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Selina Solutions Concise Maths Class 7 Chapter 2 Rational Numbers provides students with the basic idea of the method of solving them. Rational numbers are those numbers which can be expressed as the result of dividing an integer by a non zero integer. The main aim of preparing solutions is to help students solve textbook problems without any difficulty. Students who are not able to clear doubts during class hours can clear them by referring to the solutions. Students can effortlessly download the Selina Solutions Concise Maths Class 7 Chapter 2 Rational Numbers free PDF, from the links available here.

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The purpose of the present work is to construct some examples of spacetimes in f(R) gravity which admit gravitationally weak spacetime singularities [22]. The physical situation which we consider here is the following: the initial mass of the collapsing star or matter configuration is so high that the force of gravity overwhelms the thermal or quantum mechanical pressures. As a result, no stable configuration such as a neutron star or a white dwarf exists during the collapse process. Generically, such collapsing matter configurations admit diverging density and curvatures at the central singularity (we must emphasize that spherically symmetry is assumed throughout the collapse). However, we shall show below that it is possible, without violating energy conditions, that during the gravitational collapse matter is radiated away at such a rate that the matter boundary never reaches its horizon. In this case, no horizon is formed at the boundary since the star radiates off all its mass before reaching the singularity. The central singularity is naked but is gravitationally weak. During this process, all the physical quantities energy density, radial and tangential pressure, pressure anisotropy, heat flux, remain regular and positive throughout the collapse. The luminosity and adiabatic index are also regular and positive, and admits maximum value when the matter approaches the singularity. Thus, for an observer at infinity observing the collapse, the configuration shall become extremely bright, reaching its maximum luminosity before turning off, indicating that it has radiated off all its mass. Solutions of such kind are not unknown and is possible in the Newtonian gravity as well. Consider a star in the Newtonian gravity which is extremely heavy to be supported by the Pauli exclusion principle alone. So, when the gravitation contraction takes place, thermal pressure may balance to some extent. But since there is no event horizon, the star shall continue to radiate all the gravitational energy to infinity. Hence all the matter contained in the star shall be converted to thermal radiation and radiated off. We show here that configurations of similar nature are also possible in f(R) gravity.

In this paper, we shall not concern ourselves with compact objects like the white dwarf or a neutron star. We shall assume that the matter is extremely massive, and such objects continue to collapse under its own gravity. The gravitational collapse phenomenon in GR show that the collapse outcome depends upon, among other quantities, the choices of mass profiles and velocity profiles of the collapsing matter. In the context of inhomogeneous LTB models in GR, these issues have been considered in great detail for various matter models including dust and viscous fluids [30, 31]. The stellar collapse of stellar collapse in f(R) gravity using different forms of f(R) function and different matter distributions may be found in [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. Although different type of f(R) models may be considered, only the ones which are in agreement with the standard cosmological observations should be of interest. Here, we consider three models of f(R) gravity given respectively by: \((a)f(R)=(R+\lambda R^2)\) [47] , \((b) f(R)\sim R^1+\epsilon \), with \(\epsilon

a Plot of the expansion scalar \(\Theta \) (66) w.r.t. time t and radial r coordinates. At the beginning of collapse \(\Theta \) has zero value and it starts decreasing and remains negative throughout the collapse. b Plot of the mass of the collapsing star (67) w.r.t. time t, and it shows that mass radiates linearly

To show that these spacetime solutions are physically viable, we show that they satisfy the energy conditions as well. Indeed, all the energy conditions namely weak (W), null (N), dominant (D) and strong (S) hold good for the collapsing star. In the following we list these conditions [72, 74, 75]

The study of dynamical instability (stability) of spherical stellar system shows that for adiabatic index \(\Gamma 4/3\right) \) the stellar system becomes unstable (stable) as the weight of the stellar system increase much faster (remains less than) than that of its pressure [76]. Also, the causality condition imposes certain constraints on the dynamics of the stellar system such that inside the star, the radial \(V_r\) and the transverse \(V_t\) components of the speed of sound should be less than the speed of the light (\(c=1\)), so that \(0\le V_r\le 1\) and \(0\le V_t\le 1\) [77]. Thus, to check the stability/instability of the collapsing stellar system, we need to study the behavior of the important physical quantities, adiabatic index, sound of the speed which are defined as [50, 77, 78]

Although stability may be understood from the behavior of the pressure and density variables, the quantities in (51) and (52) are considered to be better to establish stability. For \(f(R)=R+\lambda R^2\) model with \(\lambda =5\), Figs. 7a and 11a shows that the total luminosity and the adiabatic index are positive and increasing. Note that the adiabatic index attains a maximum value where the luminosity is maximum. This behavior of the luminosity and adiabatic index can be interpreted as follows. Any static observer at asymptotic infinity will see an exponentially radiating radial source until a time when luminosity reaches its maximum value after which it instantaneously turn off. This is due to the fact that the total mass of the star radiates linearly as seen from the Fig. 6b and when the star reaches its maximum luminosity, there is no mass left to radiates and hence the observer at rest at infinity will see sudden turn off of the light source. The similar kind of behavior were obtained in [78]. Figure 11a shows that the effective adiabatic index is positive and less than 4/3 which implies that the considered stellar system is unstable and representing the collapsing scenario [76]. For \(f(R)=R^1+\epsilon \), and \(f(R)=R+\lambda \,\left[ \exp (-\sigma R)-1\right] \) models, similar behavior of luminosity is obtained as that of for the first model. Figure 11b shows that the effective adiabatic index is constant function of time, and is positive and less than 4/3, which implies it represents the collapsing scenario. As we have shown graphically that the star radiates all its mass before reaching at the singularity. So, there are no trapped surfaces formed during the collapse. Which implies that neither the black hole nor naked singularity are the end state of the collapse.

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