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Re: starship-design: Calculations involving self-powered spaceflight
I guess I must be nearly alone in preferring modern expositions
of relativity theory that treat mass as an invariant quantity,
and distinguish relativistic energy (which does change with
velocity) from mass.
Mass is invariant because accelerating an object does not change
its self-measured properties. If you are on a relativistic
starship you do not find that everything around you becomes more
massive as it accelerates. Neither does ejected fuel become
heavier because it's moving away from you; its energy and
momentum increase, and in more non-Newtonian ways as the exhaust
velocity becomes a higher fraction of c, but the exhaust particles
themselves are not more massive. There's a persistent problem
with people interpreting the relativistic momentum formula
p = m * v / sqrt(1 - v^2/c^2)
by regrouping it as if the object has a variable mass
m' = (m / sqrt(1 - v^2/c^2))
and then behaves as if it has pseudo-Newtoniam momentum
p = m' * v
Having worked through a fair number of relativistic dynamics
problems, I can assure that things are much easier if you
remember these things:
1. Always work from a single frame of reference.
2. Energy and momentum are always conserved in a system.
So Timothy may think it's easier to analyze an accelerating
starship in the ship's frame, but in reality there is no "ship's
frame"; one can attach an instantaneous frame to the ship, and
the ship's behavior in a self-relative sense is fairly simple,
but determining things like the ship's velocity relative to an
observer or the difference between elapsed ship time and elapsed
observer time is best done using a non-accelerating frame, at
least if you don't want the math to become really painful.
Note that in a system where objects interact via fission or
collision, mass may _not_ be conserved, while the total energy
and momentum of the interacting objects are conserved.
If this is still confusing, read chapters 7 and 8 of _Spacetime
Physics_. They (Taylor and Wheeler) also go into a lot more
detail justifying why they formulate things this way.