# Tagged Questions

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### Tensors in special relativity [duplicate]

I'm trying to understand tensors, but I've come across the following question: Let $T^{\mu\nu}$ by a $(2,0)$ tensor. Give the definitions of $T_\mu^{\,\nu}$, $T_{\mu\nu}$, and ...
For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} ... 4answers 123 views ### Difference between matrix representations of tensors and$\delta^{i}_{j}$and$\delta_{ij}$? My question basically is, is Kronecker delta$\delta_{ij}$or$\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ... 1answer 133 views ### Stress-energy tensor explicitly in terms of the metric tensor I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor$R_{\mu\nu}$and scalar curvature ... 1answer 116 views ### Hypersurface Normal Could anyone explain why $$n^{a}n_{a}=\pm1$$ where$n^{a}is the normal to the hypersurface 2answers 112 views ### Inner products in relativity In physics, the definition of a dot (inner) product is often between a vector (“contravariant vector”) and a covector (“covariant vector”). However, in mathematics, a dot product is always defined ... 0answers 54 views ### Contracting Indices in General relativity [duplicate] I was reading a book about general relativity and I came across these two equations \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ ... 1answer 311 views ### Contracting Indices Does anyone know how to get from (1) to (2) in the system \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ... 2answers 182 views ### How to prove the raising/lowering indices operation? I've read this related question, though it didn't satisfy me; I hope this complements it. I know that if I contract a covariant tensor{A_{\alpha\beta}}$with a vector${B^\beta}$, I get some other ... 2answers 225 views ### The signature of the metric and the definition of the electromagnetic tensor I've read the definition of the electromagnetic field tensor to be ... 1answer 266 views ### Weight of a tensor density Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with ... 1answer 215 views ### Metric tensor in General Relativity or otherwise [closed] What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How ... 1answer 430 views ### What are the local covariant tensors one can form from the metric? Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ... 1answer 81 views ### General expression of the redshift: explanation? In some papers, authors put the following formula for the cosmological redshift$z$:$1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$where :$S$... 1answer 285 views ### What is the Lorentz tensor with a superscript and subscript index? I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form:$\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar ... 1answer 257 views ### When a variation of a tensor is not a tensor? In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ... 1answer 93 views ### Non-diagonal elements when switching metric signature? Considering a metric tensor with the signature$(-,+,+,+)$:$g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\ g_{10} & a^2 & g_{12} & g_{13}\\ g_{20} & g_{21} ...
I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be ...