Tensor field

In mathematics and physics, a tensor field is a special kind of function that assigns a tensor to every point in a certain region of space. This space can be a mathematical space like a Euclidean space or a Riemannian manifold, or it can be the actual physical space we live in. Tensor fields are very important in many areas of science and math.

Tensor fields are used in differential geometry, algebraic geometry, and general relativity. They also help scientists understand things like stress and strain in materials. Since a tensor is a way to describe more complex information than a simple number (scalar) or a direction with strength (vector), a tensor field does the same but for every point in space.

For example, the Riemann curvature tensor is a famous tensor field. It tells us how space is curved at each point, which is a key idea in Einstein’s theory of how gravity works. Just like a scalar field gives a number to each point and a vector field gives a direction and strength to each point, a tensor field gives more detailed information. This makes tensor fields powerful tools in both theoretical and applied sciences.

Definition

A tensor field is a way to assign a tensor to every point in space. Think of it like giving each point a little package of numbers that describe something about that point.

Tensor fields are important in physics and math. They help describe things like how materials stretch or bend, and they are used in theories about space and time. They build on ideas of scalars (single numbers) and vectors (numbers with direction), making things even more detailed and powerful.

Geometric introduction

Imagine a vector field like a weather map that shows wind speed and direction at every point on Earth's surface — each point has an "arrow" representing the wind there.

Tensor fields are even more complex. They assign not just a single number or arrow, but a whole set of numbers or relationships to each point in space. These relationships can describe many things, like how distances change on curved surfaces, without needing to use specific map coordinates.

Via coordinate transitions

The idea of a tensor depends on having a reference frame, or coordinate system. This system can change in certain ways. For example, in a space with n dimensions, the coordinates can be shifted or stretched using special rules.

When we change these coordinates, different types of mathematical objects, like vectors, change in specific ways to keep certain quantities unchanged. This helps ensure that important calculations stay the same, no matter how we describe the space.

Tensor bundles

A tensor bundle is a special kind of structure called a fiber bundle, where each piece, or fiber, is made from combining copies of spaces called tangent space and cotangent space. These fibers are vector spaces, making the tensor bundle a type of vector bundle.

Tensor fields can be thought of as special sections of these tensor bundles. They assign a tensor — a mathematical object that generalizes numbers and vectors — to each point in a space. This approach keeps the math coordinate-free and focuses on the geometry of the space.

Notation

The notation for tensor fields can look similar to the notation for tensor spaces, which can be confusing. For example, the tangent bundle TM might be written in a special way to show that it relates to vector fields on a manifold M.

Sometimes, curly letters are used to represent the set of smoothly changing tensor fields on M. These tensor fields are parts of a tensor bundle and change smoothly from point to point.

Tensor fields as multilinear forms

Tensor fields can also be understood in a more abstract way. They can be seen as special kinds of maps that take vector fields and covector fields as inputs and give out smooth functions as outputs. This means that tensor fields behave in a very organized and predictable way at each point in space.

For example, a covector field, which is a type of tensor field, can act on a vector field to produce a smooth function. This action follows specific rules that make it linear and consistent across the whole space. These ideas help mathematicians and physicists describe complex relationships in geometry and physics using a clear and structured approach.

Applications

Tensor fields have many important uses in science and math. In physics, the stress–energy tensor helps explain how matter and energy curve space, as described by Einstein's theory of general relativity. In electromagnetism, electric and magnetic fields are combined into an electromagnetic tensor field. Differential forms, which are used to define integration on complex shapes, are also a type of tensor field.

Tensor calculus

In theoretical physics and other areas, differential equations that use tensor fields help describe relationships that are both geometric and tied to differential calculus. To create these equations, a special idea called the covariant derivative is needed. This idea helps explain how a tensor field changes along a vector field. The original idea, called absolute differential calculus, later became known as tensor calculus, and it helped discover the geometric idea of a connection.

Twisting by a line bundle

A tensor field can be extended by adding something called a line bundle. This helps create a special kind of tensor field called a tensor density. Tensor densities are useful in mathematics, especially when dealing with shapes that have a certain orientation.

One interesting use of tensor densities is in half-densities, where we can use values like one-half. These are important in advanced math topics like integral operators and geometric quantization.

Flat case

When we consider a special kind of space called Euclidean space and look at fields that stay the same even when we slide everything around, a tensor field acts like a tensor placed at a single starting point. This idea is often used in real-world applications, even though it has some limits when dealing with certain types of tensors known as tensor densities.

Cocycles and chain rules

The chain rule in multivariable calculus helps us understand how tensors behave when we change coordinates. It ensures that tensors are defined consistently across different coordinate systems, which is essential for working with tensor fields.

We can think of the chain rule as a special kind of mathematical object called a 1-cocycle. This helps mathematicians define important structures like the tangent bundle in a natural way. Other tensor structures also follow similar rules, making them intrinsic and geometrically meaningful.

Generalizations

Tensor densities

Main article: Tensor density

A tensor field can be extended by looking at objects that change in special ways. These are called tensor densities. They act like regular tensors but also change by a special factor depending on how we look at space.

One important example is the volume element, which helps measure space in physics, especially in theories like general relativity. It changes in a predictable way when we shift our view of space, making it useful for complex calculations.