Why Does Gamma Apply to All Object Components?

[bolognie1]

So it seems straightforward to me that if time dilated, then so would size, mass and energy, but I don't know how to prove that mathematically. I understand the concepts of special relativity and the lorentz transform, and why it is applied to time, but I can't recall or figure out why Einstein would decide to apply it to all properties of an object.
Thanks guys.

 

[robphy]

It doesn't apply to all components.
Have a look at this old thread
https://www.physicsforums.com/threads/why-not-length-contraction.298145/#post-2108296

 

[PAllen]

One way it arises, for a large class (but not all) of objects, is going from time derivatives to derivatives by proper time along a world line. Any change from time derivative to proper time derivative introduces a factor gamma. As a result, all 4-velocity (the world line tangent vector with derivative by proper time) and anything based on it get this factor. That includes 4-momentum. From the latter, it applies to energy which is the time component of 4-momentum (in standard coordinates).