Question

Show that where there is a sum over an upper index and a lower index, swapping the upper index to a lower index, and vice versa, does not change the sum. In other words, \(A^{i} B_{i}=A_{i} B^{i}\).

Short answer

The index swapping property in Einstein summation convention is true. That's because the upper and lower positions of indices are interchangeable without affecting the sum, as long as upper indices always match with lower ones and vice versa. Therefore, \(A^i B_i = A_i B^i\).

Step-by-step solution

Define Einstein Summation Convention

In Einstein's Summation Convention, whenever an index is repeated once as a superscript (upper index) and once as a subscript (lower index) in a single term, this means we sum over all possible values of this index. So for example: \(A^i B_i\) represents the sum \(\sum_{i} A^i B_i\)

Expand The Equation

Expanding this equation with a concrete example can clarify the issue. Assuming simple 3-dimensional vectors, we can expand both sides of the equation as follows:Left-hand side: \(A^i B_i = A^1 B_1 + A^2 B_2 + A^3 B_3\)Right-hand side: \(A_i B^i = A_1 B^1 + A_2 B^2 + A_3 B^3\)Note that the indices 1, 2, 3 are placeholders for any possible value.

Compare The Results

Comparing both sides, it's clear that changing the positions of the indices doesn’t change the results as long as maintaining the matching of upper and lower index. Therefore, \(A^i B_i = A_i B^i\).

Key concepts

Mathematical Physics

Mathematical physics plays a crucial role in bridging the abstract concepts of mathematics with the empirical inquiries of physics. It involves the rigorous use of mathematical methods to formulate and solve problems inspired by physical phenomena.

In the world of mathematical physics, symbols are not just placeholders for numbers; they represent physical entities like forces, fields, or particles. To understand our universe’s fabric, mathematical physicists often rely on equations that can be extremely compact yet expansive in their implications.

One such efficient tool is the Einstein Summation Convention, which is a notational shorthand that simplifies the handling of equations involving summations of vector and tensor components. It is invaluable in fields like general relativity where calculations can be immensely complex. This convention not only streamlines the math but also provides a clear and consistent framework for formulating physical laws in a tensorial form.

Tensor Notation

Tensor notation is a powerful mathematical language used for expressing a wide variety of physical and geometric phenomena. Tensors themselves extend beyond simple scalars and vectors into objects that can encapsulate direction and magnitude across multiple dimensions.

The elegance of tensor notation lies in its ability to condense information. For instance, the Einstein Summation Convention used in tensor calculations eliminates the need for writing out long summation symbols. Instead of \( \sum_{i=1}^n A_i B^i \), we simply write \( A_i B^i \), which already implies summation over the index \(i\).

Understanding tensor notation is key to working with a multitude of physical laws and transformations that involve more than just scalar quantities. Its consistent use ensures clear communication among scientists and mathematicians, particularly when dealing with complex multi-dimensional data, and is a bedrock principle for advancements in mathematical physics.

Vector Space

A vector space is a fundamental concept in mathematics and physics that provides a structured way to handle vectors, which are objects characterized by magnitude and direction. Just as numbers can be added or multiplied, vectors can be combined (added or subtracted) and scaled, obeying certain rules that define the space.

In a vector space, individual vectors can represent a multitude of physical quantities such as velocity, force, or electric field. These spaces are equipped with operations like addition and scalar multiplication that satisfy specific properties, helping to make calculations uniform and predictable.

The rules of vector spaces are deeply intertwined with tensor notation and the Einstein Summation Convention. This framework allows us to succinctly represent physical laws that are often inherently vectorial or tensorial, ensuring that concepts like force vectors in Newtonian mechanics or field tensors in electromagnetism are consistent and mathematically rigorous.