Find the number of independent components of an antisymmetric tensor of rank 2 in
dimensions
The Correct Answer and Explanation is:
To find the number of independent components of an antisymmetric tensor of rank 2 in n dimensions, we use the following principle:
Correct Answer:
The number of independent components of an antisymmetric tensor of rank 2 in nnn dimensions isn(n−1)2\frac{n(n-1)}{2}2n(n−1)
Explanation:
An antisymmetric tensor of rank 2, often written as AμνA^{\mu\nu}Aμν, is a tensor for which the components satisfy the condition:Aμν=−AνμA^{\mu\nu} = -A^{\nu\mu}Aμν=−Aνμ
This means that the component flips sign when its two indices are exchanged.
Additionally, the diagonal components must satisfy:Aμμ=−Aμμ⇒2Aμμ=0⇒Aμμ=0A^{\mu\mu} = -A^{\mu\mu} \Rightarrow 2A^{\mu\mu} = 0 \Rightarrow A^{\mu\mu} = 0Aμμ=−Aμμ⇒2Aμμ=0⇒Aμμ=0
Hence, all diagonal elements are zero.
This antisymmetry significantly reduces the number of independent components compared to a general (non-symmetric) rank 2 tensor, which would have n2n^2n2 components in nnn dimensions.
To count how many independent components remain:
- We only consider pairs of indices (μ,ν)(\mu, \nu)(μ,ν) such that μ<ν\mu < \nuμ<ν, because the components with μ>ν\mu > \nuμ>ν are determined by antisymmetry: Aμν=−AνμA^{\mu\nu} = -A^{\nu\mu}Aμν=−Aνμ.
- The number of such unordered pairs (μ,ν)(\mu, \nu)(μ,ν) with μ<ν\mu < \nuμ<ν from nnn items is given by the binomial coefficient: (n2)=n(n−1)2\binom{n}{2} = \frac{n(n-1)}{2}(2n)=2n(n−1)
This formula gives the exact count of independent elements because each independent component corresponds to one unique pair of indices (μ,ν)(\mu, \nu)(μ,ν) with μ<ν\mu < \nuμ<ν.
Examples:
- In 3 dimensions: 3(3−1)2=62=3independent components\frac{3(3-1)}{2} = \frac{6}{2} = 3 \quad \text{independent components}23(3−1)=26=3independent components
- In 4 dimensions: 4(4−1)2=122=6\frac{4(4-1)}{2} = \frac{12}{2} = 624(4−1)=212=6
- In 5 dimensions: 5(5−1)2=202=10\frac{5(5-1)}{2} = \frac{20}{2} = 1025(5−1)=220=10
Thus, for any nnn-dimensional space, the antisymmetric rank-2 tensor has n(n−1)2\frac{n(n-1)}{2}2n(n−1) independent components.
