Horace Heffner wrote:
On Jan 25, 2006, at 6:22 AM, Stephen A. Lawrence wrote:
Thank you for the summary. I had one comment (really just one, this
time!)
Horace Heffner wrote:
Mass in the conventional spacetime metric is considered invariant.
There's a semantic problem here. "An invariant" is a well-defined
mathematical concept. However, it's just that -- a mathematical
concept. Saying "this is an _invariant_" doesn't mean it's some
simple physical property which always has the same value.
Wheeler and Taylor say the mass of any isolated system is invariant. In
other words:
m^2 = E^2 + p^2
in one frame then
m^2 = (E')^2 - (p')^2
in another for that isolated subsystem.
Right. I think that's similar to what I said. I was talking about a
single body, but it's the same thing, really. If we put the whole
system in a box, then the squared magnitude of the 4-momentum of the box is
m^2 * gamma^2 * (v^2 - 1)
which is, rearranging terms,
(m^2 * gamma^2 * v^2) - (m^2 * gamma^2)
or in more familiar terms,
p^2 - E^2
But it also is equal to -m^2, since the gamma^2 and the (v^2-1) terms
cancel.
Either way it's a mathematical invariant.
And the inner product of the 4-velocity of an observer with the
4-momentum of an isolated system is _another_ invariant, and it gives
the relativistic mass of the system, which is more often referred to as
the energy these days.
Problem is, no subsystem of
mass is isolated. Stuff comes in and out of the vacuum constantly. A
significant portion of the magnetic field of the proton comes from
strange quark pairs popping in and out of the vacuum, for example.
Acceleration affects how things pop in and out of the vacuum and how
long they stick around.
It's easy to forget that relativity theory says _nothing_ about what
is "real" and what is not.
Who's relativity? Certainly not mine! You make it sound like there
is only one version! 8^)
Oh, I just meant the kind that Einstein worked on. It consists of a
mathematical model, and a bunch of points of contact with reality, which
are called "events". That theory can predict what measurements can be
made by particular observers at particular "events" but what goes on
between "events" is open to speculation, and the question of "why"
anything happens is also left open.
With all that said, when someone refers to the "invariant mass" they
mean the rest mass.
Not Wheeler and Taylor.
Really? Not everybody uses the term at all. But if they use it, _and_
they use it to refer to something else, that's a surprise.
Try googling "invariant/mass" -- it's on an awful lot of websites and as
far as I can tell it's used to mean the mass of an object in its own
rest frame.
