Tensor Calculus 0: Introduction

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Recommends Tachyon · 2021-10 · Physics Stack Exchange · on “Visualizing Tensors in the simplest possible way”

This video might help: [this video] I recommend watching the video from the beginning after you watch the part I timestamped and all of this person's videos as they help you understand tensors in a nice way (assuming you have learned and understand the math).

Cites AccidentalTaylorExpansion · 2020-05 · Physics Stack Exchange · on “How can I calculate curvature in space which can be used in formula of general theory of relativity?”

I don't know exactly what you're after but I'll try. The Schwarzschild metric [LINK] describes the metric outside of a spherically symmetric body. Like a planet or black hole. $$g=-\left(1-\frac {r_s}r\right)c^2dt^2+\left(1-\frac {r_s}r\right)^{-1}dr^2+r^2\left(d\theta^2+\sin^2(\theta)\ d\phi^2\right)$$ You can loosely write this as a matrix: $$g_{\mu\nu}=\pmatrix{\left(1-\frac {r_s}r\right)c^2&&\\ &\left(1-\frac {r_s}r\right)^{-1}&\\ &&r^2&\\ &&&r^2\sin^2\phi\\ }$$ You would then have to calculate the Christoffel symbols [LINK] . These tell you a lot about how vectors change when you move in your spacetime. $$\Gamma^\lambda_{\mu\nu}=\tfrac 1 2g^{\lambda\alpha}\left(\frac{\partial g_{\nu\alp

Cites Unmaxed · 2022-11 · Physics Stack Exchange · on “Help to understand $\mu\nu$ notation and differential forms”

Superscript vs Subscript Indices I don't understand QFT that well but I have a decent understanding of GR and tensor calc. In the case of GR, subscript and superscript indices signify covariance and contravariance respectively which are linear transformation concepts. In a vector space, tensor components (which include vectors) are considered covariant if they change using the same linear transformation as basis vectors. This can be thought of as tensor components also increasing when basis vectors increase under a change in unit length. For example: a 1-form ω (aka a covector), which is expanded the following way in a vector space V , $$\omega=\omega_\mu\epsilon^\mu$$ contain components whi