Chapter 1: Introduction

In the presence of strong gravitational fields caused by massive objects, light no longer moves in straight lines but instead follows curved paths. This phenomenon is described by general relativity. In this report, we begin by learning fundamental aspects of general relativity to understand spacetime curvature and the tensor calculus required to solve the Einstein field equations. The solutions to these equations encode the curvature of spacetime geometries, leading to an equation known as the line metric.

In Chapter 3, we examine the Schwarzschild geometry, which describes the spacetime around a non-rotating, charge-less black hole. The Schwarzschild line metric is derived using tensor calculus and is then used to formulate a system of first-order differential equations that govern the motion of photons in this geometry. These equations are implemented in a ray tracing algorithm that follows photons backwards in time, from a camera to a distant star field, producing distorted images of Schwarzschild black holes. The resulting images are analyzed through the lens of gravitational lensing, explaining the appearance of the Einstein ring and the event horizon based on photon energy levels.

Various wormhole geometries are discussed in Chapter 5, with a focus on the three-parameter wormhole created by the Double Negative Visual Effects team for the movie Interstellar ^[14]. This model was designed to produce scientifically accurate images for the film. We generate embedded diagrams to visualize the shapes of these wormholes and implement algorithms to simulate their appearances. A comparative analysis of different wormhole configurations follows, highlighting their distinct properties.

1.1 Notation

For simplification, the speed of light and the gravitational constant are set to:

\( c=1, \quad G=1 \)

so that:

\( t = ct, \quad M = \frac{GM}{c^2} \)

where both time and mass are measured in meters. This convention follows Shultz ^[1], Chapter 1.3.

Throughout this report, we use Einstein notation, where repeated Greek indices in expressions imply summation over all possible values. For instance :

\( ds^2 = g_{\alpha\beta}(x) dx^\alpha dx^\beta \)

is summed over all values of \( \alpha, \beta \), when they take values \( \{0,1,2\} \), resulting in a total of 9 terms. The first few terms are expanded as follows:

\[ ds^2 = g_{00}dx^0dx^0 + g_{01} dx^0dx^1 + g_{02}dx^0dx^2 \\ \quad + g_{10}dx^1dx^0 + g_{11}dx^1dx^1 + \dots \]