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# Questions tagged [differential-geometry]

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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### Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?

On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
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### Precise definition of a string worldsheet as a manifold in string theory

I've spent some time studying some definition in smooth manifolds theory in order to give a proper definition of a worldsheet in classical string theory at least. My attempt is the following: ...
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### Help with geometric view of conjugate momenta and Legendre transformation

I'm familiar with the ''coordinate view'' of Lagrangian and Hamiltonian mechanics where if $\pmb{q}=(q^1,\dots, q^n)\in\mathbb{R}^n$ are any $n$ generalized coordinates and $L(\pmb{q},\dot{\pmb{q}})$ ...
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### What is the formal criteria that the spacetime is curved?

We suppose we have three scenarios. We are far away from mass and energy in a spot in the universe. We put in free movement a small object $m$, for example, an apple. At the same time, we send a ...
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### What do Hawking/Ellis mean exactly by "non-rotating families of geodesics"?

In The Large Scale Structure of Space-Time, Hawking and Ellis refer twice (page 4, page 78) to non-rotating families of geodesics. I don't know what that means. Is a rotating geodesic one that ...
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### Sufficient and necessary condition for being a black hole energy-momentum tensor

Is there any necessary and sufficient mathematical condition(s) so that a (general) energy-momentum tensor can possess an assemblage of black holes? Or in other words, if I'm given a general energy ...
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### Uniqueness of the diagonal form of metric

For Schwarzschild solution, if we use the coordinates ($t$,$r$,$\theta$,$\phi$). the metric in these coordinates are diagonal, my question is, is there exist another set of coordinates ($t^{'}$,$r^{'}$...
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### Thermodynamic Potential Minimization

My textbook states that at equilibrium thermodynamic potentials are minimized. I am having trouble understanding how this minimization work and how to visualize it. For example, the Helmholtz free ...
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Imagine you have the arc-length of a curve, in spherical, coordinates:  s = \int_{\mathcal C}{d\tau \; \sqrt{f(r)^2 \left (\frac{dr}{d \tau} \right )^2 + r^2 \left (\frac{d \theta}{d \tau} \right )^...
I'm having trouble determining the connection between two covariant derivative operators. These are: the one associated with the original space-time (and thus with the metric $\tilde{g}_{ab}$) and ...