Question

Using the fact that \(d s^{2}\) is invariant, prove that \(g_{i j}\) is a second- order covariant tensor.

Short answer

By showing that \( g_{i j} = g'_{i j} \( \frac{\text{∂}x'^{i}}{\text{∂}x^k} \) \( \frac{\text{∂}x'^{j}}{\text{∂}x^l} \) \, it is proved that \( g_{i j} \) is a second-order covariant tensor.

Step-by-step solution

- Understand Invariance of ds²

The line element in a given coordinate system is expressed as: d s^{2} = g_{i j} d x^{i} d x^{j}d s^{2} is invariant under a coordinate transformation, meaning it has the same value regardless of the coordinate system used.

- Coordinate Transformation

Consider a coordinate transformation from coordinates \(x^i\) to new coordinates \(x'^{i}\). Under this transformation, differentials transform as:d x'^{i} = \( \frac{\text{∂}x'^{i}}{\text{∂}x^j} \) d x^{j}Let's denote the transformation matrix components as \( \frac{\text{∂}x'^{i}}{\text{∂}x^j} \).

- Apply Transformation to ds²

Substitute the transformed differentials into the line element:d s'^{2} = g'_{i j} d x'^{i} d x'^{j}Since \(d s^{2}\) is invariant:d s^{2} = d s'^{2}Then:g_{i j} d x^{i} d x^{j} = g'_{i j} \( \frac{\text{∂}x'^{i}}{\text{∂}x^k} \) \( \frac{\text{∂}x'^{j}}{\text{∂}x^l} \) d x^{k} d x^{l}

- Equate Coefficients

Since the terms with \(d x^{k} d x^{l}\) are invariant, equate the coefficients of the differentials on both sides:g_{k l} = g'_{i j} \( \frac{\text{∂}x'^{i}}{\text{∂}x^k} \) \( \frac{\text{∂}x'^{j}}{\text{∂}x^l} \)

- Definition of Covariant Tensor

The transformation property \( g_{k l} = g'_{i j} \( \frac{\text{∂}x'^{i}}{\text{∂}x^k} \) \( \frac{\text{∂}x'^{j}}{\text{∂}x^l} \) \) indicates that \( g_{i j} \) transforms according to the rules of a second-order covariant tensor.

Key concepts

invariant line element

The line element, denoted as \( d s^{2} \), is a fundamental concept in differential geometry and general relativity. It quantifies the infinitesimal distance between two points in a given coordinate system. Mathematically, it is expressed as:

d s^{2} = g_{i j} d x^{i} d x^{j}

Here, \( g_{i j} \) represents the metric tensor, and \(d x^{i}\) are the differentials of the coordinates.

The most crucial property of the line element is its invariance under coordinate transformations. This means that regardless of how we change or transform our coordinate system, the value of \(d s^{2}\) remains the same. This invariance is essential for ensuring that the physical distances and angles calculated are consistent in all coordinate systems.

coordinate transformation

In physics and mathematics, a coordinate transformation is the process of converting coordinates from one system to another. Consider transforming from coordinates \(x^i\) to new coordinates \(x'^{i}\). Under such a transformation, the differentials also transform. This transformation can be expressed as:

d x'^{i} = \( \frac{\text{∂} x'^{i}}{\text{∂} x^j} \) d x^{j}

Here, the matrix of partial derivatives \( \frac{\text{∂}x'^{i}}{\text{∂}x^j} \) represents the transformation matrix components. This matrix essentially describes how each coordinate in the new system is a function of the old system's coordinates.

Through such transformations, we can analyze how different physical quantities change and ensure that fundamental properties, like the line element, remain constant.

transformation matrix

At the heart of any coordinate transformation lies the transformation matrix. This matrix consists of the partial derivatives of the new coordinates concerning the old coordinates:

\( \frac{\text{∂} x'^{i}}{\text{∂} x^j} \).

The transformation matrix plays a crucial role in transforming vectors and tensors between different coordinate systems. It essentially tells us how to rewrite the old coordinates in terms of the new ones. When applied to the metric tensor and other tensors, they follow specific transformation rules.

To apply a transformation to the line element, we substitute the transformed differentials back into the original expression:

d x'^{i} = \frac{\text{∂} x'^{i}}{\text{∂} x^k} x^k

This results in the new line element, which, when simplified, reaffirms the invariance of \(d s^{2}\).

tensor transformation properties

Tensors have specific transformation properties that define how they change under coordinate transformations. For a second-order covariant tensor, like the metric tensor \( g_{i j} \), the transformation follows a particular rule:

\( g_{k l} = g'_{i j} \frac{\text{∂} x'^{i}}{\text{∂} x^k} \frac{\text{∂} x'^{j}}{\text{∂} x^l} \)

This equation states that the components of the tensor in the new coordinate system (\( g'_{i j}\)) are related to the components in the old coordinate system (\( g_{k l}\)) through the transformation matrix. Essentially, the metric tensor components adjust themselves to ensure the line element's invariance. Such transformation properties are fundamental to the definition of a tensor. They ensure that tensors are consistent representations of physical quantities in any coordinate system. Whether dealing with first-order, second-order, or higher-order tensors, understanding these transformation properties is crucial for correctly interpreting and applying tensorial mathematics in physical contexts.