Here I develop the connection between thermodynamics, entanglement, and gravity. I begin by showing that the classical null energy condition (NEC) can arise as a consequence of the second law of thermodynamics applied to local holographic screens. This is accomplished by essentially reversing the steps of Hawking's area theorem, leading to the Ricci convergence condition as an input, from which an application of Einstein's equations yields the NEC. Using the same argument, I show logarithmic quantum corrections to the Bekenstein-Hawking entropy formula do not alter the form of the Ricci convergence condition, but obscure its connection to the NEC. Then, by attributing thermodynamics to the stretched horizon of future lightcones -- a timelike hypersurface generated by a collection of radially accelerating observers with constant and uniform proper acceleration -- I derive Einstein's equations from the Clausius relation. Based on this derivation I uncover a local first law of gravity, connecting gravitational entropy to matter energy and work. I then provide an entanglement interpretation of stretched lightcone thermodynamics by extending the entanglement equilibrium proposal. Specifically I show that the condition of fixed volume can be understood as subtracting the irreversible contribution to the thermodynamic entropy. Using the AdS/CFT correspondence, I then provide a microscopic explanation of the 'thermodynamic volume' -- the conjugate variable to the pressure in extended black hole thermodynamics -- and reveal the super-entropicity of three-dimensional AdS black holes is due to the gravitational entropy overcounting the number of available dual CFT states. Finally, I conclude by providing a recent generlization of the extended first law of entanglement, and study its non-trivial 2+1- and 1+1-dimensional limits. This thesis is self-contained and pedagogical by including useful background content relevant to emergent gravity.

In this dissertation, I present the results from my recent

investigations into the interactions involving topological defects, such as

magnetic monopoles and strings, that may have been produced in the early

universe. I performed numerical studies on the interactions of twisted

monopole-antimonopole pairs in the 't Hooft-Polyakov model for a range of

values of the scalar to vector mass ratio. Sphaleron solution predicted by

Taubes was recovered, and I mapped out its energy and size as functions of

parameters. I also looked into the production, and decay modes of $U(1)$ gauge

and global strings. I demonstrated that strings can be produced upon evolution

of gauge wavepackets defined within a certain region of parameter space. The

numerical exploration of the decay modes of cosmic string loops led to the

conclusions that string loops emit particle radiation primarily due to kink

collisions, and that their decay time due to these losses is proportional to

$L^p$, where $L$ is the loop length and $p \approx 2$. In contrast, the decay

time due to gravitational radiation scales in proportion to $L$, and I

concluded that particle emission is the primary energy loss mechanism for loops

smaller than a critical length scale, while gravitational losses dominate for

larger loops. In addition, I analyzed the decay of cosmic global string loops

due to radiation of Goldstone bosons and massive scalar ($\chi$) particles.

The length of loops I studied ranges from 200-1000 times the width of the

string core. I found that the lifetime of a loop is approximately $1.4L$. The

energy spectrum of Goldstone boson radiation has a $k^{-1}$ fall off, where $k$

is the wavenumber, and a sharp peak at $k\approx m_\chi/2$, where $m_\chi$ is

the mass of $\chi$. The latter is a new feature and implies a peak at high

energies (MeV-GeV) in the cosmological distribution of QCD axions.