Tensor Algebra Rules

Understanding tensor properties requires knowledge of how specific operations affect tensor rank and type. Key operations include:

• Outer Product: Combines tensors, increasing rank. E.g., rank (1,0) x rank (0,1) = rank (1,1).
• Contraction: Sums over matched indices (one upper, one lower), decreasing rank by 2. A complete contraction yields a scalar (rank 0).
• Addition: Only possible for tensors of identical rank and type.

Option 1: $A^\alpha B_\beta$ Tensor Analysis

This is the outer product of a contravariant vector $A^\alpha$ (type (1,0)) and a covariant vector $B_\beta$ (type (0,1)). The resulting object has type (1,0) + (0,1) = (1,1), which defines a tensor. Hence, $A^\alpha B_\beta$ is a tensor.

Option 2: $\frac{A^\alpha B_\beta}{A^\alpha B_\alpha}$ Tensor Analysis

The numerator $A^\alpha B_\beta$ is a tensor of type (1,1).

The denominator $A^\alpha B_\alpha$ represents a contraction. Summing over the index $\alpha$ between the contravariant $A^\alpha$ and covariant $B_\alpha$ results in a scalar (a tensor of type (0,0)).

Dividing a tensor by a scalar yields a tensor of the same type as the original tensor. Therefore, $\frac{A^\alpha B_\beta}{A^\alpha B_\alpha}$ is a tensor of type (1,1).

Option 3: $\frac{A^\alpha}{B_\beta}$ Tensor Analysis

This expression divides a contravariant vector $A^\alpha$ (type (1,0)) by a covariant vector $B_\beta$ (type (0,1)). Generally, dividing tensors component-wise does not produce a valid tensor transformation property. Since the vectors have different types, this operation is not defined as a tensor operation.

Option 4: $A^\alpha + B_\beta$ Tensor Analysis

This expression adds a contravariant vector $A^\alpha$ (type (1,0)) to a covariant vector $B_\beta$ (type (0,1)). Tensor addition requires the operands to have the exact same type and rank. As $A^\alpha$ and $B_\beta$ differ in type and rank, their sum is not a tensor.

Summary of Tensor Expressions

The expressions identified as tensors are $A^\alpha B_\beta$ and $\frac{A^\alpha B_\beta}{A^\alpha B_\alpha}$.