The Quasi-Maxwellian Equations of General Relativity: Applications to Perturbation Theory

Abstract

A comprehensive review of the equations of general relativity in the quasi-Maxwellian (QM) formalism introduced by Jordan, Ehlers and Kundt is presented. Our main interest concerns its applications to the analysis of the perturbation of standard cosmology in the Friedman-Lemaître-Robertson-Walker framework. The major achievement of the QM scheme is its use of completely gauge-independent quantities. We shall see that in the QM-scheme, we deal directly with observable quantities. This reveals its advantage over the old method introduced by Lifshitz that deals with perturbation in the standard framework. For completeness, we compare the QM-scheme to the gauge-independent method of Bardeen, a procedure consisting of particular choices of the perturbed variables as a combination of gauge-dependent quantities.

Appendix A: An Example of Further Developments in the QM-Formalism

This appendix presents an example in which the QM-formalism is modified and the analogy with the electromagnetism is strengthened. Modifications of this kind indicate alternative, more intuitive ways to solve problems appearing in general-relativity theory. We present an enlightening example next.

Electrodynamics is the paradigm of field theory. Its theoretical and experimental properties have been simulated and sought for in many other theories and in particular in the analysis of gravitational phenomena. Much work has been done along this line, which discusses the resemblance between electrodynamics and gravidynamics.³¹ However, it seems possible to further improve this similarity, as we shall show.

In this vein, we review here a modification of Einstein’s theory of general relativity under certain special states of the geometry of the space-time. Since the original proposal of Einstein’s geometrization of gravitational processes, many physicists have discussed alternative models of gravitation. The kind of theory we shall analyze here is given by means of the metric properties—represented by a symmetrical metric tensor g _{μ ν} (x)—and by two other functions, ε(x) and μ(x), which are independent of the metric,³² but have intimate connection to the space-time.

For pedagogical reasons, we find it convenient to limit our considerations to the case in which both ε and μ are constants. The meaning one should attribute to these two constants comes from direct analogy with the dielectric and permeability constants of a given medium in electrodynamics.

We shall simplify the model by merely stating that ε and μ can be provisionally identified with the characteristics of certain states of tensions, in free space-time, due to an average procedure on (quantum) properties of gravitation.

In other words, ε and μ are interpreted as the result—in a macroscopic level—of some sort of averaging microscopic field fluctuation.³³ This is perhaps not difficult to assume if we can say exactly how the equations of motion of gravity phenomena must be modified by them, as we shall do later. We remark that we are not supporting this interpretation but merely suggesting it as a provisional sursis of the model.

We shall describe gravitational interaction³⁴ by means of a fourth-rank tensor Q _{α β μ ν} . We shall set up its algebraic properties and give its dynamics. It is possible to separate this tensor, for an observer moving with four-velocity V ^μ, into four second-order symmetric trace-free tensors E _{α β} , B _{α β} , D _{α β} and H _{α β} . The principal result is then obtained by showing that it is possible to select a class of observers with velocity ℓ ^μ in such a way as to have equations of motion for Q _{α β μ ν} similar to the Maxwell equations for the electrodynamics. That is, for E _{α β} , D _{α β} , B _{α β} and H _{α β} separated into two groups: one containing only E _{α β} and B _{α β} (and their derivatives) and the other containing only H _{α β} and D _{α β} (and their derivatives). These equations have the same formal structure of Maxwell’s equations in a given general medium. We therefore come to the conclusion that the present theory has a class of privileged observers in which gravitational field equations admit this simple separated form. Any other observer, which is in motion with respect to ℓ ^μ, mixes the terms E _{α β} , D _{α β} , H _{α β} and B _{α β} into the equations. This situation could be thought of as defining a new type of ether. However, unlike the ether of the pre-Einstein epoch, our ether is not a substance, but it is only a preferred frame of observation.

In the remainder of this presentation, we discuss in some detail a very particular situation of these tensors, that is, the case in which they can be reduced to two tensors plus two constants: ε and μ. Then, we show that Einstein’s theory is obtained from this for a particular set of values of ε and μ, that is, ε = μ = 1. It is in this sense that we can call this theory a generalization of Einstein’s gravidynamics.

1.1 A1 The Q-Field

1.1.1 A1.1 Definitions

Let us define in a four-dimensional Riemannian manifold a fourth-rank tensor Q _{α β μ ν} described by an observer V ^μ in terms of four second-order tensors E _{α β} , D _{α β} , H _{α β} and B _{α β} .We set, by analogy with the irreducible decomposition of the Weyl tensor, that

$$\begin{array}{@{}rcl@{}} Q_{\alpha\beta}{~}^{\mu\nu}=V_{[\alpha}D_{\beta ]}{~}^{[\mu} V^{\nu]}+V_{[\alpha}E_{\beta ]}{~}^{[\mu} V^{\nu]}+\delta^{[\mu}_{[\alpha} E^{\nu ]}_{\beta ]} \\-\eta_{\alpha\beta\rho\sigma}V^{\rho}B^{\sigma[\mu}V^{\nu]}-\eta^{\mu\nu\rho\sigma}V_{\rho}H_{\sigma[\alpha}V_{\beta]}.{} \end{array} $$

(A1)

The tensors E _{α β} , D _{α β} , B _{α β} and H _{α β} , represented below by X _{α β} , satisfy the following properties:

$$\begin{array}{@{}rcl@{}} &&{}X^{\alpha}{}{~}_{\alpha}=0, \end{array} $$

(A2a)

$$\begin{array}{@{}rcl@{}} &&{}X^{\alpha \beta}V_{\alpha}=0, \end{array} $$

(A2b)

$$\begin{array}{@{}rcl@{}} &&{}X_{\alpha\beta}=X_{\beta \alpha}. \end{array} $$

(A2c)

We can write D _{α β} , E _{α β} , etc. in terms of Q _{α β μ ν} and projections on V ^μ like, for instance

$$ D_{\alpha \beta}=-Q_{\varepsilon \alpha \mu \beta}V^{\varepsilon}V^{\mu} $$

and so on.

1.1.2 A1.2 Algebraic Properties

From the definition (A1) of Q _{α β μ ν} , we directly obtain the following properties:

$$\begin{array}{@{}rcl@{}} Q_{\alpha \beta}{~}^{\mu\nu}&=&-Q_{\beta\alpha}{~}^{\mu\nu}, \end{array} $$

(A3)

$$\begin{array}{@{}rcl@{}} Q_{\alpha \beta}{~}^{\mu \nu}&=&-Q_{\alpha \beta}{~}^{\nu \mu}, \end{array} $$

(A4)

$$\begin{array}{@{}rcl@{}} Q^{\alpha}_{\ \ \beta \alpha \nu}&=&E_{\beta \nu}-D_{\beta \nu}, \end{array} $$

(A5)

$$\begin{array}{@{}rcl@{}} &&{\kern-3.6pc}Q^{\alpha}{}{~}_{\alpha}=0 \end{array} $$

(A6)

1.1.3 A1.3 Dynamics

By analogy with Einstein’s equations in vacuum, we impose on Q _{α β μ ν} the equation of motion³⁵

$$ Q^{\alpha \beta \mu \nu}{}{~}_{;\nu}=0 $$

(A7)

Now, we shall use the above properties to project the system of (A7) parallel and orthogonal to the rest frame of a selected observer with four-velocity ℓ ^μ from the whole class of V ^μ. We impose that the congruence generated by ℓ ^μ satisfy the properties.

$$\begin{array}{@{}rcl@{}} &&{}\ell^{\mu}\ell_{\mu}=+1, \end{array} $$

(A8a)

$$\begin{array}{@{}rcl@{}} &&{}w_{\alpha \beta}=\frac{1}{2}\ h_{[\alpha}{\ }^{\lambda}\ h_{\beta ]}{\ }^{\varepsilon}\ell_{\lambda;\varepsilon }=0, \end{array} $$

(A8b)

$$\begin{array}{@{}rcl@{}} &&{}\theta_{\alpha \beta}=\frac{1}{2}\ h_{(\alpha}{\ }^{\lambda}\ h_{\beta )}{\ }^{\varepsilon}\ell_{\lambda;\varepsilon }=0, \end{array} $$

(A8c)

$$\begin{array}{@{}rcl@{}} &&{}\dot{\ell}^{\mu}=\ell^{\mu}{}{~}_{;\nu}\ell^{\nu}=0. \end{array} $$

(A8d)

where h _{μ ν} is the projector in the plane orthogonal to ℓ ^μ, that is

$$ h_{\mu \nu}=g_{\mu \nu}-\ell_{\mu}\ell_{\nu} $$

(A9)

So, the congruence generated by ℓ _μ is geodesic, irrotational, nonexpanding and shear-free. The reason for selecting such a particular class of observers will become clear later. Then, (A7) takes the form

$$\begin{array}{@{}rcl@{}} &&{}D_{\alpha \mu;\nu}h^{\mu \nu}h^{\alpha}_{\ \ \varepsilon}=0, \end{array} $$

(A10a)

$$\begin{array}{@{}rcl@{}} &&{}\dot{D}_{\alpha \mu}h^{\alpha}_{(\sigma}\ h{\ }^{\mu}_{\varepsilon )}+h^{\alpha}_{(\sigma}\ \eta_{\varepsilon )} {\ }^{\nu \rho \tau}\ell_{\rho}H_{\tau \alpha ;\nu}=0, \end{array} $$

(A10b)

$$\begin{array}{@{}rcl@{}}\ &&{}B_{\alpha \mu ;\nu}h^{\mu \nu}h^{\alpha}_{\varepsilon}=0, \end{array} $$

(A10c)

$$\begin{array}{@{}rcl@{}} &&{}\dot{B}^{\mu \nu}h_{\mu (\sigma}\ h{\ }_{\lambda )\nu}-h_{(\sigma}^{\ \alpha}\ \eta {\ }_{\lambda )}{~}^{\nu \rho \tau}\ell_{\rho}E_{\tau \alpha; \nu}=0 \ , \end{array} $$

(A10d)

in which a parenthesis means symmetrization.

This set of equations has a striking resemblance with Maxwell’s macroscopic equations of electrodynamics. Indeed, we can formally understand the above set as having the form [79, 80]

$$\begin{array}{@{}rcl@{}} &&{}\nabla \cdot \vec D=0\ , \end{array} $$

(A11a)

$$\begin{array}{@{}rcl@{}} &&{}\dot{\vec{D}}-\nabla\times \vec{H}=0\ , \end{array} $$

(A11b)

$$\begin{array}{@{}rcl@{}} &&{}\nabla \cdot \vec{B}=0\ , \end{array} $$

(A11c)

$$\begin{array}{@{}rcl@{}} &&{}\dot{\vec{B}}+\nabla\times \vec{E}=0\ , \end{array} $$

(A11d)

where the symbol → is put over D, E, etc. only to represent its tensorial character; the ∇ operator represents the generalizations of the usual well-known differential operators.

We can therefore understand the reason for selecting the above privileged set of observers, given by the tangential vector ℓ ^μ. Only for this class of frames does (A7) take the form (A10a–A10d). Any other observer which is in motion with respect to ℓ ^μ will mix into the equations of motion of the set of tensors (E _{α β} ,B _{α β} ) with the set of tensors (D _{α β} ,H _{α β} ). So, it is in this sense that there is a natural selection of observers, with respect to the equation of motion satisfied by Q _{α β μ ν} .

1.1.4 A1.4 ε and μ States of Tension

A particular class of states of space-time occurs in the case in which there is a linear function relating the tensors B _{α β} with H _{α β} and E _{α β} with D _{α β} by the intermediary of two constants, ε and μ.

We set

$$B_{\alpha \lambda}=\mu H_{\alpha \lambda},$$

(A12a)

$$D_{\alpha \lambda}=\varepsilon H_{\alpha \lambda}. $$

(A12b)

If we put expressions (A12b) into definition (A1) of Q _{α β μ ν} , a straightforward calculation shows that it is possible to write Q _{α β μ ν} in terms of the Weyl tensor W _{α β μ ν} and its “electric” and “magnetic” parts $\mathcal {E}_{\alpha \beta }$ and $\mathcal {H}_{\alpha \beta }$, if we identify the tensor E _{α β} with $\mathcal {E}_{\alpha \beta }$ and H _{α β} with $\mathcal {H}_{\alpha \beta }$. Then, we can write

$$\begin{array}{@{}rcl@{}} Q_{\alpha \beta} = W_{\alpha \beta}^{\ \ \ \mu \nu}+(\varepsilon -1)\ell_{[\alpha}\ \mathcal{E}_{\beta ]}^{[\mu }\ \ell^{\nu ]}\\+ (1-\mu )\eta_{\alpha \beta \rho \sigma}\mathcal{H}^{\sigma [\mu}\ \ell^{\nu ]}\ell^{\rho}\ , \end{array} $$

(A13)

where

$$\begin{array}{@{}rcl@{}} W_{\alpha \beta}^{\ \ \ \mu \nu} &=& 2\ell_{[\alpha}\ \mathcal{E}_{\beta ]}{~}^{[\mu}\ell^{\nu]}+\delta^{[\mu}_{[\alpha}\ \mathcal{E}^{\nu ]}_{\beta ]}{\kern27pt} \\ &&-\eta_{\alpha \beta \lambda \sigma}\ell^{\lambda}\mathcal{H}^{\sigma [\mu}\ell^{\nu ]}- \eta^{\mu \nu \rho \sigma}\ell_{\rho}\mathcal{H}_{\sigma [\alpha}\ \ell_{\beta ]}{} \end{array} $$

(A14)

and, consequently,

$$ \mathcal{E}_{\alpha \beta}=-W_{\alpha \mu \beta \nu}\ell^{mu}\ell^{\nu}, $$

(A15)

$$ \mathcal{H}_{\alpha \beta}=\frac{1}{2}\ \eta_{\alpha \mu}{~}^{\rho\sigma}W_{\rho \sigma \beta \lambda}\ell^{\mu}\ell^{\lambda} $$

(A16)

The resulting equations of motion (13)–(14) turn into the set:

$$\begin{array}{@{}rcl@{}} &&{}\mathcal{E}_{\alpha \mu ||\nu}h^{\mu \nu}h^{\alpha}_{\ \ \varepsilon}=0, \end{array} $$

(A17a)

$$\begin{array}{@{}rcl@{}} &&{}\varepsilon\mathcal{\dot{E}}_{\alpha \mu}h^{\alpha}_{(\sigma}\ h^{\mu}_{\varepsilon)}+h^{\alpha}_{(\sigma}\ \eta_{\varepsilon )}^{\nu \rho \tau}\ell_{\rho} \mathcal{H}_{\tau \alpha || \nu}=0, \end{array} $$

(A17b)

$$\begin{array}{@{}rcl@{}} &&{}\mathcal{H}_{\alpha \mu ||\nu}h^{\mu \nu}h^{\alpha}_{\varepsilon}=, \end{array} $$

(A17c)

$$\begin{array}{@{}rcl@{}} &&{}\mu \mathcal{\dot{H}}_{\alpha \mu}h^{\alpha}_{(\sigma}\ h^{\mu}_{\varepsilon )}-h^{\alpha}_{(\sigma}\ \eta_{\varepsilon)}^{\nu \rho \tau}\ell_{\rho}\mathcal{E}_{\tau \alpha || \nu}=0. \end{array} $$

(A17d)

By the same argument that guided us to (A10a–A10d), we see from the above set that we can identify ε as being the gravitational analogue of the dielectric constant of electrodynamics and μ as being the permeability of space-time.

Now, we recognize in (A17d) Einstein’s equations for the free gravitational field for the particular case in which ε = μ = 1.³⁶

So, it seems natural to interpret (A17d) for the general case (ε, μ different from unity) as the equations for the gravitational field on states of space-time that are macroscopically characterized (in the sense discussed in the introduction) by the two constants ε and μ.

1.1.5 A1.5 Conformal Behavior of Q _{α β μ ν}

A conformal transformation of the metric g _{α μ} is given by the map

$$\begin{array}{@{}rcl@{}} &&g_{\mu \nu}(x) \longrightarrow \tilde{g}_{\mu \nu}(x)={\Omega}^{2}(x)g_{\mu \nu}(x) \\ &&g^{\mu \nu}(x) \longrightarrow \tilde{g}^{\mu \nu}(x)={\Omega}^{-2}(x)g^{\mu \nu}(x) \end{array} $$

(A18)

Since we can set $\eta _{\alpha \beta }^{\ \ \ \mu \nu }$ as independent of the conformal transformation,

$$\tilde{\eta}_{\alpha \beta}^{\ \ \ \mu \nu}=\eta_{\alpha \beta}^{\ \ \ \mu \nu} $$

It is then easy to see that the electric and magnetic parts of Weyl tensor remain unchanged,

$$\begin{array}{@{}rcl@{}} &&\tilde{\mathcal{E}}_{\mu \nu}=-\tilde{W}_{\mu \rho \nu \sigma}\tilde{\ell}^{\rho}\tilde{\ell}^{\dot{\sigma}}= \mathcal{E}_{\mu\nu}\ , \end{array} $$

(A19a)

$$\begin{array}{@{}rcl@{}} &&\tilde{\mathcal{H}}_{\mu \nu}=-\frac{1}{2}\ \tilde{\eta}_{\mu \alpha}^{\ \ \ \rho \sigma} \tilde{W}_{\rho \sigma \nu \lambda}\tilde{\ell}^{\alpha}\tilde{\ell}^{\lambda}=\mathcal{H}_{\mu \nu}, \end{array} $$

(A19b)

where we have used conformal transformation of the velocity ℓ ^μ as usual,

$$ \tilde{\ell}^{\mu}={\Omega}^{-1}\ell^{\mu} $$

(A20)

As a consequence of the above mapping, Q _{α β μ ν} behaves, under the conformal transformation, as the Weyl tensor,

$$ \tilde{Q}_{\alpha \beta \mu \nu}(x)={\Omega}^{2}(x)Q_{\alpha \beta \mu \nu}(x). $$

(A21)

1.2 A2 Gravitational Energy in an ε−μ State of Tension

There have been many discussions, since Einstein’s [46, 47] paper, concerning the definition of the energy of a given gravitational field. We do not intend to discuss this subject here but we shall limit ourselves to considering one reasonably successful suggestion by Bel [12] for the form of the energy-momentum tensor of gravitational radiation.

The point of departure [12] comes from the similitude of the equation of motion of gravity and electrodynamics. He defines a fourth-rank tensor T ^αβμν given in terms of quadratic components of the field (identified with the Riemann tensor) and written in terms of the Weyl tensor W ^αβμν.

Bel’s super-energy tensor takes the form:

$$ T^{\alpha \beta \mu \nu}=\frac{1}{2}\left\{W^{\alpha \rho \mu \sigma}W^{\beta}{}{~}_{\rho}{~}^{\nu}{}{~}_{\sigma}+ C^{\ast \alpha \rho \mu \sigma} C^{\ast \beta}{}{~}_{\rho}{~}^{\nu}{}{~}_{\sigma}\right\}\ , $$

(A22)

where ^∗ is the dual operator. Note that the symmetry of the Weyl tensor ($C^{\,\ast }_{\alpha \beta \mu \nu } = W^{\,\,\,\,\,\,\,\,\,\ast }_{\alpha \beta \mu \nu } = \,^{\ast }W_{\alpha \beta \mu \nu }$) does not hold for Q _{α β μ ν} . This is related to the lack of Q _{α β μ ν} ≠ Q _{μ ν α β} symmetry. Indeed, we have that

$$\begin{array}{@{}rcl@{}} Q^{\ast}_{\alpha\beta}{~}^{\mu \nu}&=&W^{\ast}_{\alpha \beta}{~}^{\mu \nu}+\frac{1}{2}(\varepsilon -1) \eta_{\alpha \beta \rho \sigma}\ell^{[\rho}\ \mathcal{E}^{\sigma ]}{~}^{[\mu}\ \ell^{\nu ]}\\ &&+\frac{1}{2}\ (1-\mu )\eta_{\alpha \beta \rho \sigma}\eta^{\rho \sigma \varepsilon \sigma}\ell_{\varepsilon} \mathcal{H}_{\tau}^{[\mu}\ \ell^{\nu]}{\kern23pt} \end{array} $$

and

$$\begin{array}{@{}rcl@{}} Q_{\alpha\beta}{~}^{\mu^{\ast} \nu}&=&C^{\,\ast}_{\alpha \beta}{~}^{\mu \nu}+\frac{1}{2}(\varepsilon -1) \eta^{\mu \nu}_{\ \ \rho \sigma}\ell^{[\alpha}\ \mathcal{E}^{\beta ]}{~}^{[\rho}\ \ell^{\sigma ]}\\ &&+\frac{1}{2}\ (1-\mu )\eta^{\mu \nu}{}{~}_{\rho \sigma}\eta_{\alpha \beta \varepsilon \tau}\ell^{\varepsilon} \mathcal{H}^{\tau}{~}^{[\rho}\ \ell^{\sigma ]}{\kern17pt} \end{array} $$

Then we have that

$$\begin{array}{@{}rcl@{}} Q_{\alpha \beta \mu\nu}^{\,\,\,\,\,\,\,\,\ast}\ell^{\beta}\ell^{\nu}&=&C^{\,\ast}_{\alpha \beta \mu \nu}\ell^{\beta}\ell^{\nu}=\mathcal{H}_{\alpha \mu}\ ,\\ Q^{\,\ast}_{\alpha\beta \varepsilon \sigma}\ell^{\beta}\ell^{\sigma}&=&\mu\ \mathcal{H}_{\alpha \varepsilon}\ . \end{array} $$

This T ^αβμν tensor has properties that are very similar to the Minkowski energy-momentum tensor ofelectrodynamics. The scalar constructed with T ^αβμν and the tangent vector ℓ ^ν, for instance, takes the form

$$ U_{(T)}=T^{\alpha \beta \mu \nu}\ell_{\alpha}\ell_{\beta}\ell_{\mu}\ell_{\nu} $$

(A23)

and gives the “energy” of the field

$$ U_{(T)}=\frac{1}{2}(\mathcal{E}^{2}+\mathcal{H}^{2}), $$

(A24)

where

$$\begin{array}{@{}rcl@{}} &&\mathcal{E}^{2}=\mathcal{E}_{\alpha \beta}\mathcal{E}^{\alpha \beta}. \end{array} $$

(A25a)

$$\begin{array}{@{}rcl@{}} &&\mathcal{H}^{2}=\mathcal{H}_{\alpha \beta}\mathcal{H}^{\alpha \beta}. \end{array} $$

(A25b)

In the context of the present extended theory, for a space-time in the state ε−μ of tension, we are led to modify T ^αβμν into Θ^αβμν defined in an analogous manner by the equality

$$ {\Theta}^{\alpha \beta \mu \nu} = \frac{1}{2}\left\{Q^{\alpha \rho \mu \sigma}C^{\beta}{}{~}_{\rho}{~}^{\nu}{}{~}_{\sigma} + Q^{\alpha^{\ast}\rho\mu\sigma} {~}^{\ast}C^{\beta}{}{~}_{\rho}{~}^{\nu}{}{~}_{\sigma}\right\}. $$

(A26)

Then, the energy U _{(ε, μ)} as viewed by an observer ℓ ^μ will be given by the relation

$$ U_{(\varepsilon ,\mu )}=\theta^{\alpha \beta \mu \nu}\ell_{\alpha}\ell_{\beta}\ell_{\mu}\ell_{\nu}=\frac{1}{2} \left(\varepsilon\mathcal{E}^{2}+\mu \mathcal{H}^{2}\right) $$

(A27)

in complete analogy with the electrodynamical case in a general medium.

We would like to make an additional remark by presenting two special properties of Θ^αβμν

$$\begin{array}{@{}rcl@{}} &&{}{\Theta}^{\alpha}_{\ \ \beta \alpha \mu}=\frac{1}{2}(1-\varepsilon )\varepsilon^{\rho \sigma}C_{\beta \rho \mu \sigma}\ , \end{array} $$

(A28a)

$$\begin{array}{@{}rcl@{}} &&{}{\Theta} = {\Theta}^{\alpha \mu}{}{~}_{\alpha \mu}=0 \ . \end{array} $$

(A28b)

Property (A28a) states that not all traces of Θ^αβμν are null for a general state of tension of space-time and that the non-null parts of the contracted tensor are independent of the “permeability” μ. The second property (A28b) states that the scalar obtained by taking the trace of Θ^αβμν twice is null, independent from the state of tension of the space-time.

1.3 A3 The Velocity of Propagation of Gravitational Disturbances in States of Tension

To determine the velocity of gravitational waves in ε−μ states of space-time, let us perturb the set of equations (A10a–A10d). The perturbation will be represented by the map:

$$\begin{array}{@{}rcl@{}} &&\mathcal{E}_{\mu \nu}\longrightarrow \mathcal{E}_{\mu \nu}+\delta \mathcal{E}_{\mu \nu}\ ,\\ &&\mathcal{H}_{\mu \nu}\longrightarrow \mathcal{H}_{\mu \nu}+\delta \mathcal{H}_{\mu \nu} \, \end{array} $$

(A29a) (A29b)

where $\delta \mathcal {E}_{\mu \nu }$, $\delta \mathcal {H}_{\mu \nu }$ are null quantities. Then, (A10a–A10d) are transformed into the perturbed set of equations

$$\delta \mathcal{E}_{\alpha}{~}^{\beta}{}{~}_{;\beta}\approx 0 \ ,$$

(A30a)

$$\varepsilon \delta \dot{\mathcal{E}}_{\alpha \mu}+\frac{1}{2}\ h_{(\alpha}^{\lambda}\eta_{\mu )}{~}^{\rho\sigma \tau}\ell_{\rho}\,\delta \mathcal{H}_{\tau \lambda;\rho}\approx 0 \ , $$

(A30b)

$$\delta \mathcal{H}_{\alpha}{~}^{\beta}{}{~}_{;\beta}\approx 0 \ , $$

(A30c)

$$\mu \ \delta \dot{\mathcal{H}}-\frac{1}{2}\ h_{(\alpha}^{\lambda}\eta_{\mu )}{~}^{\rho \sigma \tau}\mathcal{E}_{\tau \lambda;\rho}\approx 0\ , $$

(A30d)

where the covariant derivative is taken in the background—and we limit ourselves to the linear terms of the perturbation.

Now, let us specialize the background to be the flat Minkowski space-time.³⁷ In this case, the covariant derivatives are the usual derivation and we can use commutative property to write:

$$ \varepsilon \delta {\mathcal{\ddot{E}}}_{\alpha \beta}+\frac{1}{2}\ h_{(\alpha}^{\lambda}\eta_{\beta)}^{\rho \sigma \tau} \ell_{\sigma}\delta \mathcal{\dot{H}}_{\tau \lambda,\rho}\approx 0 \, $$

(A31)

by taking the derivative of (A30b) projected in the privileged direction ℓ ^μ.

Multiplying (A30d) by the factor

$$\frac{1}{2 \mu}\ h_{(\alpha}^{\nu} \eta_{\beta )}{~}^{\gamma\sigma\tau}\ell_{\tau}\ \frac{\partial}{\partial x^{\sigma}}\ , $$

we find that

$$\begin{array}{@{}rcl@{}} &&\frac{1}{2}h_{(\alpha }^{\nu}\ \eta_{\beta )}{~}^{\gamma\sigma \tau}\ell_{\tau}\,\delta \dot{\mathcal{H}}_{\gamma \nu, \sigma}\\ &&{\kern6pt}-\frac{1}{4\mu}\ h_{(\alpha }^{\nu}\ \eta_{\beta )}{~}^{\sigma \tau \gamma}\ \ell_{\tau}\ell_{\rho}h^{\varepsilon}{}{~}_{(\,\gamma}\ \eta_{\nu)}{~}^{\psi \rho \phi}\delta \varepsilon_{\psi \varepsilon ,\psi ,\sigma}\approx 0.\\ \end{array} $$

(A32)

Substituting (A31) into (A31) we finally find that

$$ \delta \ddot{\mathcal{E}}_{\alpha \mu}-\frac{1}{\varepsilon \mu}\nabla^{2}\delta\mathcal{E}_{\alpha \mu}=0 \ , $$

(A33)

where ∇² is the Laplacian operator defined in the three-dimensional space orthogonal to ℓ ^μ.

In the same way, an analogous wave equation can be obtained for $\mathcal {H}_{\alpha \mu }$. From (A33), we obtain the expected result: the velocity of propagation of gravitational waves in ε, μ states of tension of space-time is equal to $1/\sqrt {\varepsilon \mu }$.

The set of privileged observers that we dealt with here may be enlarged by somehow weakening the defining conditions [see (A8a–A8d)]. This point deserves further investigation.