# Tag Info

## Hot answers tagged differential-geometry

4

Well, actually you are looking for a one-parameter group of diffeomorphisms (or isometries if referring to the boost vector field). This group is obtained by solving the differential equation $$\frac{dx}{ds}= X(x(s))\tag{1}$$ with a generic initial condition $z$ at $s=0$ in the manifold $M$ (Minkowski spacetime in your example). $X$ is your vector field on ...

3

Calculating the sum of the interior angles precisely woud be a big task as we'd need to compute the trajectory of the light ray and there isn't a convenient analytic expression for this. However we can easily calculate an upper limit for the interior angles. The key fact we need to know is that the deflection angle $\theta$ of a light ray in the ...

3

Your last expression is not valid, for two reasons: first, any given index can only occur twice per term in the Einstein convention, once as an upper index and once as a lower index. Remember that when an index is repeated, it means you sum over it with the metric: $$T^a T_a = \sum_{a,b} g_{ab} T^a T^b$$ You have terms like ...

2

Comments to the question (v1): There are three types of indices at play: (i) spinor indices, (ii) flat (vector) indices, and (iii) curved (vector) indices. The gamma matrices with flat indices are constants. They don't transform under local Lorentz transformations (LLTs). They can be viewed as intertwiners between spinor indices and flat indices. (LLTs ...

2

What you are confusing here is speed and velocity. Light speed is constant, but the velocity, which takes into account the direction as well as the speed is not. As an example of how something can accelerate without changing speed, consider the case of circular motion, where the acceleration of an object moving at a speed $v$ in a circle of radius $r$ is ...

1

Suppose you're in a coordinate system where the Christoffels don't vanish at some point. To choose a coordinate system where the Christoffel symbols vanish at a given point $p$, you must apply a Christoffel symbol change of variables: 0={\bar\Gamma}^k{}_{ij} = \frac{\partial x^p}{\partial y^i}\, \frac{\partial x^q}{\partial y^j}\, \Gamma^r{}_{pq}\, ...

1

Here is a purely geometrical way to think about this Edward says it is possible to cut a wedge out of a flat spacetime and glue the edges together. So in my mind this looks like a paper cone. A cone is flat precisely because it can be created by rolling up a flat sheet. Rolling up preserves the metric on the interior of the sheet (not on the boundaries ...

1

There's something wrong with your sign permutations in the Hodge star operator calculation. If $F = B + E \wedge dt$, then, in 2D, $F = B dx \wedge dy + E_x dx \wedge dt + E_y dy \wedge dt$, as you wrote yourself. Now, let us take our initial Hodge star as $\star dx \wedge dy = dt$. This means that $\star dt \wedge dx = dy$ and $\star dy \wedge dt = dx$, ...

1

One less well-known but great reference are the classical field theory notes by Deligne and Freed in the '99 IAS lectures. Some good things about them Very elegant treatment written for mathematicians Begins with a nice discussion of ordinary classical mechanics using principal bundles and connections Useful comments on supersymmetric gauge theories ...

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