Cross Product in E_u: Explaining Gourgoulhon's Text

In summary: If you feed two vectors into it, you get a one form. By metric duality it gives you a vector.Thank yes I did now understand, what confuse me is that it look on paper like the object ##\epsilon_u(v,w, \dots)## is a oneform ##E_u \rightarrow R## (LV tensor with a one unfilled slot for a vector), but author instead mean that this object above is g-dual ##(\in E_u##) of what I thinking before. so I simply was imagining the isomorphism wrong way round in my brain ;) ;)In summary, the author is trying to prove that
  • #1
aclaret
24
9
I study from Gourgoulhon's text 'special relativity in general frames', I have some difficulty to understanding Chapter 3 Page 84. I already learn that there exist a orthogonal projection mapping ##\bot_{u}:E \rightarrow E_u(P)## from the vector space ##E \cong R^4## to the subspace ##E_u(P)## associated with local rest space ##\mathscr{E}_u(P)## of the observer at event ##P##.

Now want to proof the proposition (3.37), that given timelike ##u \in E## and antisymmetric bilinear form ##A##, there exist unique form ##q = A(., u) \in E^*## and unique vector ##b \in E## such that ##A = u \otimes q - q \otimes u + \epsilon(u, b, \dots)##. During proof author writes "By metric duality, ##\epsilon_u## induces the cross product of two vectors of ##E_u## by $$\forall (v, w) \in {E_u}^2, \quad v \times_u w := \epsilon_u (v, w, \dots) = \epsilon(u, v, w, \dots)$$where ##\epsilon_u(v,w \dots)## stands for vector of ##E_u## associated by ##g##-duality to the linear form ##E_u \rightarrow R##, ##z \mapsto \epsilon(v, w, z)##... [and] ##\varepsilon(u,v,w, \dots)## stands for vector in ##E## that is ##g##-dual of the linear form ##E \rightarrow R, z \mapsto \epsilon(u,v,w,z)##"

I don't understand this part, please somebody can please explain how exactly this induces a cross product? (I do undertand what author mean by metric duality, that is simply the map ##\Phi_g## associating any ##u \in E## to a one-form ##\tilde{u} \in E^*## such that satisfy ## \langle \tilde{u}, v \rangle = g(u,v)## for all ##v \in E##, but I don't understand how it relate to the concept above).
 
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  • #2
Not sure which part is unclear to you. In 3D space there is one trilinear antisymmetric form (up to a constant multiple). If you feed two vectors into it, you get a one form. By metric duality it gives you a vector.
 
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  • #3
Thank yes I did now understand, what confuse me is that it look on paper like the object ##\epsilon_u(v,w, \dots)## is a oneform ##E_u \rightarrow R## (LV tensor with a one unfilled slot for a vector), but author instead mean that this object above is g-dual ##(\in E_u##) of what I thinking before. so I simply was imagining the isomorphism wrong way round in my brain ;) ;)

apologise for trivial question :), thank @martinbn
 
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  • #4
No need for apology. It is worded in an unusual way. It is easy to loose track of the notations and not see the forest because of the trees.
 
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  • #5
For me In 3D space there is one trilinear antisymmetric form
 

Related to Cross Product in E_u: Explaining Gourgoulhon's Text

1. What is the cross product in Eu?

The cross product in Eu is a mathematical operation that takes two vectors in a three-dimensional Euclidean space and produces a third vector that is perpendicular to both of the original vectors. It is denoted by the symbol "×" and is also known as the vector product or outer product.

2. How is the cross product calculated in Eu?

In Eu, the cross product is calculated by taking the determinant of a 3x3 matrix formed by the components of the two vectors. The resulting vector has a magnitude equal to the product of the magnitudes of the original vectors multiplied by the sine of the angle between them. The direction of the resulting vector is determined by the right-hand rule.

3. What is the significance of Gourgoulhon's text in relation to the cross product in Eu?

Gourgoulhon's text provides a comprehensive explanation of the cross product in Eu and its applications in mathematics and physics. It also discusses the properties and geometric interpretation of the cross product, making it a valuable resource for understanding this operation.

4. What are some common uses of the cross product in Eu?

The cross product in Eu has many practical applications, such as calculating torque in physics, determining the direction of a magnetic field in electromagnetism, and finding the normal vector to a plane in geometry. It is also used in vector calculus and 3D graphics programming.

5. Are there any limitations to using the cross product in Eu?

One limitation of the cross product in Eu is that it can only be applied to three-dimensional vectors. It also does not produce a unique solution, as the direction of the resulting vector can be reversed by changing the order of the original vectors. Additionally, the cross product is not defined for non-Euclidean spaces.

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