General relativity revolutionized our understanding of gravity, describing it as the curvature of spacetime. This chapter explores how spacetime is mathematically represented as a Lorentzian manifold, setting the stage for Einstein's groundbreaking theory.

Lorentzian manifolds provide the framework for modeling curved spacetime. We'll dive into the metric tensor, which defines distances and angles, and explore how different types of vectors represent particles and light in this four-dimensional fabric of reality.

Lorentzian Manifold Basics

Foundations of Spacetime Geometry

• Lorentzian manifold forms the mathematical foundation for describing spacetime in general relativity
• Consists of a smooth manifold equipped with a Lorentzian metric tensor
• Generalizes the concept of Minkowski space to curved spacetime
• Allows for the mathematical representation of gravitational effects as curvature in the fabric of spacetime

Metric Tensor and Signature

• Metric tensor defines the geometry and causal structure of spacetime
• Represented as a symmetric, non-degenerate bilinear form on the tangent space at each point
• Assigns a scalar product to pairs of tangent vectors, determining distances and angles
• Lorentzian metric has a signature of $(-,+,+,+)$ or $(+,-,-,-)$ in 4-dimensional spacetime
• Signature distinguishes Lorentzian manifolds from Riemannian manifolds, which have a positive-definite metric

Properties and Applications

• Enables the formulation of Einstein's field equations in general relativity
• Allows for the description of gravitational phenomena such as black holes and gravitational waves
• Provides a framework for studying the global structure of spacetime, including singularities and event horizons
• Facilitates the analysis of geodesics, which represent the paths of freely falling particles in curved spacetime

Vector Types in Spacetime

Timelike Vectors and Worldlines

• Timelike vectors have negative length under the Lorentzian metric
• Represent the direction of time in the local reference frame
• Correspond to the trajectories of massive particles moving slower than the speed of light
• Timelike curves (worldlines) connect events that can be causally related
• Future-directed timelike vectors point towards the future light cone

Spacelike Vectors and Simultaneity

• Spacelike vectors have positive length under the Lorentzian metric
• Represent spatial directions orthogonal to the time direction
• Connect events that cannot be causally related (outside each other's light cones)
• Spacelike hypersurfaces define notions of simultaneity in different reference frames
• Spacelike separated events can occur in different temporal orders for different observers

Null Vectors and Light Propagation

• Null vectors have zero length under the Lorentzian metric
• Represent the paths of light rays in spacetime
• Form the boundary between timelike and spacelike vectors
• Null geodesics describe the trajectories of massless particles like photons
• Light cones are formed by the set of all null vectors at a given event

Causal Structure and Light Cones

Causal Relationships in Spacetime

• Causal structure defines the possible cause-and-effect relationships between events
• Determines which events can influence or be influenced by other events
• Preserved under Lorentz transformations and general coordinate transformations
• Plays a crucial role in understanding the global properties of spacetime
• Helps in formulating concepts like Cauchy surfaces and global hyperbolicity

Light Cone Geometry and Causality

• Light cone represents the set of all possible light rays passing through a given event
• Divides spacetime into three regions: future, past, and elsewhere
• Future light cone contains all events that can be causally influenced by the given event
• Past light cone includes all events that could have causally influenced the given event
• Events outside both light cones (elsewhere) are causally disconnected from the given event
• Shape of light cones can be distorted in curved spacetime, leading to phenomena like gravitational time dilation