Questions tagged [differential-geometry]

Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $\mathbb{R}^n$. General relativity is written in this language.

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Hawking and Ellis Lemma 4.3.1 Proof

I have a few questions about Hawking and Ellis' proof of this lemma (pages 92-93): Write the $(2, 0)$ stress-energy tensor in coordinates as $\mathbf{T} = T^{ab} \partial_a \otimes \partial_b$ and ...
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Metric under conformal transformation

I have a question regarding the conformal factor $\Omega(x)$ when dealing with a conformal transformation. We know that under a change of coordinates $x\rightarrow x^{'}=x^{'}(x)$ our metric changes ...
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How does the Ricci tensor describe the changing separation of two airplanes flying from the equator? Conceptually understanding the Ricci tensor

I'm trying to understand the concept of the Ricci tensor and its physical implications using a concrete example involving two airplanes. Suppose two airplanes start at the equator, separated by a ...
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Notation for vector density in Lagrangian density

Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation \int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
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