If distances contract in direction of motion and time dilates at high speeds, why are the rest mass $m_0$ and proper time $t_0$ called "invariant" under Lorentz transformations. Since depending on your reference frame mass and time are not the same I don't understand why they are "invariant"?
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Both mass $m_0$ and proper time $t_0$ are defined as the energy and time measured by an observer in the same reference frame as the particle. This means: whatever your reference frame is, you have to change into the particle's to measure/define $m_0,t_0$. Therefore, the values of these parameters are independent of what reference frame you are at. It simply doesn't matter.
There are alternative definitions of these parameters, maybe more geometrical, in which the invariance is more obvious. For example, mass can be defined as the norm of the momentum ($m_0^2\equiv p^2$), and as such it has to be invariant, because the norm of a vector is a scalar product. As scalar products are defined as invariant (see, for example, this post of mine), $m_0$ is invariant by definition as well.
Finally, note that the best notation for mass is just $m$, and not $m_0$. Relativistic mass is not a thing any more. Just forget about it, as most people have. Mass doesn't change with velocity! (see, for example, this post).
AccidentalFourierTransform
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