Question

Show that a vector field that generates a conformal transformation satisfies $$ X^{k} \partial_{k} g_{i j}+\partial_{i} X^{k} g_{k j}+\partial_{j} X^{k} g_{k i}=-\psi g_{i j} $$.

Short answer

The vector field \(X^{k}\) which generates a conformal transformation indeed satisfies the given equation, when a factor of 2 is considered in the conformal transformation condition.

Step-by-step solution

Understand the variables in the equation

In the given equation, \(X^{k}\) represents a vector field, \(\partial_{k}\) is the partial derivative with respect to the k-coordinate, \(g_{ij}\) is the metric tensor, and \(\psi\) is a scalar function. The indices i, j, and k run over the number of dimensions. The object of this exercise is to show that the vector field which generates a conformal transformation satisfies this equation.

Apply the Lie derivative

In this step we compute the Lie derivative of the metric \(g_{ij}\), given a vector field \(X^{k}\), which results in \(\mathcal{L}_{X} g_{ij} = X^{k} \partial_{k} g_{i j}+\partial_{i} X^{k} g_{k j}+\partial_{j} X^{k} g_{k i} \). This is simply the application of the definition of the Lie derivative, which describes how a tensor field changes under a small displacement by the vector field.

Compute the requirement of conformal transformation

A conformal transformation is defined as a transformation that preserves angles but not necessarily lengths. For a transformation generated by a vector field \(X^{k}\) to be conformal, it must satisfy \(\mathcal{L}_{X} g_{ij} = -2 \psi g_{ij}\). This means that up to a scaling factor (\(-2\psi\), where \(\psi\) is a function of spacetime), the metric remains the same.

Compare and Conclude

Since we have deduced that \(\mathcal{L}_{X} g_{ij} = X^{k} \partial_{k} g_{i j}+\partial_{i} X^{k} g_{k j}+\partial_{j} X^{k} g_{k i}\), and for a conformal transformation, \(\mathcal{L}_{X} g_{ij} = -2 \psi g_{ij}\), equating both results, we find that \(X^{k} \partial_{k} g_{i j}+\partial_{i} X^{k} g_{k j}+\partial_{j} X^{k} g_{k i} = -2 \psi g_{i j}\). The given equation in the exercise was \(X^{k} \partial_{k} g_{i j}+\partial_{i} X^{k} g_{k j}+\partial_{j} X^{k} g_{k i} = -\psi g_{i j}\), which is the same, up to a factor of 2 compared to our result, it is evident that the equation lines up with the requirements for a vector field generating a conformal transformation.

Key concepts

Vector Fields

In mathematics and physics, a vector field is a construction in which every point in space is associated with a vector. Visualize it as a collection of arrows where each arrow has a direction and a magnitude at every point.
They are essential in vector calculus and have numerous applications in various branches of science, including electromagnetism and fluid dynamics.
Vector fields are used to model a variety of phenomena, such as:

• The direction of velocity in fluid flow.
• The strength and direction of magnetic fields.
• The gradient of potential functions in fields of force.

In the context of conformal transformations, a vector field, denoted as \(X^{k}\), helps in generating transformations that preserve angles but may alter magnitudes.
The role of the vector field is to transform the coordinates or variables, maintaining the necessary mathematical properties, such as those dictated by the Lie derivative.

Metric Tensor

The metric tensor is a fundamental concept in differential geometry and general relativity. It serves as a mathematical tool that allows one to measure distances and angles in a given space. In general relativity, it is essential for describing the curvature of spacetime.
Represented typically by \(g_{ij}\), it encodes all the geometric and causal structure of spacetime.
The properties of the metric tensor include:

• Symmetry: The metric tensor is symmetric, meaning \(g_{ij} = g_{ji}\).
• Non-degeneracy: It has a non-zero determinant, ensuring that it is invertible.
• Signature: The choice of signature of the metric tensor determines the type of geometry (i.e., Riemannian or Lorentzian).

In the exercise, the metric tensor is central to understanding how it changes under the influence of a vector field during a conformal transformation.
The transformation should preserve the angles as shown through the equation involving the Lie derivative, with a function \(\psi\) scaling the metric tensor, indicating the alteration of its length.

Lie Derivative

The Lie derivative is a concept used to describe how tensor fields, like vector fields or the metric tensor, change along the flow of another vector field. It can be thought of as a generalized derivative that incorporates both the change of the field and its movement through space.
For a metric tensor \(g_{ij}\) and a vector field \(X^{k}\), the Lie derivative \(\mathcal{L}_{X} g_{ij}\) is given by the expression:\[\mathcal{L}_{X} g_{ij} = X^{k} \partial_{k} g_{i j} + \partial_{i} X^{k} g_{k j} + \partial_{j} X^{k} g_{k i}\]Key points about the Lie derivative:

• It captures how a tensor field changes as we "drag" it along the flow of a vector field \(X^{k}\).
• It allows us to check symmetries of spacetime in general relativity.
• For conformal transformations, it facilitates the understanding of how angles are preserved while scaling occurs.

In the specific context of the exercise, the equation involving the Lie derivative shows the necessary adjustments needed for the vector field \(X^{k}\) to generate a conformal transformation.
This transformation is encapsulated by having the change in the metric tensor expressed in terms of a scaling factor \(\psi\).
Understanding these key derivatives enables us to describe precise conformal transformations in mathematical frameworks.