Arrow of Time in Gravitational Collapse

Abstract

We investigate the arrow of time problem in the context of gravitational collapse of radiating stars in higher dimensions for both neutral and charged matter. The interior spacetime is described by a shear-free, isotropic spherically symmetric metric filled with a dissipative fluid. The exterior spacetime of the radiating star is taken as the higher dimensional Vaidya metric. We establish that the arrow of time, measured by the epoch function, is opposite to the thermodynamic arrow of time for all dimensions in such spacetimes. The physical consequences of our results are considered. Our results conform with previous studies on shear-free spherical collapse, which suggests avoidance of the naked singularity as the end state results in a wrong arrow of time, indicating a fundamental problem with the local application of the epoch functions to test the Weyl curvature hypothesis, which we have demonstrated in the context of shear-free, pressure-isotropic subclass of radiating spherical collapse for dimension four and beyond.

1. Introduction

This paper deals with the arrow of time problem in a shear-free radiating gravitational collapse in N dimensions. In the subsequent portions of this section we introduce the problem with proper context and motivation.

1.1. The Context: Weyl Curvature Hypothesis

The entropy of the universe should increase with time according to the second law of thermodynamics (SLT). However, the cosmic microwave background radiation (CMBR) indicates a state of thermal equilibrium during the initial phase of the universe. This suggests that the universe began evolving from a maximum state of entropy, which is in violation of the SLT. To resolve this issue Penrose [1,2] proposed the Weyl curvature hypothesis (WCH). According to which most of the entropy of the universe is carried by the free gravitational field, which is called gravitational entropy (GE) (associated with the degrees of freedom of the free gravitational field) and it is a function of the Weyl curvature tensor. Therefore, the total entropy of the universe (which is the sum of the conventional thermodynamic entropy and the gravitational entropy) was very low at the initial phase of the cosmological evolution, and with the formation of structures in the universe, the gravitational entropy increased monotonically [3], giving rise to clumping of matter and inhomogeneity (due to gravitational tidal forces). As a result, this gives rise to the arrow of time in the universe.

1.1.1. Arrow of Time Due to Gravity: Global vs. Local

In other words, such an evolution of structures in the universe produces the observed arrow of time associated with the gravitational entropy. Largely, on a cosmological scale, the thermodynamic arrow of time associated with radiation, coincides with the gravitational arrow of time. Locally GE should reproduce the Hawking–Bekenstein entropy for black holes (BH), making BH entropy a special case of GE. This makes sense because most of the entropy of the universe is BH entropy, which is area-dependent (unlike matter entropy) [2]. Similarly, one can think that most of the entropy in the universe is GE.

However, the local arrow of time can be understood if we study the process of collapse, i.e., before the formation of BH or other astrophysical objects. It is reasonable to think that during the process of stellar collapse the interior gravitational arrow of time should match the outside radiation arrow. However, Bonnor in [4,5,6] found that this is not so. The gravitational arrow of time is oriented opposite to that of the outside radiation. This is a brief explanation of the so-called “Arrow of time problem” in collapse.

1.1.2. Gravitational Epoch Functions

The arrow of time due to WCH needs some quantitative measure. These are called gravitational epoch functions. In other words, $\mathcal{P}\left(x^i\right)$ is called a gravitational epoch function of an event $x^i$ and if it is to obey the WCH, $\mathcal{P}$ should be a non-decreasing function of time, i.e., $\partial \mathcal{P}/\partial t>0$. This means that the associated arrow of time due to gravity is measured with the direction of temporal increment of the epoch function. Essentially the $\mathcal{P}$ function measures the relative strength of the Weyl curvature (due to free gravity field through the Weyl tensor $C_{abcd}$) to the curvature due to matter (via the Ricci tensor $R_{ab}$). That is, on a cosmological scale, the Weyl part takes over the Ricci part, making the epoch function increase to give rise to the arrow of time in the universe due to increasing gravitational entropy, as the Weyl curvature dominates.

Different kinds of gravitational epoch functions were proposed to measure this phenomena. One such epoch function $\mathcal{P}$ was used by Bonnor [4,5,6] in a four-dimensional dissipative collapse, defined as

$$\mathcal{P}=\mathcal{W}{\mathcal{R}}^{-1},$$

(1)

where $\mathcal{W}\equiv C_{abcd}{C}^{abcd}$ denotes the square of the Weyl curvature tensor and $\mathcal{R}\equiv R_{ab}{R}^{ab}$ is the Ricci tensor square. In a recent study [7] we have checked the validity of Bonnor’s findings, and extended this claim using the epoch function, proposed by Rudjord et al. [8,9]. In this case the gravitational epoch function ${\mathcal{P}}_1$ is given by

$${\mathcal{P}}_1=\mathcal{W}{\mathcal{K}}^{-1},$$

(2)

where $R_{abcd}$ represents the Riemann tensor, which contains both the trace part (Ricci tensor) and the traceless part (Weyl tensor) with $\mathcal{K}\equiv R_{abcd}{R}^{abcd}$ as the Kretschmann scalar.

Through the study of shear-free collapse [7], we observe that the matter curvature dominates over the free gravity curvature (unlike cosmology). This discrepancy happens, when we bring the epoch functions to a local scale, (where Ricci dominates over the Weyl sector). The expectation, that gravitational entropy drives the structure formation to give a sense of the arrow of time in the universe therefore encounters a problem here. We also know that in a conformally flat spacetime, collapse (driven by Ricci curvature due to matter) can happen with an overall sense of the arrow of time. Therefore, without the Weyl curvature also one may associate arrow of time due to gravity with the Ricci curvature (locally). On the other hand, one may separate the free gravitational field component, e.g., the Clifton–Ellis–Tavakol (CET) proposal [10] and argue that the gravitational arrow of time should always be measured purely via the Weyl curvature. Therefore, the ratio of these two curvatures is a simple yet insightful quantitative way to capture the notion of the arrow of time due to gravity. For example, in [11] we see that during collapse, the Weyl curvature increases with time, but so does the Ricci curvature. Then one may ask, with which component one should attach the notion of the local time arrow due to gravity? It is also demonstrated transparently that the epoch functions are well behaved during collapse; however, they are decreasing. Since, both these epoch functions are not useful in conformally flat spacetimes and in vacuum, they are only meaningful where both the curvatures are competing. Therefore, if the WCH is to be applied locally for the measurement of the arrow of time due to gravity (in a general setting), an appropriate measure would be the epoch function like quantity, that includes both types of curvatures, to test whether the arrow of time is truly dominated by the Weyl component, i.e., GE. Hence, according to our understanding, $\mathcal{P}$ is a suitable epoch function, purely from the physical point of view. Mathematically, we can argue that $\mathcal{R}$ already contains contributions from the Ricci scalar, i.e.,

$$\mathcal{R}=S_{ab}{S}^{ab}+{\displaystyle \frac{1}{N}}{R}^2,\mathrm{with}{S}_{ab}=R_{ab}-{\displaystyle \frac{1}{N}}R{g}_{ab},$$

(3)

where $S_{ab}$ is the traceless Ricci tensor component (contains radiation contributions along with matter), R is the Ricci scalar (pure matter) and N is the number of spatial dimensions. Therefore, $\mathcal{P}$ is insightful for the purpose of our investigation as it contains the entire matter sector. We can think of ${\mathcal{P}}_1$ as an extension of $\mathcal{P}$, to avoid divergence in empty spacetimes and also it can be employed to compute the gravitational entropy of a black hole [8] to match it with the Hawking–Bekenstein entropy [12,13,14]. Therefore, we consider both the epoch functions. In a non-vacuum N-dimensional spacetime the two epoch functions differ from each other in the following manner

$${\mathcal{P}}_1^{-1}-{\displaystyle \frac{4}{N-2}}{\mathcal{P}}^{-1}=1-{\displaystyle \frac{2}{(N-1)(N-2)}}{\displaystyle \frac{R^2}{\mathcal{W}}}.$$

(4)

This is true for any dimension $N\ge 4$. Please refer to [7,15,16,17] for a detailed discussion on these epoch functions.

The curvature scalars not only help us to estimate the gravitational arrow of time but also are related to the notion of complexity of the system. It was shown by Herrera et al. [18] that by orthogonal splitting of the Riemann tensor with respect to the fluid velocity one can obtain the trace-free electric part of the Riemann tensor $Y_{TF}$, as the complexity factor. In spherical symmetry, the electric part of the Weyl tensor is sufficient to express the Weyl tensor and can be fully described by the scalar $\mathcal{E}$. In four dimensions ($N=4$) we can measure the complexity in the following manner

$$Y_{TF}=\mathcal{E}-4\pi {\Delta }_p.$$

(5)

We know that the evolution of shear is related to the anisotropic stress ${\Delta }_p$. In our system, the pressure anisotropy is taken to be zero, i.e., ${\Delta }_p=0$. Additionally, we also know that ${\mathcal{E}}^2\sim \mathcal{W}$. Therefore, the complexity factor becomes directly proportional to the Weyl scalar and therefore proportional to GE [10]. Hence, atleast in four dimensions ($N=4$), we can state that in an isotropic (${\Delta }_p=0$) spherically symmetric self-gravitating system, the gravitational entropy is proportional to the complexity factor. This relates the complexity factor to the epoch function also. As the arrow of time and gravitational entropy are linked intimately, we can argue that there is also a deep connection between the complexity and the gravitational arrow of time. We now proceed to ask the following question.

1.1.3. Key Question: Does the Arrow of Time, as Determined by the Gravitational Epoch Function, Depend on the Number of Dimensions During Gravitational Collapse?

It is known that the curvature scalars are dimension-sensitive [19] and therefore, there is a good possibility that the behaviour of gravitational entropy will differ in higher dimensions. The number of spacetime dimensions plays a crucial role in higher dimensional general relativity, modified gravity theories, string theories and other theories of gravity. Several authors have studied the Schwarzschild, Reissner–Nordström, de Sitter/anti de Sitter and Kerr spacetimes in higher dimensions [20,21,22,23,24]. The studies show that the higher dimensional de Sitter space is a Lorentzian analogue of a N-sphere. In [25] it is shown that the charge reduces the range of stable parameters and the damping rate of stable perturbations. Also, the N-dimensional Kerr–Newman spacetime is still not well understood.

The existence of higher dimensions also influences the dynamics of compact astrophysical objects. It has been observed that the mass–radius ratio of compact objects, and also their stability depends on the number of dimensions. Several authors have found bounds on the mass–radius ratio and its dimensional dependence in general relativity [26,27,28,29].

In this study we are interested in radiating stars and several models have been found in four dimensions. Santos [30] deduced the boundary conditions at the stellar surface that must be satisfied for such a radiating system. Subsequently, Maharaj et al. extended this condition for a generalised exterior and an interior with a composite matter distribution [31,32]. This composite matter distribution can also carry charge [33] and modifies the pressure at the stellar boundary, generalising earlier works [34,35,36]. It is clear that the number of spacetime dimensions are crucial in modelling higher dimensional radiating stars.

Consequently, researchers have tried to find a higher dimensional counterpart of the Santos condition, as in [37,38,39], where an uncharged shear-free interior is matched with the conventional pure Vaidya exterior spacetime. However, we need the higher dimensional Vaidya metric as the exterior to fully understand the higher dimensional radiating collapse. This spacetime geometry is very important in the field of relativistic astrophysics, especially for studying the end state of highly dense objects, gravitational collapse and the formation of singularities [40,41,42,43,44,45,46,47,48,49,50,51,52]. Thermodynamical studies of such radiating fluids have also been studied by Debnath [53].

Researchers have also used higher dimensional modified theories of gravity for this study. For example in [54,55,56,57], using the generalised Vaidya analogue spacetime in EGB gravity, gravitational collapse have been studied. Other researchers have also investigated collapse using extensions of the generalised Vaidya spacetime [58,59,60]. The most general study, which contains all the previous results, was done by Maharaj et al. [61], where a charged composite interior is matched to the higher dimensional generalised Vaidya spacetime at the stellar surface in higher dimensions and deduced the most general matching conditions in N dimensions in general relativity. Therefore, it was found that, in a radiating collapse of a star the number of spacetime dimensions directly affect the matching conditions at the boundary surface.

In this scheme of things the investigation of the arrow of time problem in N-dimensional gravitational collapse of a radiating star becomes extremely significant. We also mention some of the important works on GE. A detailed study on GE for Lemaître–Tolman–Bondi (LTB) dust collapse, cosmic voids and other systems was conducted by Sussman and others in [62,63,64,65,66,67].

1.2. This Paper

In this study we consider the collapse of a shear-free radiating star. We extend our previous study on gravitational entropy in collapse [7] for higher dimensions. The main idea is to examine the orientation of the gravitational arrow of time in higher dimensions with respect to the well-established radiation (thermodynamic) arrow. Because the radiation emanating from the star always moves outwards, it provides us a clear sense of the thermodynamic arrow of time. The study of such a higher dimensional system is very important to check whether the gravitational arrow of time changes direction depending upon the spacetime dimensions.

This paper is organised as follows: In the following section we briefly present the model of a radiating collapsing star in N-dimensional general relativity by taking the interior spacetime as a shear-free spherically symmetric metric and the exterior is taken as the generalised Vaidya metric. Two shear-free spherically symmetric interior spacetimes are explicitly studied for the arrow of time problem. The first spacetime has time-independent lapse function and the second one has a time-dependent lapse function. For both the cases we present our conclusions in detail. We have also studied both the spacetimes in presence of charge. Finally we conclude with a discussion on various aspects of our findings.

2. Radiating Collapse in $\mathbf{N}$ Dimensions: Overview

For a collapsing radiating star we can describe the system in two parts [30,61,68,69]: the interior metric representing a collapsing sphere of fluid with radial heat flow and the exterior metric given by the Vaidya metric with outward radiation. This provides us with a sense of the thermodynamic arrow of time.

The interior of this system can be described by the Einstein field equations with a dissipative fluid

$$\begin{array}{c}\hfill R_{ab}-{\displaystyle \frac{1}{2}}{g}_{ab}R=G_{ab}={\kappa }_N{T}_{ab},\\ \hfill T_{ab}=[(\mu +p)v_a{v}_b+p{g}_{ab}+q_a{v}_b+q_b{v}_a+{\pi }_{ab}],\end{array}$$

(6)

where ${\kappa }_N$ is the coupling constant [70,71] in higher dimensions $N\ge 4$ given by the following expression (in geometric units $G=c=1$)

$${\kappa }_N={\displaystyle \frac{2(N-2){\pi }^{\frac{N-1}{2}}}{(N-3)\left(\frac{N-1}{2}-1\right)!}}.$$

(7)

This represents a fluid with energy density $\mu$, pressure p, heat flow $q^a=(0,q,0,\dots ,0)$, four-velocity $v_a$ and anisotropic stress tensor ${\pi }_{ab}$. The interior spacetime of the star is represented by a spherically symmetric metric. The general spacetime is given by

$$d{s}_{-}^2=-A{(r,t)}^{2}d{t}^2+B{(r,t)}^{2}d{r}^2+Y{(r,t)}^{2}d{\Omega }_{N-2}^2,$$

(8)

where

$$d{\Omega }_{N-2}^2=\sum _{i=1}^{N-2}\left[\prod _{j=1}^{i-1}{sin}^2\left({\theta }_j\right)\right]{\left(d{\theta }_i\right)}^2.$$

(9)

Therefore, we can take the four-velocity of the fluid in comoving coordinates as $v^a=(1/A,0,0,\dots ,0)$. Consequently, here the kinematical quantities in N dimensions become

$$\begin{array}{cc}\hfill a^a& =\left(0,{\displaystyle \frac{A^{\prime }}{A{B}^2}},0,0,\dots ,0\right),\Theta ={\displaystyle \frac{1}{A}}\left({\displaystyle \frac{\dot{B}}{B}}+(N-2){\displaystyle \frac{\dot{Y}}{Y}}\right),\hfill \\ \hfill \sigma & ={\displaystyle \frac{1}{\sqrt{N-1}A}}\left({\displaystyle \frac{\dot{Y}}{Y}}-{\displaystyle \frac{\dot{B}}{B}}\right),{\omega }_{ab}=0,\hfill \end{array}$$

(10)

where $a^a$ is the fluid N-acceleration, $\Theta$ is the expansion scalar, $\sigma$ is the shear and ${\omega }_{ab}$ is the vorticity tensor. In the above expressions, dots represent time derivatives and primes denote radial derivatives. All the kinematical quantities are related via the following relation in $1+3$ decomposition:

$${\nabla }_b{v}_a=-v_a{a}_b+{\displaystyle \frac{1}{N-1}}{h}_{ab}\Theta +{\sigma }_{ab}+{\omega }_{ab},$$

(11)

where ∇ is the covariant derivative and $h_{ab}=g_{ab}+v_a{v}_b$ is the projection tensor on the 3-space orthogonal to $v^a$. The anisotropic stress tensor is related to the difference in radial pressure $p_r$ and tangential pressure $p_t$ as

$${\pi }_{ab}={\Delta }_p({\mathcal{N}}_a{\mathcal{N}}_b-{(N-1)}^{-1}{h}_{ab}),$$

(12)

where ${\Delta }_p\equiv p_r-p_t$ measures the anisotropic pressure. Also, the isotropic pressure in (6) can be defined for N dimensions as

$$p\equiv {(N-1)}^{-1}\left(p_r+(N-2)p_t\right).$$

(13)

Here ${\mathcal{N}}_a$ is a spacelike unit vector of the form ${\mathcal{N}}_a=(0,B^{-1},0,0,\dots ,0)$. In our study we only consider isotropic pressure cases, i.e., $p_r=p_t=p$ or ${\Delta }_p=0$.

The static interior metric with $A_0,B_0$ is like a Schwarzschild interior [72] with a perfect fluid (without heat flow) which is to match the Schwarzschild exterior spacetime (to which (15) reduces if the mass m is constant). However, in the dynamic situation the solutions become radiative (analysed in the next section (17)). To match spacetimes with heat flow we need a time-dependent extension of the Schwarzschild exterior metric. For exact functional forms of $A_0$ and $B_0$ please see [73,74].

To ensure the physicality of the collapsing matter field, we consider it to obey the usual weak energy condition (WEC), i.e., the energy density measured by any timelike observer is non-negative. We can also consider the dominant energy condition (DEC), i.e., the local energy flow is nonspacelike to a timelike observer. These two energy conditions ensure that the collapsing matter fluid remains physically reasonable. Therefore $T_{ab}{v}^a{v}^b$ has to be non-negative and $T^{ab}{v}_b$ is nonspacelike for any timelike vector $v^a$. In brief the energy conditions give us the following

$$\begin{array}{c}\hfill \mathcal{D}>0,\mu \ge p,\mu +\mathcal{D}\ge 3p,\mu +p+\mathcal{D}\ge 0,{\mathcal{D}}^2\equiv {(\mu +p)}^2-4{\left(qB\right)}^2.\end{array}$$

(14)

Here, $\mathcal{D}$ can be interpreted as the net inertial mass of the dissipative system. We also take $\mu \ge 0$ for a physically reasonable matter field. For more details please see [75].

The exterior metric represents an outgoing radiation field around a spherically symmetric star which can be assumed to be the generalised Vaidya metric in N dimensions described by

$$d{s}_{+}^2=-\left(1-{\displaystyle \frac{2m(u,R)}{(N-3)R^{N-3}}}\right)d{u}^2-2dudR+R^{2}d{\Omega }_{N-2}^2,$$

(15)

where $m(u,R)$ is the higher dimensional Misner–Sharp mass containing the gravitational energy within the radius R. Here u is the retarded null coordinate and R is the area radius. The four-dimensional mass function is given in [76,77].

We consider a timelike $N-1$-dimensional spherical hypersurface $\Sigma$, which is the surface of the star at $Y_{\Sigma }=R_{\Sigma }$. We can match both the interior and exterior spacetimes using the two junction conditions and finally get the following relation on $\Sigma$:

$$p_{\Sigma }={\left(qB\right)}_{\Sigma }.$$

(16)

Therefore, the pressure at the boundary is not zero for a radiating star. As the collapse progresses, the outer surface $\Sigma$ of the star shrinks, i.e., its area radius Y is shrinking. In a shear-free ($\sigma =0$) collapse, $Y=rB$ is the simplest condition that follows, and therefore, the diminishing spatial and temporal components of the area radius controls the physical collapse. If the star, begins collapsing from an initial static configuration, we can conclude that the temporal component starts from a constant value and diminishes rapidly towards zero. The area radius Y is helpful to understand this phenomena because, it is also related to the Gaussian curvature (K) of the spherical 2-shell as $K\sim Y^{-2}$ in four dimensions. Hence, as the collapse happens, the Gaussian curvature of the shells increase. It is also true for the outer most surface $\Sigma$. In an N-dimensional collapse, this rate of shrinking (or increase in curvature) will differ depending on the number of spatial dimensions and therefore, this is a very physically intuitive (and geometric) way of understanding the effect of higher dimensions in the process of collapse. In our subsequent sections, we demonstrate how this temporal component (lets denote it by $f\left(t\right)$ henceforth) defers in different dimensions for a shear-free collapse. For a detailed analysis and a general matching condition please refer to [30,61,78,79,80,81]. For a more graphical analysis in four dimensions, readers may refer to a recent study by one of the authors [11].

Determination of the temporal function $f\left(t\right)$ becomes important as it not only controls the gravitational arrow of time but also governs many physical processes of a radiating star. The horizon function, introduced by Ivanov and others in [82,83], has the function f in its temporal part. This horizon function is useful in the study of various stellar features like the mass, energy density, the redshift $z_{\Sigma }$, surface luminosity ${\Lambda }_{\Sigma }$ and the luminosity at infinity ${\Lambda }_{\infty }$. Moreover, in the field of causal thermodynamics also f plays an important role, where using the Maxwell–Cattaneo equation, we can get the temperature profile of the radiating star. For more details on this, please see [84,85,86,87,88]. We now proceed to our N-dimensional collapse models for further analysis.

3. Arrow of Time Problem in Spherically Symmetric Shear-Free Spacetime Collapse

In this section we consider two spherically symmetric shear-free spacetimes and study the arrow of time problem in a collapsing scenario.

3.1. The First Spacetime: Time-Independent Lapse Function

We consider the shear-free metric for the interior spacetime, which starts from a static solution and dynamically evolves with time. Consequently, we adopt the following conditions

$$A=A_0\left(r\right),Y=rB,B=B_0\left(r\right)f\left(t\right);f\left(t\right)>0,A_0>0,B_0>0,$$

(17)

where $A_0,B_0$ are solutions of the interior metric for the static perfect fluid without heat flow, having ${\mu }_0$ as the energy density and $p_0$ as the isotropic pressure. Further, $f\left(t\right)$ is a positive function to be determined [6]. A detailed study on heat conducting shear-free fluids was completed by Sussman in [89]. For recent treatments of the geometrical properties of the metric functions (17) in the context of the radiating stars see Paliathanasis et al. [90] and Ivanov [91].

Then the field Equation (6) for the interior spacetime (17) becomes

$$\begin{array}{cc}\hfill {\kappa }_N\mu A_0^2=& {\displaystyle \frac{{\dot{f}}^2}{f^2}}\left[{\displaystyle \frac{(N-1)(N-2)}{2}}\right]\hfill \\ \hfill & -{\displaystyle \frac{A_0^2}{f^2{B}_0^2}}\left[(N-2){\displaystyle \frac{B_0^{″}}{B_0}}+{\displaystyle \frac{{(N-2)}^2{B}_0^{\prime }}{r{B}_0}}+{\displaystyle \frac{(N-2)(N-5)}{2}}{\displaystyle \frac{{B_0^{\prime }}^2}{B_0^2}}\right],\hfill \\ \hfill -{\kappa }_{N}q{B}_0^2=& (N-2){\displaystyle \frac{A_0^{\prime }}{A_0^2}}\left({\displaystyle \frac{\dot{f}}{f^3}}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\kappa }_{N}p=& -{\displaystyle \frac{1}{A_0^2}}\left[(N-2){\displaystyle \frac{\ddot{f}}{f}}+{\displaystyle \frac{(N-2)(N-3)}{2}}{\left({\displaystyle \frac{\dot{f}}{f}}\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{B_0^2{f}^2}}\left[{\displaystyle \frac{(N-2)(N-3)}{2}}\{{\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)}^2+{\displaystyle \frac{2}{r}}{\displaystyle \frac{B_0^{\prime }}{B_0}}\}+(N-2)\left({\displaystyle \frac{A_0^{\prime }}{A_0}}{\displaystyle \frac{B_0^{\prime }}{B_0}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{A_0^{\prime }}{A_0}}\right)\right],\hfill \\ \hfill {\kappa }_{N}p{B}_0^2=& -{\displaystyle \frac{B_0^2}{A_0^2}}\left[(N-2){\displaystyle \frac{\ddot{f}}{f}}+(N-3){\left({\displaystyle \frac{\dot{f}}{f}}\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{f^2}}\left[{\displaystyle \frac{A_0^{″}}{A_0}}+(N-4){\displaystyle \frac{A_0^{\prime }{B}_0^{\prime }}{A_0{B}_0}}+(N-3)\{{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{A_0^{\prime }}{A_0}}+{\displaystyle \frac{B_0^{\prime }}{B_0}}\right)-{\displaystyle \frac{{B_0^{\prime }}^2}{B_0^2}}+{\displaystyle \frac{B_0^{″}}{B_0}}\}\right]\hfill \\ \hfill & -{\displaystyle \frac{(N-3)(N-4)}{2f^2}}\left[{\displaystyle \frac{B_0^2{\dot{f}}^2}{A_0^2}}-{\displaystyle \frac{B_0^{\prime }}{B_0}}\left({\displaystyle \frac{2}{r}}+{\displaystyle \frac{B_0^{\prime }}{B_0}}\right)\right].\hfill \end{array}$$

(18)

From (18), it follows that the isotropic pressure p, energy density $\mu$ and heat flow $q_a$ for the interior spacetime (8), with the conditions (17), are given by the equations

$$p={\displaystyle \frac{p_0}{f^2}}-{\displaystyle \frac{1}{{\kappa }_N{A}_0^2}}\left\{(N-2){\displaystyle \frac{\ddot{f}}{f}}+{\displaystyle \frac{(N-2)(N-3)}{2}}{\left({\displaystyle \frac{\dot{f}}{f}}\right)}^2\right\},$$

(19)

$$\mu ={\displaystyle \frac{{\mu }_0}{f^2}}+{\displaystyle \frac{(N-1)(N-2)}{2{\kappa }_N{A}_0^2}}{\left({\displaystyle \frac{\dot{f}}{f}}\right)}^2,$$

(20)

$$q^{\alpha }=q{\delta }_1^{\alpha }=-{\displaystyle \frac{(N-2)A_0^{\prime }}{{\kappa }_N{A}_0^2{B}_0^2}}\left({\displaystyle \frac{\dot{f}}{f^3}}\right){\delta }_1^{\alpha },$$

(21)

where the energy density and isotropic pressure for the static solution (without heat flow) are given by ${\mu }_0$ and $p_0$ respectively. There is no heat flow for the static solution. Consequently, imposing the condition of vanishing pressure (static solution) $p_0{|}_{\Sigma }=0$ on the boundary surface $\Sigma$, and using Equations (16), (19) and (21), we obtain the following second-order differential equation for $f\left(t\right)$ in N dimensions as

$$2f\ddot{f}+(N-3){\dot{f}}^2-2a\dot{f}=0,a={\left({\displaystyle \frac{A_0^{\prime }}{B_0}}\right)}_{\Sigma },$$

(22)

where a must be positive if the static solution generated by $A_0$ and $B_0$ is to match with the Schwarzschild exterior metric. Subsequently, we can find the first integral of (22) as

$$\dot{f}={\displaystyle \frac{1}{(N-3)}}\left(2a-C{f}^{-{\displaystyle \frac{(N-3)}{2}}}\right),$$

(23)

which integrates to give the quadrature

$$t={\displaystyle \frac{1}{a}}{\left({\displaystyle \frac{1}{2a}}\right)}^{{\displaystyle \frac{2}{N-3}}}\int {\displaystyle \frac{{(u+C)}^{\frac{2}{N-3}}}{u}}du+C_2,\mathrm{with}u\equiv 2a{f}^{{\displaystyle \frac{N-3}{2}}}-C.$$

(24)

We have tabulated all the solutions of Equation (22) in

Table 1. Nature of the differential equation for f in higher dimensions in GR.

N	Condition on $\mathbf{\Sigma }$	First Integral	Second Integral
4	$2f\ddot{f}+{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=2a-{\tilde{C}}_4{f}^{-1/2}$	$t={\displaystyle \frac{2a^{2}f+2aC\sqrt{f}-\frac{3C^2}{2}+C^{2}ln|2a\sqrt{f}-C|}{4a^3}}+C_2$
5	$2f\ddot{f}+2{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=a-{\tilde{C}}_5{f}^{-1}$	$t={\displaystyle \frac{f}{a}}+{\displaystyle \frac{C}{2a^2}}ln|2af-C|+C_2-{\displaystyle \frac{C}{2a^2}}$
6	$2f\ddot{f}+3{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=2a/3-{\tilde{C}}_6{f}^{-3/2}$	$\begin{array}{cc}& t={\displaystyle \frac{3f}{2a}}+{\displaystyle \frac{3C}{2a^2}}\int {\displaystyle \frac{\sqrt{f}}{2a{f}^{3/2}-C}}dw+C_2;\hfill \\ & w={\left(2a\right)}^{{\displaystyle \frac{1}{3}}}{f}^{{\displaystyle \frac{N-3}{6}}}\hfill \end{array}$
7	$2f\ddot{f}+4{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=a/2-{\tilde{C}}_7{f}^{-2}$	$t={\displaystyle \frac{2f}{a}}+{\displaystyle \frac{\sqrt{C}}{2\sqrt{2a}}}ln\left|{\displaystyle \frac{\sqrt{2a}f-\sqrt{C}}{\sqrt{2a}f+\sqrt{C}}}\right|+C_2$
8	$2f\ddot{f}+5{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=2a/5-{\tilde{C}}_8{f}^{-5/2}$	$\begin{array}{cc}& t={\displaystyle \frac{5f}{2a}}+{\displaystyle \frac{5C}{2a^2}}\int {\displaystyle \frac{\sqrt{f}}{(2a{f}^{5/2}-C)}}du+C_2;\hfill \\ & u=2a{f}^{5/2}-C\hfill \end{array}$
9	$2f\ddot{f}+6{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=a/3-{\tilde{C}}_9{f}^{-3}$	$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}t& ={\displaystyle \frac{3f}{a}}+{\displaystyle \frac{\sqrt{C}}{a{\left(2a\right)}^{\frac{1}{3}}}}{(ln|\left(2a\right)}^{{\displaystyle \frac{1}{3}}}f-\sqrt{C}|\hfill \\ & +{\displaystyle \frac{1}{{\omega }^2}}ln|{\left(2a\right)}^{{\displaystyle \frac{1}{3}}}f-\sqrt{C}\omega |\hfill \\ & +{\displaystyle \frac{1}{\omega }}{ln|\left(2a\right)}^{{\displaystyle \frac{1}{3}}}f-\sqrt{C}{\omega }^2\left|\right)+C_2;\hfill \\ & \omega =e^{{\displaystyle \frac{2\pi i}{3}}}\mathrm{is}\mathrm{a}\mathrm{complex}\mathrm{cube}\mathrm{of}\mathrm{unity}.\hfill \end{array}$
10	$2f\ddot{f}+7{\dot{f}}^2-2a\dot{f}=0$	$\dot{f}=2a/7-{\tilde{C}}_{10}{f}^{-7/2}$	$\begin{array}{cc}& t={\displaystyle \frac{7f}{2a}}+{\displaystyle \frac{C}{2a^2}}\int {\displaystyle \frac{\sqrt{f}}{2a{f}^{7/2}-C}}du+C_2;\hfill \\ & u=2a{f}^{7/2}-C\hfill \end{array}$

for various higher dimensions. Here, C contains the integration constant $C_1$ and for the sake of a clean presentation (in Table 1) we have dropped the subscript. The integration constant $C_2$ represents a special time moment of the system. These integration constants can be fixed by considering appropriate limits, e.g., the collapse starts with $f=1,\dot{f}=0$, that fixes C and ends towards $f\to 0$ at $t\to t_s$, fixing $C_2$ (for more details please refer to [7]). We observe that as the dimension N increases the analytical solution of the second integral becomes very difficult. Here ${\tilde{C}}_N\equiv C/(N-3)$ for all $N\ge 4$.

On physical grounds we assume pressure isotropy, i.e., ${\Delta }_p=0$. The condition of isotropy of pressure implies the following relation for the interior fluid

$$\begin{array}{c}\hfill {\displaystyle \frac{A_0^{″}}{A_0}}+(N-3){\displaystyle \frac{B_0^{″}}{B_0}}=-{\displaystyle \frac{(N-3)(N-4)}{2}}\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)\left({\displaystyle \frac{2}{r}}+{\displaystyle \frac{B_0^{\prime }}{B_0}}\right)\\ \hfill +\left({\displaystyle \frac{A_0^{\prime }}{A_0}}\right)\left({\displaystyle \frac{1}{r}}+2{\displaystyle \frac{B_0^{\prime }}{B_0}}\right)+(N-3)\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)\left[{\displaystyle \frac{(N-3)}{r}}+{\displaystyle \frac{B_0^{\prime }}{B_0}}{\displaystyle \frac{N}{2}}\right].\end{array}$$

(25)

It is interesting to note that all the temporal contributions exactly cancel and we get a time-independent relation. This means that the pressure isotropy is completely determined by the radial functions of the interior metric and the dimension; moreover, the relation holds for all time. In the above Equation (25) we can see that the first term in the RHS is zero for four-dimensional collapse, but has a nonzero contribution in higher dimensions. We can simplify this equation further to express it in a more familiar form

$${\displaystyle \frac{A_0^{″}}{A_0}}+(N-3){\displaystyle \frac{B_0^{″}}{B_0}}=\left[{\displaystyle \frac{A_0^{\prime }}{A_0}}+(N-3)\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)\right]\left({\displaystyle \frac{1}{r}}+2{\displaystyle \frac{B_0^{\prime }}{B_0}}\right).$$

(26)

The above relation holds true for ${\Delta }_p=0$ in a shear-free spherically symmetric spacetime for any dimension $N\ge 4$.

We need the higher dimensional static solutions of energy density and pressure for our further analysis. They are given by

$${\kappa }_N{\mu }_0=-{\displaystyle \frac{1}{B_0^2}}\left[(N-2){\displaystyle \frac{B_0^{″}}{B_0}}+{\displaystyle \frac{(N-2)(N-5)}{2}}{\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)}^2+{\displaystyle \frac{{(N-2)}^2}{r}}{\displaystyle \frac{B_0^{\prime }}{B_0}}\right],$$

(27)

and

$${\kappa }_N{p}_0={\displaystyle \frac{1}{B_0^2}}\left[{\displaystyle \frac{(N-2)(N-3)}{2}}\{{\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)}^2+{\displaystyle \frac{2}{r}}{\displaystyle \frac{B_0^{\prime }}{B_0}}\}+(N-2)\{{\displaystyle \frac{A_0^{\prime }}{A_0}}{\displaystyle \frac{B_0^{\prime }}{B_0}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{A_0^{\prime }}{A_0}}\}\right].$$

(28)

Please note that in (27) we get an extra contribution for dimensions $N>4$ and the second term inside the parenthesis exactly vanishes for five-dimensional collapse. We can indeed get back these expressions from Equations (19) and (20) by taking f to be a constant, indicating the initial static state of the collapsing star. Also (21) shows that for the initial static state the heat flux is zero and with time it increases. (For collapse $\dot{f}$ is negative, making the heat flux expression positive).

3.1.1. Curvature Scalars

We now proceed to derive the curvature scalars in higher dimensions. We start with the conformal Weyl tensor square curvature scalar in N dimensions

$$\begin{array}{c}\hfill \mathcal{W}={\displaystyle \frac{4(N-3)}{(N-1)B_0^4{f}^4{r}^2}}{\left[{\displaystyle \frac{A_0^{″}}{A_0}}r-{\displaystyle \frac{B_0^{″}}{B_0}}r-2{\displaystyle \frac{A_0^{\prime }}{A_0}}{\displaystyle \frac{B_0^{\prime }}{B_0}}r+2{\left({\displaystyle \frac{B_0^{\prime }}{B_0}}\right)}^{2}r-{\displaystyle \frac{A_0^{\prime }}{A_0}}+{\displaystyle \frac{B_0^{\prime }}{B_0}}\right]}^2.\end{array}$$

(29)

Notice the dimensional dependence as $(N-3)/(N-1)$, which transparently demonstrates that as the dimension of the spacetime increases the magnitude of the conformal Weyl curvature scalar increases. Also the Weyl scalar varies as $1/f^4$ with time, making it diverge as f diminishes to zero. Now, using the pressure isotropy condition (26), we rewrite the Weyl square (29) in the compact form

$$\begin{array}{c}\hfill \mathcal{W}={\displaystyle \frac{4(N-3){(N-2)}^2}{(N-1)}}{\left(B_{0}f\right)}^{-4}{X}^2,X\equiv \left({\displaystyle \frac{B_0^{″}}{B_0}}-2{\displaystyle \frac{{B_0^{\prime }}^2}{B_0^2}}-{\displaystyle \frac{B_0^{\prime }}{r{B}_0}}\right).\end{array}$$

(30)

This is the final expression for Weyl curvature scalar in N dimensions. The expression for Weyl curvature scalar in four dimensions matches with results obtained in [4], confirming the validity of the expression for N dimensions.

Now we need to calculate the Ricci scalar R. After some algebraic simplifications we obtain the Ricci scalar as

$$\begin{array}{cc}\hfill R=& {\displaystyle \frac{1}{f^2{A}_0^2}}[-{\displaystyle \frac{2A_0^{″}{A}_0}{B_0^2}}-(2N-4){\displaystyle \frac{B_0^{″}{A}_0^2}{B_0^3}}+2(N-1)f\ddot{f}\hfill \\ \hfill & -(N-2)(N-5){\displaystyle \frac{A_0^2{B_0^{\prime }}^2}{B_0^4}}-{\displaystyle \frac{2A_0}{r{B}_0^3}}\left((N-3)r{A}_0^{\prime }+{(N-2)}^2{A}_0\right)B_0^{\prime }\hfill \\ \hfill & +(N-1)(N-2){\dot{f}}^2-2(N-2){\displaystyle \frac{A_0{A}_0^{\prime }}{r{B}_0^2}}].\hfill \end{array}$$

(31)

In the above Equation (31) we notice that the fourth term inside the parenthesis vanishes in five-dimensional collapse. In this equation we separate the $\dot{f}$ and $\ddot{f}$ components from the static components. Using the relations (22), (27) and (28) we can rewrite the Ricci scalar in a concise manner as

$$\begin{array}{c}\hfill R=-{\displaystyle \frac{2{\kappa }_N}{N-2}}T={\displaystyle \frac{1}{f^2{A}_0^2}}(N-1)(2a\dot{f}+{\dot{f}}^2)+{\displaystyle \frac{2{\kappa }_N}{(N-2)f^2}}\left({\mu }_0-(N-1)p_0\right).\end{array}$$

(32)

In the above expression (32), T is the trace of the stress–energy tensor.

Similarly we can calculate the Ricci square scalar $\mathcal{R}$. We now present a general analytical form in terms of matter variables for any dimension N. Utilising the trace-reversed field equation we can express the Ricci tensor square in terms of stress–energy tensor and its trace. Consequently, we can express the Ricci tensor square as

$$\begin{array}{cc}\hfill \mathcal{R}=& {\kappa }_N^2[{\mu }^2\left(1-{\displaystyle \frac{(N-4)}{{(N-2)}^2}}\right)+p^2\{(N-1)-{\displaystyle \frac{(N-4){(N-1)}^2}{{(N-2)}^2}}\}\hfill \\ \hfill & +{\displaystyle \frac{2(N-1)(N-4)}{{(N-2)}^2}}\mu p-2q^2{B}^2]\equiv {\displaystyle \frac{1}{f^4}}\left[\sum _{\alpha =0}^4{}^N\Theta _{\alpha }\left(r\right){\dot{f}}^{\alpha }\right].\hfill \end{array}$$

(33)

Here fluid energy density, isotropic pressure and the heat flow are given by (19)–(21) respectively. The above result (33) simplifies to the expression obtained in [4] in four dimensions. We now concisely express the coefficients ${}^N\Theta _{\alpha }\left(r\right)$ in (33) in the following manner:

$$\begin{array}{cc}\hfill {}^N\Theta _0=& \left(1-{\displaystyle \frac{N-4}{{(N-2)}^2}}\right){\mu }_0^2+(N-1)\left[\left\{1-{\displaystyle \frac{(N-1)(N-4)}{{(N-2)}^2}}\right\}{p}_0^2+{\displaystyle \frac{2(N-4)}{{(N-2)}^2}}{\mu }_0{p}_0\right],\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {}^N\Theta _1=& -{\displaystyle \frac{2a(N-1)}{A_0^2}}\left[\left\{(N-2)-{\displaystyle \frac{(N-1)(N-4)}{N-2}}\right\}{p}_0+{\displaystyle \frac{N-4}{N-2}}{\mu }_0\right],\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {}^N\Theta _2=& (N-1)\left[{\displaystyle \frac{{\mu }_0}{A_0^2}}\left\{(N-2)-{\displaystyle \frac{N-4}{N-2}}\right\}+{\displaystyle \frac{a^2}{A_0^4}}\left\{{(N-2)}^2-(N-1)(N-4)\right\}\right]\hfill \\ \hfill & +{\displaystyle \frac{{(N-1)}^2(N-4)}{N-2}}{\displaystyle \frac{p_0}{A_0^2}}-2{(N-2)}^2{\displaystyle \frac{{A_0^{\prime }}^2}{A_0^4{B}_0^2}},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill {}^N\Theta _3=& -{(N-1)}^2(N-4){\displaystyle \frac{a}{A_0^4}},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {}^N\Theta _4=& \left\{N+(N-2)(N-4)\right\}{(N-1)}^2{\displaystyle \frac{1}{4A_0^4}}.\hfill \end{array}$$

(38)

Please note that we are ignoring the gravitational constant during the curvature calculations by taking ${\kappa }_N=1$, for a neater presentation. Interestingly, we observe that the fourth order coefficient ${}^N\Theta _4$ in (38) is a positive quantity during the collapse for all dimensions $N\ge 4$. We will make use of this interesting property in our next section. Subsequently, we now obtain the Kretschmann scalar in N dimensions using the following relation

$$\mathcal{K}=\mathcal{W}+{\displaystyle \frac{4}{N-2}}\mathcal{R}-{\displaystyle \frac{2}{(N-1)(N-2)}}{R}^2,$$

(39)

and obtain the following expression of the Kretschmann scalar in N dimensions

$$\begin{array}{cc}\hfill \mathcal{K}=& {\displaystyle \frac{4(N-2)}{r^2{B}_0^8{f}^4{A}_0^4}}[A_0^4{B_0^{″}}^2{B}_0^2{r}^2-2r{A}_0^2{B}_0(r{B}_0^4{\dot{f}}^2+r{A}_0^2{B_0^{\prime }}^2-A_0^2{B}_0^{\prime }{B}_0)B_0^{″}\hfill \\ \hfill & +{\displaystyle \frac{N-1}{N-2}}{f}^2{B}_0^8{\ddot{f}}^2{r}^2-{\displaystyle \frac{2r}{N-2}}\{A_0^{″}{B}_{0}r+(N-3)(r{B}_0^{\prime }+{\displaystyle \frac{N-2}{N-3}}{B}_0)A_0^{\prime }\}{B}_0^{5}f{A}_0\ddot{f}\hfill \\ \hfill & +{\displaystyle \frac{1}{N-2}}{A_0^{″}}^2{A}_0^2{B}_0^4{r}^2-{\displaystyle \frac{2}{N-2}}{A}_0^{″}{B}_0^{\prime }{A}_0^{\prime }{A}_0^2{B}_0^3{r}^2+{\displaystyle \frac{N-1}{2}}{B_0^{\prime }}^4{A}_0^4{r}^2\hfill \\ \hfill & +2(N-4){B_0^{\prime }}^3{A}_0^4{B}_{0}r+B_0^2{A}_0^2{B_0^{\prime }}^2\{-(N-5)B_0^2{\dot{f}}^2{r}^2+{\displaystyle \frac{N-1}{N-2}}{A_0^{\prime }}^2{r}^2\hfill \\ \hfill & +(2N-5)A_0^2\}-(2N-4)B_0^3{A}_0^2\left(B_0^2{\dot{f}}^2-{\displaystyle \frac{1}{N-2}}{A_0^{\prime }}^2\right)r{B}_0^{\prime }\hfill \\ \hfill & +(A_0^2{B}_0^4-2r^2{B}_0^6{\dot{f}}^2){A_0^{\prime }}^2+{\displaystyle \frac{N-1}{2}}{B}_0^8{\dot{f}}^4{r}^2].\hfill \end{array}$$

(40)

Interestingly, in this case also, the above expression (40) of the Kretschmann scalar can be expressed in a separable manner

$$\mathcal{K}={\displaystyle \frac{4(N-2)}{f^4}}\left[\sum _{\alpha =0}^4{}^N\Omega _{\alpha }\left(r\right){\dot{f}}^{\alpha }\right],$$

(41)

where ${\Omega }_{\alpha }^N$ are functions of r. The expression (41) of the Kretschmann scalar in N dimensions is sufficient for our purpose and the advantage of such a representation will be evident in the next section. The ${}^N\Omega _{\alpha }$ coefficients are given by the following expressions

$$\begin{array}{cc}\hfill {}^N\Omega _0=& [{\displaystyle \frac{{B_0^{″}}^2}{B_0^2}}+{\displaystyle \frac{2B_0^{\prime }{B}_0^{″}}{r{B}_0^2}}-{\displaystyle \frac{2{B_0^{\prime }}^2{B}_0^{″}}{B_0^3}}+{\displaystyle \frac{1}{N-2}}{\displaystyle \frac{{A_0^{″}}^2}{A_0^2}}-{\displaystyle \frac{2}{N-2}}{\displaystyle \frac{A_0^{″}{A}_0^{\prime }{B}_0^{\prime }}{A_0^2{B}_0}}+{\displaystyle \frac{N-1}{2}}{\displaystyle \frac{{B_0^{\prime }}^4}{B_0^4}}\hfill \\ \hfill & +2(N-4){\displaystyle \frac{{B_0^{\prime }}^3}{r{B}_0^3}}+{\displaystyle \frac{(2N-5){B_0^{\prime }}^2}{r^2{B}_0^2}}+{\displaystyle \frac{N-1}{N-2}}{\displaystyle \frac{{A_0^{\prime }}^2{B_0^{\prime }}^2}{A_0^2{B}_0^2}}+{\displaystyle \frac{2A_0^{\prime }{B}_0^{\prime }}{r{A}_0^2{B}_0}}+{\displaystyle \frac{{A_0^{\prime }}^2}{r^2{A}_0^2}}]{\displaystyle \frac{1}{B_0^4}},\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill {}^N\Omega _1=& -[(N-1){\displaystyle \frac{A_0^{\prime }}{A_{0}r}}+(N-1){\displaystyle \frac{A_0^{\prime }{B}_0^{\prime }}{A_0{B}_0}}+{\displaystyle \frac{(N-3)B_0^{\prime }}{B_{0}r}}+2(N-3){\displaystyle \frac{{B_0\prime }^2}{B_0^2}}\hfill \\ \hfill & -(N-3){\displaystyle \frac{B_0^{″}}{B_0}}]{\displaystyle \frac{2a}{(N-2)A_0^2{B}_0^2}},\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {}^N\Omega _2=& [-\left({\displaystyle \frac{{(N-3)}^2}{N-2}}+2\right){\displaystyle \frac{B_0^{″}}{B_0}}+{\displaystyle \frac{N-1}{N-2}}{\displaystyle \frac{a^2{B}_0^2}{A_0^2}}+{\displaystyle \frac{(N-1)(N-3)}{N-2}}({\displaystyle \frac{A_0^{\prime }{B}_0^{\prime }}{A_0{B}_0}}\hfill \\ \hfill & +{\displaystyle \frac{A_0^{\prime }}{r{A}_0}})+\left({\displaystyle \frac{2{(N-3)}^2}{N-2}}-(N-5)\right){\displaystyle \frac{{B_0^{\prime }}^2}{B_0^2}}+\left({\displaystyle \frac{{(N-3)}^2}{N-2}}-2(N-2)\right)\hfill \\ \hfill & {\displaystyle \frac{B_0^{\prime }}{r{B}_0}}-{\displaystyle \frac{2{A_0^{\prime }}^2}{A_0^2}}]{\displaystyle \frac{1}{A_0^2{B}_0^2}},\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill {}^N\Omega _3=& -{\displaystyle \frac{a(N-1)(N-3)}{(N-2)A_0^4}},\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {}^N\Omega _4=& {\displaystyle \frac{N-1}{N-2}}{\displaystyle \frac{{(N-3)}^2}{4A_0^4}}+{\displaystyle \frac{N-1}{2A_0^4}}.\hfill \end{array}$$

(46)

Here again we notice that the fourth order coefficient ${}^N\Omega _4$ is a positive quantity and remains positive for all $N\ge 4$ throughout the collapse.

3.1.2. Epoch Functions and the Arrow of Time

We now compute the gravitational epoch functions and analyse its behaviour. Following the above derived expressions (30) and (33) of the curvature scalars we can deduce the following epoch function

$$\begin{array}{c}\hfill \mathcal{P}={\displaystyle \frac{4(N-3){(N-2)}^2}{(N-1)}}{\displaystyle \frac{X^2}{B_0^4}}{\left[\sum _{\alpha =0}^4{}^N\Theta _{\alpha }\left(r\right){\dot{f}}^{\alpha }\right]}^{-1}.\end{array}$$

(47)

It is evident that the ratio of the curvature scalars depends on the dimension. It is interesting to analyse the behaviour of the epoch function with time during gravitational collapse. For this we need (23) to understand the overall behaviour of $\mathcal{P}$. In the beginning the integration constants can be chosen in such a manner that the temporal function f becomes a constant. Without any loss of generality, this constant can be assumed to be of unit value. Consequently, in this limit, the epoch function $\mathcal{P}$ is a positive quantity. However, as time passes the value of f tends to zero with $t\to t_s$, making the time derivative of f to diverge. Now we notice that in the denominator of $\mathcal{P}$ we have terms corresponding to different powers of $\dot{f}$ with their respective coefficients (34)–(38). Since $\dot{f}$ is diverging during collapse, its fourth power will diverge much faster than the other terms. Now, the corresponding coefficient of ${\dot{f}}^4$ is ${}^N\Theta _4$, given in (38) and it is a positive quantity throughout collapse for all dimensions $N\ge 4$. Therefore, the fourth order contribution will keep dominating the divergence of the denominator over other terms. As a result, as the collapse proceeds, the epoch function $\mathcal{P}$ vanishes. This means that the gravitational epoch function $\mathcal{P}$ decreases with time during collapse even in higher dimensions.

Now we investigate the epoch function ${\mathcal{P}}_1$ for higher dimensions. Taking the ratio of the Weyl square (30) and Kretschmann scalar (41) we get ${\mathcal{P}}_1$ as

$$\begin{array}{c}\hfill {\mathcal{P}}_1={\displaystyle \frac{(N-3)(N-2)}{(N-1)}}{\displaystyle \frac{X^2}{B_0^4}}{\left[\sum _{\alpha =0}^4{}^N\Omega _{\alpha }\left(r\right){\dot{f}}^{\alpha }\right]}^{-1}.\end{array}$$

(48)

In this case too when we substitute $\dot{f}$ using the first integral (23) into appropriate limits we find that initially the epoch function ${\mathcal{P}}_1$ starts from a positive value and as time progresses it goes to zero, since the fourth order coefficient ${}^N\Omega _4$ in (48) is a positive quantity during collapse for all $N\ge 4$, as can be observed in (46). This means that this epoch function also decreases with time during collapse. Consequently, the arrow of time associated with both of these gravitational epoch functions (gravitational arrow of time) are in the opposite direction of the arrow of time associated with the outgoing radiation (thermodynamic arrow of time) for all dimensions $N\ge 4$.

3.2. The Second Spacetime: Time-Modulated Lapse Function

In this section we consider another interesting case for the interior spacetime metric when the lapse function $A(r,t)=A_0\left(r\right)f\left(t\right)$ is time-dependent and separable. As a result the interior metric can be written in the following manner

$$d{s}_{-}^2=f{\left(t\right)}^2\left\{-A_0{\left(r\right)}^{2}d{t}^2+B_0{\left(r\right)}^2(d{r}^2+r^{2}d{\Omega }_{N-2}^2)\right\}.$$

(49)

The resulting metric is conformal to the shear-free static spacetime. By solving the Einstein’s field equations in N dimensions we obtain the matter variables

$$p={\displaystyle \frac{p_0}{f^2}}-{\displaystyle \frac{1}{{\kappa }_N{A}_0^2}}\left\{(N-2){\displaystyle \frac{\ddot{f}}{f}}+{\displaystyle \frac{(N-2)(N-5)}{2}}{\left({\displaystyle \frac{\dot{f}}{f}}\right)}^2\right\}{\displaystyle \frac{1}{f^2}},$$

(50)

$$\mu ={\displaystyle \frac{{\mu }_0}{f^2}}+{\displaystyle \frac{(N-1)(N-2)}{2{\kappa }_N{f}^2{A}_0^2}}{\left({\displaystyle \frac{\dot{f}}{f}}\right)}^2,$$

(51)

$$q^{\alpha }=q{\delta }_1^{\alpha }=-{\displaystyle \frac{(N-2)A_0^{\prime }}{{\kappa }_N{A}_0^2{B}_0^2}}\left({\displaystyle \frac{\dot{f}}{f^4}}\right){\delta }_1^{\alpha }.$$

(52)

It is important to notice the difference from our previous case, especially in the pressure expression the second term inside the parenthesis vanishes in five dimensions. Moreover, in each of the above expressions we have extra contributions of $1/f$. Therefore, the matching condition at the boundary hypersurface $\Sigma$ can be reduced to the following form

$$2f\ddot{f}+(N-5){\dot{f}}^2-2af\dot{f}=0.$$

(53)

Subsequently, the first integral of the above differential equation can be obtained as

$${\displaystyle \frac{\dot{f}}{f}}={\displaystyle \frac{2a}{(N-3)}}+C{f}^{-(N-3)/2}.$$

(54)

By integrating the above equation we get the explicit form of f as

$$f\left(t\right)={\left[{\displaystyle \frac{1}{2a}}(N-3)(e^{a(t+t_0)}-C)\right]}^{2/(N-3)},$$

(55)

where C and $t_0$ are the constants of integration.

From the field equations it follows that the condition of pressure isotropy remains unchanged. Subsequently, a similar analysis can be followed in this case and the curvature scalars obtained.

We begin by implementing the junction condition (53) in the isotropic pressure (50) and express it in terms of $\dot{f}/f$. Consequently, the square of the Ricci tensor can now be expressed as

$$\begin{array}{cc}\hfill \mathcal{R}=& {\displaystyle \frac{1}{f^4}}[\left(1-{\displaystyle \frac{N-4}{{(N-2)}^2}}\right){\tilde{\mu }}^2+(N-1)\left\{1-{\displaystyle \frac{(N-1)(N-4)}{{(N-2)}^2}}\right\}{\tilde{p}}^2+{\displaystyle \frac{2(N-1)(N-4)}{{(N-2)}^2}}\tilde{\mu }\tilde{p}\hfill \\ \hfill & -{\displaystyle \frac{2{(N-2)}^2{A_0^{\prime }}^2}{A_0^4{B}_0^2}}{\displaystyle \frac{{\dot{f}}^2}{f^2}}]\equiv {\displaystyle \frac{1}{f^4}}\left[\sum _{\alpha =0}^4{}^N\Xi _{\alpha }\left(r\right){\left({\displaystyle \frac{\dot{f}}{f}}\right)}^{\alpha }\right],\mathrm{with}\hfill \\ \hfill \tilde{\mu }\equiv & {\mu }_0+{\displaystyle \frac{(N-1)(N-2)}{2A_0^2}}{\displaystyle \frac{{\dot{f}}^2}{f^2}},\tilde{p}\equiv p_0-{\displaystyle \frac{a(N-2)}{A_0^2}}{\displaystyle \frac{\dot{f}}{f}}.\hfill \end{array}$$

(56)

We will not mention all the scalar coefficients explicitly for brevity of this paper; however without going into the detailed higher dimensional functional forms, we can clearly notice the temporal patterns and observe the fourth order coefficient of $\dot{f}/f$

$${}^N\Xi _4={\displaystyle \frac{{(N-1)}^2\left\{N+(N-2)(N-4)\right\}}{4A_0^4}}.$$

(57)

It is clear that the positivity of the coefficient ${}^N\Xi _4$ for all $N\ge 4$ is maintained during the collapse. Please note that the square of the Weyl tensor $\mathcal{W}$ remains same as in (30). Therefore, after a simple algebraic simplification we can reduce the epoch function $\mathcal{P}$ to

$$\mathcal{P}={\displaystyle \frac{4{(N-2)}^2(N-3)X^2}{(N-1)B_0^4}}{\left[\sum _{\alpha =0}^4{}^N\Xi _{\alpha }\left(r\right){\left({\displaystyle \frac{\dot{f}}{f}}\right)}^{\alpha }\right]}^{-1},$$

(58)

where the denominator is a fourth order polynomial of the first integral $\dot{f}/f$ with ${}^N\Xi _{\alpha }\left(r\right)$s as coefficients. Now, (from (39)) we follow a similar analysis to compute the Kretschmann scalar $\mathcal{K}$ and the epoch function ${\mathcal{P}}_1$ in the following manner

$$\mathcal{K}={\displaystyle \frac{1}{f^4}}\left[\sum _{\alpha =0}^4{}^N{\rm Y}_{\alpha }\left(r\right){\left({\displaystyle \frac{\dot{f}}{f}}\right)}^{\alpha }\right],{\mathcal{P}}_1\sim {\left[\sum _{\alpha =0}^4{}^N{\rm Y}_{\alpha }\left(r\right){\left({\displaystyle \frac{\dot{f}}{f}}\right)}^{\alpha }\right]}^{-1}.$$

(59)

Subsequently, we track the fourth order coefficient ${}^N{\rm Y}_4$ as

$${}^N{\rm Y}_4={\displaystyle \frac{4}{N-2}}{}^N\Xi _4-{\displaystyle \frac{2(N-1)}{(N-2)A_0^4}}={\displaystyle \frac{N-1}{A_0^4}}\left[{(N-3)}^2+2(N-2)\right].$$

(60)

Hence, during the collapse, the fourth order term always dominates for all $N\ge 4$ due to its positive value.

Here also, the denominator has the same pattern with coefficients ${}^N{\rm Y}_{\alpha }\left(r\right)$. In these coefficients the superscript N denotes the dimensionality, and the subscript $\alpha$ denotes the order of the coefficient in the polynomial. Now we are in a position to analyse the behaviour of the epoch functions with time evolution. From the first integral (54) it is clear that $\dot{f}/f$ diverges to infinity as f decreases to zero with time. In the denominator of both the epoch functions (58) and (59), the fourth order term dominates over all the other terms as the collapse progresses. Now, from (57) and (60) it is clear that for $N\ge 4$ these fourth order coefficients remain positive throughout the collapse. As a result the denominator in the epoch functions diverge as time progresses starting from a finite value. This conclusively demonstrates that the arrow of time due to the gravitational epoch functions is opposite to the thermodynamic arrow of time associated with radiation for all $N\ge 4$.

4. Arrow of Time Problem in Presence of Charge

We now introduce charge in the interior matter distribution and see how it affects our previous results. The idea here is to check whether the directionality of gravitational arrow of time changes due to the effect of charge in the system. Several works are available in the literature on the charged matter distributions in [33,34,35,69]. For this study we will again consider the shear-free interior spacetime (17), and the energy momentum tensor for the interior matter distribution gets modified by the electromagnetic energy tensor $E_{ab}$ to give

$$T_{ab}=(\mu +p)v_a{v}_b+p{g}_{ab}+q_a{v}_b+q_b{v}_a+{\pi }_{ab}+E_{ab}.$$

(61)

The electromagnetic energy tensor $E_{ab}$ is defined in terms of Faraday tensor $F_{ab}$ as

$$E_{ab}={\displaystyle \frac{1}{{\mathcal{A}}_{N-2}}}\left(F_a^c{F}_{bc}-{\displaystyle \frac{1}{4}}{F}^{cd}{F}_{cd}{g}_{ab}\right),{\mathcal{A}}_{N-2}={\displaystyle \frac{2{\pi }^{\frac{N-1}{2}}}{\Gamma \left(\frac{N-1}{2}\right)}},$$

(62)

where ${\mathcal{A}}_{N-2}$ is the surface area of the $(N-2)$-sphere and $\Gamma (\dots )$ is the gamma function. The Maxwell bivector or Faraday tensor is defined as

$$F_{ab}=-F_{ba}={\Phi }_{b;a}-{\Phi }_{a;b},$$

(63)

with ${\Phi }_a$ as the electromagnetic potential, which can be denoted by ${\Phi }_a=(\varphi (r,t),0,0,\dots ,0)$. That is, the charge is assumed to be at rest with respect to the observer and hence, no magnetic field is present in our comoving local coordinate system. Therefore, we consider the Einstein–Maxwell field equations as the following

$$\begin{array}{c}\hfill G_{ab}={\kappa }_N{T}_{ab},F_{[ab;c]}=0,F_{;b}^{ab}={\mathcal{A}}_{N-2}{J}^a.\end{array}$$

(64)

Here $J^a$ is the four current defined as $J^a=\zeta v^a$ with $\zeta$ as the proper electric charge density. From these sets of equations, we derive the field equations for the electromagnetic field.

$$\begin{array}{c}\hfill {\varphi }^{″}-\left[{\displaystyle \frac{A^{\prime }}{A}}+{\displaystyle \frac{B^{\prime }}{B}}-(N-2){\displaystyle \frac{Y^{\prime }}{Y}}\right]{\varphi }^{\prime }={\mathcal{A}}_{N-2}\zeta A{B}^2,\end{array}$$

(65)

$$\begin{array}{c}\hfill {\dot{\varphi }}^{\prime }-\left({\displaystyle \frac{\dot{A}}{A}}+{\displaystyle \frac{\dot{B}}{B}}-(N-2){\displaystyle \frac{\dot{Y}}{Y}}\right){\varphi }^{\prime }=0.\end{array}$$

(66)

From Equations (65) and (66) we get the total time-independent conserved charge of the star within r as

$$Q\left(r\right)={\mathcal{A}}_{N-2}{\int }_0^r\zeta B{Y}^{N-2}d\tilde{r}.$$

(67)

We will now only consider the first spacetime (17) from the previous section and introduce charge. Following the previous treatment but now including charge we arrive at the following differential equation for f as

$$2f\ddot{f}+(N-3){\dot{f}}^2-2{\left({\displaystyle \frac{A_0^{\prime }}{B_0}}\right)}_{\Sigma }\dot{f}+{\displaystyle \frac{{\kappa }_N}{(N-2){\mathcal{A}}_{N-2}}}{\left({\displaystyle \frac{A_0^2{Q}^2}{{\left(r{B}_0\right)}^{2N-4}}}\right)}_{\Sigma }=0.$$

(68)

Let us denote the last term in the LHS as

$$\varsigma \left(r_{\Sigma }\right)={\displaystyle \frac{{\kappa }_N}{(N-2){\mathcal{A}}_{N-2}}}{\left({\displaystyle \frac{A_0^2{Q}^2}{{\left(r{B}_0\right)}^{2N-4}}}\right)}_{\Sigma }.$$

(69)

To solve Equation (68), we make the substitution $v=\dot{f}$, and simplify to get the following

$$\int {\displaystyle \frac{2vdv}{-(N-3)v^2+2av-\varsigma \left(r_{\Sigma }\right)}}=lnf+C_1,$$

(70)

where $C_1$ is the constant of integration and $a={(A_0^{\prime }/B_0)}_{\Sigma }$. It is evident that the nature of the solution of Equation (70) depends on the quadratic term $-(N-3)v^2+2av-\varsigma \left(r_{\Sigma }\right)$. Therefore we compute the discriminant of the quadratic expression, which is given below

$$\Delta =4[a^2-(N-3)\varsigma \left(r_{\Sigma }\right)].$$

(71)

In

Table 2. Nature of the differential equation for f in higher dimensions with charge Q.

$\mathbf{\Delta }$	First Integral	Second Integral
>0	$\begin{array}{cc}& |\dot{f}-v_1{|}^A{|\dot{f}-v_2|}^B=e^{C_1}f;\hfill \\ & A=-{\displaystyle \frac{2v_1}{(N-3)(v_1-v_2)}},\hfill \\ & B=-{\displaystyle \frac{2v_2}{(N-3)(v_2-v_1)}},\hfill \\ & v_{1,2}={\displaystyle \frac{a\pm \sqrt{a^2-(N-3)\varsigma \left(r_{\Sigma }\right)}}{N-3}}.\hfill \end{array}$	$\begin{array}{cc}& t=\int {\displaystyle \frac{df}{v}}+C_2;\hfill \\ & \mathrm{where}\mathrm{v}\mathrm{is}\mathrm{implicitly}\mathrm{defined}\mathrm{by}\hfill \\ & \int {\displaystyle \frac{2vdv}{-(N-3)v^2+2av-\varsigma \left(r_{\Sigma }\right)}}=lnf+C_1\hfill \end{array}$
=0	$\begin{array}{cc}& -{\displaystyle \frac{2}{(N-3)}}ln\left|\dot{f}-{\displaystyle \frac{a}{N-3}}\right|+{\displaystyle \frac{2a}{{(N-3)}^2\left(\dot{f}-\frac{a}{N-3}\right)}}\hfill \\ & =lnf+C_1\hfill \end{array}$	$\begin{array}{cc}& t+C_2=\hfill \\ & \int {\displaystyle \frac{df}{\frac{a}{N-3}\left(1+\frac{1}{ln\left|\dot{f}-\frac{a}{N-3}\right|+\frac{(N-3)}{2}lnf+\frac{(N-3)}{2}{C}_1}\right)}};\hfill \end{array}$
<0	$\begin{array}{cc}& lnf+C_1=-{\displaystyle \frac{1}{N-3}}ln\left[{\left(\dot{f}-{\displaystyle \frac{a}{N-3}}\right)}^2+Q\right]\hfill \\ & -{\displaystyle \frac{a}{(N-3)\sqrt{(N-3)\varsigma \left(r_{\Sigma }\right)-a^2}}}\hfill \\ & arctan\left[{\displaystyle \frac{(N-3)\dot{f}-a}{\sqrt{(N-3)\varsigma \left(r_{\Sigma }\right)-a^2}}}\right];\hfill \\ & Q={\displaystyle \frac{(N-3)\varsigma \left(r_{\Sigma }\right)-a^2}{{(N-3)}^2}},Q>0\hfill \end{array}$	$\begin{array}{cc}& t=-\int {\displaystyle \frac{N-3}{2}}{\displaystyle \frac{u^2+Q}{u+\frac{a}{N-3}}}d(lnf+C_1)+C_2;\hfill \\ & u=\dot{f}-{\displaystyle \frac{a}{N-3}}\hfill \end{array}$

we present the first and second integrals of (68) which depends on the sign of discriminant $\Delta$.

4.1. Analysis of f and Consequences on the Arrow of Time

A careful analysis of each case is necessary to understand how these solutions behave and whether they are physically viable. For $\Delta >0$ the first integral contains the different parameters $A,B$ and $C_1$. From the expressions of $v_1$ and $v_2$ it is evident that both of them are positive as $\varsigma \left(r_{\Sigma }\right)>0$. The integration constant $C_1$ can be fixed by imposing the initial conditions $f=1$ and $\dot{f}\approx 0$. Moreover, it is evident that $v_1>v_2$, which makes $A<0$ and $B>0$. Now let us consider the case where $f\to 0$. In this limit one might think that it is only possible for $\dot{f}\to v_1,v_2$, but in a collapsing situation $\dot{f}$ is negative and therefore these cannot be the desired solutions. The only viable solution is $\dot{f}\to -\infty$ when $f\to 0$, because in this limit the LHS of the first integral is dominated by the first term, making the entire expression diverge to negative infinity. Therefore, when $\Delta >0$, the only possible and physically viable solution is $f=1$ with vanishing $\dot{f}$ in the initial phase and in the end state $f\to 0$ with $\dot{f}\to -\infty$. Now let us consider the case for $\Delta =0$. We start by taking $f=1$ and $\dot{f}\approx 0$ to fix the integration constant $C_1$. Then we consider the limit $f\to 0$ and try to find a viable solution for $\dot{f}$. As $\dot{f}<0$, it cannot take the value $a/(N-3)$ and also it makes the LHS diverge to positive infinity unlike the RHS. Therefore, the only possible solution for $\dot{f}$ is when it diverges to negative infinity. So in this case also the viable solution of f remains the same. Finally, we consider the case for $\Delta <0$. We can fix the integration constant $C_1$ by taking $f=1$ and $\dot{f}\approx 0$. Considering the end state of the collapse we take $f\to 0$. This choice leaves us with the only possible solution for RHS as $\dot{f}\to -\infty$, making the physically viable option as the only solution.

Consequently, in presence of charge, shear-free collapse with a generalised Vaidya exterior has viable solutions as identified in Table 2.

We will now proceed to investigate the nature of the gravitational epoch functions in such a system. Here the Einstein tensor is of the form (18). Therefore, we can use the epoch functions in (47) and (48). It has already been discussed that the nature of these epoch functions depends on how the function $\dot{f}$ behaves. From the above analysis it is evident that the only possible physically viable solution is when $\dot{f}$ starts from zero and decreases rapidly towards negative infinity as the collapse progresses. This means that both the epoch functions start with a positive definite value and with collapse their value diminishes to zero. As a result, the arrow of time associated with it is in the opposite direction of the thermodynamic arrow of time.

Finally, we consider the second spacetime and obtain the following junction condition in presence of charge

$$2f\ddot{f}+(N-5){\dot{f}}^2-2af\dot{f}+\varsigma \left(r_{\Sigma }\right)f^2=0.$$

(72)

The above equation simplifies to the following by taking $u\equiv \dot{f}/f$.

$$\begin{array}{c}\hfill \dot{u}=-{\displaystyle \frac{1}{2}}\left[(N-3)u^2-2au+\varsigma \left(r_{\Sigma }\right)\right]\Rightarrow \dot{u}=-{\displaystyle \frac{(N-3)}{2}}(u-u_{-})(u-u_{+}),\\ \hfill \mathrm{where}{u}_{\pm }={\displaystyle \frac{2a\pm \sqrt{\Delta }}{2(N-3)}}.\end{array}$$

(73)

Since, for a collapse we have $u<0$, the above Equation (73) automatically implies $\dot{u}<0$ for all $N\ge 4$. This means that u is a strictly decreasing function. On physical grounds, if we consider $u\le u_{-}$, then u diverges to negative infinity as the collapse progresses. In this case, the field equations only get modified in the pressure and energy density due an additional contribution of charge ${\tilde{Q}}_N(1-f^{2N-6})$. That means, all the square terms and cross terms of pressure and energy density have this quantity in it. Therefore, the functional form of $\mathcal{W}$ remains the same but the $\mathcal{K}$ and $\mathcal{R}$ gets extra charge contributions. Now, we notice

$$\begin{array}{c}\hfill {\tilde{Q}}_N(1-f^{2N-6})={\tilde{Q}}_N^0\left({\displaystyle \frac{1}{f^{2N-4}}}-{\displaystyle \frac{1}{f^2}}\right)\to -\infty ,\mathrm{for}\mathrm{all}N\ge 4,\mathrm{where}\end{array}$$

(74)

$$\begin{array}{c}\hfill {\tilde{Q}}_N\equiv {\displaystyle \frac{1}{2{\mathcal{A}}_{N-2}}}{\displaystyle \frac{Q^2}{{\left(rB\right)}^{2N-4}}}={\tilde{Q}}_N^0{\displaystyle \frac{1}{f^{2N-4}}}.\end{array}$$

(75)

Therefore, the additional charge terms diverge to negative infinity with collapse. As, the positive fourth order coefficient in $\mathcal{K}$ and $\mathcal{R}$ comes only from the ${\mu }^2$ terms, we can transparently argue that the denominator of the epoch functions will diverge much faster in this case. We also argue that during neutral collapse the square of the energy density and pressure diverge as ∼$f^{-4}$ and so does the $\mathcal{W}$ (29), however due to charge we get extra contributions in $\mu$ and p that makes them diverge much faster. Now, we observe that the curvature scalars can be expressed in the following manner:

$$\begin{array}{cc}\hfill R=& {\displaystyle \frac{2}{N-2}}\left[\mu -(N-1)p-(N-4){\tilde{Q}}_N\right],\hfill \end{array}$$

(76)

$$\begin{array}{cc}\hfill \mathcal{R}=& \left[{\mu }^2+(N-1)p^2-2q^2{B}^2-{\displaystyle \frac{(N-4)}{{(N-2)}^2}}{\left(\mu -(N-1)p\right)}^2\right]\hfill \\ \hfill & +{\displaystyle \frac{4{\tilde{Q}}_N^2}{{(N-2)}^2}}\left[7+2(N-1)\left(N-{\displaystyle \frac{9}{2}}\right)\right]+{\tilde{Q}}_N[\mu \left\{2+{\displaystyle \frac{2{(N-4)}^2}{{(N-2)}^2}}\right\}\hfill \\ \hfill & +4p\left\{1-{\displaystyle \frac{2}{{(N-2)}^2}}\right\}].\hfill \end{array}$$

(77)

The first part inside the parenthesis has the same structure of the neutral case. Notice, the extra terms appearing due to charge, and that all the coefficients are positive for $N\ge 4$, especially the coefficient of ${\tilde{Q}}_N^2$. Similarly, the coefficient of ${\tilde{Q}}_N^2$ in $\mathcal{K}$ is a positive quantity for all dimensions $N\ge 4$

$${\displaystyle \frac{4\mathcal{R}}{(N-2)}}-{\displaystyle \frac{2R^2}{(N-1)(N-2)}}\Rightarrow {\displaystyle \frac{8}{{(N-2)}^3}}\left[4+(N-4)\left\{(4N-7)+{\displaystyle \frac{3}{N-1}}\right\}\right].$$

(78)

Therefore, the divergence comes from two fronts. One is the overall temporal modulation, i.e., in the denominator of the epoch functions there are terms due to charge that are varying much faster than ∼$f^{-4}$ as the dimensions increase and secondly most of their dominant terms has positive coefficients, therefore the decrease in the epoch functions will happen more rapidly. Essentially, charge is making the Ricci sector stronger. This argument holds true for both the spacetimes. Therefore, the conclusion about the arrow of time remains same in presence of charge.

4.2. Charged Vaidya Exterior

We can also consider a special case of the generalised Vaidya metric with the following mass function

$$m(u,R)=M\left(u\right)-{\displaystyle \frac{{\kappa }_N{Q}^2}{2(N-2){\mathcal{A}}_{N-2}{R}^{N-3}}},$$

(79)

with

$$M\left(u\right)={\left[\left({\displaystyle \frac{N-3}{2}}\right)Y^{N-3}\left(1+{\displaystyle \frac{{\dot{Y}}^2}{A^2}}-{\displaystyle \frac{{Y^{\prime }}^2}{B^2}}\right)+{\displaystyle \frac{{\kappa }_N{Q}^2}{2(N-2){\mathcal{A}}_{N-2}{R}^{N-3}}}\right]}_{\Sigma }.$$

(80)

This mass function simplifies the generalised Vaidya metric into the following form

$$\begin{array}{cc}\hfill d{s}_{+}^2=& -\left(1-{\displaystyle \frac{2M\left(u\right)}{(N-3)R^{N-3}}}+{\displaystyle \frac{{\kappa }_N{Q}^2}{(N-2)(N-3){\mathcal{A}}_{N-2}{R}^{2(N-3)}}}\right)d{u}^2\hfill \\ \hfill & -2dudR+R^{2}d{\Omega }_{N-2}^2.\hfill \end{array}$$

(81)

This is the metric for the higher dimensional charged Vaidya spacetime. In this case the matching condition at the boundary hypersurface $\Sigma$ remains the same as (68) and therefore, the corresponding dynamics of $f\left(t\right)$ are the same as in Table 2. Interestingly in the presence of null strings the matching condition simplifies further as the energy density of the null string exactly cancels the contribution of the charge present [61] and the matching condition follows the differential Equation (22) and the interior dynamics follows Table 1. As a result in this case also when a null string is present the gravitational arrow of time is opposite to that of the radiation. In fact for a general composite interior matter distribution the behaviour of the gravitational arrow of time remains same.

5. Concluding Remarks

In this paper, N-dimensional shear-free dissipative spherical collapse is discussed in detail with both unmodulated and modulated lapse function. The charged collapse is also taken into account in the latter portion of the work. To keep the calculations simple, we neglected pressure anisotropy and derived the pressure isotropy relation for N dimensions. For both (uncharged and charged) systems we derived the resulting differential equation of f on the $N-1$-dimensional boundary hypersurface $\Sigma$. For the first spacetime (neutral and with charge), we derived the solutions of f and presented them in tabular forms. The higher dimensional curvature scalars were computed and their functional behaviour in terms of matter variables are also discussed. We finally deduced the higher dimensional gravitational epoch functions and found that they decrease with collapse due to the positivity of the fourth order coefficient (associated with the fourth power of $\dot{f}/f$) in the curvature expansions for all dimensions $N\ge 4$.

Therefore, the strict conclusion of this study is the following:

• The orientation of the local gravitational arrow of time due to the WCH, measured by the epoch functions $\mathcal{P}$ and ${\mathcal{P}}_1$, remains opposite with respect to the thermodynamic arrow of time, in a shear-free, spherically symmetric, isotropic subclass of radiating collapse for all spatial dimensions $N\ge 4$.

We now discuss the consequences of our conclusion.

(1)

Essentially the entire study deals with the local application of the WCH in higher dimensions using the epoch functions. We have found that the Weyl curvature scalar derived functions like $\mathcal{P}$ and ${\mathcal{P}}_1$ fail to give a properly oriented local gravitational arrow of time. This leads us to two possibilities:

• Regarding the gravitational arrow of time, local application of the WCH is not valid, at least in shear-free collapse. As it does not change the orientation in higher dimensions also. That is, we should always think of WCH as a global concept regarding the arrow of time. This does not comment on the gravitational entropy aspect of the WCH and its local validity. We are only commenting on the concept of the time arrow due to gravity; i.e., in our case, we find that when using epoch functions, the local gravitational arrow of time is not dominated by the GE (Weyl curvature); instead, the Ricci component is the dominant driver of the arrow of time (unlike cosmology). Nevertheless, the concept of GE itself, i.e., free gravitational field carries entropy, remains robust and meaningful in both the limits, as shown in [10] and the subsequent works.
  OR,
• We need a better mathematical tool to understand and align the gravitational arrow of time properly, i.e., the entire discrepancy is due to the improper choice of epoch functions.

(2)

It is important to note that we have only considered the shear-free class of solutions in our study. The effect of shear in gravitational collapse was studied before in [92], where it was shown that a sufficiently strong shearing effect can delay the formation of horizon, leading to a naked singularity. Also in [93] it was concluded that in a shear-free collapse the end state always leads to a black hole. In absence of shear (respecting the usual energy conditions), no naked singularity can form as an end state of collapse. This implies that in a shear-free situation the Weyl curvature can never diverge faster than the Ricci curvature. As a result, the gravitational epoch functions $\mathcal{P}$ and ${\mathcal{P}}_1$ will always decrease. Consequently, in such a system, the arrow of time will always be in the opposite direction. Hence our current study gets validated, and is in conformity with the previously obtained results [49], indicating the important role of shear in the arrow of time problem.

Perhaps a general setting, i.e., a sufficient amount of shear, anisotropy and dissipation may change the behaviour of the epoch functions. It can also be a property of the spherical symmetry itself, and one may need to go beyond it to understand the problem. Essentially, we are looking for scenarios where the Weyl will dominate over the Ricci part. We leave these further studies for future work. A more thorough analysis of different systems may bring more understanding on this issue.