Dr Rodal, you insist that you are here doing a study of a closed system composed of a single lump of mass, i.e. that there is no separation (separation in 2 parts from a single part, as would be the case for action/reaction) nor joining ("melting" of 2 parts in a single part, as would be the case for an inelastic collision/aggregation) nor bounce (2 parts exchanging momentum but still being separate before and after such interaction). Do I understand correctly your premise ?
If so, in a given frame, considering one lump of rest mass m1 at velocity v1 and associated γ1=γ(v1) before, and same singleton object of rest mass m2 at velocity v2 and associated γ2=γ(v2), taking both conservation of momentum and of (total) energy gives (following the equations you recall) :
γ1 m1 v1 = γ2 m2 v2 (CoM)
γ1 m1 c² = γ2 m2 c² (CoE)
⇔ (since c²≠0)
γ1 m1 v1 = γ2 m2 v2
γ1 m1 = γ2 m2
γ1 m1 v1 = γ1 m1 v2
γ1 m1 = γ2 m2
⇔ (since γ1≠0)
m1 v1 = m1 v2
γ1 m1 = γ2 m2
Now, all depends on m1
, total mass of the closed system is 0 from start
since γ2≠0 ⇒ m2=0 : it must
be that total mass stays 0
and v1 and v2 are independant
, total mass of the closed system is not 0 from start
since m1≠0 ⇒ v1=v2 : it must
be that velocity stays the same
and so γ1=γ2 and it follows that m1=m2
I don't see how you can start with a singleton lump closed system
that's supposed to respect conservation of energy and conservation of mass in the framework of SR, even assuming possibility of negative rest mass (or imaginary rest mass, whatever) and not arrive at same conclusion. There appears to be a contradiction between your premise (closed system) and the conclusions you draw from equation of conservation of momentum alone, when a closed system needs both constraints to be taken together.
Also it is not clear how you consider mass...
where, in the above equation and in the ones to follow, m is the rest mass m=mo, the mass of an object in its rest frame. Also, as already discussed here: http://forum.nasaspaceflight.com/index.php?topic=39214.msg1488362#msg1488362 the InitialVelocity must be measured with respect to the same frame where the InitialMass of the object was measured. This is an acceleration problem, hence the frame where the rest mass is measured is a privileged, non-inertial frame. If other frames of reference are used, not only the Initial Velocity will be different, but the Initial Mass will be different too, if measured in any frame other than the object's initial frame of reference to measure its mass.
<<m is the rest mass m=mo, the mass of an object in its rest frame.
All right so we are not
using so called relativistic mass mrel
, and by avoiding the traps of mrel
we are conforming to prescriptions of modern physics teaching. Fine with me. That means that whenever we talk about mass we can be confident that is not
(v) a function of velocity, i.e. a covariant value that depends on inertial frame of reference, but we talk on an invariant value, that has the exact same value for all observers. The fact that we are talking of this invariant mo
mass appears clearly in the equations, as for instance momentum is given by p=γmo
v otherwise we would have p=mrel
v. To me it then appear as a contradiction to use mo
in the equations but to discuss those very same variables as if they were a function of velocity :
<<but the Initial Mass will be different too, if measured in any frame other than the object's initial frame of reference to measure its mass.
And I fail to see the physical meaning of <<InitialVelocity must be measured with respect to the same frame where the InitialMass of the object was measured
>> that again seem to imply the use of mrel
that is a function of velocity wrt. observer, when all equations use the invariant mo
that is not a function of velocity wrt. observer
But all those question are less important than the first part of this post : unless we are talking of a 0 total (invariant) mass object from start, in SR a closed system single object with mo
≠0 just has an inertial trajectory of constant velocity and constant (invariant) mass, in this later case the hypothesis of occurrence of negative mass doesn't change the game.