# Tag Info

## New answers tagged covariance

1

If the 4-momentum were invariant then it would be a scalar. 4-vectors are defined by the way their components mix when we change coordinates. In particular when we apply a lorentz transformation to our coordinates the inverse transformation is applied to the vector. As a simple example consider what happens to the energy when we boost. If we start in the ...

1

Conservation and invariance are fundamentally different things. Conservation means "doesn't change with respect to time". While invariance means "doesn't change with respect to Lorentz transformations". Components of four-momentum transform like vector components and are thus NOT invariant under Lorentz Transformations. But that doesn't prevent them from ...

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Start by rewriting the scalar product as a covariant-contravariant contraction, like so: $${\bf u}\cdot{\bf v} = g_{ij}u^iv^j = (g_{ij}u^i)v^j = u_jv^j$$ Now transform the components with your $S$ and $T$ matrices,  u_jv^j = \left( S_j^a {\bar u}_a \right) \left( T^j_b {\bar v}^b \right) = (S_j^a T^j_b) {\bar u}_a {\bar v}^b = \delta^a_b {\bar u}_a {\bar ...

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