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Re: Mass; E=m



Jim Green wrote:

>Suppose I weigh an ice cube and then allow it to melt and then weigh
the
>resultant water.
> Are the two weights the same?

No, because the two masses are not the same as explained below.
Itdoesn't matter whether one asks about weight or mass if the two
weights are measured at the same location under the same conditions.
This follows from the equivalence of inertial and gravitational mass.

Is this question about mass?

<>Yes, because mass is energy. The melted ice (water) has more energy than
when it was frozen. By "mass" I mean what used to be called "rest
mass." John Denker prefers to call it "invariant mass." This does not
mean that it can't change when energy is transferred to it. It means
that the mass is the same in all inertial frames that can be connected
by a Lorentz transformation. This assumes that there has been no energy
transferred into or out of the system to which the mass pertains. This
is not the case for the ice cube when it melts. That the mass of the
system increases as energy is transferred to it, by heating, radiation,
or doing work on it (as in the Joule experiment), even when it is at
rest, is most clearly explained by Einstein on pp. 46-47 of his popular
book, _Relativity, The Special Theory and the General Theory_, 15th ed.
When one says, "Mass IS energy," one means that (rest) mass is
internal energy (E_0=mc2 after converting it from E_0=m, the latter
which would have follwed from his equation as Einstein wrote in _The
Meaning of Relativity_.) Einstein also states (p.47 in _Relativity, 15th
ed._), "The inertial mass of a system of bodies can even be regarded as
a measure of its energy."

Relativistic mass?

"Relativistic mass" means gamma*m, where m is the mass discussed
aboveand gamma=1/sqrt(1-v2/c2). One is usually advised not to use
"relativistic mass" these days, but some insist that it is not actually
wrong. The relativistic mass reduces to the rest mass for a body at rest
(gamma=1). Presumably the mass of the ice and the melted ice are
measured in a frame in which they are at rest, in which case the
"relativistic mass" does increase. The relativistic mass would be
greater for the encapsulated water than for the ice cube if they were
moving at the same speed v, but only because its (rest) mass m is
greater by virtue of its greater internal energy. A shorter answer to
the question, "Relativistic mass?," would have been "No, because
'relativistic mass' is a forbidden expression among many, if not most,
physicists."

The mass that I use,"m", is not "relativistic mass" in the sense of its
use in older books (though not the ones of Einstein with which I am
familiar), but how m can change with the energy transferred to it is
part of special relativity. According to Einstein, a body of mass m
that takes up energy E_1, "as judged from a co-ordinate system moving
with the body," "has the same energy as a body of mass
(m+E_1/c2) moving with velocity v." [I changed Einstein's "E_0" to
"E_1", to avoid a conflict of notation. In _Relativity, 15th ed._ "E_0
is the added contribution to the original rest energy mc2 in this book,
unlike in _The Meaning of Relativity_. In the former, Einstei writes (p.
47), "the inertial mass of a body is not a constant, but varies
according to the change in energy of the body." The energy change that
he uses as an example comes from the absorption of radiation, not a
change in the CM kinetic energy of the body.


There seem to be two groups re relativistic mass: The Pros and The Cons.

There aren't many Pros these days, but a link (or a link within a
link)that Antti posted, indicated that relativist Wolfgang Rindler
defendedthe use of relativistic mass.

Most of the discussion has been about speed but how about other causes of
energy increase? In this case heating.

The m that I use (not "relativistic mass") is the internal energy of
thesystem in question. I only related speed to the energy of a system
when it was related to the internal energy of the system. This could be
the speed of molecules . When the internal energy of the contents of
apressure cooker is increased as a result of heating on a stove, part of
the increase in internal energy is accounted for by the greater average
speed of the water molecules (although not during the vaporization
itself). The mass of the contents is increased by the increase in
thermal motion of the water molecules (as well as other contributions to
the internal energy.)

The Pros might say that as the ice melts, its energy level increases
because the warm ambient has heated it ie done work on the ice. (The
_only_ way to increase the energy level of a system is to do work on
it.)

I suppose it depends on your definition of "work." I am using
thetraditional distinction between the processes of working W and
heating Q, to which some add the energy transfer across a system
boundary by radiation R, in the statement of the first law of
thermodynamics.

And because the energy level has increased the mass has increased --
not by enough to measure it with a balance however.

(but this is the increase of mass, not relativistic mass. The
"relativistic mass," gamma*m, if one must use it, only increases because
m increases when the entire system is at rest or at some constant
non-zero speed.. The same (frozen) ice cube at the same temperature
would have a different relativistic mass if it were traveling at a speed
0.99c compared with its relativistic mass when at rest, but that is not
what we are talking about.)

So we turn to a very
large ice berg North of Greenland and we encase it in sturdy plastic and
weigh it and then tow it to NY harbor to let is melt. If we re calibrate
for the gravity and weigh it again we find that it weighs more. Hence we
conclude that the mass has increased as E=m ie E up so m up ie it is more
than E=m E _is_ m

Correct, if E is interpreted as rest energy, as it would have to be if
the measurement were done at rest in your rest frame. In such a case
E=E_0. It would probably be better to write E_0=m*c2 orE_0=m, so that
E_0 is m. This was essentially what Einstein did in his 1922 Stafford
Little Lectures delivered at Princeton University in May 1921, the
lectures forming most of the content of his book, _The Meaning of
Relativity, 5th ed._, Princeton University Press, 1956. The relevant
page is p. 46.

The Cons would say that indeed the total m is increased but the mass of
individual particles -- (They might say the atoms or maybe the electrons
and nucleus things) -- has not.

I think the Cons, regarding "relativisic mass," gamma*m, are not con
because they think it is wrong. I think they are con because they think
the 4-vector theory in which mass m is invariant in all inertial frames
connected by a Lorentz transformation, is more useful and more elegant,
and better represents the physics. In the 4-vector formulation, the
momentum-energy 4-vector (with units in which c=1) is[E, p_x, p_y, p_z]
where E is the total energy and p_x, p_y, and p_z are the components of
relativistic 3-momentum . The norm is the invariant mass m, so that
m2=E2-p2, or
E2=m2+p2=m2 + gamma2*m2^*v2=m2(1+gamma2*v2)= gamma2*m after alittle
algebra, so that E=gamma*m. I suppose one could call the energy the
relativistic mass, but it is better to think of the mass as an invariant
(in the sense mentioned above). The mass is the invariant norm of a
4-vector whose four components vary from one inertial frame to another.
In the rest frame of the object under consideration,

p_x=p_y=p_z=0, in which case E=E_0. Then the momentum-energy 4-vector is
[E_0, 0, 0, 0], and it is clear that E_0=m as was pointed out by John
Denker. (I arrived at this a different way in a previous message.)

It has never been clear to me what the
rationale is. I take it that they would claim that the electrons,
protons,
and neutrons don't increase in energy individually. I can't bring myself
to embrace this however,

Me neither. I don't think the Cons mean this. The energy of the
electrons, protons,neutrons, molecules, etc. contribute to the rest
energy of the system,no matter whether it is kinetic energy or some kind
of potential energy of interaction, etc. Of course, the speed of these
particles is relatedto their kinetic energy, usually in a
non-relativistic way because thespeeds involved are usually not that
great. The v that occurs in the familiar relativistic equations is
usually the speed of the object as awhole -- not its consituent parts.

Perhaps someone could tutor me here. How can the
particles not increase in energy level but the ice berg does?

When ice melts but is not raised to a temperature above the melting
point, the potential energy of interaction of the system of water
molecules increases. The kinetic energy of the water molecules
doesn't change on the average unless the temperature increases. The
temperature is an indication of the average kinetic energy of the
molecules. If the water is heated until the temperature is above the
melting point, the average kinetic energy of the molecules increases.
During vaporization at atmospheric pressure, the water does not change
temperature -- thus no change in average KE of water molecules, but the
PE again increases. If the steam released as water boils can be kept at
atmospheric pressure (or even if not as in the pressure cooker) further
heat added once all the water becomes steam will increase the average
kinetic energy as indicated by the increased temperature. Whatever
increases the energy of the system of water molecules also increases the
(rest) mass of the system, albeit immeasurably.

They say
that the particles are vibrating faster therefore the total energy has
increased.... Maybe I just don't understand the Cons.

The Cons (and also the Pros) say that anything that increases theenergy,
kinetic or otherwise, of the system of water molecules, increases the
rest mass of the system as a whole. For the Cons, those against
"relativistic mass," rest mass is the only game in town, so it is just
"mass."I think the confusion between what is meant by the Pros and the
Cons hasto do with the difference in the meaning of "invariant" and
"constant." The Pro "relativistic mass" people say that relativistic
mass is not invariant in going from one inertial frame to another. The
Cons say that mass (meaning what the the Pros mean by "rest mass") is
invariant. Both Pros and Cons say that rest mass, which the Cons call
"mass," is not constant when energy istransferred to or from it in its
rest frame. There is a section (Box 7-3) on pp. 208-209 of _Spacetime
Physics, 2nd ed._ by Taylor and Wheeler entitled "Invariant? Conserved?
Constant." For them, mass is invariant. They only mention "relativistic
mass" on p. 250 to discourage its use, because it leads to
misunderstanding. They give two reasons -- one being that mass, as the
norm of the monmentum-energy 4-vector, is a very different concept from
the total energy, E=gamma*m, the time component of the energy momentum
4-vector,[E, p_x, p_y, p_z], whereas the mass m is the norm (magnitude)
of the energy momentum 4-vector. E is the "time component" because it
corresponds to the time component of the proper time
4-vector(displacement for a timelike interval) __d tau__=[dt, dx, dy,
dz]. (d tau)= dt only in a frame of reference in which the object is is
at rest where dx=dy=dz=0. It is easy to show that dt/(d tau)=gamma=
1/sqrt(1-v2). The 4-vector version of velocity is __u__= [dt/(d tau),
dx/(d tau), dy/(d tau), dz/(d tau)]
=[dt/(d tau), dx/dt*dt/(d tau), dy/dt*dt(d tau), dz/dt*dt/(d tau)]

=[gamma, u_x, u_y, u_z] where u_x=dx/dt*dt/(d tau)= gamma*v_x, etc.
Defining the momentum-energy 4-vector __P__ as __P__=m*__u__, we get
__P__=[gamma*m, p_x, p_y, p_z] where p_x= gamma*u_x=gamma*m*v_x, etc.
__P__=[E, p_x, p_y, p_z] where E=gamma*m=m*dt/(d tau) is the time
component, obviously not an invariant as previously mentioned.

More importantly, T&W caution that calling gamma*m "relativistic mass"
might give the incorrect impression that the change of energy E with
velocity (of the object as a whole) might result from
some change in internal structure of the object. They claim that the
increase of energy with v originates in the geometric properties of
spacetime itself. (p. 251). They elaborate on this in Sec. 5 in Chap. 1
starting on p. 1-7 of _Exploring Black Holes_, Addison Wesley Longman, 2000.

The question as to whether mass changes with velocity is discused in
detail under the FAQ's at
<http://math.ucr.edu/home/baez/physics/index.html>, a reference that
Antti Savinainen provided.

In conclusion, one might say that most pros in relativity are Cons in
that they don't allow the use of "relativistic mass."

Hugh Logan