advanced_tools:hodge_dual
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advanced_tools:hodge_dual [2023/03/04 20:40] edi [Concrete] |
advanced_tools:hodge_dual [2023/03/12 17:02] edi [Intuitive] |
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| - | The Hodge dual, also known as the Hodge star operator ($\star$), in $n$-dimensional space takes an antisymmetric rank-$p$ tensor and maps it to an antisymmetric rank-($n-p$) tensor. | + | The Hodge dual, also known as the Hodge star operator ($\star$), in $n$-dimensional space takes an antisymmetric rank-$p$ tensor and maps it to an antisymmetric rank-($n-p$) tensor. If we picture the rank-$p$ tensor as a $p$-dimensional volume element, the Hodge dual can be pictured as its orthogonal complement. |
| - | Example #1: In Euclidean 3D space ($n=3$), the Hodge dual of the antisymmetric rank-2 tensor $a \wedge b$ is a vector, thus mapping rank $p=2$ to $p=3-2=1$. In this case, the resulting vector is just the ordinary cross product: $\star (a \wedge b) = a \times b$. | + | Example #1: In Euclidean 3D space ($n=3$), the Hodge dual of the antisymmetric rank-2 tensor $a \wedge b$ is a vector, thus mapping rank $p=2$ to $p=3-2=1$. In this case, the resulting vector is just the ordinary cross product: $\star (a \wedge b) = a \times b$. (For the meaning of $a \wedge b$, see [[advanced_tools:exterior_product|Exterior Product]].) |
| Example #2: In Euclidean 3D space ($n=3$), the Hodge dual of the antisymmetric rank-3 tensor $a \wedge b \wedge c$ is a scalar, thus mapping rank $p=3$ to $p=3-3=0$. In this case, the resulting scalar is just the scalar triple product: $\star (a \wedge b \wedge c) = a \times b \cdot c$. | Example #2: In Euclidean 3D space ($n=3$), the Hodge dual of the antisymmetric rank-3 tensor $a \wedge b \wedge c$ is a scalar, thus mapping rank $p=3$ to $p=3-3=0$. In this case, the resulting scalar is just the scalar triple product: $\star (a \wedge b \wedge c) = a \times b \cdot c$. | ||
| - | Example #3: In 4D space ($n=4$), the Hodge dual of an antisymmetric rank-2 tensor is again an antisymmetric rank-2 tensor ($4-2=2$). This is the reason why self-dual and anti-self-dual tensors exist in 4D. | + | Example #3: In 4D space ($n=4$), the Hodge dual of an antisymmetric rank-2 tensor is again an antisymmetric rank-2 tensor ($p=4-2=2$). This is the reason why self-dual ($\star T=T$) and anti-self-dual ($\star T = -T$) tensors exist in 4D. |
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| In we increase the rank of a general tensor is has more and more components. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 4x4=16 components, a rank-3 tensor has 4x4x4=64 components, and a rank-4 tensor has 4x4x4x4=256 components. | In we increase the rank of a general tensor is has more and more components. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 4x4=16 components, a rank-3 tensor has 4x4x4=64 components, and a rank-4 tensor has 4x4x4x4=256 components. | ||
| - | The situation changes completely if we consider (totally) antisymmetric tensors instead of general tensors. Now the number of independent components increases as first but the decreases again. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 6 components, a rank-3 tensor has 4 components, and a rank-4 tensor has just 1 component. | + | The situation changes completely if we consider (totally) antisymmetric tensors instead of general tensors. Now the number of independent components increases at first but then decreases again. For example in 4D space a rank-0 tensor (= scalar) has 1 component, a rank-1 tensor (= vector) has 4 components, a rank-2 tensor has 6 components, a rank-3 tensor has 4 components, and a rank-4 tensor has just 1 component. |
| This pattern suggests a relationship between antisymmetric rank-$p$ tensors and and antisymmetric rank-($4-p$) tensors in 4D space. This is the idea behind the Hodge dual! The diagram below shows this pattern for 2, 3, and 4-dimensional spaces. | This pattern suggests a relationship between antisymmetric rank-$p$ tensors and and antisymmetric rank-($4-p$) tensors in 4D space. This is the idea behind the Hodge dual! The diagram below shows this pattern for 2, 3, and 4-dimensional spaces. | ||
advanced_tools/hodge_dual.txt ยท Last modified: 2023/03/19 21:35 by edi
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