Dynamical Wormholes and their Thermodynamics

Abstract

This thesis deals with static and dynamical traversable wormholes. We study both charged and uncharged versions of these wormholes and analyse them in general relativity and alternate theories of gravity. We investigate thermodynamic properties of these objects including the unified first law and generalized surface gravity. A two way traversable wormhole is a tunnel-like object comprising of trapped surfaces between horizons, defined as temporal outer trapping horizons. Usually, in a spacetime there are trapped, untrapped or marginal surfaces. On trapped surfaces both of the ingoing and outgoing light rays either converge or diverge, on untrapped surfaces one of the ingoing or outgoing light rays converges and the other diverges while on the marginal surfaces one or both of the ingoing or outgoing light rays remain constant (i.e., neither converge nor diverge but travel parallely). Trapping horizons are the hyper-surfaces foliated by marginal surfaces that may be past, future or bifurcating and further, outer, inner or degenerate. These trapping horizons coincide at the throat in static wormholes. For the purpose of studying thermodynamics, we have used a technique which was first developed in the literature for studying spherically symmetric black hole spacetimes. This technique uses a 2+2 formalism to derive the generalized surface gravity at a trapping horizon which becomes part of the first law of wormhole dynamics which is obtained from the unified first law by taking its projection along the trapping horizon. This unified first law is the rearrangement of Einstein field equations which can easily be generalized to 𝑓(𝑅, 𝑇) gravity, where 𝑅 is the Ricci scalar and 𝑇 is the trace of the stress-energy tensor, and non-minimal curvature-matter coupling where the equations, when written in the form of the Einstein tensor, replace the role of stress-energy tensor with an effective stress-energy tensor. Chapter 1 is about some basic concepts that are related with the main subject of the thesis. In Chapter 2 we have reviewed the Hayward formalism and its application to the Morris-Thorne wormholes in Einstein’s gravity. The generalized surface gravity, iii unified first law of thermodynamics and wormhole dynamics have been studied at (bifurcating) trapping horizons. We work out the generalized surface gravity for wormholes of different shapes as well. Thermodynamic stability of Morris-Thorne wormholes has been discussed in GR. We have also investigated thermodynamics in non-minimal curvature-matter coupling which produces very complex equations. The extension of this work to 𝑓(𝑅, 𝑇) gravity has been done for Morris-Thorne wormholes. Chapter 3 deals with thermodynamics of charged wormholes, which are static as well as spherically symmetric. The electric charge acts as additional matter to the Morris-Thorne wormhole which is already constructed by exotic matter. All the anal ysis (unified first law, thermodynamic stability and generalized surface gravity) done in Chapter 2 for Morris-Thorne wormholes is generalized to the charged wormholes in this chapter. In the absence of electric charge the results that have been derived in Chapter 2 can be recovered. In Chapter 4 we study thermodynamics of dynamical traversable wormholes. We considered uncharged dynamical wormholes which are the time generalization of static Morris-Thorne wormholes. These wormholes are investigated in the background of different cosmological models, with and without the cosmological constant, and which include the power-law and exponential cosmologies also. The generalized surface grav ity is evaluated at the trapping horizon and the unified first law of thermodynamics is set up. The trapping horizon in this case is not bifurcating but a past trapping horizon which does not coincide with the throat of the wormhole and it corresponds to the expanding universe. The thermodynamic stability of these wormholes has also been investigated. Some cases of asymptotically flat, de Sitter and anti-de Sitter wormholes have been considered as well. We have also extended the results from un charged to charged dynamical wormholes. All the work done for uncharged dynamical wormholes has been generalized to charged dynamical wormholes. In the absence of charge the results derived for uncharged dynamical wormholes can be recovered. We summarize our results and conclude the thesis in Chapter 5