Transformation law Definition and 16 Threads

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I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...

I'm reading the article https://www.researchgate.net/publication/267938119_ON_THE_GALILEAN_COVARIANCE_OF_CLASSICAL_MECHANICS (pdf link here), in which the authors want to establish the transformation rule for momentum, assuming only that ##\vec{F}=d\vec{p}/dt## and notwithstanding the relation...

We have 4-tensor of second rank. For example energy-momentum tensor ##T_μν## , which is symmetric and traceless. Then ##T_{μν}=x_μx_ν+x_νx_μ## where ##x_μ## is 4-vector. Every 4- vector transform under Lorentz transform as (12,12). If we act on ## T_{μν}## , by representation( with...

Hello PF, in Carroll’s “Spacetime and Geometry”, he works out the transformation law for connection coefficients in his introduction to covariant derivatives, and I’m wondering if there is a typo in the final equation. He starts with$$\nabla_{\mu} V^{\nu} = \partial_{\mu} V^{\nu} +...

In Carroll's GR book (pg. 96), the transformation law for Christoffel symbols is derived from the requirement that the covariant derivative be tensorial. I think I understand that, and the derivation Carroll carries out, up until this step (I have a very simple question here, I believe-...

Typically an element of a vector space is called a vector, but Carroll's GR book repeatedly refers to elements of tangent spaces as "transforming as a vector" when they change coordinates as Vμ = ∂xμ/∂xν Vν. However, dual vectors are members of vector spaces (cotangent space) but obey ωμ =...

I am having a hard time understanding vector transformations. I know that vectors must transform a certain way and that dual vectors (or covectors) transform the "opposite" way. What is strange to me is that the basis vectors transform like dual vectors and the basis dual vectors transform like...

I just have a quick question on which order around the numerator and denominator should be in the jacobian matrix that multiplies the expression. As in general Lecture Notes on General Relativity by Sean M. Carroll, 1997 he has the law as ## \xi_{\mu'_{1}\mu'_{2}...\mu'_{n}}=|\frac{\partial...

I really don't understand what it is and what is the use of constant, like in this equation of transformation. x=k(x' + vt). The equation can also be good if it is just like this, x=x' + vt Thank you.

I would just like to start off by saying the problem comes from Intro to electrodynamics, 3rd edition, griffiths. the problem is number 1.9. Question: Find the transformation matrix R that descries a rotation by 120 degrees about an axis from the origin through the point (1,1,1) the...

Hey everyone, This formula was just provided in a book and I was trying to prove it but I'm having a hard time understanding what it's saying. The formula is attached, along with the definitions given for the Christoffel symbols. In the definitions the i's are the standard basis vectors and...

Hi, I am wondering what are the transformation laws (Lorentz transformations) for a general metric, if they exist.

Supposed we are given a set of SUSY transformation law, the way to get the BPS equation is by requiring that \delta \psi = 0 where \psi is a fermion field. Could somebody explain why this is the BPS equation? Thanks!

What's the transformation law for the permutation (or Levi-Civita) symbols?

Hi, I have met a problem, that is how to prove transformation law for Christoffel symbol of first kind. I have read books about that, but many of them just state: cyclic permutation of the 3 indices and substitution. When I tried to work out, I could not elimate some terms... Can anyone show...