General Relativity Lecture 2

The first paragraph emphasizes the importance of proper notation in physics, particularly in the context of general relativity. It likens understanding notation to knowing which end of a stick fits into a hole, suggesting that once the rules are learned, physics can be done almost mindlessly. The rules in question pertain to tensor analysis and algebra, which are crucial for distinguishing between flat and non-flat (curved) geometries. The paragraph also touches on the concept of extrinsic curvature, which relates to how a space is embedded in a higher-dimensional space, as opposed to intrinsic geometry, which a 'tiny bug' moving on the surface would experience.

This paragraph delves into geometry, focusing on intrinsic properties of a space that are independent of its embedding in a higher-dimensional space. It discusses how the geometry is defined by specifying distances between points and how certain geometries can be flattened without changing these distances. The paragraph uses the example of a triangular lattice to illustrate how altering certain lengths can cause a space to curve. It also mentions the goal of understanding whether a gravitational field is real or an artifact of space-time coordinates, and how tensors and the metric tensor play a role in this understanding.

The third paragraph introduces tensors as objects that transform between coordinate systems and are crucial for describing quantities that vary from point to point in space. Scalars, which are special tensors, are introduced as quantities that do not change their value when coordinates are changed. The concept of contravariant and covariant vectors is also explained, with the former being related to the expansion of arbitrary vectors in terms of basis vectors, and the latter to the dot product of vectors with basis vectors. The paragraph highlights the importance of understanding these concepts to study the properties of space and space-time.

The fourth paragraph discusses the metric tensor in the context of defining the geometry of a space. It explains how the metric tensor is constructed from the basis vectors through dot products and how it can be used to calculate the length of a vector. The paragraph also touches on the idea of the metric tensor being used to determine whether a space is flat or curved by examining its components.

This paragraph explores the dot product of vectors and how it relates to contravariant components. It explains that in a coordinate system where vectors are not orthogonal and have non-unit separations, the coefficients of a vector's expansion are not the same as the dot products. The paragraph also introduces the concept of covariant components as being related to the gradient of a scalar field and how they transform under coordinate changes.

The sixth paragraph expands on the concept of tensors, focusing on higher-rank tensors, which are tensors with more indices. It explains that these tensors can be formed by multiplying vectors and that their transformation properties are a key characteristic. The paragraph also discusses how tensors transform under coordinate changes and how this transformation is indicative of their rank and the nature of their indices.

The seventh paragraph discusses the operations that can be performed on tensors, including multiplication by a scalar, addition, and contraction. It emphasizes that tensor equations have the property of being true in one frame if and only if they are true in every frame, highlighting the invariance of tensor operations. The paragraph also touches on covariant differentiation, which is a type of differentiation that will be further explored in the context of tensor calculus.

The eighth paragraph focuses on the metric tensor, proving that it is indeed a tensor by demonstrating its transformation properties. It shows that the metric tensor transforms as a tensor with two covariant indices, confirming its status as a legitimate tensor. The paragraph also discusses the metric tensor as a matrix and its symmetric properties.

The ninth paragraph introduces the concept of the inverse of the metric tensor, which is also a tensor. It explains how the inverse metric tensor is defined and its role in the multiplication of matrices. The paragraph also establishes that the metric tensor, with its inverse, allows for the construction of equations that hold true across different coordinate systems.

The final paragraph is a discussion that clarifies certain aspects of tensors and metric tensors. It addresses questions about the nature of the metric tensor, its eigenvalues, and the conditions under which it is positive definite. The paragraph also touches on the concept of a tangent space, where tensors live, and the implications of the metric tensor for different geometries, including its role in general relativity.