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 Feb 19 awarded Nice Answer Jan 13 answered A manifold question: Why smooth functions and what is a Jacobian? Jan 13 comment What is the metric tensor for? It may mean curved space; it is not simple to tell if space is curved or not just by looking at the metric. The metric may be flat and still not identity (for instance, a multiple of the identity is still flat); conversely, even in polar coordinates the space is still flat. But in general, the space isn't necessarily flat. Consider, for example, the metric for a Schwarzschild black hole. Even if you considered the coordinates there to be related to Cartesian ones by the usual transformation, the resulting metric for those Cartesian coordinates would not be the identity. Jan 13 comment What is the metric tensor for? Sure, that computes the new metric (in polar coordinates) when the old metric (in Cartesian coordinates) is the identity matrix. That formula would not be correct if the metric in Cartesian coordinates were some other symmetric matrix (positive definite as well, for Euclidean spaces). Jan 13 comment What is the metric tensor for? Is that your question--whether the metric plays a role in the transformation from one set of coordinates to another? If so, then I would ask what you're trying to compare against. Do you want to look at the expressions for the spherical basis vectors in terms of the Cartesian ones? Do you want to look at the new metric compared to the old one? Jan 13 answered What is the metric tensor for? Jan 11 answered How to understand the continuity equation $i\omega \rho=\nabla\cdot\mathbf{J}$? Jan 11 comment How to understand the continuity equation $i\omega \rho=\nabla\cdot\mathbf{J}$? Try $\rho(r',t) = \rho(r') e^{-i \omega t}$. Dec 10 comment Why isn't $G=1$ as common as $c=\hbar=1$? $G=1$ is standard practice in numerical relativity. Dec 3 comment Notation: tetrad indices Yeah, that's correct. Dec 3 comment Notation: tetrad indices Common practice is to compute it one component at a time. From the function standpoint (the standpoint of tensors as "multilinear maps"), $U_{ab}$ means a function of two vector arguments, so you plug in basis vector for each argument and evaluate. You would need the tetrad vectors' components with respect to that basis. You could also use the tetrad itself as a basis--the inner products of the tetrad vectors (at least, those used in Newman-Penrose) are known values by construction. Dec 3 comment Notation: tetrad indices Well, therein lies the problem with trying to think about them as matrices: you're not equipped to deal with something that converts a vector to a covector, let alone two such somethings. - But a general concept of tensors as functions of vectors and covectors (functions that are linear in their arguments) makes it easy: you have a function of four vectors, which is some other function taking the first two vectors as arguments, multiplied by that same function taking the second two vectors as arguments. Dec 3 answered Notation: tetrad indices Nov 11 comment How to derive the relation satisfied by “gravitational magnetic field” from an equation of the Weyl tensor? Isn't $\eta$ a timelike vector field? Oct 19 awarded Yearling Oct 1 awarded Mortarboard Sep 24 answered Reconciling Minkowski and 3+1 view of special relativity Sep 3 comment 4-vectors in special relativity and spacetime interval You said you've done dot products before; a dot product is just something you can do with elements of a normed vector space. So if you already know how dot products work, you've already been working with vector spaces. The big difference here is merely that the dot product is not always nonzero when the vector is nonzero. That has a lot of consequences--all thoroughly explored and derived by now, and many of those have been used to model physical phenomena. Have you worked with dot products before? And in what context? Sep 3 answered 4-vectors in special relativity and spacetime interval Jul 24 comment 9-point stencil “equivalent” for advection equation Can you write the particular equation you are trying to solve?