from worm holes to a bullet train
__________________________________________________________E Paramasivan
இந்த விண்வெளியில் ஈர்ப்பு கழன்ற வெளிகள் உண்டு.அவை ஈர்ப்பு தப்பிய வேகங்கள் கொண்டிருக்கும்.(எஸ்கேப் வெலாசிட்டி)
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Figure 9: Subcase 2(iv): Plots for NEC, WEC, SEC, DEC, ω & 4 with β = −0.5 18 (a) WEC (b) NEC (c) NEC (d) SEC (e) DEC (f) DEC (g) ω (h) 4 Figure 10: Subcase 2(v): Plots for NEC, WEC, SEC, DEC, ω & 4 with α = 0.5 4 Results and Perspectives The important energy conditions are the Null Energy Condition (NEC), Weak Energy Condition (WEC), Strong Energy Condition (SEC) and Dominated Energy Condition (DEC). These conditions are expressed as follows: (I) ρ + pr ≥ 0, ρ + pt ≥ 0 (II) ρ ≥ 0, ρ + pr > 0, ρ + pt > 0 (III) ρ + pr ≥ 0, ρ + pt ≥ 0, ρ + pr + 2pt ≥ 0 (IV) ρ ≥ 0, ρ − |pr| ≥ 0, ρ − |pt | ≥ 0 The wormholes are non-vacuum solutions of Einstein’s field equations and according to Einstein’s field theory, they are filled with a matter which is different from the normal matter and is known as exotic matter. This matter does not validate the energy conditions. In this paper, the wormhole solutions are explored with two different shape functions in f(R) theory of gravity. The function f(R) is taken as f(R) = R + αRm − βR−n , where m and n are arbitrary constants. The two shape functions are chosen as b(r) = r0 log(r + 1) log(r0 + 1) and b(r) = r0( r r0 ) γ . Both of the functions satisfy all the conditions discussed in Section 2. In this work, the energy conditions are investigated and, anisotropy parameter & equation of state parameter are analyzed based on two different shape functions in two cases. Since the function f(R) contains two parameters α and β, so according to the values of these parameters, both of the cases have been dealt with the following five subcases: (i) α = 0, β = 0, (ii) α-variable, β = 0, (iii) α = 0, β-variable, (iv) α-variable, β-non-zero constant, (v) α-non-zero constant, β-variable. For each shape function, Subcase (i) reduces to the general relativity, Subcase (ii), reduces to the Starobinsky model [11] with m = 2 which was introduced for the explanation of early 19 time inflation, Subcase (iii) leads to the form f(R) = R − βR−n used by Cao et al. [84] for the investigation of late time acceleration in the context of FRW universe, Subcase (iv) converts to f(R) = R + αR− 1 2 − βR1 2 with m = n = − 1 2 , α as variable and β as a non-zero constant, studied in [84], finally in Subcase (v) it takes the form f(R) = R + αR− 1 2 − βR1 2 for case 1 and f(R) = R + αR1 2 − βR1 2 for case 2 with α as constant and β as variable. The energy density ρ, various combinations of energy density and pressure i.e. ρ + pr, ρ + pt , ρ + pr + 2pt , ρ − |pr| and ρ − |pt |, equation of state parameter and anisotropy parameter are plotted for all five subcases of Case 1 in Figures 1-5 and for Case 2 in Figures 6-10 with respect to the radial coordinate r. For Case 1, in Subcase 1(i), the function adopts the form f(R) = R, which is free from parameters α and β. Therefore, two dimensional graphs are obtained in Fig. (1). In rest subcases, either α is variable or β is variable. That is why, three dimensional graphs are obtained in all other four subcases. In Subcase 1(i), ρ and ρ+pt are obtained to be positive and decreasing to zero with the variation of r (Figures 1(a) & 1(c)). However, ρ + pr and ρ − |pr| are come out be negative, increasing and tending towards zero with the increment in r (Figures 1(b) and 1(e)). One DEC term ρ − |pt | is decreasing for small range of r and then it is increasing, negative and tending to zero (Fig. 1(f)). One SEC term ρ + pr + 2pt is fluctuating between positive and negative values and going to zero with the increment of r (Fig. 1(d)). The anisotropy parameter is obtained to be positive (Fig. 1(h)). Thus, all the energy conditions are violated and geometry is repulsive in Subcase 1(i). In Subcase 1(ii), f(R) = R + αR2 . Varying the parameter α along with r, only the energy density is obtained to be positive and tending to zero (Fig. 2(a)). All other energy condition terms are found to negative with the variation of α and r (Figures 2(b)-2(f)). The anisotropy parameter possesses negative values with respect to radial coordinate (Fig. (2(h))). This subcase shows the violation of all energy conditions and the geometry to be of attractive in nature. In Subcase 1(iii), f(R) = R − βR−n . Taking n = −0.5 and varying β and r, ρ and ρ + pt are found to be positively decreasing functions of r (Figures 3(a), 3(c)). Other energy condition terms are found to be negatively decreasing functions of r (Figures 3(b), 3(d)-3(f)). The anisotropy parameter is negative value with respect to radial coordinate (Fig. 3(h)). This shows attractive nature of geometry and violation of all energy conditions. In Subcases 1(iv) and 1(v), the form of f(R) is f(R) = R+αR− 1 2 −βR1 2 . In Subcase 1(iv), taking β = −0.5 and α as variable, ρ and ρ+pt are found to be positive and decreasing functions with respect to r (Figures 4(a), 4(c)). All other combinations of pressure and energy density are found to be negative (Figures 4(b), 4(d)-4(f)). In Subcase (v), α is taken to be equal to -0.5 and β is varied. Then, again both ρ and ρ + pt are found to be positive and decreasing functions with respect to r (Figures 5(a), 5(c)) and other energy condition terms are come out to be negative (Figures 5(b), 5(d), 5(e), 5(f)). The anisotropy parameter has positive values with the increment in radial coordinate in both subcases 1(iv) and 1(v) (Figures 4(h), 5(h)). This represents that the geometry is repulsive in nature and all energy conditions are violated in both Subcases 1(iv) and 1(v). For Case 2, in Subcase 2(i), the graphs are drawn for f(R) = R. In Figures 6(a), 6(c) & 6(f), ρ, ρ+pt and ρ−|pt | are positive, decreasing and tending to zero as r tends to infinity. In Figures 6(b) and 6(e), ρ+pr and ρ−|pr| are found to be negative, increasing and tending towards zero with the increment in r. Like Subcase 1(i), SEC term ρ + pr + 2pt fluctuates between positive and negative values and then tends to zero as r increases (Fig. 6(d)). The anisotropy parameter is come out to be positive with radial coordinate (Fig. (6(h))). Thus, the geometry has repulsive nature and all the energy conditions are violated in Subcase 2(i). In Subcase 2(ii), for f(R) = R + αR2 , the 20 parameter α is varied with r. In this subcase, only the energy density is obtained to be positively decreasing function and tending to zero (Fig. 7(a)). However, all other energy condition terms are found to be negative with the change in α and r (Figures 7(b)-7(f)). The anisotropy parameter has negative values (Fig. (7(h))). Hence, the geometry is attractive and all the energy conditions are dissatisfied. In Subcase 2(iii), the function f(R) is read as f(R) = R−βR−n . Taking n = −0.5 and varying β and r, ρ, ρ + pt & ρ + pr + 2pt are found to be positively decreasing functions of r (Figures 8(a), 8(c), 8(d)). Other energy condition terms are found to have negative values with the increment in r (Figures 8(b), 8(e), 8(f)). The anisotropy parameter is obtained to be positive (Fig. 8(h)). This again shows the violation of all energy conditions and the geometry to be repulsive in nature. In Subcases 2(iv), the function f(R) has the form f(R) = R + αR− 1 2 − βR1 2 with β = −0.5 and α as a variable. Here, only ρ possesses positive values and decreases with respect to r (Figures 9(a)). However, all other combinations take negative values (Figures 9(b)-9(f)). The anisotropy parameter is obtained to negative with the radial coordinate (Fig. 9(h)). Thus, all energy conditions are dissatisfied and geometry is attractive in this subcase. In Subcase 2(v), f(R) = R + αR1 2 − βR1 2 with α = 0.5 and β as a variable. Here also, only ρ is positive and decreasing functions of r (Figure 10(a)) and other energy condition terms are negative (Figures 10(b)-10(f)). The anisotropy parameter is found to have negative values with the change in radial coordinate (Fig. 10(h)). This represents an attractive nature of geometry and violation of all energy conditions. In each subcase of both cases, the equation of state parameter is observed to have values less than -1 with the increment in the radial coordinate (Figures (1(g), 2(g), 3(g), 4(g), 5(g), 6(g), 7(g), 8(g), 9(g), 10(g))). Since the energy density is obtained to be positive in every subcase (Figures (1(a), 2(a), 3(a), 4(a), 5(a), 6(a), 7(a), 8(a), 9(a), 10(a))), therefore the value of the mass function, defined in Eq. 10, is also positive. Thus, no subcase satisfies any energy condition and hence strongly shows the presence of exotic matter which confirms the existence of wormholes in the universe. In this work, wormhole models are presented in f(R) gravity. The structure of wormholes is characterized by the choice of the shape function and this shape function needs to satisfy various conditions. Here, two shape functions satisfying all the desired properties are taken to investigate wormholes with the background of f(R) model in which the function f(R) is taken to be dependent on two parameters. Depending upon the values of these parameters, its five forms are dealt with for each type of shape function. The energy conditions are calculated and plotted. All these are found to be violated (Table 1). Since the existence of exotic or abnormal matter demands this violation, therefore the framed wormholes strongly confirm their existence in the universe. The value of equation of parameter is obtained to be less than -1 which specify the wormholes to be filled with phantom fluid. In both cases, the value of anisotropy parameter, that shows the nature of geometry, is found to have same nature for GR and Starobinsky models for both types of shape functions. In GR case, it shows repulsive nature, while in Starobinsky case, it represents attractive nature. In rest cases, the behavior of anisotropy parameter is different for different shape functions. Thus, every shape function has a significant role in determining the geometry, the type of filled fluid and the type of present matter in the context of wormholes.
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