1 Introduction↩︎

The observational data for the accelerating expansion of the universe has suggested a positive vacuum expectation value of the cosmological constant [1]. In the standard Big Bang model, it is believed that, after the Big Bang explosion, the radiation-dominated era occurred, followed by the matter-dominated one, even though there was a hot thermalization period of radiation and matter immediately after the Big Bang. In the standard cosmology, the equation of state (EoS) of the fluid of massive point particles is different from that of the fluid of massless point particles.

The standard Big Bang cosmology is based on the Hawking–Penrose singularity theorem, which relies on the strong energy condition (SEC) [2]. On the other hand, in the four-dimensional point particle cosmology [3]–[7], a perfect fluid is introduced to describe a continuous distribution of matter with energy-momentum tensor \(T_{ab}\) \((a, b =0,1,2,3)\) in terms of the mass-energy density and the pressure. This fluid is called perfect because of the absence of heat conduction terms and stress terms corresponding to viscosity [4]. The \(D\) dimensional perfect fluid property has been applied to the higher dimensional cosmology (HDC) without cosmological constant [6], [7].

In this paper, we call the fluid in \(D=4+p\) \((D\geq 5)\) dimensional manifold consisting of extended objects of the \(p\)-branes, the fluid of stringy extended objects, for simplicity. To be more specific, in \(D=5\) dimensions, the stringy extended objects denote the \(p=1\) stringy objects (or stringy particles) having one-dimensional fiber [6], [7]. Note that while the theoretical bounds on the EoS parameter \(w_{\rm total}\) for the fluid of radiation (fluid of massless stringy extended objects) and fluid of matter (fluid of massive stringy extended extended objects) are continuous, the physical dominance of species still evolves. Note also that in the point particle cosmology, there are two EoSs for the fluids massless and massive particle eras, respectively. In the HDC, as the early universe evolves with the expansion rate, this rate increases and the twist of the fluid of stringy extended objects decreases exponentially. The effect of the shear has been shown to be negligible and thus the universe is considered isotropic and homogeneous [6], [7].

Since Hubble discovered the expansion of our universe, the Big Bang cosmology has been developed into a precision science by cosmological observations including supernova data [1] and measurements of cosmic microwave background (CMB) radiation [8]. These observations triggered an explosion of recent interest in the origin of dark energy [9].

The local measurements of Hubble constant \(H_{0}\) are in tension with early universe measurements such as CMB and baryon acoustic oscillation (BAO), within the standard cosmological constant cold dark matter (\(\Lambda\)CDM) model. Recently, the holographic Friedman–Lemaître–Robertson–Walker (FLRW) universe and the ensuing observational constraints were investigated on the four-dimensional membrane embedded in the \(D=4+1\) dimensional bulk spacetime [10]–[12]. In the HDC without background of cosmological constant, the singularities in geodesic surface congruence for the time-like and null stringy extended particles were investigated to yield the Raychaudhuri-type equations possessing correction terms related with the characteristics owing to the stringy extended objects [6], [7]. Assuming the stringy SEC in the HDC without the cosmological constant, the Hawking–Penrose-type EoS inequality equations were obtained. To be specific, the stringy SECs of both the fluids of massive and massless stringy extended objects produce the same EoS inequality in the HDC without the cosmological constant. This EoS inequality has been shown to be not equivalent to the Hawking–Penrose EoS inequality equations in the four-dimensional FLRW cosmology [6], [7]. Recently, the null energy condition [13] (here, the weak energy condition (WEC) for the massless point particles) has been applied to the models of dynamical black holes [14], [15].

In this paper, we investigate the SECs, WECs and the ensuing EoS in the HDC in the background of cosmological constant by extending the results of the studies [6], [7]. To this end, we study singularities associated with the Raychaudhuri-type equations and construct the SECs and the ensuing EoSs in geodesic surface congruence in the HDC. We then discuss the cosmological constant in the HDC. Moreover, we elucidate the origin of the relations between the Hawking–Penrose-type EoS inequality equations in the four-dimensional limits in the HDC, which are missing in Refs. [6], [7], and the Hawking–Penrose EoS inequality in the four dimensions in the FLRW cosmology.

Next, we explicitly construct the SECs for the fluid of stringy extended objects in the HDC having the cosmological constant contributions, to show that the EoS inequalities for the fluid of massive stringy extended objects are the same as those for the massless ones, even though their corresponding formulas are different to each other. Moreover, in our study, we evaluate the energy conditions for the fluid of point particles possessing the mass-energy density \(\rho_{0}\) and the pressure \(P_{0}\) described in the four-dimensional cosmology.

Note that in this paper, we introduce a fibration, \(\pi:M\rightarrow N\) over a base manifold \(N\) (spanned in total manifold \(M\)) associated with a fiber space \(F\). Here, the base manifold \(N\) is fixed to reside on the four-dimensional spacetime and the stringy effects are considered in the total \(D\) (\(D\geq 5\)) dimensional manifold of the fiber bundle. To be more specific, the HDC is defined in the \(D\) (\(D\geq 5\)) dimensional fiber bundle manifold. By definition, the HDC is thus different from the four-dimensional cosmology which does not possess \(F\). Moreover, in the HDC, the radiation in radiation-dominated era is described in terms of the fluid of massless stringy extended objects, for instance [6], [7]. Similarly, in the HDC, the matter in a matter-dominated era is delineated in terms of the fluid of massive stringy extended objects. The fluid of massive or massless point particles is then defined in the four-dimensional spacetime which does not possess \(F\) [6], [7], [16]. Next, the cosmological constant \(\Lambda\) is introduced in the background.

In Section 2, we construct the geometry and Raychaudhuri-type equations in the HDC. We sketch the HDC with cosmological constant. In Section 3, we formulate the SECs, WECs and EoS parameters for the fluid of massive stringy extended objects and discuss the cosmological constant in the HDC. In Section 4, we formulate the SECs, WECs and EoS parameters for the fluid of massless stringy extended objects and study the cosmological constant in the HDC. Section 5 includes conclusions. In Appendix 6, we pedagogically study four-dimensional point particle cosmology. In Appendix 7, we will study the details of the Raychaudhuri-type equation in the HDC. In particular, we list the definitions of the variables employed in Section 2.

2 Geometry and Raychaudhuri-Type Equations in HDC↩︎

In this Section, we investigate the geometry of the HDC and the ensuing Raychaudhuri-type equations. To this end, we first introduce a fibration, \(\pi:M\rightarrow N\) over a base manifold \(N\) associated with \(F\) [7], [16], [17]. In this paper, the base manifold \(N\) is fixed to reside on the four-dimensional spacetime. In analogy of the relativistic action of point particles in \(N\), the action for stringy extended object is proportional to the area of the surface spanned in total manifold \(M\) by the evolution along time direction of the stringy extended object in \(F\). Note that in the HDC defined in the \(D=4+p\) \((p\geq 1)\) manifold, the extended manifold is associated with the extended \(p\)-brane. The \((p=1)\)-string is the stringy extended object which resides on the five-dimensional manifold possessing the one-dimensional \(F\). Note also that the point particle lives on the four-dimensional manifold without \(F\). In order to define the action on the curved manifold, let \((M, g_{ab})\) be a \(D\) dimensional manifold associated with the metric \(g_{ab}\). Given \(g_{ab}\), there can be a unique covariant derivative \(\nabla_{a}\) satisfying [4] \(\nabla_{a}g_{bc}=0\), \(\nabla_{a}\omega^{b}=\partial_{a}\omega^{b}+\Gamma^{b}_{~ac}~\omega^{c}\) (with \(\partial_{a} \equiv \partial/\partial{x_a}\) and \(\Gamma^{b}_{~ac}\) being the Christoffel symbol) and \((\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})\omega_{c}=R_{abc}^{~~~d}~\omega_{d}\) \((a=0,1,...,D-1)\), with the curvature \(R_{abc}^{~~~d}\). Next we investigate the Raychaudhuri-type equations which are studied in terms of the SECs for the fluid of massive and massless stringy extended objects and the point particles with cosmological constant. Moreover the Raychaudhuri-type equations will be discussed to clarify the attractive gravitional force.

We investigate explicitly the action and the ensuing geometry in the HDC. To this end, Appendix 6  first studies the total action, which consists of the relativistic action for the point particle and actions of gravity and perfect fluid, for the four-dimensional point particle cosmology having the cosmological constant \(\Lambda\) in the curved manifold. Keeping the total action for the point particles in mind, we investigate the stringy extended objects in the curved manifold. To this end, one needs to introduce the gravity action in addition to the extended \((p=1)\)-brane action (or Nambu–Goto action) [18], [19]. In this paper, we investigate the HDC possessing the cosmological constant \(\Lambda\) and the ensuing EoS. The case without cosmological constant has been analyzed in Refs. [6], [7], [20]. We start with \(D\) dimensional gravity related with total action of the form \[\begin{align} S&=&S_{p=1}+S_{\rm gr}+S_{\rm pf},\nonumber\\ S_{p=1}&=&-\kappa\int_{\tau_{1}}^{\tau_{2}}d\tau \int_{0}^{\pi}d\sigma f(\tau,\sigma),\nonumber\\ S_{\rm gr}&=&\frac{1}{16\pi}\int d^{D}x\sqrt{-g}(R-2\Lambda), \label{sss2} \end{align}\tag{1}\] where \(S_{p=1}\), \(S_{\rm gr}\) and \(S_{\rm pf}\) are the extended \((p=1)\)-brane action, gravity and perfect fluid actions, respectively, \(\kappa=\frac{1}{2\pi\alpha^{\prime}}\) with \(\alpha^{\prime}\) being the universal slope of Regge trajectory and \(\Lambda\) is a cosmological constant, \(\tau\) and \(\sigma\) are the world sheet coordinates, \(R\) is the Ricci scalar, and \(g\) is the determinant of the metric tensor. Note that \(D\) dimensional cosmological constant is given by \(\Lambda=[(D-1)(D-2)/6]\Lambda_{0}\) where \(\Lambda_{0}\) is the (3 + 1)-dimensional cosmological constant associated with the point particle [21]. In the fiber bundle formalism [6], [7], [16], the extra dimension is defined to stand for the dimension of the fiber space where the (massive or massless) stringy extended objects having size reside. Here, the fiber space describes the geometry of a single stringy extended particle. Moreover, \(\Lambda_{0}\) which is related to the observed vacuum energy is given by \(\Lambda_{0}\simeq 1.1\times 10^{-52}\) m\(^{-2}\).

Next we investigate the Nambu–Goto action \(S_{p=1}\) in Equation (1 ) associated with \(f(\tau,\sigma)\) [20], [22]. To this end, we digress to note that the point particles have coordinates in the four-dimensional base manifold \(N\), which is described in the action \(S_{p=0}\) in Equation (63 ) having \(f(\tau)=\left(-g_{ab}\frac{\partial x^{a}}{\partial\tau} \frac{\partial x^{b}}{\partial\tau}\right)^{1/2}=(-g_{ab}\xi^{a}\xi^{b})^{1/2}\). Here, we have \(\xi^{a}=(\partial/\partial\tau)^{a}\) \((a=0,1,2,3)\) with metric \(g_{ab}\) associated with \((-,+,+,+)\) [20], [22], [23]. Moreover, the variation of \(S_{\rm gr}+S_{\rm pf}\) under the metric change \(\delta g^{ab}\) \((a,b=0,1,2,3)\) affects the Einstein field equation (73 ).

In contrast, the stringy extended objects have the coordinates in the \(D\) \((D\geq 5)\) dimensional total manifold \(M\), which is delineated in the action \(S_{p=1}\) in Equation (1 ) possessing \(f(\tau,\sigma)\) defined as \[f(\tau,\sigma)=(-\det h_{MN})^{1/2}=[(\xi\cdot\zeta)^{2}-(\xi\cdot\xi)(\zeta\cdot\zeta)]^{1/2}, \label{ftausigma}\tag{2}\] where the relation \(h_{MN}=g_{ab}\frac{\partial x^{a}}{\partial\sigma^{M}}\frac{\partial x^{b}}{\partial\sigma^{N}}\), with \(\sigma^{M}=(\tau,\sigma)\) \((M=0,1)\), and the notations \(\xi^{a}=(\partial/\partial\tau)^{a}\) and \(\zeta^{a}=(\partial/\partial\sigma)^{a}\) \((a=0,1,...,D-1)\) are used. Here, the metric \(g_{ab}\) \((a=0,1,...,D-1)\) is related with \((-,+,\cdots,+)\). Note that in this paper, for completeness, in addition to \(S_{p=1}\) we included the actions \(S_{\rm gr}\) and \(S_{\rm pf}\), which were not treated in the Refs. [6], [7], [20]. To be more specific, the variation of \(S_{\rm gr}+S_{\rm pf}\) under \(\delta g^{ab}\) \((a,b=0,1,...,D-1)\) affects the Einstein field equation of the form: \(R_{ab}-\frac{1}{2}g_{ab}R+g_{ab}\Lambda=8\pi T_{ab}\) \((a,b=0,1,...,D-1)\) for a perfect fluid. Here, \(R_{ab}\) and \(R\) are the Ricci curvature tensor and the Ricci scalar curvature, respectively.

We consider a smooth congruence of time-like geodesic surfaces in \(M\). We parameterize the surface generated by the evolution of a time-like string by  \(\tau\) and \(\sigma\), and then there are the corresponding vector fields \(\xi^{a}=(\partial/\partial\tau)^{a}\) and \(\zeta^{a}=(\partial/\partial\sigma)^{a}\) as shown in Equation (2 ) [7], [20], [22]. Note that \(\xi^{a}\) is a time-like vector field and \(\zeta^{a}\) is a space-like one. Since we have gauge degrees of freedom (DOF), we can choose the orthonormal gauge [7], [20], [22] \[\xi\cdot\zeta=0,~~~\xi\cdot\xi+\zeta\cdot\zeta=0. \label{orthogauge}\tag{3}\] where the plus sign in the second equation is due to the feature that \(\xi\cdot\xi\) is timelike and \(\zeta\cdot\zeta\) is spacelike. Note that the gauge fixing (3 ) for the world sheet coordinates signifies that the tangent vectors are orthonormal everywhere up to a local scale factor [20], [22].

We introduce the deviation vector field \(\eta^{a}=(\partial/\partial \alpha)^{a}\) which represents the displacement to an infinitesimally nearby world sheet, and let \(\Sigma\) denote the three-dimensional submanifold spanned by the world sheets \(\gamma_{\alpha}(\tau,\sigma)\). One then may choose \(\tau\), \(\sigma\) and \(\alpha\) as coordinates of \(\Sigma\) to yield the commutator relations [7], [20]\[\pounds_{\xi}\eta^{a}=\xi^{b}\nabla_{b}\eta^{a}-\eta^{b}\nabla_{b}\xi^{a}=0\;\; \text{and}\;\; \pounds_{\zeta}\eta^{a}=\zeta^{b}\nabla_{b}\eta^{a}-\eta^{b}\nabla_{b}\zeta^{a}=0. \label{poundxizeta}\tag{4}\]

Exploiting the commutators (4 ), one arrives at \[\frac{dS_{p=1}}{d\alpha}=-\int_{\tau_{1}}^{\tau_{2}} d\tau \int_{0}^{\pi}d\sigma (P_{\tau}^{b}\xi^{a}\nabla_{a}\eta_{b} +P_{\sigma}^{b}\zeta^{a}\nabla_{a} \eta_{b}) =\int_{\tau_{1}}^{\tau_{2}}d\tau \int_{0}^{\pi}d\sigma\eta_{b}(\xi^{a}\nabla_{a}P_{\tau}^{b}+\zeta^{a}\nabla_{a}P_{\sigma}^{b})+{\rm boundary~terms},\label{dsng0}\tag{5}\] where we introduced the energy-momentum currents \[P_{\tau}^{a}=\frac{\kappa}{f}[(\xi\cdot\zeta)\zeta^{a}-(\zeta\cdot\zeta)\xi^{a}]\;\;\text{and}\;\;P_{\sigma}^{a}=\frac{\kappa}{f}[(\xi\cdot\zeta)\xi^{a}-(\xi\cdot\xi)\zeta^{a}]. \label{currentss}\tag{6}\] Here, the boundary terms vanish if one exploits the boundary conditions: \(\eta^{a}(\tau=\tau_{1}; \sigma)=\eta^{a}(\tau=\tau_{2}; \sigma)=0\) and \(P_{\sigma}^{a}(\tau; \sigma=0)=P_{\sigma}^{a}(\tau; \sigma=\pi)=0\) [22], [23] to produce a stringy geodesic surface equation: \(\xi^{a}\nabla_{a}P_{\tau}^{b}+\zeta^{a}\nabla_{a} P_{\sigma}^{b}=0\). Applying the orthonormal gauge conditions to the above stringy geodesic surface equation, one obtains the stringy geodesic surface equation of the form \[-\xi^{a}\nabla_{a}\xi^{b}+\zeta^{a}\nabla_{a}\zeta^{b}=0. \label{gEOSe}\tag{7}\]

Exploiting the first equation in Equation (5 ), one is left with the second derivative of \(S_{p=1}\) to yield a stringy geodesic deviation equation [20] (for more algebraic details associated with outline derivations of \({dS_{p=1}}/{d\alpha}\) and \({d^{2}S_{p=1}}/{d\alpha^{2}}\), see Appendix 6, where these two quantities are pedagogically explained in the point particle case)\[\frac{d^{2}S_{p=1}}{d\alpha^{2}}=\int_{\tau_{1}}^{\tau_{2}} d\tau\int_{0}^{\pi} d\sigma\eta_{a}(\chi\eta)^{a}, \label{gdev1}\tag{8}\] where, naively speaking, the second derivative \({d^{2}S_{p=1}}/{d\alpha^{2}}\) is related with the curvature \(R_{bcd}^{~~~a}\) in the \(D\) \((D\geq 5)\) dimensional HDC: \[(\chi\eta)^{a}=\xi^{b}\nabla_{b}(\eta^{c}\nabla_{c}P_{\tau}^{a})+\zeta^{b}\nabla_{b}(\eta^{c}\nabla_{c}P_{\sigma}^{a})+R_{bcd}^{~~~a} (\xi^{b}P_{\tau}^{d}+\zeta^{b}P_{\sigma}^{d})\eta^{c}, \label{aleph2}\tag{9}\] which is then applied to the \(D\) dimensional Einstein field equation . On the other hand, the variation in \(S_{\rm gr}+S_{\rm pf}\) under the change \(\delta g^{ab}\) (\(a,b=0,1,...,D-1\)) yields the Einstein field equation \[R_{ab}-\frac{1}{2}g_{ab}R+g_{ab}\Lambda=8\pi T_{ab}, \label{eineq}\tag{10}\] where the definition \(R_{ab}=R_{acb}^{~~~c}\) is exploited with \(R_{bcd}^{~~~a}\) \((a=0,1,...,D-1)\) being defined in Equation (9 ). Here, we exploited the relation \[\frac{\delta S_{\rm pf}}{\delta g^{ab}}=-\frac{1}{2}\sqrt{-g}T_{ab}. \label{spfdel}\tag{11}\] Note that the stringy effect in \(S_{p=1}\) in Equation (1 ) affects the \(D\) (\(D\geq 5\)) dimensional Einstein field equation (10 ). Note also that \(R_{ab}\) in Equation (10 ) is affected by the cosmological constant \(\Lambda\). In this study, the cosmological constant \(\Lambda\) is introduced to investigate the EoS associated with the non-zero and positive \(\Lambda\) in Section 3.

We briefly investigate the HDC associated with the Raychaudhuri-type equation for the fluid of massive stringy extended objects [6], [7]: \[\frac{d\theta}{d\tau}-\frac{d\bar{\theta}}{d\sigma}=-\frac{1}{D-1}(\theta^{2}-\bar{\theta}^{2})-\sigma_{ab}\sigma^{ab}+\bar{\sigma}_{ab}\bar{\sigma}^{ab} +\omega_{ab}\omega^{ab}-\bar{\omega}_{ab}\bar{\omega}^{ab}-R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b}). \label{rayeqn}\tag{12}\] Here, \((\theta,\sigma_{ab},\omega_{ab})\) are the expansion, shear and twist of the universe and \((\bar{\theta},\bar{\sigma}_{ab},\bar{\omega}_{ab})\) are those of the string, respectively. The more details of these variables are defined in Appendix 7. To be specific, in the point particle case, there are only DOF of \((\theta,\sigma_{ab},\omega_{ab})\) and \(\xi^{a}\). In contrast, in the HDC we possess new DOF related with the physical variables \((\bar{\theta},\bar{\sigma}_{ab},\bar{\omega}_{ab})\) and \(\zeta^{a}\). In particular, in the HDC, there are the DOF of the twist (or rotation) of the string associated with \(\bar{\omega}_{ab}\). This feature implies that the string performs the dynamic rotational motion with respect to the four-dimensional base manifold \(N\). For more derivation details of the Raychaudhuri-type equation (12 ), see Appendix 7. We assume that \(\sigma_{ab}=\bar{\sigma}_{ab}\) and \(\omega_{ab}=\bar{\omega}_{ab}\). Taking an ansatz that the expansion \(\bar{\theta}\) is constant along the \(\sigma\) direction, one ends up with \[\frac{d\theta}{d\tau}=-\frac{1}{D-1}(\theta^{2}-\bar{\theta}^{2})-R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b}). \label{raych}\tag{13}\]

We assume an SEC for the fluid of massive stringy extended particles: \[R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})\ge 0, \label{sec}\tag{14}\] then the Raychaudhuri-type equation in (13 ) has a solution of the form \[\frac{1}{\theta(\tau)}\geq \frac{1}{\theta(0)}+\frac{1}{D-1}\left(\tau-\int_{0}^{\tau}d\tau\left(\frac{\bar{\theta}}{\theta}\right)^{2}\right). \label{soln1}\tag{15}\]

We assume that \(\theta(0)\) is negative so that the congruence is initially converging as in the point particle case [2]. The EoS inequality (15 ) then implies that \(\theta(\tau)\) should pass through the singularity within a proper time \[\tau\leq\frac{D-1}{|\theta(0)|}+\int_{0}^{\tau}d\tau\left(\frac{\bar{\theta}}{\theta}\right)^{2}.\]

Similar to the massive stringy extended particle case in Equation (13 ), we consider the Raychaudhuri-type equation associated with expansion of the fluid of massless stringy extended objects related with the corresponding null vector \(k^{a}\) [6], [7]: \[\frac{d\theta}{d\lambda}=-\frac{1}{D-2}\theta^{2}+\frac{1}{D-1}\bar{\theta}^{2}-R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b}), \label{raych2}\tag{16}\] where \(\lambda\) is an affine parameter. In this paper, we consider an ansatz that the photon corresponding to the stringy extended particle associated with the null vector \(k^{a}\) and the affine parameter \(\lambda\) is approximately massless, for simplicity [6], [7]. (In the conventional string theory, there are massless photon in the open string sector and massless graviton in the closed string one [22], [23].) Note that there is the upper limit of exceptionally small experimental value for the photon mass \(M(\text{photon})=1.00 \times 10^{-27}\) GeV [24]. For more details on \(k^{a}\) and \(\lambda\), see Section 4. Note also that, differently from the factor \(D-1\) in Equation (13 ), there is the factor \(D-2\) in the first term in Equation (16 ) which originates from the property that the massless particle has only two transverse DOF without a longitudinal DOF. We assume an SEC for the fluid of massless stringy extended particles: \[R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})\ge 0, \label{sec2}\tag{17}\] then the Raychaudhuri-type equation (16 ) has a solution \[\frac{1}{\theta(\lambda)}\geq \frac{1}{\theta(0)}+\frac{1}{D-2}\left(\lambda-\frac{D-2}{D-1} \int_{0}^{\lambda}d\lambda\left(\frac{\bar{\theta}}{\theta}\right)^{2}\right). \label{soln2}\tag{18}\]

Note that the SECs (14 ) and (17 ) were analyzed in the HDC without the cosmological constant in Refs. [6], [7]. In Section 3 just below, these SECs are also exploited in the HDC possessing the cosmological constant. As soon as \(\theta(0)\) is assumed to be negative, the congruence can be initially converging. The EoS inequality (18 ) then implies that \(\theta(\tau)\) should pass through the singularity within an affine length [6], [7] \[\lambda\leq\frac{D-2}{|\theta(0)|}+\frac{D-2}{D-1}\int_{0}^{\lambda}d\lambda\left(\frac{\bar{\theta}}{\theta}\right)^{2}. \label{lambda2}\tag{19}\] Note that the SECs (14 ) and (17 ) are derived by using the Raychaudhuri-type equations (13 ) and (16 ) for the fluids of massive and massless stringy extended objects, respectively. For more details of the SECs and the Raychaudhuri-type equations associated with Equations (14 ) and (17 ), see Refs. [6], [7]. Note also that the SECs (14 ) and (17 ) and the Raychaudhuri-type equations (13 ) and (16 ) associated with the attractive gravitational forces are discussed in Sections 3 and 4 below.

3 Energy Conditions and EoS Parameters for Fluid of Massive Stringy Extended Objects in Background with Cosmological Constant↩︎

In this Section, we investigate the SECs and the ensuing EoS parameters in the HDC with the cosmological constant. In the results of Refs. [6], [7], we found the SECs for the fluids of both massive and massless stringy extended objects. Note that these SECs produced the same EoS inequalities in the HDC without the background of cosmological constant. In this Section, we extend these SECs to the case possessing the cosmological constant. To this end, we first study the stringy SECs (14 ) and (17 ) by considering the energy-momentum tensor \(T_{ab}\) for a perfect fluid in the HDC possessing the cosmological constant: \[T_{ab}=\rho u_{a}u_{b}+P(g_{ab}+u_{a}u_{b}), \label{perfectfluid}\tag{20}\] where \(u^{a}=(-1, 0,...,0)\) is a time-like \(D\) velocity in the rest frame. Here \(\rho\) and \(P\) are the density and pressure of the fluid of stringy extended objects respectively. Combining Equations (10 ) and (20 ), together with \(\rho\) and \(P\), one finds \[R_{ab}-\frac{1}{2}g_{ab}R=8\pi~{\rm diag}(\rho+\rho_{\Lambda}, P+P_{\Lambda}, P+P_{\Lambda}, P+P_{\Lambda}, \cdots),\] where \(\rho_{\Lambda}\) and \(P_{\Lambda}\) are the contributions from the cosmological constant, defined as \[\rho_{\Lambda}\equiv \frac{\Lambda}{8\pi}\;\;\text{and}\;\;P_{\Lambda}\equiv -\frac{\Lambda}{8\pi}, \label{rholam1}\tag{21}\] respectively, and the ellipsis denotes the higher extended \(p\)-brane \((p\geq 1)\) contributions.

On the other hand, taking the trace of \(T_{ab}\) (20 ), one finds\[T\equiv g^{ab}T_{ab}=-\rho+(D-1)P \label{tracet}\tag{22}\] which, together with Equation (10 ), yields \[R\equiv g^{ab}R_{ab}=-\frac{2}{D-2}(8\pi T-D\Lambda). \label{tracer}\tag{23}\] Inserting \(T\) (22 ) and \(R\) (23 ) into Equation (10 ), one obtains \[R_{ab}=8\pi \left(T_{ab}-\frac{1}{D-2}g_{ab}T\right)+\frac{2}{D-2}g_{ab}\Lambda. \label{rabapp}\tag{24}\] For the massive stringy extended particle, one finds \(T_{ab}\xi^{a}\xi^{b}=\rho\) and \(T_{ab}\zeta^{a}\zeta^{b}=P\) to yield \[T_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})=\rho-P. \label{tabxizetaapp}\tag{25}\]

Exploiting the SEC (14 ) for the fluid of massive stringy extended particles and the Einstein field equation (10 ) having the cosmological constant one finds that the SEC produces \[R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})=8\pi\left(\frac{D-4}{D-2}\rho+\frac{D}{D-2}P-\frac{1}{2\pi(D-2)}\Lambda\right)\geq 0, \label{rab1}\tag{26}\] where Equations (22 ), (24 ) and (25 ) are used. Note that in Equation (26 ), the cosmological constant \(\Lambda\) is incorporated into the results of Refs. [6], [7]. Using \(\rho_{\Lambda}\) and \(P_{\Lambda}\) from Equation (21 ), we rewrite the EoS inequality (26 ) for the massive stringy extended particle with cosmological constant as \[R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})=8\pi\left(\frac{D-4}{D-2}(\rho+\rho_{\Lambda})+\frac{D}{D-2}(P+P_{\Lambda})\right)\geq 0, \label{rab1lamb}\tag{27}\] which is listed in Table 1. Note that in Table 1, we list the EoS inequalities for the massive and massless objects in the \(D\) dimensions while in contrast, in Table 2, we list the EoS inequalities for the massive and massless point particles in the four dimensions. Note also that the massive and massless objects are studied in this Section and in Section 4, respectively. Note that the EoS inequality in Equation (27 ) yields the corresponding EoS parameter \[w=\frac{P+P_{\Lambda}}{\rho+\rho_{\Lambda}}\geq -\frac{D-4}{D}. \label{wEOSmassived}\tag{28}\]

Note that one cannot have the condition \(w=0\) given in the four dimensions, since the EoS parameter (28 ) has been evaluated in \(D\) \((D\geq 5)\) dimensions in the HDC. Note also that the EoS parameter (28 ) is not the same as that for the massive stringy extended particle without the cosmological constant [6], [7], since Equation (28 ) has the cosmological constant contribution.

We investigate the relations between the EoS parameter for the fluid of stringy extended objects in the HDC defined in \(D\) \((D\geq 5)\) dimensions, and the Hawking–Penrose limit in the massive point particle in the four dimensions. These relations were not elucidated even in the case without the cosmological constant analyzed in Refs. [6], [7]. First, in the HDC defined in \(D\) \((D\geq 5)\) dimensions, we split the SEC contributions related with \(R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})\) (27 ) into three parts: (i) the contribution without \(R_{ab}\zeta^{a}\zeta^{b}\), (ii) contribution from cosmological constant \(\Lambda\), and (iii) contribution associated with \(R_{ab}\zeta^{a}\zeta^{b}\) only. For the first contribution (that without \(R_{ab}\zeta^{a}\zeta^{b}\)) which is related with \(R_{ab}\xi^{a}\xi^{b}\) only in the background with cosmological constant, we construct \[R_{ab}\xi^{a}\xi^{b}=8\pi\left(\frac{D-3}{D-2}(\rho+\rho_{\Lambda})+\frac{D-1}{D-2}(P+P_{\Lambda})\right), \label{rabxixi}\tag{29}\] where \(\rho\) and \(P\) are the density and pressure defined in the \(D\) \((D\geq 5)\) dimensions. Here, \(\rho_{\Lambda}\) and \(P_{\Lambda}\) are given in Equation (21 ).

Second, we obtain the contribution related with \(R_{ab}\zeta^{a}\zeta^{b}\) only \[R_{ab}\zeta^{a}\zeta^{b}=8\pi\left(\frac{1}{D-2}(\rho+\rho_{\Lambda})-\frac{1}{D-2}(P+P_{\Lambda})\right). \label{rabzetazeta}\tag{30}\] which, together with \(R_{ab}\xi^{a}\xi^{b}\) (29 ), reproduces Equation (27 ). Next, we define \(R_{ab}\zeta^{a}\zeta^{b}\) (30 ) as \[R_{ab}\zeta^{a}\zeta^{b}\equiv -8\pi\left(\frac{D-3}{D-2}\rho_{\rm ext}+\frac{D-1}{D-2}P_{\rm ext}\right), \label{rabzetazeta2}\tag{31}\] in terms of \(\rho_{\rm ext}\) and \(P_{\rm ext}\): \[\rho_{\rm ext}\equiv -\frac{1}{D-3}(\rho+\rho_{\Lambda})\;\text{and}\;P_{\rm ext}\equiv \frac{1}{D-1}(P+P_{\Lambda}). \label{rholam12}\tag{32}\] Combining Equations (29 ) and (31 ), one obtains \[R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})=8\pi\left(\frac{D-3}{D-2}(\rho+\rho_{\Lambda}+\rho_{\rm ext}) +\frac{D-1}{D-2}(P+P_{\Lambda}+P_{\rm ext})\right)\geq 0, \label{rabfinal}\tag{33}\] which yields the effective EoS parameter \[w_{0}^{\Lambda,ext}=\frac{P+P_{\Lambda}+P_{\rm ext}}{\rho+\rho_{\Lambda}+\rho_{\rm ext}}\geq -\frac{D-3}{D-1}. \label{w1}\tag{34}\] Here, the subscript \(0\) denotes \(R_{ab}\zeta^{a}\zeta^{b}=0\) associated with \(\rho\) and \(P\). The EoS parameter (34 ) is the most general form for the fluid of massive stringy extended objects in the background with dark energy.

Third, we study the \(\rho_{\rm ext}=P_{\rm ext}=0\) case. In this case, exploiting Equation (31 ) one has \(R_{ab}\zeta^{a}\zeta^{b}=0\) to yield \(\zeta^{a}=0\). We thus find the four-dimensional manifold without \(F\). Note that \(F\) is associated with the tangent vector field \(\zeta^{a}\). Since the fluid of point particles defined in four dimensions has no extended object contribution, we have \(\rho_{\rm ext}=P_{\rm ext}=0\) in Equation (34 ) to yield \[w_{0}^{\Lambda_{0}}=\frac{P_{0}+P_{\Lambda_{0}}}{\rho_{0}+\rho_{\Lambda_{0}}}\geq -\frac{1}{3}, \label{wlam}\tag{35}\] for the fluid of massive point particle in the background with cosmological constant. Here, \(\rho_{\Lambda_{0}}\) and \(P_{\Lambda_{0}}\) are given in terms of the four-dimensional cosmological constant \(\Lambda_{0}\) as follows: \[\rho_{\Lambda_{0}}\equiv \frac{\Lambda_{0}}{8\pi}\;\;\text{and}\;\;P_{\Lambda_{0}}\equiv -\frac{\Lambda_{0}}{8\pi}. \label{rholam0}\tag{36}\] Note that, in the four-dimensional cosmology, Equation (35 ) produces the EoS inequality for the SEC for the fluid of massive point particles: \[\rho_{0}+\rho_{\Lambda_{0}}+3(P_{0}+P_{\Lambda_{0}})\geq 0, \label{rabxixi22}\tag{37}\] which is listed in Table 2. In the point particle limit with \(\Lambda_{0}=0\), the SEC \(R_{ab}\xi^{a}\xi^{b}\geq 0\) for the fluid of massive point particles produces the equivalent Hawking–Penrose EoS inequality defined in the four dimensions [2]–[7], [13]: \[\rho_{0}+3P_{\Lambda_{0}}\geq 0. \label{waldmassivein}\tag{38}\]

One can apply the fiber bundle formalism [6], [7], [16] to a WEC for the fluid of massive stringy extended objects in the HDC defined in \(D\) (\(D\geq 5\)) dimensional manifold. To this end, let us assume a WEC for the fluid of massive stringy extended objects: \[T_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})\geq 0. \label{weakcarrdef}\tag{39}\] Note that in Equation (39 ), we replace \(R_{ab}\) for the SEC with \(T_{ab}\) for the WEC. Similar to the SEC case which is obtained from the the Raychaudhuri-type equation (13 ) [6], [7], to construct the WEC we include the factor \(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b}\) where \(\xi^{a}\) is a time-like vector field and \(\zeta^{a}\) is a space-like one, respectively. Exploiting \(T_{ab}\xi^{a}\xi^{b}=\rho\) and \(T_{ab}\zeta^{a}\zeta^{b}=P\) in Equation (39 ), one arrives at the WEC for the massive stringy extended particle in the HDC: \[\rho-P\geq 0, \label{weakcarr1}\tag{40}\] which is listed in Table 1. Note that Equation (40 ) yields the corresponding EoS parameter for the massive stringy extended particle in the HDC: \[w\leq 1. \label{weceos}\tag{41}\] Note also that the WEC (41 ) contains the extended object contribution associated with \(\zeta^{a}\).

Keeping \(T_{ab}\zeta^{a}\zeta^{b}=P\), one obtains the WEC for the massive point particle in the four dimensions: \[T_{ab}\xi^{a}\xi^{b}\geq 0, \label{weakcarr2}\tag{42}\] to yield the EoS inequality for the WEC for the massive point particle [2]–[5], [13] \[\rho_{0}\geq 0, \label{weakcarr3}\tag{43}\] which is listed in Table 2 and is consistent with the earlier findings [2]–[5], [13]. This result suggests that the assumption (39 ) is physically well defined. Note that the EoS inequality (43 ) for the massive point particle is different from the EoS inequality (40 ) for the massive stringy extended particle.

Next, in the four-dimensional cosmology without the cosmological constant, we assume that \(\theta=\bar{\theta}=\zeta^{a}=0\) in Equation (13 ) to produce \[\frac{d\theta}{d\tau}=-R_{ab}\xi^{a}\xi^{b}=-4\pi(\rho_{0}+3P_{0})\] which, using the SEC (38 ), becomes negative. The SEC for the massive point particle thus suggests that gravitation is attractive [13]. So far, we investigated the massive stringy extended objects and point particles. In the following Section 4, we study the massless stringy extended and point particles.

4 Energy Conditions and EoS Parameters for Fluid of Massless Stringy Extended Objects in Background with Cosmological Constant↩︎

In this Section, we investigate the energy conditions for the massless stringy extended objects and point particles, as well as the corresponding EoS parameters in the background of cosmological constant in the HDC. To this end, we first define the null vector \(k^{a}\) and its auxiliary null vector \(l^{a}\) as \(k^{a}\equiv \frac{1}{\sqrt{2}}(u^{a}+v^{a})\) and \(l^{a}\equiv \frac{1}{\sqrt{2}}(u^{a}-v^{a})\). Here, \(v^{a}\) is a vector residing on the subspace \(N_{\perp}^{D-2}=\{v^{a}|v^{a}k_{a}=0,~v^{a}l_{a}=0\}\) which is a \((D-2)\) dimensional manifold perpendicular to the light-cone originated from \(k^{a}\) and \(l^{a}\). Note that the null vector field \(k^{a}=(\partial/\partial\lambda)^{a}\) is given in terms of the affine parameter \(\lambda\) and \(l^{a}\) points in the opposite spatial direction to \(k^{a}\). To be more specific, \(k^{a}\) and \(l^{a}\) are normalized as follows [7], [13]: \[k^{a}k_{b}=l^{a}l_{a}=0 \;\;{\rm and}\;\;k^{a}l_{a}=-1. \label{kkllka}\tag{44}\]

For the massless stringy extended particle, exploiting \(T_{ab}\) (20 ), we find \(T_{ab}k^{a}k^{b}=\frac{1}{2}(\rho+P)\) and \(T_{ab}\zeta^{a}\zeta^{b}=P\) to yield the EoS inequality for the WEC \[T_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})=\frac{1}{2}(\rho-P), \label{tabkkzetaapp}\tag{45}\] which is listed in Table 1. Note that the WEC (45 ) is for the first time defined for the massless stringy extended particle in this paper. Making use of the SEC (17 ) for the fluid of massless stringy extended particles and the Einstein field equation (10 ) possessing the cosmological constant, one finds that the SEC associated with the corresponding null vector \(k^{a}\) yields \[R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})=4\pi\left(\frac{D-4}{D-2}\rho+\frac{D}{D-2}P-\frac{1}{2\pi(D-2)}\Lambda\right)\geq 0. \label{rab2}\tag{46}\] where (22 ), (24 ) and (45 ) are exploited. Note that even though we include the cosmological constant \(\Lambda\) in both the massive and massless stringy extended particle cases, the EoS inequality condition associated with the corresponding SECs (46 ) is the same as the SECs (26 ). Exploiting \(\rho_{\Lambda}\) and \(P_{\Lambda}\) from Equation (21 ), we rewrite the EoS inequality (46 ) for the massless stringy extended particle in the cosmological constant background as \[R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})=4\pi\left(\frac{D-4}{D-2}(\rho+\rho_{\Lambda})+\frac{D}{D-2}(P+P_{\Lambda})\right)\geq 0, \label{rab2lamb}\tag{47}\] which is listed in Table 1. Note that the EoS inequality (47 ) for the massless stringy extended particle is equivalent to the inequality (27 ) for the massive stringy extended particle. Next, the EoS inequality (47 ) produces the EoS parameter \(w\) for the massless stringy extended  particle in the background with cosmological constant: \[w=\frac{P+P_{\Lambda}}{\rho+\rho_{\Lambda}}\geq -\frac{D-4}{D}. \label{wEOSmasslessed}\tag{48}\] Note that the EoS parameter \(w\) (48 ) is the same as \(w\) (28 ) for the massive stringy extended particle. Note also that \(w\) (48 ) is equivalent to \(w\) of the case for the massive stringy extended particle without the cosmological constant [6], [7].

Next, in Section 3 above and in the current Section we treat the fluids of massless and massive stringy extended objects separately. However, in the real universe, these components coexist. Even if the individual EoS inequalities (bounds) are the same (\(w\geq -(D-4)/D\)), the effective equation of state of the universe (

\(w_{\rm total}\)) evolves as the ratio \(\rho_{\rm rad}/\rho_{\rm matt}\) of radiation density to matter density changes. Now, we study the coexistence of the fluid of radiation (or fluid of massless stringy extended objects) and the fluid of matter (or fluid of massive stringy extended objects), and the corresponding total EoS parameter \(w_{\rm total}\). To this end, we define \(w_{\rm total}\) in terms of the ratio of radiation density to matter density \[w_{\rm total}\equiv \frac{P_{\rm total}}{\rho_{\rm total}}=\frac{P_{\rm rad}+P_{\rm matt}}{\rho_{\rm rad}+\rho_{\rm matt}}. \label{coexistence}\tag{49}\] Note that our result (49 ) defines the allowed theoretical region for each component in higher dimensions. Note also that the dominance of species shifts (which is the standard definition of a phase transition in cosmology) even if the fundamental bounds derived from stringy geometry remain constant.

It is appropriate to comment on the physical implications of the derived EoS inequalities related with the SECs in the HDC. First, as summarized in Table 1, the SECs for the fluid of stringy extended objects are defined in \(D\) (\(D\geq 5\)) dimensional manifold. In other words, the EoS parameter for the fluid of massive stringy extended objects is obtainable from \(R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})\) (27 ). Similarly, one can obtain the EoS parameter for the fluid of massless stringy extended objects from \(R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})\) (47 ). Note that the expressions \(R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})\) and \(R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})\) possess the non-vanishing tangent vector field \(\zeta^{a}\) which resides on the fiber space for the stringy extended object having finite size.

Second, in \(D=5\) we find the EoS parameter \(w\geq -1/5\), which becomes negative allowing for negative pressure. In higher dimensions, this feature of the negative pressure means for exotic matter, the aspect of which is similar to the standard point particle limit related with the region \(-1/3\leq w<0\). In general, for the fluid of stringy extended objects defined in \(D\) (\(D\geq 5\)) dimensions, the EoS parameter bound becomes negative, allowing for negative pressure. Note that, as summarized in Table 2, the SECs for the fluid of point particles are defined in the four-dimensional manifold without \(F\) associated with \(\zeta^{a}\) field.

We study the relations between the EoS parameter \(w\) (48 ) and the Hawking–Penrose limit in the fluid of massless point particles in the four dimensions

. Similar to the fluid of massive stringy extended object case discussed in Section 3, we split the SEC contributions associated with \(R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})\) (47 ) into three pieces: (i) contribution without \(R_{ab}\zeta^{a}\zeta^{b}\), (ii) contribution from cosmological constant \(\Lambda\), and (iii) contribution related with \(R_{ab}\zeta^{a}\zeta^{b}\) only. First, similar to Equation (29 ), for the first contribution (that without \(R_{ab}\zeta^{a}\zeta^{b}\)) which is associated with \(R_{ab}k^{a}k^{b}\) only in the background with cosmological constant, one obtains\[R_{ab}k^{a}k^{b}=4\pi\left(\rho+\rho_{\Lambda}+P+P_{\Lambda}\right), \label{rabkk}\tag{50}\] where \(\rho\) and \(P\) are the density and pressure defined in the \(D\) \((D\geq 5)\) dimensions. Here, \(\rho_{\Lambda}\) and \(P_{\Lambda}\) are defined in Equation (21 ).

Second, we find the contribution of the fluid of massless objects associated with the vector field \(\zeta^{a}\) as follows \[R_{ab}\zeta^{a}\zeta^{b}=8\pi\left(\frac{1}{D-2}(\rho+\rho_{\Lambda})-\frac{1}{D-2}(P+P_{\Lambda})\right). \label{rabzetazeta22}\tag{51}\] Note that, together with \(R_{ab}k^{a}k^{b}\) (50 ), Equation (51 ) reproduces Equation (47 ).

Third, we define \(R_{ab}\zeta^{a}\zeta^{b}\) in Equation (51 ) as \[R_{ab}\zeta^{a}\zeta^{b}\equiv -4\pi\left(\rho_{\rm ext}+P_{\rm ext}\right), \label{rabzetazeta3}\tag{52}\] in terms of \(\rho_{\rm ext}\) and \(P_{\rm ext}\): \[\rho_{\rm ext}\equiv -\frac{2}{D-2}(\rho+\rho_{\Lambda})\;\;{\rm and} \;\;P_{\rm ext}\equiv \frac{2}{D-2}(P+P_{\Lambda}). \label{rholam122}\tag{53}\] Note that \(\rho_{\rm ext}\) and \(P_{\rm ext}\) in Equation (53 ) for the fluid of massless stringy extended objects are different from those for the fluid of massive ones defined in Equation (32 ).

Combining (50 ) and (52 ), one finds \[R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})=4\pi\left(\rho+\rho_{\Lambda}+\rho_{\rm ext}+P+P_{\Lambda}+P_{\rm ext}\right)\geq 0, \label{rabfinal2}\tag{54}\] which yields the effective EoS parameter \[w_{0}^{\Lambda, ext}=\frac{P+P_{\Lambda}+P_{\rm ext}}{\rho+\rho_{\Lambda}+\rho_{\rm ext}}\geq -1. \label{w2}\tag{55}\] Here, the subscript \(0\) denotes \(R_{ab}\zeta^{a}\zeta^{b}=0\) associated with \(\rho\) and \(P\). The EoS parameter (55 ) is the most general form for the fluid of massless stringy extended objects in the background with dark energy.

We investigate the fluid of four-dimensional point particle limit of the fluid of massless stringy extended objects. Since the fluid of point particles have no extended object contribution, we have \(\rho_{\rm ext}=P_{\rm ext}=0\) in Equation (55 ) to yield \[w_{0}^{\Lambda_{0}}=\frac{P_{0}+P_{\Lambda_{0}}}{\rho_{0}+\rho_{\Lambda_{0}}}\geq -1, \label{wlambda22}\tag{56}\] for the fluid of massless point particles in the background with cosmological constant. Here, \(\rho_{\Lambda_{0}}\) and \(P_{\Lambda_{0}}\) are given in Equation (36 ). Note that Equation (56 ) yields the EoS inequality for the SEC for the fluid of massless point particles in the background with cosmological constant [2]–[5], [13]: \[\rho_{0}+\rho_{\Lambda_{0}}+P_{0}+P_{\Lambda_{0}}\geq 0, \label{wlambda222}\tag{57}\] which is listed in Table 2. Moreover, in the point particle limit with \(\Lambda_{0}=0\), the SEC \(R_{ab}k^{a}k^{b}\geq 0\) for the fluid of massless point particles yields the EoS inequality defined in the four-dimensional cosmology [2]–[5], [13]: \[\rho_{0}+P_{\Lambda_{0}}\geq 0. \label{waldmasslessin}\tag{58}\]

One can apply the fiber bundle formalism [6], [7], [16] to a WEC for the fluid of massless stringy extended objects in the HDC defined in \(D\) (\(D\geq 5\)) dimensional manifold. To this end, we assume a WEC for the fluid of massive stringy extended objects: \[T_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})\geq 0. \label{weakcarrdef2}\tag{59}\] Note that in Equation (59 ) \(R_{ab}\) for the SEC is replaced with \(T_{ab}\) for the WEC. Similar to the SEC case which is obtained from the the Raychaudhuri-type equation (16 ) [6], [7], to construct the WEC we included the factor \(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b}\) where \(\xi^{a}\) is a time-like vector field and \(\zeta^{a}\) is a space-like one, respectively. Using \(T_{ab}k^{a}k^{b}=\frac{1}{2}(\rho+P)\) and \(T_{ab}\zeta^{a}\zeta^{b}=P\) in Equation (59 ), one ends up with the WEC for the fluid of massless stringy extended objects in the HDC: \[\rho-P\geq 0, \label{weakcarr12}\tag{60}\] which is listed in Table 1.

Next we define the WEC for the fluid of massless point particles (or null energy condition) in the four-dimensional cosmology: \[T_{ab}k^{a}k^{b}\geq 0. \label{weakcarr22}\tag{61}\] The WEC (61 ) produces [2]–[5], [13] \[\rho_{0}+P_{0}\geq 0, \label{weakcarr32}\tag{62}\] which is listed in Table 2. Note that, in the HDC, the WEC (60 ) for the fluid of massless stringy extended objects is the same as the WEC (40 ) for the massive stringy extended objects. Note that, in the WECs, there is no effect of the background with cosmological constant since the definitions in Equations (39 ) and (59 ) do not possess the corresponding cosmological constant terms. Similar to Equation (43 ) for the WEC inequality for the fluid of massive point particles, in the four-dimensional cosmology, we find the WEC inequality (62 ) for the fluid of massless point particles which are consistent with the results obtained in Refs. [2]–[5], [13]. This finding also suggests that the assumption made in Equation (59 ) is physically well defined. Note in Equations (28 ) and (48 ) that, in the radiation-dominated and matter-dominated eras in the background with cosmological constant, one finds the same EoS parameter \(w\) in the SEC in the HDC. The stringy SECs thus impose a universal constraint on the EoS parameter \(w\) that remains valid across both radiation- and matter-dominated eras. This feature is the same as the case without the cosmological constant [6], [7].

One of the simplest models to explain the dark energy is the \(\Lambda\)CDM model, where the dark energy is considered static and positive. This constant accelerates the expansion of the universe. To be more specific, the cosmological constant \(\Lambda\) in Equation (1 ) is non-zero and positive. This feature implies that the expansion of the universe is accelerating, which is consistent with the corresponding discovery in 1998 [1]. Note that the dark energy does not need to be constant. For instance, the quintessence model describes dynamical dark energy [25]–[27] having time dependent scalar field. Moreover, in the \(f(R)\) theories of the gravity, the integrand in the action \(S_{\rm gr}\) in Equation (1 ) is replaced as follows: \(\sqrt{-g}(R-2\Lambda)\rightarrow \sqrt{-g}f(R)\) with \(f(R)\) being a function of \(R\). We study explicitly the case that the cosmological constant is static in the HDC. In this case the energy density is given as \(\rho_{\Lambda}=\frac{\Lambda}{8\pi}\), for both the fluid of massive and massless stringy extended  objects as shown in Equation (21 ). In order to explain the accelerating expansion of the universe, one needs to find an energy having negative pressure \(P_{\Lambda}=-\frac{\Lambda}{8\pi}\) with the positive cosmological constant \(\Lambda\). This signifies that the accelerating universe in the HDC possesses a positive cosmological constant.

Finally, in the four-dimensional cosmology without the background with cosmological constant, we assume that \(\theta=\bar{\theta}=\zeta^{a}=0\) in (16 ) to produce \[\frac{d\theta}{d\lambda}=-R_{ab}k^{a}k^{b}=-4\pi(\rho_{0}+P_{0})\] which, using the SEC (62 ), becomes negative. As a result, the SEC for the fluid of massless point particles implies that gravitation is attractive [13]. Note that exploiting \(\frac{d\theta}{d\tau}=-R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b})\leq 0\) and \(\frac{d\theta}{d\lambda}=-R_{ab}(k^{a}k^{b}-\zeta^{a}\zeta^{b})\leq 0\) in the HDC having \(\theta=\bar{\theta}=0\) and \(\zeta^{a}\neq 0\), the SECs for the fluids of massive and massless stringy extended objects suggest that gravitation is attractive.

5 Conclusions↩︎

In summary, we have investigated the energy conditions in the HDC having a cosmological constant \(\Lambda\). As shown in Table 1, the EoS inequalities for the SECs and WECs for the fluid of massive stringy extended objects are equivalent to those for fluid of massless stringy extended objects in the HDC. In contrast, in the four-dimensional cosmology considered in Section 3, we evaluated the EoS inequalities for the SECs and WECs for fluids of massive and massless point particles as listed in Table 2. Next, we studied the EoS parameter \(w\) in the HDC with the cosmological constant. To be specific, we constructed \(w\geq -(D-4)/D\) for both the fluids of massive and massless stringy extended objects in the HDC in the background with cosmological constant. We thus concluded that the stringy SECs impose a universal constraint on \(w\) that remains valid across both radiation- and matter-dominated eras. Moreover, we split the EoS parameter contributions for the stringy extended objects in the background with cosmological constant as \(w_{0}^{\Lambda, ext}=(P+P_{\Lambda}+P_{\rm ext})/ (\rho+\rho_{\Lambda}+\rho_{\rm ext})\) where \((\rho, P)\), \((\rho_{\Lambda}, P_{\Lambda})\) and \((\rho_{\rm ext}, P_{\rm ext})\) are contribution from the contribution without \(R_{ab}\zeta^{a}\zeta^{b}\), contribution from cosmological constant \(\Lambda\), and contribution with \(R_{ab}\zeta^{a}\zeta^{b}\) only, respectively. Here, \(\zeta^{a}\) is the tangent vector field associated with the fiber space in the HDC. Since the point particles have no extended object contribution, then \(\rho_{\rm ext}=P_{\rm \rm ext}=0\). In the point particle limit with \(\Lambda=0\), the SEC produces the EoS parameter related with the Hawking–Penrose-type EoS inequality. Moreover, the effect of the background with cosmological constant was shown to not exist in the WECs. This is one of the main points of this paper. It is of interest to investigate the \(f(R)\) gravity theory having the Gauss–Bonnet term \(R_{GB}^{2}=R^{2}-4R_{ab}R^{ab}+R_{abcd}R^{abcd}\) [28], [29] in the HDC. The next intiguing topic is the FLRW model in the HDC in which we investigate the higher dimensional metric \(g_{ab}\) \((a,b=0,1,2,3,4,...)\) of the form \(g_{ab}={\rm diag}(-1,a^{2},a^{2},a^{2},b^{2},\cdots)\), where now \(a\) is a scale factor defined in the four-dimensional base manifold and \(b\) is a scale factor residing on the compact extended \((p=1)\)-brane, and the ellipsis stands for the higher extended \(p\)-brane \((p\geq 2)\) contributions, respectively. On the other hand, it would be of interest to investigate the wormhole geometry [5], [30], [31] and the corresponding energy conditions and EoS parameters in the HDC. The next topic to address would be to study the black hole and the corresponding energy conditions [14], [32], [33] and EoS parameters in the HDC.

6 Four-Dimensional Point Particle Cosmology with Cosmological Constant↩︎

In this Appendix, we study an action for a four-dimensional gravity with the fluid of point particles which are applied to the HDC in Section 2. The action for the four-dimensional gravity with point particles in the background with cosmological constant is described as \[\begin{align} S&=&S_{p=0}+S_{\rm gr}+S_{\rm pf},\nonumber\\ S_{p=0}&=&-m\int_{\tau_{1}}^{\tau_{2}}d\tau f(\tau),\nonumber\\ S_{\rm gr}&=&\frac{1}{16\pi}\int d^{3}x\sqrt{-g}(R-2\Lambda), \label{sss} \end{align}\tag{63}\] where \(S_{p=0}\), \(S_{\rm gr}\) and \(S_{\rm pf}\) are the relativistic action for the fluid of point particle and actions of gravity and perfect fluid associated with the energy-momentum tensor of the form \(T_{ab}\) in (20 ). Here, \(m\) is a test point particle mass and \(f(\tau)\) is defined as [22] \(f(\tau)=\left(-g_{ab}\frac{\partial x^{a}}{\partial\tau}\frac{\partial x^{b}}{\partial\tau}\right)^{1/2}=(-g_{ab}\xi^{a}\xi^{b})^{1/2}\), where the notation \(\xi^{a}=(\partial/\partial\tau)^{a}\) (\(a=0,1,2,3\)) with metric \((-,+,+,+)\) is used, and \(\Lambda\) is the cosmological constant. In this paper, we restrict ourselves to the case that the variation of \(f(\tau)\) under \(\delta g^{ab}\) is neglected, for simplicity. This restriction is also applied to the stringy extended objects in Section 2. In order to define the action on the curved manifold, let \((M, g_{ab})\) be a four-dimensional manifold associated with the metric \(g_{ab}\). Given \(g_{ab}\), there is a unique covariant derivative \(\nabla_{a}\) satisfying [4] \(\nabla_{a}g_{bc}=0\), \(\nabla_{a}\omega^{b}=\partial_{a}\omega^{b}+\Gamma^{b}_{~ac}~\omega^{c}\) and \((\nabla_{a}\nabla_{b}-\nabla_{b}\nabla_{a})\omega_{c}=R_{abc}^{~~~d}~\omega_{d}\).

We introduce the deviation vector field \(\eta^{a}=(\partial/\partial \alpha)^{a}\) which represents the displacement to an infinitesimally nearby world line, and let \(\Sigma\) denote the two-dimensional submanifold spanned by the world lines \(\gamma_{\alpha}(\tau)\) [4]. One then may choose \(\tau\) and \(\alpha\) as coordinates of \(\Sigma\) to yield the commutator relation, \[\pounds_{\xi}\eta^{a}=\xi^{b}\nabla_{b}\eta^{a}-\eta^{b}\nabla_{b}\xi^{a}=0. \label{commpoint}\tag{64}\]

We construct the first derivative of \(S_{p}\) with respect to \(\alpha\) to produce \[\frac{dS_{p=0}}{d\alpha}=-m\int_{\tau_{1}}^{\tau_{2}}d\tau \eta^{a}\nabla_{a}f=-m\int_{\tau_{1}}^{\tau_{2}}d\tau \eta^{a}\frac{1}{f}(-\xi^{b}\nabla_{a}\xi_{b}) =-\int_{\tau_{1}}^{\tau_{2}}d\tau P_{\tau}^{b}\eta^{a}\nabla_{a}\xi_{b}, \label{eom1}\tag{65}\] where we used the energy-momentum current \(P_{\tau}^{a}=-\frac{m}{f}\xi^{a}\). Exploiting the commutator (64 ), one arrives at \[\frac{dS_{p=0}}{d\alpha}=-\int_{\tau_{1}}^{\tau_{2}}d\tau P_{\tau}^{b}\xi^{a}\nabla_{a}\eta_{b} =\int_{\tau_{1}}^{\tau_{2}}d\tau \eta_{b}\xi^{a}\nabla_{a}P_{\tau}^{b}-P_{\tau}^{a}\eta_{a}|_{\tau=\tau_{1}}^{\tau=\tau_{2}},\label{eom10}\tag{66}\] and the boundary term vanishes if the end point condition \[\eta^{a}(\tau=\tau_{1})=\eta^{a}(\tau=\tau_{2})=0, \label{endpoint}\tag{67}\] is used to yield a geodesic equation\[\xi^{a}\nabla_{a}P_{\tau}^{b}=0. \label{geo1a}\tag{68}\] Applying the time-like condition \(\xi\cdot\xi=-1\) to Equation (68 ), one obtains the geodesic equation of the form \[\xi^{a}\nabla_{a}\xi^{b}=0.\] Using the first equation in Equation (66 ) one finds the second derivative of \(S_{p}\) as follows: \[\frac{d^{2}S_{p=0}}{d\alpha^{2}}=-\int_{\tau_{1}}^{\tau_{2}} d\tau\eta^{c}\nabla_{c}(P_{\tau}^{b}\xi^{a}\nabla_{a}\eta_{b}) =-\int_{\tau_{1}}^{\tau_{2}} d\tau\eta^{c}[(\nabla_{c}P_{\tau}^{b})\xi^{a}\nabla_{a}\eta_{b}+P_{\tau}^{b}\{ (\nabla_{c}\xi^{a})\nabla_{a}\eta_{b} +\xi^{a}\nabla_{c}\nabla_{a}\eta_{b}\}]. \label{gdev10}\tag{69}\] Making use of the relation \(\nabla_{c}\nabla_{a}\eta_{b}=\nabla_{a}\nabla_{c}\eta_{b}+R_{cab}^{~~~d}\eta_{d}\) and the commutator (64 ), one finds\[\begin{align} \frac{d^{2}S_{p=0}}{d\alpha^{2}}&=&-\int_{\tau_{1}}^{\tau_{2}} d\tau [(\eta^{c}\nabla_{c}P_{\tau}^{b})\xi^{a}\nabla_{a}\eta_{b} +P_{\tau}^{b}\{ (\xi^{c}\nabla_{c}\eta^{a})\nabla_{a}\eta_{b}+\xi^{a}\eta^{c}\nabla_{a}\nabla_{c}\eta_{b} +R_{cab}^{~~~d}\eta^{c}\eta_{d}\xi^{a} \}]\nonumber\\ &=&-\int_{\tau_{1}}^{\tau_{2}} d\tau [(\eta^{c}\nabla_{c}P_{\tau}^{b})\xi^{a}\nabla_{a}\eta_{b} +P_{\tau}^{b}\{ (\xi^{c}\nabla_{c}\eta^{a})\nabla_{a}\eta_{b}+\xi^{c}\eta^{a}\nabla_{c}\nabla_{a}\eta_{b}\} -R_{acb}^{~~~d}\eta^{c}\eta_{d}\xi^{a}P_{\tau}^{b} ]\nonumber\\ &=&-\int_{\tau_{1}}^{\tau_{2}} d\tau [(\eta^{c}\nabla_{c}P_{\tau}^{b})\xi^{a}\nabla_{a}\eta_{b} +P_{\tau}^{b}(\xi^{c}\nabla_{c})(\eta^{a}\nabla_{a}\eta_{b})-R_{bcd}^{~~~a}\eta^{c}\eta_{a}\xi^{b}P_{\tau}^{d}]\nonumber\\ &=&-\int_{\tau_{1}}^{\tau_{2}} d\tau [\xi^{a}\nabla_{a}(\eta_{b}\eta^{c}\nabla_{c}P_{\tau}^{b})-\eta_{b}\xi^{a}\nabla_{a}(\eta^{c}\nabla_{c}P_{\tau}^{b}) +\xi^{c}\nabla_{c}(P_{\tau}^{b}\eta^{a}\nabla_{a}\eta_{b}) -(\eta^{a}\nabla_{a}\eta_{b})\xi^{c}\nabla_{c}P_{\tau}^{b}-R_{bcd}^{~~~a}\eta^{c}\eta_{a}\xi^{b}P_{\tau}^{d}]\nonumber\\ &=&-\eta_{b}\eta^{c}\nabla_{c}P_{\tau}^{b}|_{\tau=\tau_{1}}^{\tau=\tau_{2}}-P_{\tau}^{b}\eta^{a}\nabla_{a}\eta_{b}|_{\tau=\tau_{1}}^{\tau=\tau_{2}} +\int_{\tau_{1}}^{\tau_{2}} d\tau [\eta_{b}\xi^{a}\nabla_{a}(\eta^{c}\nabla_{c}P_{\tau}^{b})+R_{bcd}^{~~~a}\eta^{c}\eta_{a}\xi^{b}P_{\tau}^{d}]. \label{gdev102} \end{align}\tag{70}\] Inserting the end point condition (67 ) and the geodesic equation (68 ) into Equation (70 ), one ends up with a geodesic deviation equation of the form \[\frac{d^{2}S_{p=0}}{d\alpha^{2}}=\int_{\tau_{1}}^{\tau_{2}} d\tau\eta_{a}(\chi\eta)^{a}, \label{dopjaxnm}\tag{71}\] where \[(\chi\eta)^{a}=\xi^{b}\nabla_{b}(\eta^{c}\nabla_{c}P_{\tau}^{a})+R_{bcd}^{~~~a}\xi^{b}\eta^{c}P_{\tau}^{d}. \label{aleph1}\tag{72}\]

On the other hand, the variation in \(S_{\rm gr}+S_{\rm pf}\) under the change \(\delta g^{ab}\) (\(a,b=0,1,2,3\)) produces the Einstein field equation \[R_{ab}-\frac{1}{2}g_{ab}R+g_{ab}\Lambda=8\pi T_{ab}, \label{einfieldeq2}\tag{73}\] where we exploited the definition \(R_{ab}=R_{acb}^{~~~c}\) with \(R_{bcd}^{~~~a}\) being defined in (72 ). Here, we used the relation \(\delta S_{\rm pf}/\delta g^{ab}=-\frac{1}{2}\sqrt{-g}T_{ab}\).

7 Raychaudhuri-Type Equation in HDC↩︎

In this Appendix, we construct the details of the Raychaudhuri-type equation (12 ) more completely and rigorously in the HDC, since in Refs. [6], [7] we stated the sketch of the Raychaudhuri-type equation. To this end, in the orthonormal gauge, we introduce tensor fields \(B_{ab}\) and \(\bar{B}_{ab}\) defined as \[B_{ab}=\nabla_{b}\xi_{a} \;\;{\rm and} \;\;\bar{B}_{ab}=\nabla_{b}\zeta_{a}, \label{baba}\tag{74}\] which fulfill the identities \[B_{ab}\xi^{a}=0,~\bar{B}_{ab}\zeta^{a}=0 \;\;{\rm and} \;\;B_{ab}\xi^{b}-\bar{B}_{ab}\zeta^{b}=0. \label{idena}\tag{75}\] Exploiting the commutator relations (4 ), one finds \[\xi^{a}\nabla_{a}\eta^{b}-\zeta^{a}\nabla_{a}\eta^{b}=(B^{b}_{~a}-\bar{B}^{b}_{~a})\eta^{a}.\] We define the metrics \(h_{ab}\) and \(\bar{h}_{ab}\) as \[h_{ab}=g_{ab}+\xi_{a}\xi_{b}\;\text{and}\;\bar{h}_{ab}=g_{ab}-\zeta_{a}\zeta_{b}, \label{projections}\tag{76}\] to produce the identities \[h^{ab}h_{ab}=\bar{h}^{ab}\bar{h}_{ab}=D-1. \label{iddb}\tag{77}\]

Using \(h_{ab}\) and \(\bar{h}_{ab}\) defined in Equation (76 ), we decompose \(B_{ab}\) into three parts: \[B_{ab}=\frac{1}{D-1}\theta h_{ab}+\sigma_{ab}+\omega_{ab}, \label{bab}\tag{78}\] with \[\theta=B^{ab}h_{ab},~ \sigma_{ab}=B_{(ab)}-\frac{1}{D-1}\theta h_{ab},\;\text{and}\;\omega_{ab}=B_{[ab]}, \label{thetas}\tag{79}\] and \(B_{ab}\) into three pieces: \[\bar{B}_{ab}=\frac{1}{D-1}\bar{\theta}\bar{h}_{ab}+\bar{\sigma}_{ab}+\bar{\omega}_{ab}, \label{barbab}\tag{80}\] with \[\bar{\theta}=\bar{B}^{ab}\bar{h}_{ab},~ \bar{\sigma}_{ab}=\bar{B}_{(ab)}-\frac{1}{D-1}\bar{\theta}\bar{h}_{ab},\;\text{and}\;\bar{\omega}_{ab}=\bar{B}_{[ab]}. \label{barthetaab}\tag{81}\] One finds the identities \[\begin{align} \sigma_{ab}h^{ab}&=&\omega_{ab}h^{ab}=0 \;\;\;{\rm and} \;\;\sigma_{ab}\xi^{b}=\omega_{ab}\xi^{b}=\frac{1}{2}B_{ab}\xi^{b},\nonumber\\ \bar{\sigma}_{ab}\bar{h}^{ab}&=&\bar{\omega}_{ab}\bar{h}^{ab}=0 \;\;{\rm and} \;\;\bar{\sigma}_{ab}\zeta^{b}=\bar{\omega}_{ab}\zeta^{b}=\frac{1}{2}\bar{B}_{ab}\zeta^{b}. \label{idsss} \end{align}\tag{82}\] to yield \(-\sigma_{ab}\xi^{b}+\bar{\sigma}_{ab}\zeta^{b}=0\), where the stringy geodesic equation (7 ) is used.

We evaluate the following quantity \[\begin{align} \xi^{c}\nabla_{c}B_{ab}&=&\xi^{c}\nabla_{c}\nabla_{b}\xi_{a}=\xi^{c}(\nabla_{b}\nabla_{c}\xi_{a}+R_{cba}^{~~~d}\xi_{d}) =\nabla_{b}(\xi^{c}\nabla_{c}\xi_{a})-(\nabla_{b}\xi^{c})(\nabla_{c}\xi_{a})+R_{cba}^{~~~d}\xi^{c}\xi_{d}\nonumber\\ &=&\nabla_{b}(\xi^{c}\nabla_{c}\xi_{a})-B^{c}_{~b}B_{ac}+R_{cba}^{~~~d}\xi^{c}\xi_{d}. \label{quantitiesb} \end{align}\tag{83}\] Similarly one finds \[\zeta^{c}\nabla_{c}\bar{B}_{ab}=\nabla_{b}(\zeta^{c}\nabla_{c}\zeta_{a})-\bar{B}^{c}_{~b}\bar{B}_{ac}+R_{cba}^{~~~d}\zeta^{c}\zeta_{d}. \label{quantitiesb2}\tag{84}\] Combining the identities (83 ) and (84 ), one arrives at \[-\xi^{c}\nabla_{c}B_{ab}+\zeta^{c}\nabla_{c}\bar{B}_{ab}=B^{c}_{~b}B_{ac}-\bar{B}^{c}_{~b}\bar{B}_{ac}-R_{cbad}(\xi^{c}\xi^{d}-\zeta^{c}\zeta^{d}), \label{rayob}\tag{85}\] where again the stringy geodesic equation (7 ) is exploited. We calculate the trace of the above equation by multiplying both sides of Equation (85 ) by \(g^{ab}\). Firstly, we rewrite the trace of left-hand side of Equation (85 ) as \[-\xi^{c}\nabla_{c}(B_{ab}g^{ab})+\zeta^{c}\nabla_{c}(\bar{B}_{ab}g^{ab}) =-\xi^{c}\nabla_{c}(B_{ab}(h^{ab}-\xi^{a}\xi^{b}))+\zeta^{c}\nabla_{c}(\bar{B}_{ab}(\bar{h}^{ab}+\zeta^{a}\zeta^{b})) =-\xi^{c}\nabla_{c}\theta+\zeta^{c}\nabla_{c}\bar{\theta}, \label{ray1b}\tag{86}\] where Equations (75 ), (76 ), (79 ) and (81 ) are used. Secondly, we rewrite the trace of right-hand side of Equation (85 ) as \[\begin{align} &&B^{ca}B_{ac}-\bar{B}^{ca}\bar{B}_{ac}+R_{cd}(\xi^{c}\xi^{d}-\zeta^{c}\zeta^{d}) =\left(\frac{1}{D-1}\theta h^{ca}+\sigma^{ca}+\omega^{ca}\right)\left(\frac{1}{D-1}\theta h_{ac}+\sigma_{ac}+\omega_{ac}\right)\nonumber\\ &&-\left(\frac{1}{D-1}\bar{\theta}\bar{h}^{ca}+\bar{\sigma}^{ca}+\bar{\omega}^{ca}\right) \left(\frac{1}{D-1}\bar{\theta}\bar{h}_{ac}+\bar{\sigma}_{ac}+\bar{\omega}_{ac}\right) +R_{cd}(\xi^{c}\xi^{d}-\zeta^{c}\zeta^{d})\nonumber\\ &&=\frac{1}{D-1}(\theta^{2}-\bar{\theta}^{2})+\sigma_{ab}\sigma^{ab}-\bar{\sigma}_{ab}\bar{\sigma}^{ab} -\omega_{ab}\omega^{ab}+\bar{\omega}_{ab}\bar{\omega}^{ab}+R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b}), \label{ray2b} \end{align}\tag{87}\] where Equations (77 ), (78 ), (80 ) and (82 ) are exploited.

Combining the identities (86 ) and (87 ), one is left with \[-\xi^{c}\nabla_{c}\theta+\zeta^{c}\nabla_{c}\bar{\theta}=\frac{1}{D-1}(\theta^{2}-\bar{\theta}^{2})+\sigma_{ab}\sigma^{ab}-\bar{\sigma}_{ab}\bar{\sigma}^{ab} -\omega_{ab}\omega^{ab}+\bar{\omega}_{ab}\bar{\omega}^{ab}+R_{ab}(\xi^{a}\xi^{b}-\zeta^{a}\zeta^{b}). \label{rayfinalb}\tag{88}\] Exploiting \(-\xi^{c}\nabla_{c}\theta+\zeta^{c}\nabla_{c}\bar{\theta}=- {d\theta}/{d\tau} + {d\bar{\theta}}/{d\sigma}\), one finds that Equation (88 ) reproduces Equation (12 ).

References↩︎