Question

A useful form of the quotient rule for three-dimensional vectors is this: Suppose that a and b are known to be three-vectors and suppose that for every orthogonal set of axes there is a \(3 \times 3\) matrix T with the property that \(\mathbf{b}=\mathbf{T}\) a for every choice of \(\mathbf{a},\) then \(\mathbf{T}\) is a tensor. (a) Prove this. (b) State and prove the corresponding rule for four-vectors and four-tensors.

Short answer

The matrix \( \mathbf{T} \) is a tensor because it obeys transformation rules in both three and four dimensions.

Step-by-step solution

Define the Problem

We need to prove that if there exists a matrix \( \mathbf{T} \) such that \( \mathbf{b} = \mathbf{T}\mathbf{a} \) for every choice of the vector \( \mathbf{a} \) in a three-dimensional space, then \( \mathbf{T} \) is a tensor. Additionally, we have to extend this result to four-vectors and four-tensors.

Use Tensor Transformation Properties

Recall that for a matrix to be a tensor, it must transform according to tensor transformation rules under changes of coordinates. Specifically, under a rotation of axis represented by the orthogonal matrix \( \mathbf{R} \), the vector \( \mathbf{a} \) becomes \( \mathbf{R}\mathbf{a} \), and \( \mathbf{b} = \mathbf{T}\mathbf{a} \) becomes \( \mathbf{R}\mathbf{b} \).

Prove for Three-dimensional Vectors

For \( \mathbf{b} = \mathbf{T}\mathbf{a} \), we must have \( \mathbf{R} \mathbf{b} = \mathbf{R} \mathbf{T} \mathbf{a} \). By continuity, this transforms as \( \mathbf{b}' = \mathbf{R}\mathbf{b} = \mathbf{R} \mathbf{T} \mathbf{R}^T \mathbf{R} \mathbf{a} \), suggesting \( \mathbf{T} \) is indeed a tensor because it transforms linearly with multiplication of orthogonal matrices. This proves \( \mathbf{T} \) obeys the transformation rules for three-tensors.

Extend the Concept to Four-dimensions

For four-vectors in a four-dimensional space (such as spacetime in relativity), the same principle extends. A four-vector \( \mathbf{A} \) transforms with a transformation matrix \( \mathbf{ar{T}} \) as \( \mathbf{B} = \mathbf{ar{T}} \mathbf{A} \). The condition is \( \mathbf{R}\mathbf{T} \mathbf{R}^T \) for the transformation in four-space.

Prove for Four-vectors and Four-tensors

With the same reasoning as for three-vectors, if \( \mathbf{B} = \mathbf{ar{T}} \mathbf{A} \) holds for all \( \mathbf{A} \), applying a Lorentz transformation matrix \( \mathbf{ar{R}} \) implies \( \mathbf{R}\mathbf{B} = \mathbf{R} \mathbf{ar{T}} \mathbf{R}^T \mathbf{R} \mathbf{A} \), proving that \( \mathbf{ar{T}} \) is also a four-tensor under such transformations.

Key concepts

Tensor Transformation Rules

Tensors are powerful mathematical objects that extend the concept of scalars and vectors. In order for a matrix to be classified as a tensor, it must adhere to specific transformation rules under a change of coordinates.

• For three-dimensional space, when coordinates are rotated by an orthogonal matrix, a true tensor transforms by sandwiching it between the rotation matrix and its inverse (transpose in the orthogonal case).
• This means if you start with a vector \(\mathbf{a}\) and apply a tensor \(\mathbf{T}\) resulting in vector \(\mathbf{b}\), under rotation by \(\mathbf{R}\), the resulting vector becomes \(\mathbf{R}\mathbf{T}\mathbf{R}^T\).
• Similarly, in four-dimensional spacetime, a four-tensor transforms utilizing Lorentz transformations which ensure the structure and relationships between components remain intact in spacetime.

Understanding these transformation rules is crucial in physics, notably in contexts like relativity, where the physical laws must be consistent across different observers and coordinate systems.

Three-dimensional Vectors

Vectors in three-dimensional space are represented using three components corresponding to the three axes (x, y, and z). They are a foundational element in both physics and engineering problems. Here's a closer look:

• The transformation of a 3D vector involves linear algebra operations, which means its components interact systematically when coordinates are rotated.
• For example, multiplying a vector \(\mathbf{a}\) by matrix \(\mathbf{T}\) changes it into another vector \(\mathbf{b}\). The tensor \(\mathbf{T}\) ensures the relationship between the original and new vectors stays consistent even if we rotate our reference frame.
• This principle is especially visible in mechanical systems or electromagnetic fields, where directions and magnitudes of phenomena must agree across orientations.

Three-dimensional vector mathematics is essential in modeling a vast array of real-world motion and forces.

Four-dimensional Vectors

Four-dimensional vectors extend the idea of three-dimensional vectors into an additional dimension, often time in a spacetime context. These vectors are central to the field of relativity.

• A four-vector encompasses three spatial components and one temporal component, representing quantities that are intrinsic to an event in space and time.
• Transformations involving four-vectors are executed using Lorentz matrices which maintain the spacetime intervals invariant, despite relative motion or coordinate system rotations.
• For instance, in special relativity, the energy and momentum of particles are represented as a four-vector. Ensuring transformation rules are met guarantees the laws of physics are uniform for all observers.

Four-dimensional vectors provide essential tools for describing interactions in the realm of high-speed and gravitationally intense environments.