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Coordinate transformations on the Minkowski metric
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[QUOTE="Antarres, post: 6248686, member: 643619"] Well, finding the eigenvalues of g isn't really the best thing to do. This is because even though we represent ##g## by matrix, we have to keep in mind that ##g## is a (0,2) tensor. What does this mean? It means that when we act with ##g## on some vector, we don't get another vector. Instead, we get a dual vector. Or in otherwords, we need to act with ##g## on two vectors to get a scalar(this action is called scalar product). So, since we don't get a vector when acting with ##g##, we can't talk about eigenvectors, since those don't exist. This is a confusion that can arise when we're not taking into account that this matrix is just a convenient representation of ##g##, but we must know how to act with it on vectors in order to make use of it. Now on to your question about the transformation of ##g##. Law of transformation of ##g## under coordinate transformations, reads: $$g'_{\mu\nu} = \frac{\partial x^\rho}{\partial x'^\mu}\frac{\partial x^\sigma}{\partial x'^\nu}g_{\rho\sigma}$$ Here indices represent different components of ##g##, so in your case you can take ##g'## to be standard Minkowski metric, and take ##g## to be your own metric. Summation convention is in use, which means we're summing over indices that are repeating. I will give you example of the first component, so you get the gist of how to use this, in case you haven't used this before. $$g'_{00} = \frac{\partial x^\rho}{\partial x'^0}\frac{\partial x^\sigma}{\partial x'^0}g_{\rho\sigma} = \frac{\partial x^0}{\partial x'^0}\frac{\partial x^0}{\partial x'^0}g_{00} +2\frac{\partial x^0}{\partial x'^0}\frac{\partial x^1}{\partial x'^0}g_{01} + \frac{\partial x^1}{\partial x'^0}\frac{\partial x^1}{\partial x'^0}g_{11} + \frac{\partial x^2}{\partial x'^0}\frac{\partial x^2}{\partial x'^0}g_{22} + \frac{\partial x^3}{\partial x'^0}\frac{\partial x^3}{\partial x'^0}g_{33} $$ Now you proceed the same thing for the other components that should be transformed, and from the system of equations that you get, you find what the coordinate transformation is. This is like the most straightforward(brute-force) method that you get, so in case you have no other intuition on how to do it, it should work. Either way, even when guessing the transformation, this is how you check if it is right or wrong. [/QUOTE]
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Coordinate transformations on the Minkowski metric
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