# Tensor equations in General Relativity

In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent.

Question: why is this true?

A flavour of why this is true: a vector, say a normal one in 3d, is really a physical object $\mathbf{v}.$ You can of course project unto a certain basis $\{\mathbf{e}_i\}_i$, so $\mathbf{v} = \sum v_i \mathbf{e}_i$, and then the components $v_i$ are frame-dependent numbers, but as long as you don't choose a particular basis you're safe. You can then manipulate vectors and tensors (take sums, multiply with scalars, contract them etc.) in a way that doesn't depend on any basis chosen. –  Vibert May 24 at 6:42