Conformally flat travelling plane wave solutions of Einstein equations

TL;DR

This paper explores conformally flat plane wave solutions to Einstein's equations, incorporating various matter sources like scalar fields, electromagnetic fields, and relativistic particles, providing explicit metrics for these scenarios.

Contribution

It introduces explicit conformally flat solutions of Einstein equations with diverse matter sources depending on the wave phase, expanding the class of known exact solutions.

Findings

Explicit conformally flat metrics for scalar, electromagnetic, and particle sources.

Solutions depend on the wave phase with specific frequency conditions.

Models describe massless scalar fields, electromagnetic waves, and relativistic particles.

Abstract

We discuss conformally flat plane wave solutions of Einstein equations depending on the plane wave phase $\xi=\omega\tau-{\bf qx}$, where $\tau$ is the conformal time. We show that ideal fluid Einstein equations and scalar fields with exponential self-interaction have solutions of this form. We consider in more detail the source depending on $\xi$ with $\omega=\vert{\bf q}\vert$ describing models of a massless scalar field, electromagnetic field and relativistic particles with space-time depending mass density. We obtain explicit conformally flat metrics solving Einstein equations with such a source of the energy-momentum.

Equations

$$d{s}^2=a(x{)}^2(d{\tau }^2-d{\mathbf{x}}^2)=a(x{)}^2{\eta }^{\mu \nu }d{x}_{\mu }d{x}_{\nu },$$

$$G_{00}=3(a^{-1}{\partial }_{\tau }a{)}^2+(a^{-1}\nabla a{)}^2-2a^{-1}△a,$$

$$G_{0j}=4a^{-2}{\partial }_{\tau }a{\partial }_{j}a-2a^{-1}{\partial }_{\tau }{\partial }_{j}a,$$

$$G_{jk}=4a^{-2}{\partial }_{j}a{\partial }_{k}a+{\delta }_{jk}{a}^{-2}(({\partial }_{\tau }a{)}^2-(\nabla a{)}^2)-2a^{-1}({\partial }_j{\partial }_{k}a+{\delta }_{jk}({\partial }_{\tau }^2-{\nabla }^2)a).$$

$$\xi =\omega \tau -\mathbf{qx}={\eta }^{\mu \nu }{q}_{\mu }{x}_{\nu },$$

$$G_{00}=(3{\omega }^2+q^2)a^{-2}(\frac{da}{d\xi }{)}^2-2q^2{a}^{-1}\frac{d^{2}a}{d{\xi }^2}$$

$$G_{0j}=\omega q_{j}a^{-2}\Big{(}4(\frac{da}{d\xi})^{2}-2a\frac{d^{2}a}{d\xi^{2}}\Big{)},$$

$$G_{jk}=a^{-2}\Big{(}4q_{j}q_{k}+\delta_{jk}(\omega^{2}-q^{2})\Big{)}(\frac{da}{d\xi})^{2}-2a^{-1}\Big{(}q_{j}q_{k}+\delta_{jk}(\omega^{2}-q^{2})\Big{)}\frac{d^{2}a}{d\xi^{2}}.$$

$$(8\pi G{)}^{-1}{G}_{\mu \nu }=(\rho +p)u_{\mu }{u}_{\nu }-p{g}_{\mu \nu }+{\Pi }_{\mu \nu },$$

$$u_{\mu}=\partial_{\mu}a\Big{(}\partial_{\mu}a\partial^{\mu}a\Big{)}^{-\frac{1}{2}}.$$

$$8\pi G\rho =3a^{-2}{g}^{\mu \nu }{\partial }_{\mu }a{\partial }_{\nu }a-g^{\mu \nu }{\Pi }_{\mu \nu }$$

$$8\pi Gp=a^{-2}{g}^{\mu \nu }{\partial }_{\mu }a{\partial }_{\nu }a+g^{\mu \nu }{\Pi }_{\mu \nu },$$

$$8\pi G{\Pi }_{\mu \nu }=-2a^{-1}{\partial }_{\mu }{\partial }_{\nu }a.$$

$$u_{\mu}=\partial_{\mu}a\Big{(}-\partial_{\mu}a\partial^{\mu}a\Big{)}^{-\frac{1}{2}}.$$

$$8\pi G\rho =-3a^{-2}{g}^{\mu \nu }{\partial }_{\mu }a{\partial }_{\nu }a+g^{\mu \nu }{\Pi }_{\mu \nu }$$

$$8\pi Gp=a^{-2}{g}^{\mu \nu }{\partial }_{\mu }a{\partial }_{\nu }a+g^{\mu \nu }{\Pi }_{\mu \nu },$$

$$(8\pi G{)}^{-1}{G}_{\mu \nu }=(\rho -p)u_{\mu }{u}_{\nu }-p{g}_{\mu \nu }+{\Pi }_{\mu \nu },$$

$$(8\pi G{)}^{-1}{G}_{\mu \nu }=(\tilde{\rho }+\tilde{p}){\tilde{u}}_{\mu }{\tilde{u}}_{\nu }-\tilde{p}{g}_{\mu \nu },$$

$${\tilde{u}}_{\mu }=q_{\mu }a({\omega }^2-q^2{)}^{-\frac{1}{2}},$$

$$8\pi G\tilde{p}=(a^{-4}(\frac{da}{d\xi }{)}^2-2a^{-3}\frac{d^{2}a}{d{\xi }^2})({\omega }^2-q^2),$$

$$8\pi G\tilde{\rho }=3a^{-4}(\frac{da}{d\xi }{)}^2({\omega }^2-q^2).$$

$${\partial }_{\nu }\tilde{\rho }+3(\tilde{\rho }+\tilde{p})=0$$

$$(8\pi G{)}^{-1}{G}_{\mu \nu }=(\tilde{\rho }-\tilde{p}){\tilde{u}}_{\mu }{\tilde{u}}_{\nu }-\tilde{p}{g}_{\mu \nu },$$

$${\tilde{u}}_{\mu }=q_{\mu }a(q^2-{\omega }^2{)}^{-\frac{1}{2}},$$

$$8\pi G\tilde{p}=(3a^{-4}(\frac{da}{d\xi }{)}^2+2a^{-3}\frac{d^{2}a}{d{\xi }^2})(q^2-{\omega }^2),$$

$$8\pi G\tilde{\rho }=(-a^{-4}(\frac{da}{d\xi }{)}^2+2a^{-3}\frac{d^{2}a}{d{\xi }^2})(q^2-{\omega }^2).$$

$$8\pi G{t}_{\mu \nu }=G_{\mu \nu }+Q_{\mu \nu }$$

$${\partial }_{\mu }{t}^{\mu \nu }=0.$$

$$8\pi G{t}_{\mu \nu }=(a^{-2}(\frac{da}{d\xi }{)}^2-2a^{-1}\frac{d^{2}a}{d{\xi }^2})(q_{\mu }{q}_{\nu }-{\eta }_{\mu \nu }({\omega }^2-q^2))$$

$$8\pi G{Q}_{\mu \nu }=3a^{-2}(\frac{da}{d\xi }{)}^2{q}_{\mu }{q}_{\nu }.$$

Full text

1 Introduction

On a large scale the universe looks isotropic and homogeneous. Then, the dynamics involves only the expansion scale factor $a$. The resulting $\Lambda$CDM model describes well [1] the observational data.. There is however some tension concerning the value of the Hubble constant resulting from (local) CMB and supernova observations. It may be that the problem can be explained by inhomogeneities observed on a local scale [2][3][4]. An assumption of the isotropy and homogeneity allows to derive explicit solutions of Einstein equations [5]. Inhomogeneous cosmological models with a spherical symmetry have been extensively studied (see the review in [6]). At an intermediate scale there are some phenomena (voids, walls ) which disturb the homogeneous isotropic picture [7][4]. For a recent review of anisotropic Bianchi cosmologies see [8]. Particular solutions of Einstein equations may have a cosmological meaning reflecting some observed inhomogeneities and anisotropies in galaxy distribution and in CMB. In this paper we find anisotropic and inhomogeneous solutions of Einstein equations resulting from scalar and electromagnetic fields as a source of plane waves. We make an assumption that the metric is determined by the energy-momentum evolving like a plane wave in a direction ${\bf q}$, i.e., that it depends on $\xi=\omega\tau-{\bf qx}$, where $\tau$ is the conformal time. We consider conformally flat metrics ( see [9] for their cosmological relevance). Then, its scale factor $a$ also depends on $\xi$. We did not encounter such an explicit assumption in general relativity although the plane-wave Ansatz is a standard tool in classical theory of scalar waves (solitons)[10]. Einstein equations with a given lhs could be treated as a definition of the energy-momentum on the rhs. This is the way the Riemannian geometry is exploited in the plasma physics [11]. However, without a local Lagrangian defining the energy-momentum on the rhs we would in general get non-local and acausal theories. Plane-wave solutions of Einstein equations are discussed (and classified) from the group-theoretical point of view in [5] (sec.37). We think that such plane-waves, when their velocity is less than the velocity of light, may describe idealized thin walls encountered in astronomical observations [7]. If $\omega^{2}>{\bf q}^{2}$ then we show that the plane waves are Lorentz transformations of homogeneous solutions of Einstein equations with an ideal fluid as a source (Lorentz boosts to an arbitrary frame of the well-known homogeneous solutions obtained in the frame moving with the fluid, see [12] for an application of such boosts). If $\omega^{2}<{\bf q}^{2}$ then the phase velocity is less than the velocity of light. The plane waves can be considered as Lorentz transformations of static solutions of Einstein equations (with a static fluid) . Another explicit travelling wave solution results when the source is a free scalar field or a scalar field with an exponential interaction.

We discuss in more detail the case $\omega^{2}={\bf q}^{2}$. In this case the travelling plane wave moves with the velocity of light. We show that a massless free field or an electromagnetic field of a plane wave can be a source of the gravitational travelling plane wave. As another source we consider a particle with a space-time dependent mass. We believe that the travelling gravitational waves moving with the velocity of light may be relevant near the strong sources where the linear approximation to Einstein equations is not sufficient ( the well-known plane fronted exact gravitational waves are solutions of sourceless Einstein equations [5] ). In general, it is not simple (see [13]) to divide the metric resulting from various sources into the radiative and non-radiative parts. Nevertheless, both parts influence the geodesic motion of a test body, hence are measurable. The conformally flat metrics play a distinguished role because of their Lorentz covariance [9]. In such a metric the electromagnetic and gravitational radiation always propagates with the velocity of light. For this reason the conformally flat metrics are an interesting object of investigation.

The plan of the paper is the following. In sec.2 we discuss the conformal flat metrics. In sec.3 we show that if the source is an ideal fluid then there are solutions of Einstein equations in the form of a travelling plane wave. In sec.4 we discuss scalar fields as a source of the energy-momentum tensor. We show that in the case of the travelling wave there is a (dispersion) relation between $\omega$ and $|{\bf q}|$. In sec.5 we solve the geodesic equation in a gravitational field of the travelling plane-wave. In sec.6 we discuss Lagrangian models of the energy-momentum leading to plane wave solutions with $\omega=|{\bf q}|$. In sec.7 we obtain some explicit formulas for the metric of the plane wave with $\omega=|{\bf q}|$. In sec.8 we summarize the results.

2 The conformally flat metric

We consider the conformally flat metric in four space-time dimensions (we set the velocity of light $c=1$)

[TABLE]

where $\eta$ is the Minkowski metric. Then, the components of the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ are [14][5]

[TABLE]

[TABLE]

[TABLE]

We assume that the fields as well as the scale factor depend only on the plane wave phase

[TABLE]

where $(q_{\mu})=(\omega,{\bf q})$. Then (where $q^{2}={\bf q}^{2}$)

[TABLE]

The remaining components of the Einstein tensor as functions of $\xi$ are

[TABLE]

[TABLE]

If $\partial_{\mu}a$ is not a null vector then Einstein equations can be treated as an identity defining an energy-momentum tensor for a relativistic viscous fluid [15][16](this way Einstein equations are treated in the quark-gluon plasma [11]). We write $T_{\mu\nu}=(8\pi G)^{-1}G_{\mu\nu}$ (where $G$ is the Newton constant) then $G_{\mu\nu}$ can be expressed as

[TABLE]

where the fluid velocity is defined by

[TABLE]

The energy density is

[TABLE]

the pressure

[TABLE]

where

[TABLE]

The conservation law $(G^{\mu\nu})_{;\mu}=0$ can be decomposed into a continuity equation for the fluid $u_{\mu}(T^{\mu\nu})_{;\nu}=0$ and a Navier-Stokes type equation $(g_{\sigma\mu}-u_{\sigma}u_{\mu})(T^{\mu\nu})_{;\nu}=0$.

For space-like $q^{\mu}$ the square root in eq.(10) makes no sense. Then, we define

[TABLE]

[TABLE]

[TABLE]

Hence

[TABLE]

It follows from eq.(9) that for general $a$ the Einstein tensor $G_{\mu\nu}$ has the form of the energy-momentum tensor of a viscous fluid. However, if $a$ depends only on $\xi$ then we can write $G_{\mu\nu}$ as the energy-momentum of an ideal fluid

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

From eqs.(20)-(21) we can obtain the continuity equation conveniently expressed in terms of the e-fold time $\nu=\ln(a)$. Then

[TABLE]

We have two equations (20)-(21) for one function $a$.If we define $\rho(a)$ in eq.(21) then we obtain $a(\xi)$ and inserting it in eq.(20) we determine $p(a)$ . It follows from eq.(22) that an equation of state $p=f(\rho)$ determines $\rho(a)$ and $p(a)$ which are consistent with eqs.(20)-(21). With $p(a)$ and $\rho(a)$ satisfying eqs. (20)-(21) Einstein equations with $T_{\mu\nu}$ defined on the rhs of eq.(18)(with $u_{\mu}$ of eq.(19)) will be satisfied.

Let us still mention a modification of eqs.(18)-(21) if $q>\omega$. From eqs.(14)-(17)

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

The energy-momentum $T_{\mu\nu}$ of the gravitational field (proportional to $G_{\mu\nu}$) is covariantly conserved. We introduce the energy-momentum of the matter field $Q_{\mu\nu}$ so that

[TABLE]

satisfies

[TABLE]

We obtain

[TABLE]

and

[TABLE]

If $(q^{\mu})=(1,{\bf 0})$ then $8\pi GQ_{00}=3(a^{-1}\partial_{\tau}a)^{2}$ as in the model with the energy-momentum $Q_{00}$ of a homogeneous fluid.

3 Ideal fluids

We have ten Einstein equations

[TABLE]

for a single conformal factor $a$. Eqs.(31) are not of the same type. $G_{jk}$ equations are hyperbolic whereas equations for $G_{00}$ and $G_{0j}$ are parabolic (first order in time derivatives). For $G_{jk}$ we can insert arbitrary initial conditions for time derivatives of $a$ whereas the initial value for the time derivative of $a$ in the equations for $G_{00}$ and $G_{0j}$ is determined by the initial conditions for $T_{\mu\nu}$. The distinction between the $G_{\mu\nu}$ equations is not explicit for the $\xi$ equations (which are of the second order in $\xi$). However, if the $G_{00}$ equation is to have the $q=0$ limit then we have to choose a proper initial condition for $\partial_{\xi}a$. If we solve one of Eqs.(31) then the remaining equations determine the other components of the energy-momentum tensor. It remains an open problem whether the energy-momentum tensor defined this way follows from a Lagrangian field theory. As an example let us consider the $\delta_{jk}$ part of $G_{jk}$ equation

[TABLE]

Choosing

[TABLE]

we can integrate eq.(32) with the result

[TABLE]

where $k_{0}$ is an arbitrary constant. By differentiation of eq.(34) we obtain

[TABLE]

We can then calculate all terms $G_{\mu\nu}$ and $T_{\mu\nu}$ as functions of $a$. In particular, from eq.(21)

[TABLE]

As an equation of state we obtain

[TABLE]

where

[TABLE]

We have the linear relation

[TABLE]

if and only if

[TABLE]

The result (34) shows that a power-law for $\tilde{p}$ admits only the equation of state (37) as a solution of eq.(22) (with eq.(38) as a particular case).

If we solve eq.(21) with the Ansatz

[TABLE]

and insert the solution in eq.(20) then we obtain the result (38) and the initial condition (39). On the other hand if we assume eq.(38) then from eq.(22) we obtain the power-law behaviour (33) and (40). Then, the solution of eq.(31) is a standard extension $\tau\rightarrow\xi$ of the homogeneous solution. We obtain this way a Lorentz transformation of the homogeneous solution if $\tau$ and $\xi$ are related by a Lorentz transformation (see the end of the next section).

4 Scalar field as a source

Let us consider the Lagrangian for scalar fields

[TABLE]

The energy-momentum tensor is

[TABLE]

We investigate a soluble model of the power-law inflation [17]

[TABLE]

where $V_{0}$ is a constant. We assume that the fields depend only on $\xi$. Then, the Lagrange equation for $\phi$ reads

[TABLE]

We solve the $G_{0j}$ Einstein equations first with (from eq.(42))

[TABLE]

assuming

[TABLE]

where $\kappa$ is a constant. Let us note that if $V=V_{0}$ ($\lambda=0$ in eq.(43) ) then $r=4$. Eq.(31) for $G_{0j}$ can be integrated (with the Ansatz (46); we choose the initial condition $a(0)=1$)

[TABLE]

with an integration constant $k_{0}$. It follows from eq.(47) that

[TABLE]

We can insert the results (47)-(48) into the remaining Einstein equations and into the Lagrange equations for $\phi$. If $\lambda=0$ then the remaining Einstein equations have the solution if

[TABLE]

whereas the Lagrange equations (44) require $r=4$ in eq.(46) ( solutions $a(\xi)$ of eq.(47) are discussed in sec.7 below eq.(93)) .

When $\lambda\neq 0$ then we must have $V_{0}=0$ (hence there is zero on the rhs of eq.(49)). Then, the Lagrange equations (44) and the remaining Einstein equations are solved if

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then, in addition the dispersion relation

[TABLE]

must be satisfied. We obtain

[TABLE]

$G_{\mu\nu}$ is covariant with respect to Lorentz transformations $L$. Hence, if the energy-momentum tensor on the rhs of eq.(31) is also Lorentz covariant then

[TABLE]

Note that $\xi=\eta^{\mu\nu}q_{\mu}x_{\nu}$ is Lorentz invariant. In a special Lorentz frame it may take the form $\xi=\omega_{0}\tau$ where $\omega_{0}^{2}=\eta^{\mu\nu}q_{\mu}q_{\nu}$ if $q_{\mu}$ is time-like or $\xi=-q_{10}x^{1}$ where $q_{10}^{2}=-\eta^{\mu\nu}q_{\mu}q_{\nu}$ if $q_{\mu}$ is space-like. We may solve Einstein equations (31)in a special frame (with a special choice of $q$). Then, on the basis of eq.(56) they will hold true for a general $\xi$.

As an example we apply the Lorentz transformation when $V=0$ and $\omega^{2}\neq q^{2}$ in eq.(44). Then

[TABLE]

with a certain constant $\kappa$. Choose $\omega=1$ and ${\bf q}=0$. We solve eq.(6) for $G_{00}$ ($a\simeq\tau^{\frac{1}{2}}$). In the resulting solution $a(\tau)$ we replace $\tau$ by $\xi=\gamma\tau-v\gamma x_{1}$ where $(q_{\mu})=(\gamma,v\gamma,0,0)$ with $\gamma=(1-v^{2})^{-\frac{1}{2}}$ coming from the Lorentz transformation (an application of such a boost of a homogeneous solution is discussed in [12]). We check that

[TABLE]

solves all of 10 Einstein equations (31) ( $T_{\mu\nu}$ of eq.(42) with $V=0$ ) transformed to an arbitrary frame by eq.(56).

In the case with an exponential interaction if $r<4$ and $\lambda>0$ we have from eq.(54) $\omega>q$. We could solve the homogeneous Einstein equations ($q=0$, as in [17]). Then, by a boost we obtain the inhomogeneous solution (55) in the form of the plane wave with the phase velocity $\frac{\omega}{q}>1$ larger than the velocity of light. If $r>4$ (large $\alpha$) and $\lambda>0$ then $\omega<q$. We can look for static one dimensional solutions of Einstein equations with only $x^{1}\neq 0$. Then, by a boost we obtain a general plane wave solution (55) with the phase velocity $\frac{\omega}{q}<1$ (less than the velocity of light). This is the standard way how solitons are derived from static solutions in field theory of a scalar field [10]. It is well-understood that if the plane wave moves with the velocity ${\bf u}$ less than the velocity of light then in the frame moving with the velocity ${\bf u}$ the wave looks static.

5 Test-body motion

We consider a geodesic motion of a body under the influence of the conformally flat gravity. The geodesic equation is

[TABLE]

where

[TABLE]

Inserting the Christoffel symbols $\Gamma$ we derive a simple equation for $\xi$

[TABLE]

Denoting $\nu=\frac{d\xi}{ds}$ we can integrate eq.(60) with the result

[TABLE]

where $K_{0}$ and $a_{0}$ are constants of integration. From eq.(61) we can calculate $a(s)$

[TABLE]

where $\frac{da}{d\xi}$ is determined by eq.(34) or (47).Subequently, we can obtain $\xi(s)$ when $a(\xi)$ is known .

The velocity $\frac{d\xi}{ds}$ is expressed by the phase velocity ${\bf u}$ ( for the velocity $|{\bf u}|<1$)

[TABLE]

with

[TABLE]

The phase velocity is related to the coordinate velocity in proper time

[TABLE]

In order to determine the motion beyond the $\xi$ plane let us define

[TABLE]

Inserting $v^{\mu}$ into the geodesic equations (59) we obtain

[TABLE]

Hence,

[TABLE]

where $\xi(s)$ is determined from the solution of eq.(60). In principle, we could detect the source of gravity observing its action upon a test body. However, it would be difficult to separate the astrophysical sources from the local ones.

6 Einstein

equations for $\omega=q$

When $\omega=q$ then the travelling plane waves generated by the energy-momentum $T_{\mu\nu}$ move with the velocity of light. We consider various sources of such waves: massless scalar fields, electromagnetic fields and a relativistic particle with a continuous spectrum of mass. These fields produce a deformation of the space-time which propagates with the velocity of light. The waves will interact with massive test-bodies and charged particles. They carry an energy and momentum which are conserved because if $\omega=q$ then in eqs.(6)-(8) $\partial_{\mu}G^{\mu\nu}=0$ hence also $\partial_{\mu}T^{\mu\nu}=0$.

As a source of the energy-momentum in this section we consider a scalar field with the Lagrangian

[TABLE]

where $h$ is an arbitrary function and

[TABLE]

The electromagnetic field with the Lagrangian

[TABLE]

provides a source moving with a velocity of light. We discuss also a relativistic particle with $x$-dependent energy density $m$ with the Lagrangian

[TABLE]

From the Lagrangian $-\frac{R}{8\pi G}+L$ we obtain Einstein equations (31). The scalar field Lagrangian defines the energy-momentum tensor

[TABLE]

From the Lagrangian (64) we obtain

[TABLE]

The energy-momentum of the relativistic particle (65) is defined as

[TABLE]

The Lagrangian equations for the scalar field (under the assumption that the fields depend only on $\xi$) are

[TABLE]

It follows from eq.(69) that if $\omega^{2}=q^{2}$ then for every function $h$ any $\phi(\xi)$ satisfies the scalar wave equation. Note that if $\omega^{2}=q^{2}$ then $W=0$. Hence, $(h(W)=h(0)$ and $h^{\prime}(W)=h^{\prime}(0)$ are constants in the energy-momentum (66) . The electromagnetic field satisfies the Maxwell equations

[TABLE]

(together with $(\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta})_{;\mu}=0$). The Lagrange equations for the particle read

[TABLE]

where $x=\xi$. If $q_{0}=|{\bf q}|$ then eq.(71) and the constraint $x=\xi$ are solved by

[TABLE]

Then, $\eta^{\mu\nu}X_{\mu}q_{\nu}=\eta^{\mu\nu}x_{\mu}q_{\nu}=\xi$ . Under the assumption that $a$ depends only on $\xi$ (5) with $\omega^{2}=q^{2}$ eqs.(2)-(4) can be expressed in the form

[TABLE]

where $(q_{\mu})=(q,{\bf q})$.

The components of the scalar field energy-momentum tensor (on solutions (69)) are

[TABLE]

[TABLE]

[TABLE]

Comparing eqs.(74)-(76) with eq.(73) we can see that if 00 component of Einstein equations is satisfied then the remaining components will be satisfied if

[TABLE]

We assume the normalization $h^{\prime}(0)=1$ as for the free massless scalar field (when $h(W)=W$).

After an insertion of the solution (72) the particle energy-momentum (68) is

[TABLE]

In the case of the electromagnetic field (64) it is known (see ,e.g.,[18] ) that for the plane wave solutions

[TABLE]

where $\rho_{em}$ is the electromagnetic energy density. Summarizing, it follows from eqs.(73) and (74)-(79) that in the Lagrangian models of this section it is sufficient to solve the 00 component of Einstein equations. After the solution of the $00$ component the remaining equations will be satisfied.

We have no a priori restriction on the density $\rho(\xi)q_{\mu}q_{\nu}=a^{-2}T_{\mu\nu}$ in the formulas (66)-(71) for $T_{\mu\nu}$. $T_{\mu\nu}(\xi)$ can be expressed as a function of $a$. For a comparison with standard models we choose this $a$-dependence in a power-law form as it is obtained for ideal fluids with the equation of state $p=w\rho$ (where $p$ is the pressure and $w$ is a constant). It is sufficient to restrict ourselves to one of the models (74)-(79). In the scalar field model the 00 component of Einstein equations reads

[TABLE]

where we express $\frac{d\phi}{d\xi}$ as a function of $a$.

Eq.(80) is solved with the initial conditions $a(\xi=0)=a_{0}$ and $\frac{da}{d\xi}(\xi=0)=u_{0}$. Let $K=u^{2}$ then eq.(80) can be expressed as

[TABLE]

Eq.(81) can be integrated

[TABLE]

where we introduced an integration constant $k_{0}\geq 0$ related to $u_{0}=\pm a_{0}^{2}\sqrt{k_{0}}$.

If $T_{00}=0$ then $a=a_{0}=const$ is a solution of eq.(82) with $k_{0}=u_{0}=0$. However, there is also a non-trivial solution. It can be checked that

[TABLE]

is a solution of the Einstein equation (31) with the energy-momentum tensor $T_{\mu\nu}=0$. This is a special case of the plane fronted gravitational waves (flat polarization) when they are conformally flat (as noted in [19][20]).

We assume that $T_{00}$ is a power of $a$

[TABLE]

We may assume the $a$-dependence because a function of $\xi$ can be expressed as a function of $a$ (if $a(\xi)$ is invertible). So, we may assume that $\phi(\xi)=\tilde{\phi}(a)$ (we skip ”tilde” further on). Then

[TABLE]

Hence,

[TABLE]

where from eqs.(80)-(82)

[TABLE]

where

[TABLE]

and we set $a_{0}=1$ ($a_{0}>0$ has no physical meaning it just rescales coordinates). Taking the square root in eq.(87) and integrating we can calculate $\phi$ as a function of $a$.

Note that if we know $\phi(a)$ then eq.(85) can be expressed in another form suitable for integration

[TABLE]

Eq.(82) can be expressed as

[TABLE]

where $V=-K$ is a potential of a particle moving with the kinetic energy $K$ on the half-line $a\geq 0$ with the total energy $E=0$ . The motion is possible in the range of $a$ such that $-V\geq 0$. The evolution $a(\xi)$ based on eqs.(82) and (90) is discussed in the next section.

7 Elementary solutions of Einstein equations

Taking the square root of eq.(87) and integrating we obtain the equation for $a$

[TABLE]

We consider the cases when the integral (91) can be expressed by elementary functions or elliptic functions (such integrals have been discussed in [21][22][23]).

Let $w=1$ in eq.(84) (stiff matter [24]). Denote

[TABLE]

and consider $A=a^{2}$. Then

[TABLE]

If $\sigma_{s}>0$

[TABLE]

then the integral is expressed by the Weierstrass elliptic function ${\cal P}(8\sqrt{\sigma_{s}}(\xi+\xi_{0}),0,-\frac{\alpha^{2}}{48\sigma_{s}})$ [21][22]. We can extend the upper limit in eq.(93) to infinity showing that if $\sigma_{s}>0$ then $a$ achieves infinity for a finite $\xi$ (then the energy density tends to zero). If $\sigma_{s}<0$ then the range of $a$ is bounded by the requirement $K\geq 0$.

For $w=\frac{1}{3}$ (relativistic matter)

[TABLE]

Hence, again the integral (91) is expressed by an elliptic function. If $k_{0}-\frac{1}{2}\alpha^{2}>0$ then there is an explosion at finite $\xi$, whereas if $k_{0}-\frac{1}{2}\alpha^{2}<0$ then $a$ varies in a bounded interval.

Let us consider now negative $w$. First, $w=-\frac{1}{3}$ (cosmic string or coasting cosmology [25])

[TABLE]

Let

[TABLE]

Then, the integral (91) gives

[TABLE]

where $\xi_{0}$ is chosen is such a way as to satisfy the initial condition $a(0)=1$. If $\sigma<0$ then $a\rightarrow\infty$ at finite $\xi$ (then the energy density tends to zero ) . If $\sigma>0$ then $a$ is a bounded function ($a\rightarrow 0$ when $\xi\rightarrow\pm\infty$).

If $w=-\frac{2}{3}$ (domain wall) then

[TABLE]

Eq.(91) gives

[TABLE]

If $2\alpha^{2}<k_{0}$ then $a$ explodes at finite $\xi$ (the energy density tends to 0).If $2\alpha^{2}>k_{0}$ then $a$ is a bounded function.

Let $w=-\frac{4}{3}$ (phantom matter)

[TABLE]

From the integral (91) we obtain

[TABLE]

$\xi_{0}$ is chosen to satisfy the initial condition $a(\xi=0)=1$. $a$ varies in the interval $[1,1+\frac{k_{0}}{2\alpha^{2}}]$.

If $w=-\frac{5}{3}$ then

[TABLE]

The integral is again an elementary function. This problem is similar to the previous one so we do not write down (rather complicated ) explicit formula. If $w=-2$ then $a$ varies in a bounded interval, it can be expressed by an elliptic function. If $w<-2$ then the integral (91) can be discussed by means of the methods of ref.[26] which allow to detect some elementary solutions for integer values of $w$.

8 Summary

We have discussed some exact inhomogeneous solutions of Einstein equations in the form of conformally flat travelling plane waves. Such gravitational deformations of space-time can propagate with the velocity smaller than the velocity of light. They can disturb measurements of some quantities with a global meaning such as the Hubble constant.We can test the presence of inhomogeneities on the basis of a geodesic motion of a test body . If the source of the disturbance is an ideal fluid with time-like velocity then the plane waves can be considered as a transformation of the homogeneous expanding solution to the moving frame. If the fluid has space-like velocity (a static source) then the plane wave has the phase velocity smaller than the velocity of light. In the frame moving with the wave its amplitude and energy is static. It can describe a measurable wave propagation. In the case when the phase velocity is equal to the velocity of light the travelling waves may contain gravitational waves produced by a massless scalar field or an electromagnetic field. The metric is not asymptotically flat. Hence, the gravitational field is not of the type of pure radiation (it may contain a non-radiation background). The electromagnetic field which is produced when neutron stars merge can be a source of plane conformally flat gravitational waves. These waves could possibly be detected in a different experimental set-up than the one so far prepared for a detection of the transverse (TT) gravitational waves. From the mathematical point of view the conformally flat solutions are distinguished by their covariance with respect to the Lorentz transformations. Using this covariance we can obtain non-homogeneous solutions from homogenous solutions of Einstein equations (with Lorentz covariant sources) or time-dependent solutions from static ones.