Sunday, May 15, 2016

special relativity - Will 1 gram of matter moving at relativistic speeds completely annihilate a larger quantity of stationary antimatter?


This is a question about the relativistic mass concept which I am having trouble understanding, mainly because of the scenario below.


Simple scenario:


Suppose 1 gram of matter is accelerated to 99% the speed of light. At this speed, the relativistic mass is 7 times greater than the rest mass. If this matter collides with a stationary quantity of 7 grams of antimatter, will the two masses annihilate completely with each other? Or will the matter just annihilate 1 grams worth of antimatter?


If the latter is true then what exactly am I overlooking about the relativistic mass concept that makes the former incorrect?




Answer



A sophisticated, yet easy way to see that this the answer must be "No." is to recall that velocity is relative — that there is no absolute notion of velocity.


You said the matter was moving and the antimatter still, but that point of view (AKA frame of reference) is not privileged in any way. An observer at rest with respect to the matter has just as much right to conclude that the anti-matter is in motion as you have to conclude that the matter is moving.


So you can't rely on a velocity dependent notion of mass to work out the consequences of the scenario.




The modern approach to relativity is to define the (only!) mass of a particle or system as the square of its energy-momentum four vector (with appropriate factors of $c$): $$ m = \frac{\sqrt{E^2 - (pc)^2}}{c^2} \,. $$


The thing that you you've been taught to call "relativistic mass", $\gamma m$, is (to within two factors of $c$) described as the "total energy" of the particle or system.


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