Thermodynamics of shearing massless scalar field spacetimes is inconsistent with the Weyl curvature hypothesis

Daniele Gregoris, Yen Chin Ong, and Bin Wang

Phys. Rev. D 102, 023539 – Published 27 July 2020

Abstract

Our Universe has an arrow of time. In accordance with the second law of thermodynamics, entropy has been increasing ever since the big bang. The fact that matter is in thermal equilibrium in the very early Universe, as indicated by the cosmic microwave background, has led to the idea that gravitational entropy must be very low in the beginning. Penrose proposed that gravitational entropy can be quantified by the Weyl curvature, which increases as structures form. A concrete realization of such a measure is the Clifton-Ellis-Tavakol gravitational entropy, which has been shown to be increasing in quite a number of cosmological models. In this work, we show a counterexample involving a class of inhomogeneous universes that are supported by a chameleon massless scalar field and exhibit anisotropic spacetime shearing effects. In fact, in our model the Clifton-Ellis-Tavakol gravitational entropy is increasing although the magnitude of the Weyl curvature is decreasing; this is due to the growth of the spacetime shear. The topology and the values of the three free parameters of the model are constrained by imposing a positive energy density for the cosmic fluid, and the thermodynamical requirements which follow from the cosmological holographic principle and the second law. It is shown that a negative deceleration parameter and a time-decreasing Weyl curvature automatically follow from those conditions. Thus, we argue that our model can account for the formation of some primordial structures, like the large quasar groups, which has required a nonstandard evolution of the spatial anisotropies.

Received 30 April 2020
Accepted 7 July 2020

DOI:https://doi.org/10.1103/PhysRevD.102.023539

Physics Subject Headings (PhySH)

Evolution of the Universe Relativistic aspects of cosmology

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Daniele Gregoris ^*, Yen Chin Ong ^†, and Bin Wang^‡

Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou City, Jiangsu Province 225002, People’s Republic of China and School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

^*danielegregoris@libero.it
^†ycong@yzu.edu.cn
^‡wang_b@sjtu.edu.cn

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Vol. 102, Iss. 2 — 15 July 2020

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Images

Figure 1
The figure depicts the time evolution of the quantity $\tilde{σ} ≔ 3 r^{2} σ^{2}$ computed from (18) for the universe (4) at the location $r = 1$ , and in units such that $c = 1$ . In panel (a) we choose $A = 0.50$ and $B = 0.01$ (red line), $B = 0.03$ (black line), $B = 0.05$ (green line), $B = 0.07$ (yellow line), and $B = 0.09$ (purple line); in panel (b) we choose $B = 0.50$ and $A = 0.01$ (red line), $A = 0.03$ (black line), $A = 0.05$ (green line), $A = 0.07$ (yellow line), and $A = 0.09$ (purple line). We remark that our choices for the numerical values of the free parameters are consistent with (15). In Sec. 4b we show that our model is physically acceptable in light of the second law of thermodynamics only at times $t > \frac{1}{2 c} \ln \frac{B}{A}$ , for which we can see that the shear would be monotonically increasing.
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Figure 2
The figure depicts the time evolution of the quantity $\tilde{W} ≔ 3 r^{2} | Ψ_{2} |$ computed from (24) for the universe (4) at the location $r = 1$ , and in units such that $c = 1$ . In panel (a) we choose $A = 0.50$ and $B = 0.01$ (red line), $B = 0.03$ (black line), $B = 0.05$ (green line), $B = 0.07$ (yellow line), and $B = 0.09$ (purple line); in panel (b) we choose $B = 0.50$ and $A = 0.01$ (red line), $A = 0.03$ (black line), $A = 0.05$ (green line), $A = 0.07$ (yellow line), and $A = 0.09$ (purple line). We remark that our choices for the numerical values of the free parameters are consistent with (15). In Sec. 4b we show that our model is physically acceptable in light of the second law of thermodynamics only at times $t > \frac{1}{2 c} \ln \frac{B}{A}$ , for which we can see that the Weyl curvature would be monotonically decreasing. Comparing with Fig. 1 we can understand that in this model of the universe the shear is increasing when the Weyl curvature is decreasing and vice versa.
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Figure 3
The figure depicts the time evolution of the quantity $\tilde{S} ≔ T_{grav} {\dot{s}}_{grav} / d V$ computed from (61) for the universe (4) at the location $r = 1$ , and in units such that $c = 1 = 8 π G$ . In panel (a) we choose $A = 0.50$ and $B = 0.01$ (red line), $B = 0.03$ (black line), $B = 0.05$ (green line), $B = 0.07$ (yellow line), and $B = 0.09$ (purple line); in panel (b) we choose $B = 0.50$ and $A = 0.01$ (red line), $A = 0.03$ (black line), $A = 0.05$ (green line), $A = 0.07$ (yellow line), and $A = 0.09$ (purple line). We remark that our choices for the numerical values of the free parameters are consistent with (15). In Sec. 4b we have shown that our model is physically acceptable in light of the second law of thermodynamics only at times $t > \frac{1}{2 c} \ln \frac{B}{A}$ , for which we can see that the gravitational entropy would be monotonically increasing (because its first time derivative is positive).
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Physical Review D

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