Energy conditions in gravity and BransDicke theories
Abstract
The equivalence between gravity and scalartensor theories is invoked to study the null, strong, weak and dominant energy conditions in BransDicke theory. We consider the validity of the energy conditions in BransDicke theory by invoking the energy conditions derived from a generic theory. The parameters involved are shown to be consistent with an accelerated expanding universe.
1 Introduction
Recent observations have revealed that the present state of the universe is undergoing an accelerated expansion [1]. There are, in general, a number of different approaches towards explaining this acceleration. One such approach utilizes what is known as modification of gravity which, in effect is equivalent to BransDicke (BD) type theories. The assumption of the existence of dark energy is another approach often used in this respect. In all the above theories, energy conditions impose stringent constraints whose validity should be studied in the light of their ability to explain the observational data. From a theoretical viewpoint, energy conditions in their various forms, namely strong energy condition (SEC), weak energy condition (WEC), dominant energy condition (DEC), and null energy condition (NEC) have been used in different contexts to derive general results that would hold for a variety of situations [2]. For example, the HawkingPenrose singularity theorems invoke the WEC and SEC [3], whereas the proof of the second law of black hole thermodynamics requires the NEC [4]. Another example comes from cosmology [5] where energy conditions are studied by using red shifts.
The equivalence between BD theories and gravity is a subject that has been studied by various authors, as an example see [6]. The study of energy conditions may thus benefit from such equivalence in that knowing the energy conditions in one theory would point to the energy conditions in the other. For example, in the BD theory case [7] where cannot be found easily, one can use the equivalent theory to facilitate the calculation of which, could then be used in the BD theory.
Even though the goal of this paper is to study energy conditions in modified theories of gravity and consequently in BransDicke theory, much of the techniques will be borrowed from the analysis of energy conditions in Einstein’s gravity. Therefore, we shall briefly review the derivation of energy conditions in general relativity. For a pedagogical review, see for example [3].
2 Energy conditions in general relativity
Let be a tangent vector to a congruence of time like geodesics. For a hypersurface orthogonal congruence, Raychaudhuri’s equation reads
(1) 
where and are the expansion and sheer of two nearby tangent vectors, respectively. In Einstein’s gravity, SEC and the ensuing singularities theorem follow from requiring that
(2) 
for all timelike for which . A pair of nearby timelike geodesic vectors converge and will eventually intersect. The stressenergy tensor at each point obeys the inequality for any time like vector of an observer whose worldline at has the unit tangent vector v and the local energy density appears to be . This assumption is thus equivalent to the energy density being nonnegative as measured by any observer which, of course, is physically reasonable.
3 Energy condition in gravity
In this section we use the metric formalism in gravity and derive the strong, weak and dominant energy conditions for a general form of . In doing so we will follow the formalism recently developed in [8]. We take the FreedmanRobertsonWalker (FRW) metric to study the cosmological implications of the models studied here.
The action for gravity is [9]
(3) 
where we have set . The field equations resulting from this action in the metric approach, assuming the connections are that of the LeviCivita, are given by
(4) 
where represents the energymomentum tensor of ordinary matter considered as perfect fluid given by
(5) 
and is the stress energy tensor of the gravitational fluid
(6) 
where a prime represents differentiation with respect to . The field equation (4) now reads
(7) 
Contracting the above equation we obtain
(8) 
Now, let us briefly review the energy conditions in gravity. We begin by defining an effective stressenergy tensor using equation (7) as follows
(9) 
with
(10) 
Therefore, we can write in the terms of an effective stressenergy tensor and its trace, that is
(11) 
Using the spatially flat FRW metric as
(12) 
the effective energy density and pressure are given by
(13) 
and
(14) 
where and is the Hubble parameter. Now, using these equations, we can write the NEC and SEC, given by and respectively as
(15) 
and
(16) 
To compare our results here with that of general relativity for a given , we use the FRW metric which, for WEC () leads to
(17) 
For DEC (), we find
(18) 
4 Energy conditions in BransDicke theory
Let us now investigate a nonminimally coupled self interacting scalartensor field theory such as the BransDicke (BD) theory and find the various energy conditions for this type of modified gravity. In the context of BD theory [10] with a self interacting potential and a matter field, the action is given by
(19) 
where is the usual BD parameter and we have chosen units such that . The gravitational field equations can be derived from action (18) by varying the action with respect to the metric
(20) 
where is the stressenergy tensor of the normal matter as expressed in equation (5). Variation of action (19) with respect to gives
(21) 
where the expression , using (12) is given by
(22) 
Using the equation of motion, we can write where . The stressenergy tensor and its trace for the BD theory may now be calculated with the result
(23) 
and
(24) 
Comparison of these equations with equation (2) leads to a similar equation for the SEC,
(25) 
We can now write the relations for NEC and SEC analogously as and respectively [3] so that we may first derive and for the spatially flat FRW metric as follows
(26) 
and
(27) 
Thus, the NEC and SEC for the BD theory are given by
(28) 
and
(29) 
Now, following and expanding on the GR approach to include WEC and DEC, as has been employed in gravity theories, we may obtain similar equations in the BD theory. Therefore, the WEC and DEC in the BD theory are respectively given by
(30) 
and
(31) 
5 Equivalence of the energy conditions in gravity and BransDicke theory
Considering action (3) within the context of the metric formulation of gravity, one can introduce a new field and write a dynamically equivalent action [6]
(32) 
Variation with respect to leads to equation provided , which reproduces action (3). Redefining the field by and setting
(33) 
the action takes the form
(34) 
Comparison with action (19) reveals that this is the action of a BD theory with . Therefore, metric theories, as has been observed long ago, are fully equivalent to a class of BD theories with a vanishing kinetic term [6]. Now, taking and we have
(35) 
Substituting these relations into equations for NEC, SEC, WEC and DEC in BD theory, namely equations (28), (29), (30) and (31) with , one can easily derive respectively NEC, SEC, WEC and DEC for the modification of gravity, that is equations (15), (16), (17) and (18).
5.1 Examples
To see how equation (17) can be used to put constraints on a given and equivalently on the BD potential, let us examine two examples. First, we consider as having a general powerlaw form, given by
(36) 
Let us now concentrate on the vacuum sector i.e. . Substituting in equation (17) we have the following condition for WEC
(37) 
where , jerk and snap
parameters for the presentday values are defined in [8]. I
what follows, we examine two values of the exponent , namely
and
which satisfy the inequality (37).
Case I:
Takeing and given in
[8], so that . Equation (37) for
reduces to and the
deceleration
This relation is satisfied for which in turn satisfies equation (37) with . The potential in BD theory with for , corresponding to can thus be obtained simply as
where
, so that for the corresponding BD potential
is .
Case II:
As a second case we consider , so that ,
leading to a WEC given by which requires
. The corresponding BD potential is
with being negative.
One can therefore come to the conclusion that must also
have a negative value.
6 Redshift and energy conditions
Let us now take the potential in BD theory with and use the following powerlow ansätze
(38) 
Inserting these relations into equation (21) in the vacuum sector, one then finds that
(39) 
Substituting the spatially flat FRW metric (12) in the field equations (4) we get
(40) 
and
(41) 
Now, from equations (38) and (40) we can write
(42) 
where we have used the relations , and where is the redshift of a luminous source [11]. From this equation one may conclude that in the very early universe would have been small and therefore large. Now, the WEC is given by
(43) 
This equation for the effective energy density clearly satisfies the WEC. If we follow the same method as in section 3 or in [8] and substitute the total energy density and pressure by and , we will find the same relation between the distance modulus and redshift parameter which is studied in [11] where the energy conditions have been used. Now let us write the NEC, DEC and SEC with respect to the redshift parameter as follows
(44) 
(45) 
(46) 
respectively. From equation (44) one can see that it is not possible to extract more information from NEC. As far as the DEC and SEC are concerned however, they require and respectively. The first is in agreement with the observation that the universe is undergoing an accelerated expansion phase, . A glance at the last equation reveals that it is completely in contradiction with an accelerated expanding universe, but we know that the SEC ensures gravity to be always attractive. Violation, as discussed in [12], allows for the late time accelerated cosmic expansion as suggested by the combination of recent astronomical observations.
7 Energy condition in the Einstein frame
What we have done so far in the pervious sections has been in the socalled Jordan frame. However, it would also be instructive to study these relations in the Einstein frame. As is well known, the usual procedure, going from one frame to the other, is to use a conformal transformation. A problem then arises in that whether the tensor representing the physical metric structure of spacetime is the one belonging to the Jordan frame or to the Einstein frame. However, what we are concerned with in this work is the relation between the energy conditions in these frames and will not deal with the question posed above. Under the conformal transformation
(47) 
and taking
(48) 
the action (3) is rewritten as
(49) 
Here
(50) 
As a result, one finds the field equations for the metric in the form
(51) 
Also, the equation of motion for becomes
(52) 
For the FRW metric we find
(53) 
In order to solve these equations for the case we use the following powerlow ansätze
(54) 
where and are arbitrary constants. The Hubble parameter, , then reads
(55) 
Now, using equations (48) and (50) we have
(56) 
Matching the exponents of in equation (51) we arrive at the following expression for
(57) 
If we define as
(58) 
then we obtain the following relations for and
(59) 
Now let us write the WEC, SEC, DEC, and NEC
(60) 
(61) 
(62) 
(63) 
Using our ansätze (54), we find the following expression for the WEC
(64) 
As we can see in equation (64), the WEC always holds. However, in the case that either or , we have a constraint on , namely that must be even. For the other energy conditions at late times, , we have
(65) 
(66) 
Equation (65) tells us that there is no guarantee that it remains positive. As can be seen, when the SEC and NEC hold, the DEC does not. For these equations we also have a constraint on , that is, when either or , then must be even. In the case of another constraint appears which is important for both equations. There are ranges of for which either SEC or NEC do not hold. Figure 1 shows this situation. Also, for the case , we see that the DEC does not hold at all, regardless of the sign of .
8 Conclusions
In this work we have studied the energy conditions in the BD theory and compared the results with that of gravity, benefiting from the ease with which the parameters of interest can be derived in the latter and subsequently used in the former. This would help us to check the validity of the energy conditions in BD theory by invoking the energy conditions derived from a generic theory. The parameters involved were shown to be consistent and compatible with an accelerated expanding universe.
Acknowledgment
One of us, YT, is grateful to S. Jalalzadeh for valuable discussions.
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