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6

That's because you are forgetting that $A$ has a Yang-Mills index. You better write this in components, which reads $\epsilon^{\mu\nu\rho} g_{IJ} \Big( A^I_\mu \partial_\nu A_\rho^J + \frac{1}{3} f^J{}_{KL} A^I_\mu A^K_\nu A^L_\rho \Big) $ This component notation also answers your second question.


3

OP's proposal (v2) is a special case of Finsler geometry with $n=3$. The main idea is to replace the quadratic metric tensor $g^{(2)}_{\mu_1\mu_2}$ for pseudo-Riemannian manifolds, which defines (infinitesimal, possibly imaginary) distance on the manifold via $$ds ~=~ \sqrt[2]{g^{(2)}_{\mu_1\mu_2}dx^{\mu_1}dx^{\mu_2}},$$ with (possibly a sequence of) ...


2

The (Lie-)group $U(1)$ is the topological space $S^1$ (what we call a circle together with its standard open subsets) together with a rule how to multiply its points. In its representation as numbers in ${\mathbb C}$ with absolute value $1$, we have ${\mathrm e}^{{\mathrm i}\alpha}\bullet{\mathrm e}^{{\mathrm i}\beta}:={\mathrm e}^{{\mathrm ...


2

The metric being a rank $(0,2)$ tensor transforms under general coordinate transformations $x^\mu \to x'^\mu(x)$ as $$ g'_{\mu\nu} (x') = \frac{ \partial x^\rho}{ \partial x'^\mu } \frac{ \partial x^\sigma }{ \partial x'^\nu } g_{\rho\sigma} (x) $$ Now set $x'^\mu (x) = x^\mu + \alpha k^\mu(x)$ in the above expression and take a limit of small $\alpha$. ...


2

We know that the Levi-Civita connection satisfies $\nabla_a g_{bc} = 0$ and the product rule. The definition of the inverse metric $g^{ab}$ is $g^{ab}g_{bc} = \delta^a_c$. Therefore, we have: $$\begin{align} 0 &= \nabla_a \delta^b_c \\ &= \nabla_a (g^{bd}g_{dc}) \\ &= (\nabla_a g^{bd}) g_{dc} + g^{bd} \nabla_a g_{dc} \\ &= (\nabla_a g^{bd}) ...


2

I think you have all the right pieces to answer the question, here are a few hints that should be of some use. You say that you picked coordinates $ \{v^{\mu} \}$. It seems to me that they should instead be called $ \{ x^{\mu} \}$, as that is what you're taking partial derivatives with respect to. As you correctly pointed out, you are working with ...


2

Writing $\vec{e}_r = \partial_r$, $\vec{e}_x = \partial_x$ transforms the "$\vec{e}$-notation" into the partial-derivative-notation, so the "relation" is just that $\vec{e}_{x^\mu} = \frac{\partial}{\partial x^\mu}$.


1

Globally hyperbolic refers to the fact that hyperbolic equations always have locally a well defined Cauchy problem, that is, a unique development given initial conditions. Which means that, given a matter field at a time $t_1$, there exists a unique solution of that field at a time $t_2$. It is boosted up to globally hyperbolic if that property holds ...


1

Killing vector fields correspond to infinitesimal isometry generators of the spacetime manifold and any physical action including the Polyakov action should be preserved under it. In fact, any physical action should be invariant under the (infinitely) larger group of diffeomorphisms of a manifold. Isomotry transformations are just a finite subset of these ...


1

Here's a paper for you to ponder on: Teaching electromagnetic field theory using differential forms Excerpt from the abstract: computational simplifications result from the use of forms: derivatives are easier to employ in curvilinear coordinates, integration becomes more straightforward, and families of vector identities are replaced by ...


1

I will use the notation $\theta^i$ for the dual basis since $\omega$ is reserved for another important form. Since $\mathrm{d}\theta^i$ is a 2-form, we may expand it in the basis $\theta^i$ itself: $$\mathrm{d}\theta^i=-\frac{1}{2}C^i{}_{jk}\theta^j\wedge\theta^k$$ But the first structure equation gives $$\mathrm{d}\theta^i=-\omega^i{}_j\wedge\theta^j$$ ...


1

I don't have the book, but I would guess that it's referring to the maximally extended Schwarzschild metric. If you write the metric in Kruskal-Szekeres coordinates you find it in effect consists of two copies of the Schwarzschild coordinate. This is because the KS $u$ coordinate (in the exterior region) is defined by: $$ u = ...



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