Question

Show that the Jacobi identity can be written $$ \mathcal{L}_{[x, r]} Z=\mathcal{L}_{x} \mathcal{L}_{r} Z-\mathcal{L}_{r} \mathcal{L}_{x} Z $$ and this property extends to all tensors \(T\) :

Short answer

The Jacobi identity for Lie derivatives is verified through application of the properties of Lie brackets of vector fields and the Leibniz rule for the Lie derivative. This result extends to all tensors due to the derivation property of the Lie derivative on the algebra of tensor fields.

Step-by-step solution

Define tensors

First, identify the tensors in the given equation. In this case, x and r are vector fields and Z is a generic tensor.

Apply the Jacobi identity

The Jacobi identity is a property of the Lie bracket of vector fields, and it is commonly given by: \([x, [y, z]] + [z, [x, y]] + [y, [z, x]] = 0\). Based on this, we express the left-hand side in the following way: \(\mathcal{L}_{[x, r]} Z = \mathcal{L}_{x} \mathcal{L}_{r} Z-\mathcal{L}_{r} \mathcal{L}_{x} Z = \mathcal{L}_{x} \mathcal{L}_{r} - \mathcal{L}_{r} \mathcal{L}_{x} Z\).

Apply the properties of the Lie derivative

The Lie derivative obeys the Leibniz rule, which means \(\mathcal{L}_{x}(fg) = (\mathcal{L}_{x}f)g + f(\mathcal{L}_{x}g)\), where f and g can be any tensors. Apply this property for each term: \(\mathcal{L}_{[x, r]} = \mathcal{L}_{x} \mathcal{L}_{r} - \mathcal{L}_{r} \mathcal{L}_{x}\). Hence, \(\mathcal{L}_{[x, r]} = 0\), demonstrating The Jacobi identity.

Extension to other tensors

The Lie derivative \(\mathcal{L}\) is a derivation on the algebra of tensor fields. It therefore follows that the same properties that hold for Lie derivatives of vector fields also hold for Lie derivatives of arbitrary tensors. Thus, the result given by the proof extends to all tensors.

Key concepts

Lie derivative

The Lie derivative is a fundamental concept in differential geometry and is often used to examine how a tensor field changes along the flow of a vector field. It is particularly important in the study of symmetries and invariants in manifold theory. The Lie derivative of a tensor field can be thought of as a generalization of the directional derivative you might know from calculus, but specifically tailored for use on manifolds. Understanding this derivative allows us to see how one piece of geometric data changes when translated along the direction of a specified vector field.

The Lie derivative \( \mathcal{L}_x T \) measures the change of a tensor field \( T \) when moved along a vector field \( x \). This operation preserves the rank of the tensor, ensuring that a tensor of a particular type remains the same type after differentiation.

• It respects the Leibniz rule, which ensures that differentiation over products of functions behaves predictably, as seen with functions \( f \) and \( g \): \( \mathcal{L}_x(fg) = (\mathcal{L}_x f)g + f(\mathcal{L}_x g) \).
• It is linear, meaning \( \mathcal{L}_{x+y} T = \mathcal{L}_x T + \mathcal{L}_y T \).
• It gives us a tool to define the symmetry of vector fields.

The versatility of the Lie derivative means it appears in various contexts within physics and geometry, often underpinning the laws governing momentum, energy, and other pivotal physical quantities.

Vector fields

Vector fields are crucial in mathematics and physics as they assign a vector to each point of a space, such as regions of the plane, surfaces, or known spaces like three-dimensional Euclidean space. They provide a way to describe directional magnitudes or flow of various phenomena, like wind speeds across a surface or electromagnetic fields.

Mathematically, a vector field \( X \) on a manifold \( M \) is a function that assigns a tangent vector \( X_p \) to each point \( p \in M \). Understanding how vector fields interact and transform is essential to grasping complex systems in physics and geometry.

• Vector fields can be added: \( (X+Y)(p) = X(p) + Y(p) \).
• They can be multiplied by scalars, allowing control over their effect on their vector magnitude and direction.
• They can commute under certain conditions, expressed by the Lie bracket, \([X, Y]\).

In essence, vector fields allow us to encode more information about the structure of the space, and when coupled with operations such as the Lie derivative or Lie bracket, they enable deeper insights into geometric and physical frameworks.

Tensor algebra

Tensor algebra is the study of tensors, which are geometric entities that describe linear relations between geometric vectors, scalars, and other tensors. It generalizes scalars and vectors to higher dimensions, providing a framework to encode multivariate data and transformations. The algebra of tensors is vital for fields like physics, where describing complex interactions within a continuum framework requires an understanding that goes beyond simple vector arithmetic.

Key aspects of tensors include:

• Tensors are classified by type and rank, where the type indicates the number of vector and covector (dual vectors) components.
• Tensor operations adhere to rules consistent with vector spaces, such as addition, scalar multiplication, and contraction.
• They transform according to specific rules under coordinate changes, ensuring that tensor equations reflect consistent properties regardless of the reference frame.

The ability to manipulate tensors with algebraic operations makes them indispensable tools in formulating physical laws and studying the natural behavior of systems under deformation or other constraints.

Lie bracket

The Lie bracket is a vital operation in differential geometry and algebra, strongly connected to the study of smooth manifolds and Lie groups. Essentially, it measures the non-commutativity of vector fields and provides a way to investigate their structure. The Lie bracket \( [X, Y] \) of two vector fields \( X \) and \( Y \) produces a new vector field, representing the infinitesimal commutator of the flow generated by these fields.

This operation is key to several advanced concepts:

• The Lie bracket is anti-symmetric, meaning \( [X, Y] = -[Y, X] \).
• It satisfies the Jacobi identity: \( [X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = 0 \).
• It is used to define whether different vector fields commute, which is important in understanding integrable systems.

The Lie bracket operation not only helps in advancing our understanding of vector field interactions but also aids in creating symmetries in a manifold's differential structure, offering pathways to simplifying complex problems.