# Hydrodynamics in Class B Warped Spacetimes

###### Abstract

We discuss certain general features of type B warped spacetimes which have important consequences on the material content they may admit and its associated dynamics. We show that, for Warped B spacetimes, if shear and anisotropy are nonvanishing, they have to be proportional. We also study some of the physics related to the warping factor and of the underlying decomposable metric. Finally we explore the only possible cases compatible with a type B Warped geometry which satisfy the dominant energy conditions. As an example of the above mentioned consequences we consider a radiating fluid and two non-spherically symmetric metrics which depend upon an arbitrary parameter , such that for spherical symmetry is recovered.

###### pacs:

04.20.Jb, 04.20.Cv, 95.30.SfMarch 30, 2021

## I Introduction.

Given two metric manifolds and and given a
smooth real function , (*warping
function*), one can build a new metric manifold by setting
and

where above are the canonical projections onto and
respectively, such an structure is called *warped product
manifold*, and in the case in which is a spacetime (i.e.:
and a Lorentz metric) it is called a warped product spacetime (or simply
*warped spacetime*). One of the simplest examples of warped spacetime is
provided by the Friedman-Robinson-Walker universe. But the warped structure
accommodates a large number of metrics in General Relativity, such as
Bertotti-Robinson, Robertson-Walker, Schwarzschild, Reissner-Nordstrom, de
Sitter, etc. (see CarotdaCosta1993 and references therein). Also warped
spacetimes can be regarded, in some sense, as generalizations of locally
decomposable spacetimes in the sense usually meant in general relativity
(HallKay1988 ).

The importance of warped spacetimes is that their geometry and, as we will presently show, also its physics, is directly related to the properties of their lower-dimensional factors, which are generally easier to study. The warped product construction provides a useful method for studying large classes of spacetimes. If the warping factor is constant the spacetime is decomposable, such as the Bertotti-Robinson spacetime or the Einstein static universe. Warped product spacetimes with non-constant warping factors are much richer and include such well known examples such as all the spherically, plane and hyperbolically symmetric spacetimes (therefore including Schwarzshild solution), Friedmann Robertson Walker cosmologies, all the static spacetimes, etc. (see CarotTupper2002 and references therein).

Anisotropy and shear properties of fluids in General Relativity have been extensively studied. Shearfree and non shearfree spacetimes have been widely considered in the literature (see for example StephaniEtal2004 ). On the other hand, the assumption of local anisotropy of pressure (i.e. non pascalian fluids where radial and tangential pressures are different, ), has been proven to be very useful in the study of relativistic compact objects. Although the perfect pascalian fluid assumption (i.e. ) is supported by solid observational and theoretical grounds, an increasing amount of theoretical evidence strongly suggests that, for certain density ranges, a variety of very interesting physical phenomena may take place giving rise to local anisotropy (see HerreraSantos1997 and references therein).

The purpose of this paper is twofold, on the one hand we present and discuss in detail certain general features of type B warped spacetimes which have important consequences on the material content they may admit and its associated dynamics; on the other hand, a thorough study of the dissipative anisotropic fluid dynamics in such spacetimes is carried out, with particular emphasis put on a set of geometrical and physical variables which appear to play a special role in the evolution of such systems.

In this paper, it will be shown how the local anisotropy of pressures and the shear of relevant velocity fields are closely related; in fact, and for warped spacetimes, we will show that if both, shear and anisotropy, are non-vanishing they have to be proportional. We shall also show how isotropic and anisotropic physics are related to the warping factor or to the conformally related decomposable metric. Further, we will explore the possible material contents that are compatible with a type warped geometry and satisfy the dominant energy condition. As an example of the above mentioned consequences we shall consider a radiating fluid. Radiative hydrodynamics is a theory of fundamental importance in astrophysics, cosmology, and plasma physics. It has became a very active field with a wide variety of application areas ranging from Plasma laboratory physics, to astrophysical and cosmological scenarios (see the comprehensive treatise of D. Mihalas MihalasMihalas1984 and his recently updated bibliography Mihalas2004 ).

The mathematical model of radiative hydrodynamics consists of the equations for a two-component medium: matter and radiation, which interact by exchanging energy and momentum; i.e.: anisotropic matter plus radiation (photon/neutrinos) which can be described by the total stress-energy tensor where the material part is described by , and is then the corresponding term for the radiation field. The interaction between matter and radiation is described by a radiative transfer equation through the absorption and emission terms, describing the rate at which matter absorbs and emits photons, and by integral terms describing the scattering of radiation (photons/neutrinos) off matter.

The paper is organized as follows: the first section contains a brief account of the definitions and most immediate properties of warped spacetimes, especially those of the type B, and introduces the notation and conventions used throughout the paper. In Section III some general results regarding the energy-momentum tensor of this class of spacetimes are proven and their implications on the physical content and material dynamics are pointed out relating them to the issues discussed in section II; these results complement and extend those in IshakLake2004 . Section IV displays some of the consequences of the geometric structure of this type of spacetimes in a simple and, we believe, useful form for the case of a radiating fluid; thus generalizing the results by Herrera et al. HerreraEtall2004 in the spherically symmetric case. The restrictions imposed by the energy conditions are explored and illustrated in this case. In section V we provide two examples of quasi-spherical but non-spherically symmetric metrics showing that the departure from sphericity is controlled by a single parameter that can be arbitrarily large and such that when , spherical symmetry is recovered. Finally, in section VI we summarize the main results and conclusions.

## Ii Preliminary results, notation and conventions.

In this section we set up the notation and summarize some of the results to be used in the remainder of the paper. We recall the basic definitions regarding warped spacetimes and introduce the concepts of adapted observers and adapted tetrads. We also explore the structure of the energy-momentum tensors which are compatible with a type warped geometry.

### ii.1 Warped and decomposable spacetimes

As mentioned in the previous section, given two metric manifolds and and a smooth real function , (warping function), a new metric manifold (warped product manifold) can be built where and

(1) |

with the canonical projections onto and respectively (see ONeill1983 , BeemErhlich1981 ). Where there is no risk of confusion; we shall omit the projections and write from now on:

(2) |

Notice that by pulling out the warping factor, we can always rewrite the metric as

(3) |

where is also a metric on ; thus, a warped manifold is always conformally related to a decomposable one (see StephaniEtal2004 ).

If and has Lorentz signature (i.e.: one of the manifolds is Lorentz and the other Riemann), is usually referred to as a warped spacetime; see CarotdaCosta1993 and HaddowCarot1996 where (local) invariant characterizations are provided along with a classification scheme and a detailed study of the isometries that such spacetimes may admit. If one has either or , the spacetime is said to be of class A, whereas if it is said to be of class B, which is the class we shall be interested in. Class is further subdivided into four classes according to the gradient of the warping function: if it is non-null and everywhere tangent to the Lorentz submanifold, if it is null (hence also tangent to the Lorentz submanifold), if it is tangent to the Riemann submanifold, and if it is zero, i.e.: constant which corresponds to being locally decomposable.

Of all the above possibilities we shall only be concerned in this paper with class . Thus, and without loss of generality we shall assume that is Lorentz (coordinates ) and is Riemann (coordinates ); the warping function then being . An adapted coordinate chart for the whole spacetime manifold will be denoted as where and are those defined previously. We shall always use such adapted charts, furthermore and in order to ease out the notation, we shall use the following coordinate names: . At this point, it is worthwhile noticing that spherically, plane and hyperbolically symmetric spacetimes are all special instances of warped spacetimes.

In what follows, we shall write the spacetime metric in the form (3), i.e.: explicitly conformally decomposable, and we shall put for convenience; further, we shall drop primes in (3) as well as the subscripts and in the metrics of the submanifolds and where there is no risk of confusion, thus the line element will be written as

(4) |

i.e.

(5) |

where is the underlying conformally related, decomposable metric with line element

(6) |

Since and are two two-metrics, one can always choose the coordinates and so that both take diagonal forms (even explicitly conformally flat); thus, and in order to fix our notation further, we shall most often use in our calculations the following form of the metric:

(7) |

We shall denote the covariant derivative with respect to the connection associated with by a semicolon (or also ), whereas that associated with will be noted by a stroke (or alternatively ); accordingly, tensors defined in or referred to the metric will be noted with a hat ‘’.

### ii.2 Adapted observers and tetrads

A further important remark concerns observers (congruences of timelike curves) in these spacetimes. A future directed unit timelike vector field will be said to be an adapted observer in if it is hypersurface orthogonal and everywhere tangent to . These requirements are equivalent to saying that, in an adapted coordinate chart its components are . It is easy to see that these observers always exist and that the coordinates may be chosen so that while the metric preserves its diagonal form. We shall construct an adapted tetrad in by choosing a unit spacelike vector field which is everywhere tangent to and orthogonal to ; i.e.: , and two other unit spacelike vector fields, , which are also hypersurface orthogonal, tangent to and mutually orthogonal (hence in an adapted chart: , and something similar for , also note that ). In terms of this adapted tetrad one has

(8) |

and a trivial calculation now shows that

(9) |

(10) |

Notice that and are respectively, the expansion and the acceleration of in . Using the above expressions, the shear associated with turns out to be (recall that ):

(11) |

We next define an adapted observer in , to be , where is any adapted observer in the decomposable spacetime as defined above. Note that is also hypersurface orthogonal and tangent everywhere to , and its components, in any adapted chart, will be functions of the coordinates alone. We construct the rest of an adapted tetrad in simply as , , and , where the hatted vectors form an adapted tetrad in as defined above. In terms of an adapted tetrad:

(12) |

Regarding the shear and vorticity of one has StephaniEtal2004 :

(13) |

¿From a geometric point of view, adapted observers and tetrads, seem very natural in both warped and conformally related decomposable spacetimes. As we shall see early on in the next section, they also arise very naturally from physical considerations.

Notice that one could have observers that, while being tangent to they are not hypersurface orthogonal, e.g.: in the coordinates introduced in (7), consider

(14) |

where depends on all four coordinates, it is immediate to check that this vector field has non-vanishing vorticity (indeed its components depend on coordinates in both and in any adapted chart). We shall briefly return to this point later on, but as already hinted above, such observers are somehow unnatural from a physical viewpoint.

### ii.3 Einstein Tensor and Warped Spacetimes

The geometry of the decomposable spacetime imposes certain restrictions that will become important later on in our study of hydrodynamics in warped spacetimes of this class and that have to do with the natural occurrence of the adapted tetrads and observers discussed above.

With the conventions and notation introduced so far, it turns out (see e.g. Wald1984 ) that the Einstein tensor in can be written as

(15) |

Note that is such that

(16) |

where and are the Ricci scalars associated with the two-metrics and respectively. The Ricci scalar is , hence

(17) |

Furthermore

(18) |

and taking (15) into account, it follows that has box diagonal form:

(19) |

with

(20) |

where (and therefore ) is non-diagonal in the general case.

At this point, it is interesting to notice that, on account of the form of , it follows that any vector field tangent to that is an eigenvector of (or equivalently of ) will automatically be an eigenvector of and viceversa; and that any vector field tangent to that is an eigenvector of (or equivalently of ) will also automatically be an eigenvector of and viceversa; in the next section we will show that all eigenvectors of the Einstein tensor are necessarily tangent to or to , as the block diagonal structure suggests.

Also notice that almost all the physical properties of the spacetime under consideration are somehow encoded in the warping factor , since the contribution to the energy momentum tensor of the underlying decomposable spacetime is simply a shift in the eigenvalues.

We shall dedicate the next section to study the allowed algebraic types of the Einstein tensor, which through Einstein’s field equations will provide information on the material content allowed for such spacetimes.

## Iii Material content of warped spacetimes.

### iii.1 Observers and Matter content

Given a second order symmetric tensor such as the energy-momentum tensor in an arbitrary spacetime , and given an arbitrary unit timelike vector field (which we shall assume future oriented) defined on , one can always decompose as follows

(21) |

where is the projector orthogonal to , that is: , and the rest of quantities appearing above are

(22) |

If represents the material content of the spacetime and is the four-velocity of some observer, then is the energy density as measured by such an observer, is called the isotropic pressure (measured by that observer), and and are, respectively, the momentum flux and the anisotropic pressure tensor that the observer measures. Notice that

(23) |

Recalling now (19), one has that in the case of warped spacetimes and working in an adapted (but otherwise arbitrary) chart, the Einstein tensor has got this box diagonal form. A direct inspection of the functional dependence of the components of above shows that given any adapted tetrad to , the Einstein, or equivalently, the energy-momentum tensor , may be written as

(24) |

for some functions

(25) |

Moreover, if one defines the null vector , the above expression can be rewritten as

(26) |

Physically, this can be interpreted by saying that the material content of one such spacetime can always be represented either as an anisotropic fluid with four-velocity (comoving with an adapted observer), density , pressures and and momentum flow (equation (24)); or else (equation (26)) as the sum of an anisotropic fluid with the same four-velocity , density , pressures and , plus a null radiation field directed along carrying an energy density . This splitting of the energy momentum-tensor (especially the last one (26)) has been extensively used in the spherically symmetric context: see HerreraEtall2004 and references cited therein.

It is also interesting to note that the above decompositions are highly non-unique in the sense that or can be split in a similar manner for all observers whose world lines are tangent to everywhere (be they adapted, i.e.: hypersurface orthogonal, or not), that is; whose four velocity is for an arbitrary function , then and also . If depends on (i.e.: the observer is non-adapted) the corresponding density , pressures , etc. will not have the functional form (25), but if alone the resulting observer and tetrad are also adapted and then (25) holds for the primed quantities , etc.

### iii.2 The anisotropic pressure tensor and the shear tensor

Writing in equations (24, 26) in the form of equation (21) and using the adapted observer to perform the decomposition, one has:

(27) |

where

(28) |

¿From (13) and the expression of given above, it is now immediate to see that the shear tensor of is proportional to the anisotropic pressure tensor , whenever both tensors are non vanishing:

(29) |

If , it can be interpreted as a shear viscosity coefficient: , being the so called kinematic viscosity coefficient, and then viscosity can be seen as the source of anisotropy in the pressure.

For any other adapted observer , with

where one obtains expressions similar to those above:

where the primed magnitudes are those measured by . Notice that one also has ; thus, for all the adapted observers the anisotropic pressure tensor is proportional to their shear tensor. This proportionality can be tracked back to the decomposable spacetime ; to this end consider the adapted tetrad and adapted observer in which are conformally related to those in ; i.e.: (see previous section); from (17) we get

(30) |

which may also be decomposed with respect to the observer as in (21) thus getting

(31) |

with

(32) |

where . From the above expression for and (11) one has , and recalling that and , one finally concludes

(33) |

The true equation of state that describes the properties of matter at densities higher than nuclear ( ) is essentially unknown due to our inability to verify the microphysics of nuclear matter at such high densities Glendening2000 . Having this uncertainty in mind, it seems reasonable to explore some possible equations of state for the local anisotropy starting from a simple geometrical object as is the shear tensor The proportionality of the anisotropic and the shear tensors opens the possibility to devise such equations of state .

Needless to say, a decomposable spacetime of these characteristics does not represent itself any reasonable physical content (notice that ), however, it is still interesting to realize how this decomposable structure somehow ‘generates’ anisotropy in the pressures in the physically realistic warped spacetime. This is in contrast with the warping factor , that contributes to what one could roughly call the ‘isotropic physics’, namely: the energy density and the isotropic pressure .

### iii.3 Eigenvector structure and energy conditions.

Let us next see how the assumed geometry (warped spacetime) imposes certain restrictions on the material content, and how this shows up in the algebraic (eigenvector/eigenvalue) structure of the Einstein tensor.

We begin by noting that the eigenvectors of the Einstein tensor are the same as those of the Ricci tensor , their corresponding eigenvalues being ‘shifted’ by an amount , where is the Ricci scalar associated with , furthermore, on account of the form of (see the remarks at the end of the preceding section) and equation (15), it follows that these eigenvectors coincide with those of the tensor .

Thus, the three tensors and all have the same Segre type HallNegm1986 with the same eigenvectors. For convenience we shall work with the Ricci tensor in an adapted coordinate chart, thus we have

(34) |

with

The characteristic polynomial of is then

(35) |

and therefore there is one repeated eigenvalue that corresponds to two spacelike eigenvectors tangent to that can be chosen unit and mutually orthogonal, say and ; one therefore has in an adapted chart: and (furthermore: in a chart in which takes diagonal form and ). The remaining eigenvalues are the roots of the second degree polynomial

(36) |

where is the trace of the matrix and is its determinant. Some elementary algebra considerations lead to the following three possibilities:

#### The polynomial has two real roots.

If has two real roots, say and , they will be functions on (i.e.: functions of the coordinates ) since are also functions on . The necessary and sufficient condition for this to happen is that

(37) |

or, on account of our previous considerations on eigenvector/eigenvalue structure of and , that

(38) |

with

which involves only covariant derivatives of the warping function taken with respect to the connection of the decomposable metric. This corresponds to being of the diagonal Segre type or equivalently to the existence of two non-null, mutually orthogonal eigenvectors of (and therefore eigenvectors of ), say and that may be chosen unit timelike and unit spacelike respectively, which are tangent to at every point and such that, in the basis of the tangent space to formed by and , the Jordan form of the matrix is

(39) |

In the adapted coordinate chart under consideration, these two eigenvectors are part of an adapted tetrad (i.e.: and with ), and in particular corresponds to an adapted observer.

¿From the conditions it is easy to see that a function exists such that, for the coordinate gauge introduced in (7)

(40) |

and the eigenvector equations for and readily imply that

(41) |

Further, a coordinate change in exists such that and the metric still retains its diagonal form (i.e.: it still can be written as in (7)). Such an specific coordinate gauge will be called comoving; at this point though, we shall not assume it yet.

¿From our previous remarks, it follows that the Ricci tensor and hence the Einstein tensor, are of the diagonal Segre type with a double spacelike eigenvalue degeneracy , that is:

(42) |

which amounts to saying that it takes a diagonal matrix form in the (pseudo-orthonormal) adapted tetrad . The quantities are given by

(43) |

(44) |

(45) |

where , and , . In the comoving gauge alluded to above, the coordinate components of the Einstein tensor also take a matrix diagonal form (i.e.: , etc.), and , . Einstein’s field equations imply then that the energy-momentum tensor takes that same form. The dominant energy condition is satisfied if and only if and for .

Physically, this can be interpreted by saying that there exists one (adapted) observer that moves with four-velocity such that measures a vanishing momentum flow, energy density and pressures in the direction (which we shall call radial direction/pressure), and in any other spatial direction perpendicular to (tangential directions/pressures). The use of the names ‘radial’ and ‘tangential’ is justified by thinking of the situation arising in spherically symmetric spacetimes (which are particular instances of those studied here), where the direction is perpendicular to the orbits (spheres) and can therefore be identified with the radial direction, whereas the spatial directions perpendicular to that one are necessarily tangent to the spheres, hence the name ‘tangential’.

Note that perfect fluids are included into this class and they are those
solutions satisfying . From the above expressions (44,45) it is immediate to see, on account of the functional
dependence of and , that a necessary condition for
this to happen is that . Thus we have the
result that *Perfect fluid type B warped spacetimes are
necessarily spherically, plane or hyperbolically symmetric*.

If the matter content is described by the energy momentum tensor (28) this implies, again on account of our considerations on the eigenvector/eigenvalue structure of , etc., that

(46) |

with

or, in terms of the physical quantities introduced in (28):

(47) |

These results can also be arrived at from (24) by writing

(48) |

and then demanding that is such that the term in containing the mixed terms vanishes. This is equivalent to saying that there exists a privileged observer that measures zero momentum flow. Such an observer is moving with four-velocity

(49) |

Notice that from the remarks following equation (24), it follows that is a function of the coordinates , and so is , hence which is the condition for being an adapted observer.

The quantities and in (28) and and in (42) are related through:

or equivalently

and therefore the dominant energy condition reads in these variables (recall we are assuming that (47) holds):

(50) |

(51) |

(52) |

and

(53) |

with

(54) |

Notice that the second inequality (51) above implies the first one (50), therefore only the four last inequalities need be taken into account.

#### The polynomial has only one real root.

If has just one real root, then it must be that

(55) |

where the definitions are the same as in the previous case. The Ricci (Einstein, , etc.) tensor has then a null eigenvector with corresponding eigenvalue (in the case of the Ricci tensor) , and the Jordan form of the matrix is

(56) |

The whole tensor is then of the Segre type and therefore may be written as

(57) |

where and , thus form a null tetrad, and may be chosen so that their components are functions on (i.e.: depend only on the coordinates ). The functions and are given by: