obeyed by the action. In this vein, the time translation invariance leads to the energy conservation, whereas the position translation makes the linear momentum to be conserved, as it happens to the angular momentum when the action exhibits invariance under space rotations. This feature reveals a universal aspect of conservative systems in nature, and its validation is evoked in all physical domains ranging from the quantum world to the cosmos. There is also a recent attempt to provide a unified view on the conservation laws in gravitational systems Obukhov:2014nja .

The above discussion can not be seen as an argument to refute any alternative to equation (14) as long as $g_{\mu\nu}$ is the only field variable sourced by $T_{\mu\nu}$. In the next sections, we are going to show some remarkable examples in the literature.

Although we refer to Equation (4) in most of this work as a conservation law for the energy–momentum tensor, led mainly by the common usage of the term, it is important to emphasize that, roughly speaking, this classification may be misleading and not strictly correct if we think of how to extract in practice from $T_{\mu\nu}$ the physical quantities that will follow conservation laws, namely, the energy and momentum. This discussion is made by taking different paths in the various GR textbooks. As it is shown in landau in the Minkowski spacetime, the quantity below ¹¹1Although we consider $c=1$ throughout this work, particularly in this section we show $c$ explicitly, in order to match the Ref. landau that is used in the present discussion.

$$P^{\mu}=\frac{1}{c}\int T^{\mu\nu}dS_{\nu}$$

(25)

is a conserved quantity identified with the $4-$momentum of the system. The integration is taken on the hypersurface $S$ which contains all the three-dimensional space. It is well known that the conservation of $P^{\mu}$ can be expressed in terms of the null divergence of $T^{\mu\nu}$:

$$\partial_{\mu}T^{\mu\nu}=0,$$

(26)

which makes it clear why the statement of (26) as a conservation law is totally consistent.

It is natural to think of the extension of (25) in a curved spacetime as follows:

$$P^{\mu}=\frac{1}{c}\int_{S}\sqrt{-g}T^{\mu\nu}dS_{\nu}.$$

(27)

We may wonder if does lead to conservation of energy and momentum in a curved manifold as (25) does in the Minkowski spacetime. If this is true, the integral (27) would be conserved if the following condition applied:

$$\frac{\partial(\sqrt{-g}T^{\mu}_{\;\;\nu})}{\partial x^{\mu}}=0.$$

(28)

However, a mismatch is verified when we rewrite the equation (4) in the form below

$$T^{\mu}_{\;\;\nu\;;\mu}=\frac{1}{\sqrt{-g}}\frac{\partial(\sqrt{-g}T^{\mu}_{\;\;\nu})}{\partial x^{\mu}}-\frac{1}{2}\frac{\partial g_{\mu\lambda}}{\partial x^{\nu}}T^{\mu\lambda}=0.$$

(29)

The above expression is different from (28), unless we are in a particular coordinate system, $x^{*}$, where $\partial g_{\mu\lambda}(x^{*})/\partial x^{\nu}=0$ holds. This result points to the need of reformulating the relation (27) in order to properly describe conservation of the energy and momentum in a curved spacetime. In this vein, Landau and Lifshitz call our attention to the fact that, although Equation (4) indeed expresses a local covariant conservation of $T^{\mu\nu}$, it is not true globally, since the gravitational field itself carries energy, whose contribution is missing in (27). This point is also illustrated by S. Weinberg in weinberg , by comparing the gravity with the electromagnetism. While in the latter case the electromagnetic field itself does carry charge, in the former, the gravitational field indeed carries energy and momentum that works as a source for gravity. This explains partially both the well-known linear nature of the Maxwell electromagnetic theory and the fully nonlinear character of GR. Thus, in landau , the author handles this issue by including into the model the contribution of the gravitational field. Such a contribution comes up in the form a pseudo-tensor $t^{\mu\nu}$, which is constructed with the aid of the geometric terms present in the Einstein equations. Thereby, they show that, instead of Equation (27), we shall have the following relation for the $4$-momentum in order for us to properly describe a conservation law for energy and momentum:

$$P^{\mu}=\frac{1}{c}\int_{S}(-g)(T^{\mu\nu}+t^{\mu\nu})dS_{\nu},$$

(30)

where the quantity $t^{\mu\nu}$ is given by

			$\displaystyle t^{\sigma\kappa}=\frac{c^{4}}{16\pi G}[(2\Gamma^{\nu}_{\lambda\mu}\Gamma^{\theta}_{\nu\theta}-\Gamma^{\nu}_{\lambda\theta}\Gamma^{\theta}_{\mu\nu}-\Gamma^{\nu}_{\lambda\nu}\Gamma^{\theta}_{\mu\theta})(g^{\sigma\lambda}g^{\kappa\mu}-g^{\mu\kappa}g^{\lambda\mu})$		(31)
		$\displaystyle+$	$\displaystyle g^{\sigma\lambda}g^{\mu\nu}(\Gamma^{\kappa}_{\lambda\theta}\Gamma^{\theta}_{\mu\nu}+\Gamma^{\kappa}_{\mu\nu}\Gamma^{\theta}_{\lambda\theta}-\Gamma^{\kappa}_{\nu\theta}\Gamma^{\theta}_{\lambda\mu}-\Gamma^{\kappa}_{\lambda\mu}\Gamma^{\theta}_{\nu\theta})$
		$\displaystyle+$	$\displaystyle g^{\kappa\lambda}g^{\mu\nu}(\Gamma^{\kappa}_{\lambda\theta}\Gamma^{\theta}_{\mu\nu}+\Gamma^{\sigma}_{\mu\nu}\Gamma^{\theta}_{\lambda\theta}-\Gamma^{\sigma}_{\nu\theta}\Gamma^{\theta}_{\lambda\mu}-\Gamma^{\sigma}_{\lambda\mu}\Gamma^{\theta}_{\nu\theta})+g^{\kappa\lambda}g^{\mu\nu}\left(\Gamma^{\sigma}_{\lambda\mu}\Gamma^{\kappa}_{\mu\theta}-\Gamma^{\sigma}_{\lambda\mu}\Gamma^{\kappa}_{\nu\theta}\right)].$

The expression above for $t^{\mu\nu}$ is achieved by means of a lengthy calculation which is provided step by step in the aforementioned reference Landau $\&$ Lifshitz landau , to which we refer the interested reader for full details. Given (30), the equivalent equation that indeed leads to a conservation law shall be

$$\partial_{\mu}\left[(-g)(T^{\mu\nu}+t^{\mu\nu})\right]=0.$$

(32)

Notice that the nontensorial nature of $t^{\mu\nu}$ is evident due to its explicit dependence on products of Christoffel symbols. However, it does behave as a tensor under linear coordinate transformations, of which the Lorentz transformation is a particularly interesting case. In addition, notice that it vanishes in the locally inertial frame $x^{*}$, where $\Gamma^{\alpha}_{\mu\nu}(x^{*})=0$, at which the special relativity relation (27) is recovered. This enhances the fact that even the procedure leading to (32) fails in providing a meaningful local conservation law that we could associate with the energy conservation, due to the absence in GR of a local meaning for the gravitational field.

In the R. Wald textbook wald , the physical meaning of (4) is also discussed, although in a bit of a different way. The author shows that, in special relativity, the meaning of $\partial_{\mu}T^{\mu\nu}=0$ is unambiguous as a conservation expression for the energy–momentum tensor. To show this, let us consider a family of inertial observers with parallel worldlines, $v^{\alpha}$, which means

$$\partial_{\beta}v^{\alpha}=0.$$

(33)

If we assume, for instance, a perfect fluid (although the reasoning actually holds for any matter and fields) described by $T^{\mu\nu}$, it is known that the quantity

$$J_{\alpha}=-T_{\alpha\beta}v^{\beta}$$

(34)

shall correspond to the mass–energy current density $4-$vector as measured by these observers. Given (26), from (34), it follows that

$$\partial_{\alpha}J^{\alpha}=0.$$

(35)

The Gauss law tells us that the integration of (35) over a four-dimensional volume, $V$, is equal to the surface integral below

$$\int_{S}J^{\alpha}n_{\alpha}dS=0,$$

(36)

where $n_{\alpha}$ is the unit normal vector to $S$. The null flux defined by (36) clearly indicates a conservation of energy, as it leads to the vanishing of the time variation of such a quantity inside the volume $V$. In other words, we can say that (26) is a requirement for the energy conservation as measured by a family of inertial observers.

On the other hand, in a curved spacetime, as it is known, the relation (26) is replaced by (4). However, the presence of curvature spoils the interpretation of such a equation as a conservation law, since in this case there is no well-defined notion of parallel vectors at different points, which harms the introduction of a global family of inertial observers able to measure the energy of a distant particle. Let us see this feature in more detail. It is natural to think of the curvature-dependent extension of the condition (33) as

$$\nabla_{(\beta}v_{\alpha)}=0.$$

(37)

Thus, the energy–momentum four vector (34) now is the one measured by observers satisfying (37). Nevertheless, in this case, the covariant derivative of $T^{\mu\nu}$ does not lead naturally to the covariant divergence of the current $J^{\mu}$,

$$\nabla^{\alpha}(T^{\mu\nu}v_{\nu})=0,$$

(38)

which, by Gauss law, as we see before, this could ensure the conservation of energy. Because in the curved spacetime, in general, one is not able to define a family of observers satisfying $v^{\alpha}v_{\alpha}=-1$ and (37). However, notice that an exception occurs if $v^{\mu}$ is a Killing vector that generates a one-parameter group of isometries in the spacetime. In this case, as it is well known, $v^{\mu}$ shall obey Equation (37), which, in this context, is called the Killing equation. Thus, this inconsistency between (37) and (38) prevents $\nabla_{\mu}T^{\mu\nu}=0$ to be interpreted as a requirement for a global energy conservation. In fact, let us recall that gravitational field itself, by means of tidal effects, can do work on a material system, thus altering locally its energy content. However, if we consider a small region of the space where such effects can be neglected, the energy of the system shall be conserved in a reasonable approximation. Thus, within such a small spacetime region, it is possible to define a vector field such that $\nabla_{\beta}v_{\alpha}\approx 0$. Thus, Equation (4) would indeed reflect an approximate conservation of energy as seen by these observers. Therefore, it is plausible to say that Ref. (4) represents a local conservation of the energy content of a physical system over small regions of spacetime.

For this model, the gravity is endowed with a spacetime-dependent cosmological constant. The total action of this theory is given by

$$S=\int d^{4}x\sqrt{-g}\left\{\frac{1}{2\kappa}[R-2\Lambda(x)]+{\cal L}_{m}\right\}.$$

(56)

By comparing (56) with (49), we can set the mapping among the corresponding variables. Here, the fixed background is $\bar{k}_{\chi}=\Lambda(x)$, and the Lorentz-invariant Lagrangian is ${\cal L}_{LI}={\cal L}_{m}$, whereas the symmetry-violating piece is ${\cal L}_{LV}=-\Lambda(x)/\kappa$.

Notice that, when $\Lambda(x)\not=0$ and is non-constant, the condition (53) applies, as the theory is endowed with a fixed, non-dynamical, background field given by $\Lambda(x)$ that breaks explicitly the diffeomorphism, since this field appears explicitly in the action. In fact, for this case, Equation (53) becomes

$\displaystyle(\delta S_{LV})_{\textrm{diff}}=\int d^{4}x\sqrt{-g}\frac{\delta{\cal L}_{LV}}{\delta\Lambda(x)}{\cal L}_{V}\Lambda(x)=-\int d^{4}x\sqrt{-g}\frac{1}{\kappa}V^{\mu}\partial_{\mu}\Lambda(x)\not=0,$

(57)

since the Lie derivative on $\Lambda(x)$ is ${\cal L}_{V}\Lambda(x)=V^{\mu}\partial_{\mu}\Lambda(x)$. The corresponding field equations are

$$G^{\mu\nu}=-\Lambda(x)g^{\mu\nu}+\kappa T^{\mu\nu}_{m},$$

(58)

where the symmetry-breaking energy–momentum tensor in the present example is $T^{\mu\nu}_{LV}=-g^{\mu\nu}\Lambda(x)/\kappa$. By using the contracted Bianchi identity on (58), we obtain for this case the corresponding form for (55), which is

$$\nabla_{\mu}T^{\mu\nu}_{m}=(1/\kappa)g^{\mu\nu}\partial_{\mu}\Lambda(x).$$

(59)

Therefore, the presence of a non-constant $\Lambda(x)$ in the description of the gravity leads to a deadlock: if the standard conservation for $T^{\mu\nu}_{m}$ is required, Equation (59) tells us that necessarily $\partial_{\mu}\Lambda(x)=0$ implying in $\Lambda(x)=\textrm{const.}$, thus restoring the diffeomorphism invariance and contradicting the a priori assumption that $\Lambda(x)$ is non-constant. Notice that, in this case, the aforementioned conflict between the dynamics and the geometrical identities (in the present case, the Bianchi identity) is not evaded unless one assumes the non-trivial conservation law above (59). One may wonder if such a conflict is unavoidable, by raising the following question: is it possible to ensure the standard conservation law for $T^{\mu\nu}_{m}$ without necessarily restoring the diffeomorphism invariance? The next example provides an affirmative answer for this question.

The so-called Chern–Simons term was first introduced in the context of three-dimensional gauge field theory and gravitational models CS3d ; CS3dA . Some years later, this same model was extended in order to represent a theory of gravity in the four-dimensional spacetime CSgravity1 ; CSgravity2 . The action for the Chern–Simons gravity in four dimensions can be written as follows:

$$S_{CS}=\int d^{4}x\left[\frac{1}{2\kappa}\left(\sqrt{-g}R+\frac{1}{4}\theta^{\;*}RR\right)+\sqrt{-g}{\cal L}_{m}\right],$$

(60)

where ${}^{*}RR\equiv\;$${{{}^{*}R}^{\kappa}_{\;\lambda}}$${{}^{\mu\nu}R}^{\lambda}_{\;\;\kappa\mu\nu}$ is the gravitational Pontryagin density and ${{{}^{*}R}^{\kappa}_{\;\lambda}}^{\mu\nu}$$=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}$$R^{\kappa}_{\;\;\lambda\alpha\beta}$. The explicit breaking of diffeomorphism invariance occurs due to the embedding coordinate, $v^{\mu}$, which is related to the non-dynamical scalar $\theta(x)$ through $v^{\mu}\equiv\partial^{\mu}\theta$. The variation of the action (60) gives

$$G^{\mu\nu}+C^{\mu\nu}=\kappa T^{\mu\nu}_{m},$$

(61)

where $C^{\mu\nu}$ is the four-dimensional Cotton tensor which has the following form:

$$C^{\mu\nu}=-\frac{1}{2\sqrt{-g}}\left[v_{\sigma}\left(\epsilon^{\sigma\mu\alpha\beta}\nabla_{\alpha}R^{\nu}_{\;\;\beta}+\epsilon^{\sigma\nu\alpha\beta}\nabla_{\alpha}R^{\mu}_{\;\;\beta}\right)+\nabla_{\sigma}v_{\tau}\left({}^{*}R^{\tau\mu\sigma\nu}+^{*}R^{\tau\nu\sigma\mu}\right)\right].$$

(62)

Due to the dependence of $C^{\mu\nu}$ upon the embedding coordinate $v^{\mu}$, the Cotton tensor encodes the information of diffeomorphism breaking in the field equations, so that we can set the following correspondence:

$$T^{\mu\nu}_{LV}=\frac{C^{\mu\nu}}{\kappa}.$$

(63)

By computing the covariant divergence of $C^{\mu\nu}$, we will have

$$\nabla_{\mu}C^{\mu\nu}=\frac{1}{8\sqrt{-g}}(\partial^{\nu}\theta)^{\;*}RR.$$

(64)

Looking at the action (60) and considering diffeomorphism transformations like (15), Equation (53) shall be

$$(\delta S_{LV})_{\textrm{diff}}=\int d^{4}x\sqrt{-g}\frac{1}{4}(^{\;*}RR)V^{\mu}\partial_{\mu}\theta.$$

(65)

Thus, by the equation above, we find a twofold condition for the explicit breaking of diffeomorphism. The first condition is $\partial^{\mu}\theta\not=0$ and the second one is that the Pontryagin density is non-zero. We can examine the occurrence (or not) of the dynamics-geometry conflict by taking the covariant divergence of (61). If $\nabla_{\mu}T^{\mu\nu}_{m}=0$ is required, the Bianchi identity then obliges the relation below:

$$\nabla_{\mu}C^{\mu\nu}=0$$

(66)

to hold on shell. Since the divergence of $C^{\mu\nu}$ was also computed in (64), the condition (66) imposes that ${}^{\;*}RR\;\partial_{\mu}\theta=0$. Thus, the conflict is evaded not only in the trivial situation when $\partial_{\mu}\theta=0$, implying in the restoration of the diffeomorphism invariance. It is evaded as well when the Pontryagin density vanishes, obeying the so-called Pontryagin constraint

$$^{\;*}RR=0.$$

(67)

This is a constraint on the geometry that indicates a way to avoid the mentioned dynamics-geometry conflict by restricting the class of allowed geometries. It is known, for example, that any spacetime of Petrov types III, N, and O automatically satisfy (67) (See yunes ).

On the other hand, if the usual conservation law $\nabla_{\mu}T^{\mu\nu}_{m}=0$ was somehow relaxed, when we took the covariant divergence of (61), we would have

$$\nabla_{\mu}T^{\mu\nu}_{m}=\frac{1}{8\kappa\sqrt{-g}}\partial^{\nu}\theta^{\;*}RR,$$

which means a deviation from the standard conservation law sourced by both the presence of $\partial^{\nu}\theta$ and the Pontryagin density.

The formulation showed here for the Chern–Simons gravity is not the only one found in the literature. In yunes , one can see an alternative one, where it is possible to render dynamics to the scalar field $\theta$. In this case, such a field shall obey a Klein–Gordon like equation of motion that is sourced both by stress-energy tensor and the curvature of spacetime.

To end this section, let us mention an important attempt of setting experimental bounds on the non-dynamics Chern-Simons gravity. In smith , by studying gravitomagnetic effects within this theory, the authors place important bounds on the parameter $m_{cs}$. In that work, they define such a parameter as $m_{cs}^{-1}\propto\dot{\theta}$ and assume the scalar field $\theta$ as being time varying but spatially homogeneous. They compute orbits of test bodies and the precession of gyroscopes in the linearized Chern–Simons gravity around a massive spinning bod. Then, they use observation from the LAGEOS lageos and Gravity Probe B gpb satellites to restrict $m_{cs}^{-1}$ to be less than 1000 km, which corresponds to $m_{cs}\geq 2\times 10^{-22}$ GeV.