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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"?

AccidentalFourierTransform
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LoveScience
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    Because they are defined as the mass and time as measured in the rest frame. They're invariant by definition. – Javier Mar 15 '16 at 22:02
  • If our ages increase overtime, why is my age at marriage always the same? – WillO Mar 16 '16 at 02:34

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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).

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AccidentalFourierTransform
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  • I learned that mass does indeed change with velocity. As v tends to the speed of light, mass tends to infinity? – LoveScience Mar 15 '16 at 22:13
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    @LoveScience well, I'm afraid that is (somewhat) wrong. Almost a century ago, people thought that it would be useful to use other definition of mass, in which it increases with velocity. But we soon realised that definition was not useful, so we discarded it. Nowadays, you won't hear no physicists saying that mass increases with velocity, only inexperts would say that. My advice is: don't you say it either :) [in the link above, this is discussed more thoroughly] Long story short: mass does not change with velocity: that is just a (common) misconception! – AccidentalFourierTransform Mar 15 '16 at 22:18
  • @LoveScience If you think mass changes with velocity, then you must also think that mass is different depending which direction you want to apply a force. Either that, or you'll make a mistake. There are equations in Newtonian physics that are equivalent to each in Newtonian physics that are not equivalent to each other in relativity. So you have to learn to abandon some of them as wrong. Not every equation from Newtonian physics can survive. And surely you realize how silly it is that I even have to say that, but apparently people do need the reminder sometimes. – Timaeus Mar 16 '16 at 00:05