The gravitational energy-momentum pseudo-tensor of Higher-Order Theories of Gravity

TL;DR

This paper derives a pseudo-tensor for gravitational energy-momentum in higher-order gravity theories, analyzing its properties, weak field limit, and gravitational wave energy flux, to explore beyond-standard gravitational radiation modes.

Contribution

It provides a general derivation of the gravitational energy-momentum pseudo-tensor for arbitrary order Lagrangians in gravity, including nonlocal terms, and analyzes its properties and implications.

Findings

Pseudo-tensor is generally affine, not covariant.

Derived in the weak field limit up to second order in perturbations.

Calculated gravitational wave energy flux in specific gauges.

Abstract

We derive the gravitational energy momentum tensor $\tau^{\eta}_{\alpha}$ for a general Lagrangian of any order $L=L\left(g_{\mu\nu}, g_{\mu\nu,i_{1}}, g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots, g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ and in particular for a Lagrangian such as $L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$. We prove that this tensor, in general, is not covariant but only affine, then it is a pseudo-tensor. Furthermore, the pseudo-tensor $\tau^{\eta}_{\alpha}$ is calculated in the weak field limit up to a first non-vanishing term of order $h^{2}$ where $h$ is the metric perturbation. The average value of the pseudo-tensor over a suitable spacetime domain is obtained. Finally we calculate the power per unit solid angle $\Omega$ carried by a gravitational wave in a direction $\hat{x}$ for a fixed wave number $\mathbf{k}$ under a suitable gauge. These results are useful in view of searching for further modes of gravitational radiation beyond the standard two modes of General Relativity and to deal with nonlocal theories of gravity where terms involving $\Box R$ are present. The general aim of the approach is to deal with theories of any order under the same standard of Landau pseudo-tensor.

Equations

$$I={\int }_{\Omega }{d}^{4}xL\to \tilde{\delta }I={\int }_{{\Omega }^{\prime }}{d}^4{x}^{\prime }{L}^{\prime }-{\int }_{\Omega }{d}^{4}xL={\int }_{\Omega }{d}^{4}x\left[\delta L+{\partial }_{\mu }\left(L\delta x^{\mu }\right)\right]$$

$$x^{\prime \mu }=x^{\mu }+{ϵ}^{\mu }\left(x\right)$$

$$\delta g_{\mu \nu }=g_{\mu \nu }^{\prime }\left(x\right)-g_{\mu \nu }\left(x\right)=-{ϵ}^{\alpha }{\partial }_{\alpha }{g}_{\mu \nu }-g_{\mu \alpha }{\partial }_{\nu }{ϵ}^{\alpha }-g_{\nu \alpha }{\partial }_{\mu }{ϵ}^{\alpha }$$

$$\begin{split}0=\delta L+\partial_{\mu}\left(L\delta x^{\mu}\right)=\biggl{(}\frac{\partial L}{\partial g_{\mu\nu}}-\partial_{\rho}\frac{\partial L}{\partial g_{\mu\nu,\rho}}+\partial_{\rho}\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda}}-\partial_{\rho}\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi}}\\ +\partial_{\rho}\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\xi\sigma}}\biggr{)}\delta g_{\mu\nu}+\partial_{\eta}\left(2\chi\sqrt{-g}\tau_{\alpha}^{\eta}\right)\epsilon^{\alpha}\end{split}$$

$$\frac{\partial L}{\partial g_{\mu \nu }}-{\partial }_{\rho }\frac{\partial L}{\partial g_{\mu \nu ,\rho }}+{\partial }_{\rho }{\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}-{\partial }_{\rho }{\partial }_{\lambda }{\partial }_{\xi }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}+{\partial }_{\rho }{\partial }_{\lambda }{\partial }_{\xi }{\partial }_{\sigma }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}=0$$

$${\partial }_{\eta }\left(\sqrt{-g}{\tau }_{\alpha }^{\eta }\right)=0$$

$$\begin{split}\tau_{\alpha}^{\eta}=\frac{1}{2\chi\sqrt{-g}}\biggl{[}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta}}-\partial_{\lambda}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda}}+\partial_{\lambda}\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda\xi}}-\partial_{\lambda}\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\eta\lambda\xi\sigma}}\right)g_{\mu\nu,\alpha}\\ +\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\eta}}-\partial_{\xi}\frac{\partial L}{\partial g_{\mu\nu,\rho\eta\xi}}+\partial_{\xi}\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\eta\xi\sigma}}\right)g_{\mu\nu,\alpha\rho}+\left(\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\eta}}-\partial_{\sigma}\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\eta\sigma}}\right)g_{\mu\nu,\rho\lambda\alpha}\\ +\frac{\partial L}{\partial g_{\mu\nu,\rho\lambda\eta\sigma}}g_{\mu\nu,\rho\lambda\xi\alpha}-\delta^{\eta}_{\alpha}L\biggr{]}\end{split}$$

$$\tau_{\alpha}^{\eta}=\frac{1}{2\chi\sqrt{-g}}\biggl{[}\sum_{m=0}^{3}\left(-1\right)^{m}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{m}}g_{\mu\nu,\alpha}\\ +\sum_{j=0}^{2}\sum_{m=j+1}^{3}\left(-1\right)^{j}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}-\delta_{\alpha}^{\eta}L\biggr{]}$$

$${\left(\right)}_{,i_0}=1;{\left(\right)}_{,i_0\cdots i_m}=\{\begin{array}{ll}{\left(\right)}_{,i_1}& \text{if}m=1\\ {\left(\right)}_{,i_1{i}_2}& \text{if}m=2\\ {\left(\right)}_{,i_1{i}_2{i}_3}& \text{if}m=3\\ \text{and so on}& \end{array};{\left(\right)}_{,i_k{\dot{i}}_k}={\left(\right)}_{,i_k}$$

$$\delta L=\sum _{m=0}^n\frac{\partial L}{\partial g_{\mu \nu ,i_0\cdots i_m}}\delta g_{\mu \nu ,i_0\cdots i_m}=\sum _{m=0}^n\frac{\partial L}{\partial g_{\mu \nu ,i_0\cdots i_m}}{\partial }_{i_0\cdots i_m}\delta g_{\mu \nu }$$

$$\sum _{m=0}^n{\left(-1\right)}^m{\partial }_{i_0\cdots i_m}\frac{\partial L}{\partial g_{\mu \nu ,i_0\cdots i_m}}=0$$

$$\boxed{\begin{split}\tau_{\alpha}^{\eta}=\frac{1}{2\chi\sqrt{-g}}\biggl{[}\sum_{m=0}^{n-1}\left(-1\right)^{m}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{m}}g_{\mu\nu,\alpha}\\ +\Theta_{\left[2,+\infty\right[}\left(n\right)\sum_{j=0}^{n-2}\sum_{m=j+1}^{n-1}\left(-1\right)^{j}\left(\frac{\partial L}{\partial g_{\mu\nu,\eta i_{0}\cdots i_{m}}}\right)_{,i_{0}\cdots i_{j}}g_{\mu\nu,i_{j+1}\cdots i_{m}\alpha}-\delta_{\alpha}^{\eta}L\biggr{]}\end{split}}$$

$${\Theta }_{\left[a,+\infty \right[}\left(n\right)=\{\begin{array}{ll}1& \text{if}n\in \left[a,+\infty \right[\\ 0& \text{otherwise}\end{array}$$

$$\delta I=\delta {\int }_{\Omega }{d}^{4}xL\left(g_{\mu \nu },g_{\mu \nu ,\rho },g_{\mu \nu ,\rho \lambda },g_{\mu \nu ,\rho \lambda \xi },g_{\mu \nu ,\rho \lambda \xi \sigma }\right)=0$$

$$\updownarrow$$

$$\frac{\partial L}{\partial g_{\mu \nu }}-{\partial }_{\rho }\frac{\partial L}{\partial g_{\mu \nu ,\rho }}+{\partial }_{\rho }{\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}-{\partial }_{\rho }{\partial }_{\lambda }{\partial }_{\xi }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}+{\partial }_{\rho }{\partial }_{\lambda }{\partial }_{\xi }{\partial }_{\sigma }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}=0$$

$$\begin{array}{rl}\frac{\partial L}{\partial x^{\alpha }}=& \frac{\partial L}{\partial g_{\mu \nu }}\frac{\partial g_{\mu \nu }}{\partial x^{\alpha }}+\frac{\partial L}{\partial g_{\mu \nu ,\rho }}\frac{\partial g_{\mu \nu ,\rho }}{\partial x^{\alpha }}+\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}\frac{\partial g_{\mu \nu ,\rho \lambda }}{\partial x^{\alpha }}+\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}\frac{\partial g_{\mu \nu ,\rho \lambda \xi }}{\partial x^{\alpha }}+\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}\frac{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{\partial x^{\alpha }}\\ =& {\partial }_{\rho }\frac{\partial L}{\partial g_{\mu \nu ,\rho }}{g}_{\mu \nu ,\alpha }-{\partial }_{\rho }{\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}{g}_{\mu \nu ,\alpha }+{\partial }_{\rho }{\partial }_{\lambda }{\partial }_{\xi }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}{g}_{\mu \nu ,\alpha }-{\partial }_{\rho }{\partial }_{\lambda }{\partial }_{\xi }{\partial }_{\sigma }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{g}_{\mu \nu ,\alpha }\\ & +\frac{\partial L}{\partial g_{\mu \nu ,\rho }}{g}_{\mu \nu ,\rho \alpha }+\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}{g}_{\mu \nu ,\rho \lambda \alpha }+\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}{g}_{\mu \nu ,\rho \lambda \xi \alpha }+\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{g}_{\mu \nu ,\rho \lambda \xi \sigma \alpha }\\ =& {\partial }_{\rho }\left(\frac{\partial L}{\partial g_{\mu \nu ,\rho }}{g}_{\mu \nu ,\alpha }\right)-{\partial }_{\rho }\left({\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}{g}_{\mu \nu ,\alpha }\right)+{\partial }_{\lambda }\left(\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda }}{g}_{\mu \nu ,\rho \alpha }\right)+{\partial }_{\rho }\left({\partial }_{\lambda }{\partial }_{\xi }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}{g}_{\mu \nu ,\alpha }\right)\\ & +{\partial }_{\lambda }\left(\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}{g}_{\mu \nu ,\rho \xi \alpha }\right)-{\partial }_{\xi }\left({\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi }}{g}_{\mu \nu ,\alpha \rho }\right)-{\partial }_{\rho }\left({\partial }_{\lambda }{\partial }_{\xi }{\partial }_{\sigma }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{g}_{\mu \nu ,\alpha }\right)\\ & +{\partial }_{\lambda }\left(\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{g}_{\mu \nu ,\rho \xi \sigma \alpha }\right)-{\partial }_{\xi }\left({\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{g}_{\mu \nu ,\rho \sigma \alpha }\right)+{\partial }_{\sigma }\left({\partial }_{\xi }{\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\rho \lambda \xi \sigma }}{g}_{\mu \nu ,\rho \alpha }\right)\end{array}$$

$${\partial }_{\eta }\left(\sqrt{-g}{\tau }_{\alpha }^{\eta }\right)=0$$

$$T^{\eta \alpha }=\frac{2}{\sqrt{-g}}\frac{\delta \left(\sqrt{-g}{\mathcal{L}}_m\right)}{\delta g_{\eta \alpha }}$$

$$P^{\eta \alpha }=-\frac{1}{\sqrt{-g}}\frac{\delta L_g}{\delta g_{\eta \alpha }},\text{mbox}withthecoupling\chi =\frac{8\pi G}{c^4}$$

$${\left(2\chi \sqrt{-g}{\tau }_{\alpha }^{\eta }\right)}_{,\eta }=-\sqrt{-g}{P}^{\rho \sigma }{g}_{\rho \sigma ,\alpha }=-\chi \sqrt{-g}{T}^{\rho \sigma }{g}_{\rho \sigma ,\alpha }=2\chi \sqrt{-g}{T}_{\alpha ;\eta }^{\eta }-{\left(2\chi \sqrt{-g}{T}_{\alpha }^{\eta }\right)}_{,\eta }$$

$${\partial }_{\eta }\left[\sqrt{-g}\left({\tau }_{\alpha }^{\eta }+T_{\alpha }^{\eta }\right)\right]=\sqrt{-g}{T}_{\alpha ;\eta }^{\eta }$$

$$\delta L+{\partial }_{\mu }\left(L\delta x^{\mu }\right)=-P^{\mu \nu }\sqrt{-g}\delta g_{\mu \nu }+{\partial }_{\eta }\left(2\chi \sqrt{-g}{\tau }_{\alpha }^{\eta }\right){ϵ}^{\alpha }=\left[\sqrt{-g}{P}^{\mu \nu }{g}_{\mu \nu ,\alpha }+{\partial }_{\eta }\left(2\chi \sqrt{-g}{\tau }_{\alpha }^{\eta }\right)\right]{ϵ}^{\alpha }=0$$

$$\sqrt{-g}{T}_{\alpha ;\eta }^{\eta }={\left(\sqrt{-g}{T}_{\alpha }^{\eta }\right)}_{,\eta }-\frac{1}{2}{g}_{\rho \sigma ,\alpha }{T}^{\rho \sigma }\sqrt{-g}$$

$$G_{;\eta }^{\eta \alpha }=0\leftrightarrow P_{;\eta }^{\eta \alpha }=0\leftrightarrow T_{;\eta }^{\eta \alpha }=0\leftrightarrow {\partial }_{\eta }\left[\sqrt{-g}\left({\tau }_{\alpha }^{\eta }+T_{\alpha }^{\eta }\right)\right]=0$$

$${\partial }_0{\int }_{\Sigma }{d}^{3}x\sqrt{-g}\left(T^{\mu 0}+{\tau }^{\mu 0}\right)=-{\int }_{\partial \Sigma }d{\sigma }_i\sqrt{-g}\left(T^{\mu i}+{\tau }^{\mu i}\right)=0$$

$$P^{\mu }={\int }_{\Sigma }{d}^{3}x\sqrt{-g}\left(T^{\mu 0}+{\tau }^{\mu 0}\right)$$

$${\tau }_{\alpha }^{\eta }=\frac{1}{2\chi \sqrt{-g}}\left[\left(\frac{\partial L}{\partial g_{\mu \nu ,\eta }}-{\partial }_{\lambda }\frac{\partial L}{\partial g_{\mu \nu ,\eta \lambda }}\right)g_{\mu \nu ,\alpha }+\frac{\partial L}{\partial g_{\mu \nu ,\eta \xi }}{g}_{\mu \nu ,\xi \alpha }-{\delta }_{\alpha }^{\eta }L\right]$$

$${\tau }_{\alpha }^{\prime \eta }\left(x^{\prime }\right)\ne {\text{J}}_{\sigma }^{\eta }{\text{J}}_{\alpha }^{-1\tau }{\tau }_{\tau }^{\sigma }\left(x\right)$$

$${\text{J}}_{\sigma }^{\eta }=\frac{\partial x^{\prime \eta }}{\partial x^{\sigma }}{\text{J}}_{\alpha }^{-1\tau }=\frac{\partial x^{\tau }}{\partial x^{\prime \alpha }}\text{det}\left({\text{J}}_{\beta }^{\alpha }\right)=|J|=\frac{1}{{\text{J}}^{-1}}$$

$$x^{\prime \mu }={\Lambda }_{\nu }^{\mu }{x}^{\nu }+a^{\mu }{\text{J}}_{\nu }^{\mu }={\Lambda }_{\nu }^{\mu }|\Lambda |\ne 0$$

Full text

1 Introduction

It is well know that there are many procedures for the calculation of energy momentum tensor of the gravitational field in General Relativity and many possible its definitions. This object behaves like a tensor under a linear coordinate transformation, in this case it is an affine tensor or a pseudo-tensor, but it does not behaves like a tensor under general coordinate transformations: in this case we are dealing with a non-covariant tensor. In general, the energy and the momentum of the gravitational field plus matter fields are conserved in an appropriate spatially infinite region even if the gravitational energy cannot be localized. Landau and Lifshitz pointed out these features of the gravitational stress-energy tensor that, in General Relativity is a pseudo-tensor [1].

However any alternative theory of gravity can present the same problem so a general method to deal with gravitational energy-momentum pseudo-tensor is needed for several practical issues ranging from the investigation of further gravitational modes in gravitational radiation, up to the identification and classification of nonlocal gravitational terms. In particular, such terms are now assuming a fundamental role starting from quantization of gravity up to dark energy cosmology [2].

Up to now, there is no final quantum theory of gravity, but several proposals indicate the existence of intrinsic extended structures in spacetime geometry [3, 4]. Such features are always related to effective nonlocal behaviors of spacetime and imply, in general, higher-order terms in the effective interaction Lagrangian [5, 6, 7, 8]. In string theory, for example, the measure of spacetime is impossible below the string scale and then the theory gives rise, intrinsically, to effective nonlocal behaviors [9, 10]. A similar situation comes out in loop-quantum gravity where minimal areas have to be taken into account [11]. In general, any theory of quantum gravity presents intrinsic extended structures comparable to the Planck length. These structures prevents to probe geometry below a given scale [12, 13].

Quantum gravitational effects produce nonlocality and then higher order terms in the effective gravitational Lagrangian[3, 4, 14, 15]. In cosmology, nonlocality could be related to the cosmic acceleration [16, 17]. In astrophysics, dark matter phenomena and the same Modified Newtonian Dynamics could be explained in view of nonlocality [18, 19].

Specifically, any nonlocal term could be represented as some $\Box R$ or $\Box^{k}R$ terms or similar terms in the effective gravitational Lagrangian and this feature give rise to extensions of General Relativity. In view of these facts, it is extremely important to fix and classify general properties of nonlocal terms, in particular the features of gravitational stress-energy pseudo-tensor where they are involved.

The aim of this paper is to generalize the Landau Lifshitz pseudo-tensor to Extended Theories of Gravity [20, 21] where Lagrangians depend on metric tensor derivatives up to $n^{th}$ order. We will show that such a tensor, despite depending on metric tensor derivatives higher than second order and therefore cannot vanish in an appropriate chart, is an affine non-covariant tensor, then it is a pseudo-tensor in the Landau sense. At first glance, it might sound strange that using covariant objects, such as the tensor $g_{\mu\nu}$ or scalar densities of weight $w=-1$, like the Lagrangian $L$ and metric determinant $\sqrt{-g}$, we obtain an affine object $\tau^{\eta}_{\alpha}$ that transforms like a tensor under affinities but not under general diffeomorphisms. However there are other examples of affine object in General Relativity, e.g. affine connections $\Gamma$, that we can call pseudo-tensorial field [22], that not being a covariant tensor, transforms like a tensor under affinity transformations. The reason of this behavior is in the definition of the gravitational energy momentum tensor $\tau^{\eta}_{\alpha}$ and affine connection $\Gamma$: both of them are functions of ordinary derivatives of the metric $g_{\mu\nu}$ and then are non-covariant objects. Furthermore the energy of the gravitational field still is not localizable.

However, in the weak-field limit, after a suitable gauge choice, the gravitational energy-momentum pseudo-tensor for a Lagrangian of $n^{th}$ order becomes a more manageable object after it has been averaged over a suitable spacetime domain. In fact, after an accurate average procedure, it would be possible to calculate the power of emitted gravitational radiation by some astrophysical source. The approach can be relevant in order to investigate possible additional polarization states of gravitational waves besides the standard two of General Relativity [23, 24].

The paper is organized as follows. In Sec. 2 we obtain the gravitational energy momentum tensor for a general Lagrangian of $n^{th}$ order in two ways: by locally varying the gravitational Lagrangian and by adopting the Landau-Lifshitz procedure [1]. Hence in Sec. 3, we prove that this tensor is an affine tensor and not a covariant one. In Sec. 4 we calculate the stress energy pseudo-tensor of gravitation field for a particular Lagrangian $L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p}a_{k}R\Box^{k}R)\sqrt{-g}$. Sec. 5, is devoted to the weak-field limit of the gravitational stress-energy pseudo-tensor. We expand the gravitational energy-momentum pseudo-tensor in the metric perturbation $h$ up to the $h^{2}$ order by providing two simple cases where the index $p$ is equal to 0 and 1. In Sec. 6, the average value of gravitational energy-momentum pseudo-tensor on a 4-dimensional region is calculated. We assume that the region is large enough that any integral asymptotically vanishes. Explicit calculations of emitted power are performed in view of applications to the gravitational waves physics. Conclusions are drawn in Sec.7. Finally in Appendix A, we give the demonstration that the additive terms related to the symmetries of $g_{\mu\nu}$ and its derivatives are on average equal to zero, i.e. $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle=\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle=0$, and, in Appendix B, we explicitly show the six polarization tensors related to the gravitational waves derived from higher-order theories.

2 The gravitational energy-momentum tensor of Fourth-Order Gravity

Let us calculate the stress-energy tensor for a gravitational Lagrangian depending on metric tensor $g_{\mu\nu}$ and its derivatives111 The metric signature of $g_{\mu\nu}$ is $(+\ \ ,-\ \ ,-\ \ ,-)$, Ricci tensor is defined as $R_{\mu\nu}=R_{\ \ \mu\rho\nu}^{\rho}$ and Riemann tensor as $R_{\ \ \beta\mu\nu}^{\alpha}=\Gamma_{\beta\nu,\mu}^{\alpha}$+… up to fourth order $L=L\left(g_{\mu\nu},g_{\mu\nu,\rho},g_{\mu\nu,\rho\lambda},g_{\mu\nu,\rho\lambda\xi},g_{\mu\nu,\rho\lambda\xi\sigma}\right)$. This choice is due to the fact that considering all the possible curvature invariants, without $\Box$ operators into the gravitational action, the field equations results of fourth order in metric formalism (see also [25, 26]). After, we will generalize the approach to a gravitational Lagrangian depending on metric tensor derivatives up to $n^{th}$ order. We will derive the stress-energy tensor both adopting the procedure by Landau [1] and varying locally the Lagrangian.

Let us consider the variation of the action integral with respect to metric $g_{\mu\nu}$ and coordinates $x^{\mu}$ [20, 27, 28]:

[TABLE]

where $\tilde{\delta}$ means the local variation while $\delta$ means the total variation because keeps the value of coordinate $x$ fixed. For an infinitesimal transformations like:

[TABLE]

we obtain the total variation of the metric tensor:

[TABLE]

The functional variation of the metric under global transformation $\partial_{\lambda}\epsilon^{\mu}=0$ is $\delta g_{\mu\nu}=-\epsilon^{\alpha}\partial_{\alpha}g_{\mu\nu}$ and if we require the action to be invariant under this transformation, that is $\tilde{\delta I}=0$, for a arbitrary volume of integration $\Omega$, we get:

[TABLE]

By imposing the constrain on metric tensor $g_{\mu\nu}$ that satisfies the Euler-Lagrange equations for a gravitational Lagrangian:

[TABLE]

we get a continuity equation:

[TABLE]

for an arbitrary $\epsilon^{\alpha}$ and $\tau_{\alpha}^{\eta}$ is the stress-energy tensor for a gravitational field defined as:

[TABLE]

In a more compact form:

[TABLE]

where we used the following notation:

[TABLE]

Let us now consider a general Lagrangian density depending up to $n^{th}$ derivative of $g_{\mu\nu}$ that is $L=L\left(g_{\mu\nu},g_{\mu\nu,i_{1}},g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots,g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$. The most general total variation of $L$ and the Euler-Lagrange equations for this Lagrangian are:

[TABLE]

[TABLE]

where $\delta g_{\mu\nu,i_{0}\cdots i_{m}}=\partial_{i_{0}\cdots i_{m}}\delta g_{\mu\nu}$ because we are varying keeping $x$ fixed. Proceeding in the same way as for fourth order gravity, we find a conserved quantity that satisfies a more general conservation law which allows us to define the energy momentum pseudo-tensor (which is an affine tensor as it will be proved later) for the gravitational field of $n^{th}$ order gravity

[TABLE]

where $\Theta$ is the step function:

[TABLE]

An alternative way to obtain the tensor (2.11) is the procedure developed by Landau [1]. We will consider, as example, the tensor derived from forth gravity (2.8): the generalization to higher order Lagrangians is formally identical. First of all, let us impose the stationary condition and vary the action with respect to the metric to find the field equations under the hypothesis that both $\delta g_{\mu\nu}$ and the variation of derivative $\delta\partial^{n}g$ vanishing on the boundary of integration domain to cancel the surface integrals. Hence we have:

[TABLE]

Now we calculate the derivative and then substitute into the field equations. We have:

[TABLE]

Grouping together terms and renaming dumb indices, we get:

[TABLE]

where $\tau^{\eta}_{\alpha}$ is the tensor defined in (2.8).

If we consider also the material Lagrangian $L_{m}=2\chi\sqrt{-g}\mathcal{L}_{m}$ whose stress-energy tensor is defined as:

[TABLE]

the field equations in presence of matter become $P^{\eta\alpha}=\chi T^{\eta\alpha}$ where

[TABLE]

From these field equations, we obtain:

[TABLE]

[TABLE]

since

[TABLE]

and also because for a symmetric tensor $T^{\eta}_{\alpha}$, one has

[TABLE]

The contracted Bianchi identities imply the conservation law of the sum of two stress-energy tensors, i.e. matter plus gravitational field, and conversely:

[TABLE]

where ${\displaystyle G^{\eta\alpha}=R^{\eta\alpha}-\frac{1}{2}g^{\eta\alpha}R}$ is the Einstein tensor. In the other worlds, the contracted Bianchi identities involve the conservation both of matter and gravitational field or, viceversa, the conservation of matter and gravitational field involves the Bianchi identities for the Einstein tensor (see also [29] for a detailed discussion in alternative gravity).

From the continuity equation (2.23), it is possible to derive some conserved quantities, such as the total 4-momentum and the total angular momentum of matter plus gravitational field, if we assume the metric tensor derivatives up to the $n^{th}$ order vanishing on the 3-dimensional space-domain $\Sigma$, the surface integral cancels out over the boundary $\partial\Sigma$, that is

[TABLE]

where $\Sigma$ is a slice of 4-dimensional manifold of spacetime at $t$ fixed and $\partial\Sigma$ its boundary. Hence the total 4-momentum conserved is [30]

[TABLE]

This quantity is extremely useful for practical applications in relativistic astrophysics [31].

3 Non-covariance of gravitational energy-momentum tensor

It is possible to prove that the tensor $\tau^{\eta}_{\alpha}$ is not generally covariant but it is a tensor only under affine transformations [22], that is a pseudo-tensor. Firstly we consider the particular case of the tensor (2.11) for $n=2$:

[TABLE]

It is possible to show that, under a general diffeomorphism transformation $x^{\prime}=x^{\prime}\left(x\right)$, it is

[TABLE]

where the Jacobian matrix and determinant are defined as

[TABLE]

On the other hand, under linear or affine transformations:

[TABLE]

the tensor transforms as

[TABLE]

In general, it is possible to obtain the following identities:

[TABLE]

being $A^{(\alpha\beta)}B_{\alpha\beta}=A^{\alpha\beta}B_{\alpha\beta}$, if $B_{\alpha\beta}$ is symmetric, it is $B_{\alpha\beta}=B_{\beta\alpha}$. Hence we get:

[TABLE]

[TABLE]

[TABLE]

Finally, considering the previous relations we obtain:

[TABLE]

This result proves the non-covariance of the stress-energy tensor of the gravitational field which is invariant under affine transformations because of additional terms containing derivatives of order higher than or equal to two vanish for any non-singular linear transformation. This result stems from the non-covariance of the derivatives of the metric tensor $g_{\mu\nu}$. Such derivatives give rise to an affine tensor. In general, considering

[TABLE]

and

[TABLE]

the non-covariance of tensor $\tau^{\eta}_{\alpha}$ comes out. On the other hand, under affine transformations, we have:

[TABLE]

[TABLE]

that is, the energy momentum tensor of gravitational field is an affine pseudo-tensor. This result generalize to Extended Theories of Gravity the result in [1]. The affine character of the stress-energy tensor $\tau^{\eta}_{\alpha}$ is related to the nonlocality of gravitational energy, namely, gravitational energy in a finite-dimensional domain, at a given time, depends on the choice of coordinate system [32, 30].

It is worth noticing that the existence of particular Lagrangians, giving rise to energy momentum tensors ${\tau_{\alpha}^{\eta}}$ where terms depending on derivatives in the parentheses of (3.6) as ${\frac{\partial^{2}x}{\partial x^{\prime 2}},\cdots,\frac{\partial^{m+2}x}{\partial x^{\prime m+2}}}$ elide each other cannot be excluded a priori. In such cases, the energy momentum tensor ${\tau_{\alpha}^{\eta}}$ would be covariant as well as affine, becoming an effective tensor and not a pseudo-tensor. However, due to the structure of (3.6), ${\tau_{\alpha}^{\eta}}$ is, in general, a pseudo-tensor.

4 The gravitational energy-momentum pseudo-tensor of Higher than Fourth-Order Gravity

Let us consider now higher than fourth order theories of gravity where nonlocal terms containing $\Box$ operators are present in the action. Such theories are becoming extremely interesting in supergravity and, in general, in gauge theories dealing with gravity [33, 34, 35]. These theories are relevant not only as effective field theories, but also as fundamental theories. Indeed, there is at least a subclass of local higher derivative theories, the so called Lee-Wick theories, that is unitary and super-renormalizable or finite at quantum level as demonstrated in [7, 8].

Specifically. we want to calculate the energy momentum pseudo-tensor $\tau_{\alpha}^{\eta}$ for a gravitational Lagrangian

[TABLE]

where $\overline{R}$ is the linear part of the Ricci scalar $R$. Such a Lagrangian has been first considered in [36]. In fact, it is possible to split the scalar curvature $R$ in linear and quadratic part: the former $\overline{R}$ depends only on first derivative of metric tensor $g_{\mu\nu}$ while the latter depends linearly on second derivative of metric tensor as follows [1, 22, 37]

[TABLE]

[TABLE]

[TABLE]

In order to obtain the pseudo-tensor $\tau^{\eta}_{\alpha}$, we have to calculate the following derivatives

[TABLE]

[TABLE]

[TABLE]

where $\lambda=i_{1}$, $n=2p+4$ and

[TABLE]

Hence, one gets

[TABLE]

So, considering that $j+1\leq m\leq 2h+1$ $\rightarrow$ $j\leq 2h$, we finally obtain:

[TABLE]

By substituting these expressions into (2.11), we get the gravitational stress-energy pseudo-tensor for the Lagrangian (4.1)

[TABLE]

where we used the notation $\partial_{i_{0}}=\mathbb{1}$ and where, with $\tau_{\alpha|GR}^{\eta}$, we indicate the energy momentum pseudo-tensor of General Relativity [32] as

[TABLE]

We replaced the term $\sqrt{-g}R$, that is a density scalar, with the effective one $\sqrt{-g}\overline{R}$ that is no longer a density scalar. This because only the effective term of the curvature $R$ contributes to the field equations and not the one that depends on the second derivatives of the metric tensor. This consideration makes stress-energy pseudo-tensor easier to be integrated at fixed time $t$ over 3-dimensional spatial domains and then allows independent calculations of energy and momentum. These results are a straightforward generalization of those in [27].

An important generalization of local Lagranian (4.1) can be achieved for $p\rightarrow\infty$. In such a case, it becomes nonlocal. Studying this kind of nonlocal Lagrangians, exact or in weak approximation, is a fundamental issue for several reasons: for example, it is worth for regularizing local terms which present problems with divergences. Under suitable assumptions for the coefficients $a_{k}$ (e.g. $\sum_{k=0}^{\infty}|a_{k}|<\infty$ ) and for the domain of the linear differential operator $D^{p}$

[TABLE]

which guarantee the weak or strong convergence:

[TABLE]

our local action becomes non local, i.e.

[TABLE]

In this case, nonlocality makes differential operators of infinite order and then integral operators like

[TABLE]

Let us perform now the limit $n\rightarrow\infty$ for the energy momentum pseudo-tensor in (2.11) derived from a $n$-order Lagrangian. We want to study the behavior of the pseudo-tensor (2.11) in a nonlocal theory, that is:

[TABLE]

Under the hypothesis that $\tau_{\alpha}^{\eta}\left(x\right)$ transforms as an affine object, let us show that also its limit for $n\rightarrow\infty$, i.e. $\overline{\tau}_{\alpha}^{\eta}\left(x\right)$, is an affine pseudo-tensor. For an affine transformation

[TABLE]

the following affine pseudo-tensor transforms as :

[TABLE]

Substituting (4.18) in (4.16), we have:

[TABLE]

which means that $\overline{\tau}^{\sigma}_{\tau}\left(x\right)$ transforms as an affine object also in the limit $n\rightarrow\infty$.

5 The weak-field limit of energy-momentum pseudo-tensor

The above energy-momentum pseudo-tensor (4.10) derived from the Lagrangian (4.1) can be expanded in order to derive the weak-field limit. For this purpose, we can write the spacetime metric as

[TABLE]

where $\eta_{\mu\nu}$ is the Minkowski metric and $h=\eta^{\mu\nu}h_{\mu\nu}$ is the trace of perturbation. We expand the energy momentum pseudo-tensor in power of $h$ to lower order considering terms up to $h^{2}$. Let us start to expand the pseudo-tensor (4.11) in harmonic coordinates where $g^{\mu\nu}\Gamma^{\sigma}_{\mu\nu}=0$. The linear part of the Ricci scalar $\overline{R}$ becomes:

[TABLE]

that is

[TABLE]

Approximating up to second order $h^{2}$, we have:

[TABLE]

being:

[TABLE]

and also

[TABLE]

Thus we substitute these terms into (4.11) and we find the explicit expression for the stress-energy pseudo-tensor in General Relativity up to order $h^{2}$

[TABLE]

Now in order to expand the extended gravity part of the pseudo-tensor (4.10) to second order in $h$, it is sufficient to consider the following terms approximated to $h^{2}$ in harmonic gauge

[TABLE]

[TABLE]

[TABLE]

[TABLE]

given that to lower order in $h$ we have:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

In Eqs. (5.10), (5.11) and (5.15), we explicitly calculated only the term deriving from (A.1) without considering the index permutations ($\mu\nu$) and $\left(\eta i_{1}\cdots i_{2h+1}\right)$. This because, taking into account terms obtained from permutations in $\left(A_{p}\right)_{\alpha}^{\eta}$ and $\left(B_{p}\right)_{\alpha}^{\eta}$, averaged on a suitable spacetime region, we obtain that are equal to zero as we will see below. This mathematical trick is essential to calculated the averaged gravitational energy-momentum pseudo-tensor and the emitted power.

Finally, by substituting equalities (5.8), (5.9), (5.10) and (5.11) into (4.10), we obtain the further extended gravity term of pseudo-tensor $\tau^{\eta}_{\alpha}$ to second order, that we call $\tilde{\tau}^{\eta}_{\alpha}$, it is

[TABLE]

where we have used the conventions:

[TABLE]

Summing up we can split the total energy-momentum pseudo-tensor in the General Relativity part and in the Extended Gravity part, that is

[TABLE]

As simple examples of corrections related to $\tilde{\tau}^{\eta}_{\alpha}$, let us consider the cases where the index $p$ runs up to [math] and up to $1$.

For $p=0$, it is $L_{g}=\left(\overline{R}+a_{0}R^{2}\right)\sqrt{-g}$ as in the case discussed in [27]. We have

[TABLE]

with

[TABLE]

For $p=1$, that is $L_{g}=\left(\overline{R}+a_{0}R^{2}+a_{1}R\Box R\right)\sqrt{-g}$, one gets

[TABLE]

where

[TABLE]

Clearly, the iteration can be performed to any $p$ introducing new contributions into dynamics.

6 Averaging the energy-momentum pseudo-tensor

Let us now consider the solutions of the linearized field equations in the weak-field limit in vacuum coming from the dynamics given by (4.1). From a physical point of view, such solutions are extremely important in order to calculate quantities related to the gravitational radiation. In general, it is

[TABLE]

where

[TABLE]

Here "c.c." stands for the complex conjugate. The trace is

[TABLE]

and $k_{m}^{\mu}=\left(\omega_{m},\mathbf{k}\right)$ with $k_{m}^{2}=\omega_{m}^{2}-|\mathbf{k}|^{2}=\text{M}^{2}$ where $k_{1}^{2}=0$ and $k_{m}^{2}\neq 0$ for $m\geq 2$. At fixed $\mathbf{k}$, we obtain the following relations:

[TABLE]

By using the following identities:

[TABLE]

[TABLE]

we can average the energy-momentum pseudo-tensor $\tau_{\alpha}^{\eta}$ over a region of spacetime $\Omega$ such that $|\Omega|\gg\frac{1}{|k|}$ [37]. It is worth noticing that all integrals, including terms like $e^{i\left(k_{i}-k_{j}\right)_{\alpha}x^{\alpha}}$, approach to zero. If we assume the harmonic gauge, after averaging and some algebra, we get (see Appendix A):

[TABLE]

A base for the linearized solutions $h_{\mu\nu}$ is given in Appendix B. Using these equalities, we can calculate the average value of the energy momentum pseudo-tensor :

[TABLE]

where, as above, ${\displaystyle\chi=\frac{8\pi G}{c^{4}}}$. Assuming the TT gauge for the first mode concerning $k_{1}$ and only harmonic gauge for residual modes $k_{m}$, it is:

[TABLE]

With these considerations in mind, we can consider a gravitational wave traveling in the $+z$-direction at $\mathbf{k}$ fixed, with 4-wave vector given by $k^{\mu}=\left(\omega,0,0,k_{z}\right)$ where $\omega_{1}^{2}=k_{z}^{2}$ if $k_{1}^{2}=0$ and $k_{m}^{2}=m^{2}=\omega_{m}^{2}-k_{z}^{2}$ otherwise with $k_{z}>0$. Therefore we get the averaged tensorial component

[TABLE]

As an application of these results, we can compute the emitted power per unit solid angle $\Omega$, radiated in a direction $\hat{x}$ at a fixed $\mathbf{k}$. Under a suitable gauge, it is:

[TABLE]

Specific cases can be considered according to the index $p$ of the pseudo-tensor (5.16). We have the cases

p=0

[TABLE]

p=1

[TABLE]

p=2

[TABLE]

where we have explicitly indicated the coupling $\chi$. By a rapid inspection of the above formulas, it is clear that the first term is the General Relativity contribution while the corrections strictly depends on $p$. In any context where corrections to General Relativity can be investigated, this approach could constitute a paradigm to search for higher order effects.

7 Conclusions

Corrections to the standard Hilbert-Einstein Lagrangian and, in particular nonlocal terms, are gaining more and more interest in view of addressing gravitational phenomena at ultraviolet and infrared scales. However most of the main features of General Relativity should be retained to get self-consistent theories. In particular, the properties of the gravitational energy momentum tensor need a detailed consideration in view of both foundation and applications of any gravitational theory.

Here, we derived the gravitational energy momentum tensor $\tau^{\eta}_{\alpha}$ for a general Lagrangian of the form $L=L\left(g_{\mu\nu},g_{\mu\nu,i_{1}},g_{\mu\nu,i_{1}i_{2}},g_{\mu\nu,i_{1}i_{2}i_{3}},\cdots,g_{\mu\nu,i_{1}i_{2}i_{3}\cdots i_{n}}\right)$ of any order in metric derivatives, showing that it is, in general, a pseudo-tensor . In particular, we considered a Lagrangians like $L_{g}=(\overline{R}+a_{0}R^{2}+\sum_{k=1}^{p}a_{k}R\Box^{k}R)\sqrt{-g}$, in the weak field limit up to the order $h^{2}$ in metric perturbations. Under suitable gauge conditions, we averaged it over a suitable four-dimensional domain. The averaged tensor (6.8) depends on the free parameters $a_{m}$ and $p$, on the amplitudes $\text{A}_{j}\left(\mathbf{k}\right)$, $C_{11}\left(\mathbf{k}\right)$ and $C_{22}\left(\mathbf{k}\right)$ other then on the wave numbers $k_{m}^{2}=\text{M}^{2}$ that, in turn, are linked to $a_{m}$. As a general result, the gravitational wave (6.1) associated with higher order Lagrangians can be expressed, under a suitable gauge choice for a wave propagating along the $+z$-direction, in terms of six polarization tensors (see Appendix B) as

[TABLE]

where $v_{G_{m}}$ is the group velocity of the $m_{th}$ massive mode (see also [23, 25]). By using the total conservation of gravitational and matter energy momentum tensor (2.23), the amplitudes $\text{A}_{j}\left(\mathbf{k}\right)$, $C_{11}\left(\mathbf{k}\right)$ and $C_{22}\left(\mathbf{k}\right)$ and then the averaged gravitational tensor (6.8) can be expressed in terms of components of matter energy momentum tensor $T^{\mu\nu}$ (as $T^{00}$) and then to the source generating gravitational waves. In principle, it would be possible to find the gravitational power emitted by a local astrophysical source in slow-motion approximation in terms of $T^{00}$, and then observationally fix the range of parameters $a_{m}$ and $p$ of compatible models. The decreasing of timing of the Hulse-Taylor binary pulsar [38] is an example in this sense. See also [39, 40] for $f(R)$ gravity. This procedure could lead to fix the derivative degree $2p+4$ of the theory of gravity [36, 41] and could constitute a test bed for any Extended Theory of Gravity222It is worth noticing that the index $p=0$ gives, in general, a fourth-order theory like $f(R)$. The only second-order theory is the General Relativity which is the particular case $f(R)=R$.. In general, this procedure allows to investigate additional polarization states of gravitational waves apart the standard polarizations $\epsilon_{\mu\nu}^{\left(+\right)}$ and $\epsilon_{\mu\nu}^{\left(\times\right)}$ of General Relativity. Furthermore, the **pseudo-tensor ** $\left\langle\tau_{\alpha}^{\eta}\right\rangle$ depends on $k_{m}^{2}$ which are related to massive modes: this fact could enable to fix lower and upper limits for massive modes. Finally, considering the average value $\left\langle\tau_{\alpha}^{\eta}\right\rangle$, it is possible to calculate the power carried by gravitational waves emitted by a binary black hole merger and to compare the results with measurements already performed by LIGO-VIRGO collaboration [42]. Specifically, this procedure allows to fix the degree $2p+4$ of the theory and may provide the "signature" for further polarizations. In a forthcoming paper, astrophysical applications of the above results will be considered.

As a final remark, it is important to emphasize again that the gravitational energy momentum tensor $\tau^{\eta}_{\alpha}$ is a pseudo-tensor. This object, as the affine connections $\Gamma^{\mu}_{\nu\lambda}$, behaves like tensor components only under linear coordinate transformations but not under the general group of coordinate transformations. Even if the class of diffeomorphisms is more general than the one of affine transformations, that is affinities are a subset of diffeomorphisms, the set of affine objects in more general than the one of covariant objects: this means that an object that behaves like a tensor under general diffeomorphism behaves as a tensor also under affine transformations while the reverse is not always true. In other words, the pseudo-tensor $\tau^{\eta}_{\alpha}$ loses the general covariance although it is defined starting from covariant objects. This fact is related to the presence of partial derivatives of the metric tensor that are not tensors. Finally, the affine properties of the gravitational energy momentum pseudo-tensor could be relevant to discriminate between the teleparallel and metric formulations of theories of gravity. For a detailed discussion of this point see Ref.[43].

Acknowledgements

SC acknowledges INFN Sez. di Napoli (Iniziative Specifiche QGSKY and TEONGRAV) and the COST Action CA15117 (CANTATA).

Appendix A The average of $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle$ and $\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle$ terms

Let us now demonstrate the last two relations in (6.7), that is $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle=\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle=0$. The general formula for $\Box^{h}R$ -derivative, taking into account symmetries of $g_{\mu\nu}$ and its derivatives, is [41]:

[TABLE]

We have to verify the condition $\langle\left(B_{p}\right)_{\alpha}^{\eta}\rangle=0$. Inserting (A.1) in the l.h.s. of (5.11) that, in the weak field limit up to the order $h^{2}$ becomes

[TABLE]

with

[TABLE]

and

[TABLE]

where $\sigma{\left(\mu\nu\right)}$ and $\sigma{\left(\eta i_{1}\cdots i_{2h+1}\right)}$ represent the set of index permutations in the brackets. Averaging (A.2) by fixing $\mathbf{k}$ over a suitable spacetime region adopting a harmonic gauge, we get

[TABLE]

The average of (A.3) is independent of index permutations in the lower and upper cases of $Q_{\mu\nu}^{\ \ \ \eta i_{1}\cdots i_{2h+1}}$, that is

[TABLE]

By substituting (A.4) in (A.3), we get

[TABLE]

Averaging the right term in (5.11), we have

[TABLE]

Finally, by averaging in the weak field limit Eq. (5.11) and from (A.5) and (A.6), we obtain:

[TABLE]

A similar argument gives $\langle\left(A_{p}\right)_{\alpha}^{\eta}\rangle=0$. It is

[TABLE]

[TABLE]

and then averaging Eq. (5.10) on the l.h.s. and r.h.s., in the weak field limit, we have

[TABLE]

that completes our demonstration.

Appendix B The polarizations of gravitational waves

The six polarizations in the solution (7.1) can be defined in a suitable matrix base. It is

[TABLE]

The $+$ and $\times$ are the two standard of General Relativity. The other are related to the position of non-null terms with respect to the trace (T). See also [25] for another derivation in fourth order gravity.