How do you find a metric tensor given a coordinate transformation, $(t', x', y', z') \rightarrow (t, x, y, z)$? Our textbook gives a somewhat vague example as it skips some steps making it difficult to understand. What's the general definition for a metric tensor of a given transformation? The closest I could find was http://en.wikipedia.org/wiki/Metric_tensor#Coordinate_transformations, but I'm having trouble understanding that.
Answer
You look at the distance between two infinitesimally different points. Let the two coordinate systems be x and y, where x is four numbers and y is four numbers. Consider an infinitesimal displacement from y to y+dy. You know this distance in the x coordinates, so you find the two endpoints of the displacement
$$x(y)$$ $$x^i(y + dy) = x^i(x') + {\partial x^i \over \partial y^j} dy^j $$
This is using the Einstein summation convention--- repeated upper/lower indices are summed automatically, and an upper index in the denominator of a differential expression becomes a lower index, and vice-versa. The distance between these two infinitesimally separated points is:
$$ g_{ij}(x) {\partial x^i \over \partial y^k} {\partial x^j \over \partial y^l} dy^k dy^l $$
And from this, you read off the metric tensor coefficients--- since this is the quadratic expression for the distance between y and y+dy.
$$ g'_{kl}(y) = g_{ij}(x(y)) {\partial x^i \over \partial y^k} {\partial x^j \over \partial y^l}$$
This is a special case of the tensor transformation law--- every lower index transforms by getting contracted with a Jacobian inverse, and every upper index by getting contracted with a Jacobian.
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