Question

Show that \(g^{i j} g^{s r} c_{i k s}\) is antisymmetric in \(j\) and \(r\).

Short answer

The tensor \(T^{j r} = g^{i j} g^{s r} c_{i k s}\) has been shown to be antisymmetric in \(j\) and \(r\), fulfilling the antisymmetry condition \(T^{j r} = -T^{r j}\)

Step-by-step solution

Write down the given tensor components

We are given the tensor components \(g^{ij}\), \(g^{sr}\) and \(c_{iks}\)

Formulate the combined tensor

Let's combine these components to form a new tensor component \(T^{j r} = g^{i j} g^{s r} c_{i k s}\)

Exchange the indices

Now let's exchange \(j\) and \(r\) in the tensor \(T^{j r}\) to form \(T^{r j} = g^{i r} g^{s j} c_{i k s}\)

Use properties of metric tensor and Levi-Civita symbol

We recognize that metric tensors \(g^{ij}\) and \(g^{sr}\) are symmetric, hence \(g^{ir} = g^{ri}\) and \(g^{sj} = g^{js}\). However, the Levi-Civita symbol \(c_{iks}\) is antisymmetric under exchange of indices, so we get \(- c_{iks} = c_{iks}\). Therefore, \(T^{rj} = g^{r i} g^{j s} (-c_{iks}) = - T^{j r}\)

Conclude antisymmetry

We have shown that upon exchanging \(j\) and \(r\) in \(T^{j r}\) gives us the negation of it, hence demonstrating that \(T^{j r}\) is antisymmetric in \(j\) and \(r\)

Key concepts

Metric Tensor

In the world of differential geometry and general relativity, the metric tensor is a central concept that helps in understanding the geometry of space-time. It's a mathematical object that describes how distances and angles are measured in a given space. You can think of it as the ruler of the space that provides quantitative data concerning the curvature and layout of the space you're examining.

• Nature: The metric tensor is a symmetric tensor, which means the order of its indices doesn’t matter. For example, for a metric tensor denoted as \( g^{ij} \), it holds that \( g^{ij} = g^{ji} \).
• Purpose: Metric tensors allow us to calculate both the arc length of curves and the shortest distance between two points, known as geodesics.

Understanding the symmetry property here is important because it simplifies many calculations within the context of tensor algebra. When you hear about the 'raising' or 'lowering' of indices, the metric tensor is involved. It enables the conversion between covariant and contravariant components, which are critical in tensor calculations used throughout physics.

Levi-Civita Symbol

The Levi-Civita symbol, also known as the permutation symbol, is a mathematical tool that's quite useful in physics and engineering, especially when dealing with cross products and determinants. It's particularly critical in the field of tensor calculus.

• Nature: This symbol is an antisymmetric tensor, meaning swapping any two indices changes its sign. For instance, if you take \( \epsilon_{123} \), swapping any two of the indices will result in the opposite sign: \( \epsilon_{213} = -\epsilon_{123} \).
• Properties: The Levi-Civita symbol can be used to express vector products and rotational transformations. This antisymmetry is what makes it so powerful: if any two indices are the same, the symbol is zero, effectively eliminating unwanted components in certain operations.

Understanding antisymmetry in the Levi-Civita symbol is key, as it relates directly to the antisymmetric nature of other tensors or operations, such as the one discussed in this exercise.

Tensor Algebra

Tensor algebra is a broad and rich area of study that deals with operations over tensors, which are generalizations of scalars, vectors, and matrices. These operations are incredibly vital in areas such as physics, engineering, and even computer graphics.

• Basic Operations: You can add and multiply tensors much like how you handle matrices, but with added complexity because tensors can have many indices. It's crucial to keep track of these indices and their order.
• Index Gymnastics: Also known as tensor manipulation, this involves summing over repeated indices, a process that’s essential for simplifying equations. In tensor algebra, raising and lowering indices are typical operations performed by the innate action of the metric tensor.
• Applications: Tensor algebra is everywhere, from stress and strain analysis in materials to the description of electromagnetic fields.

In our original problem, understanding the combination of metric tensors with the Levi-Civita symbol within the framework of tensor algebra is essential for proving antisymmetry. It involves recognizing how the properties of these tensors and symbols come together to illustrate a broader concept.