## On Emergent Gravity

Summary and overview of the main ideas in Erik Verlinde’s paper on emergent gravity.

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## On Emergent Gravity

## The Kerr Spacetime

## More on Schwarzschild Black Holes

## Schwarzschild Spacetime And Black Holes

## The Maths of General Relativity – An Overview

## General Relativity for Laypeople – A First Primer

## Manifolds and Curvature

Summary and overview of the main ideas in Erik Verlinde’s paper on emergent gravity.

A brief and very basic introduction is given to Kerr spacetime. The metric is presented, and the various horizon surfaces and their physical meanings are discussed. An overview of the geometric and topological structure of maximally extended Kerr spacetime is given, using the waterfall analogy. Frame dragging, closed time-like curves, naked singularities, and the cosmic censorship hypothesis are introduced.

We take a closer look at the maximally extended version of Schwarzschild spacetime, and investigate the geometry and topology beyond the event horizon, leading us to the notion of “white holes”. We discuss the concept of wormhole, its implications, and choices of coordinate system which allow us to quantify them. We then consider the laws of thermodynamics in the presence of event horizons, and find that Schwarzschild black holes have entropy and temperature, and undergo an evaporation process through the emission of thermal radiation. We introduce the holographic principle, and briefly discuss the implications it has for our understanding of the interior region enclosed by the horizon surface.

We introduce and discuss the Schwarzschild metric, at a level suitable for beginners with basic calculus knowledge. First, the exterior vacuum metric is examined, and several techniques about how to work with metrics are demonstrated, and their physical significance explained. A geometric understanding of the various terms in the metric is developed, and some fully worked examples are given. We then progress on to the interior metric, and present its peculiarities and physical consequences. This leads on to a discussion of Schwarzschild black holes, the meaning of event horizons, and what physically happens if we allow a test particle to freely fall into such black holes, from the points of view of different observers. Animations and a video are presented that help visualise these principles.

In this article we will be presenting an overview of the mathematical formulation and foundations of General Relativity. Basic objects such as the metric tensor and the connection are introduced, and given a geometric interpretation. The structure and meaning of the Einstein Field Equations will be discussed in more detail, a recipe for solving them will be presented, and the calculation of geodesics will be explained.

Prerequisits : Special Relativity for Laypeople, Special Relativity : The Rest Of The Story Recommended : Manifolds and Curvature The purpose of this article is to introduce the basic concepts of what GR is about, without getting lost in mathematics, but also without succumbing to the “easy way out”, as given by seemingly simple yet highly misleading […]

We introduce the basic concepts of differential geometry on manifolds. We define manifold, and explain the reasoning behind connections. This leads us on to the covariant derivative, and eventually to the Riemann curvature tensor, as well as the Ricci tensor. The geometric meaning of these objects is explained.