- #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).
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).
Last edited: