As I said, I had written up a reply to some things you said earlier.
So, with extensive revisions, here it is.
First, as an aside, I don't think Einstein originated the idea of the
interchangeability of mass and energy. I have been told (by someone
sort of reliable, IIRC) that there had been at least one prior
published derivation, and I have the general impression that more than
one person had worked on it; an open question at the time was what the
coefficient should be. IIRC previous derivations had led to
coefficients around (1/2)c^2, and certainly less than 1, unlike
Einstein's result, which put it at exactly 1(c^2). Don't ask me for
the details of the earlier work, though; I don't have them.
Before I go on to reply to the points in your earlier post, I should
also mention that, for all the time I spent studying relativity, I had
never looked all that closely at the derivation of e=mc^2. Over the
past day or so I've gone back to Einstein's papers (in Dover's
"Principle of Relativity") to see how he actually did it. As much for
my benefit as anyone else's, here's a capsule review, as I understand
it. I don't claim this is a detailed proof itself; I'm just reviewing
the high points of Einstein's derivation in simple English. You may
ignore it or pick it apart, as you wish. Comments on your post
begin after this summary.
(In what follows, by "Electro" I mean his paper "On the
Electrodynamics of Moving Bodies"; by "Inertia" I mean his very short
paper, "Does the Inertia of a Body Depend Upon its Energy Content?",
both published in 1905.)
In "Electro" Einstein derived the relativistic Doppler shift formula,
and produced a formula to show how the intensity of an electromagnetic
wave varies in different reference frames (end of section 7 -- I don't
see where he got it which is why I don't say he "derived" it), and
from those determined that the energy carried by a "given quantity" of
light varies in different reference frames by the same formula as the
relativistic Doppler shift. (This foreshadows the later conclusion
that the energy of a photon goes as h*nu, of course; all that needed
to be added were the quantization of light, and the fact that the
coefficient is "h".)
Given that much, the existence of "radiation pressure" follows from
conservation of momentum; he also works that through in "Electro", by
looking at light striking a mirror in a moving frame.
Once you have radiation pressure and the relativistic Doppler shift,
the change in mass for a radiating body follows pretty easily, which
is why "Inertia" is such a short paper. He makes one additional
assumption, which is that the total energy of an object in a
particular frame is some constant plus its kinetic energy. Given
that, using the previously derived transformation rules for energy of
a photon (term not yet coined in 1905, of course!), he shows with
simple arithmetic that if energy is to be conserved, the mass of a
radiating body must decrease.
To summarize, the "heavy lifting" was done in "Electro", where the
fact that light carries energy and momentum was established. Once
that is given, conservation of momentum leads almost inevitably to the
conclusion that radiation must _also_ carry away some mass, which is
really all the "E=mc^2" formula says.
Anyhow I had a few comments on your response.
David Thomson wrote:
[ snip ]
>
>>> E=mc^2
>>> mc^2=mc^2
>>> for c=1; m=m
>
>> OK, so if E=mc^2 and our units are such that c=1, then m=m. Is that
>> a contradiction? Do you want to say m is _not_ equal to itself?
>
> My presentation of the equation is not a contradiction. The standard
> presentation is:
>
> E=mc^2
> for c=1; E=m
>
> The standard presentation for the equivalence of energy and mass is
> where the contradiction lies, not my view.
>
> If you are going to change one side of an equation, you have to
> change the other side, too, in order to maintain the equality. Do
> you disagree?
I think I disagree, but I don't understand what you're getting at
here. Setting "c" to 1 _does_ change both sides of the equation,
after all, as it changes the units in which energy is measured.
It is common to choose units such that c=1 when working with
simple problems in special relativity, as it simplifies a lot of
equations. Working in units of light years (distance) and years
(time), for instance, makes c equal to 1.
But of course, that implicitly changes the units of "E", and if the
equation balanced to start with, it'll still balance afterwards, using
those new units for "E". By the same token, if you write the kinetic
energy as "K = (1/2)mv^2" in Newtonian mechanics, the equation is
still valid, and still balances, if you change your measure of
velocity from miles per hour to kilometers per hour, or lightyears per
year (or furlongs per fortnight).
Setting "c" to 1 that way doesn't help with the dimensions, of course,
and it's still not really valid to actually drop out the coefficients
of "c". However, in more formal treatments, such as, e.g., in Misner,
Thorne and Wheeler's "Gravitation", time and space are recorded using
the same dimensions, applying the "c" conversion factor at the point
of measurement. The axes are subsequently labeled using the same
units for everything. IIRC, MTW use meters for both time and space,
but they could just as well use seconds for both.
This is not invalid, but it does occasionally lead to confusion.
Among other things it makes dimensional analysis almost useless for
checking results. The main reason it's done is that the equations
simplify enormously when "c" is dropped out. In particular, the
Lorentz transforms are symmetric if "c" is left out, but they're not
apparently symmetric if "c" is included. From the point of view of
the math, it doesn't make much sense to label the axes with different
units; from the point of view of the physics, using the same units for
time and space makes it easy to make mistakes.
Whatever, it's a tradeoff...
And whatever whatever, I still don't see the problem you have with the
algebra. Perhaps you can clarify it a bit, and show the exact
operation you feel is illegal, or show a contradiction?
>>> There is no equivalence of mass and energy, except if you make
>>> special provisions for breaking the rules of algebra.
>
>> So I gather you feel e=mc^2 is false.
>
>> I have the impression that would come as a surprise to a lot of
> particle physicists.
>
> I have argued this point with many physicists, and yes, it is a
> point of contention with them.
I actually had the impression particle physicists used E=mc^2 in their
day to day work. But I can't back that up with anything solid.
> In order to pass their classes involving SR, they had to accept that
> one side of the equation could be altered, while not altering the
> other side. For them, there is no questioning the "fact" that
> energy is equivalent to mass, even when it is pointed out the basic
> math is wrong.
>
>
>> Actually modern QM is based rather heavily on SR, or so I've been led
>> to believe in conversations with quantum physicists.
>
> And your point is what? That just because people use SR that it has
> to be correct?
No, not at all. I thought you had claimed SR and QM produced
contradictory results. I was pointing out that the modern treatment
of QM is based on SR -- they're not separate fields.
> What about the rules of math, do they count for anything?
In my little world they count for almost everything. I'm a lot more
interested in the math than in physics experiments, to tell the truth.
>>> Further, with regard to SR, if we use the equation as it is given,
>>> then the energy of a photon should be zero, because it has zero
>>> mass (unless you try to fix the problem by inventing a new kind of
>>> "thought mass").
>
>> The photon has no _rest_ mass. It carries energy and can be said
>> -- and is said, by some physicists -- to carry mass as well.
>
> Nonsense!!! Absolute nonsense!!!
OK things are getting a little out of hand here.
The "rest mass" is the mass of an object when it's standing still. Do
you agree with that? When I step on the scale in the morning I
measure my "rest mass".
I think your objection is to the claim that inertial mass increases
with velocity. Is that correct?
Photons never stand still (let's wait at least a few minutes before we
start arguing about standing waves!) so the "rest mass" of a photon
isn't exactly well defined. If we try to extrapolate the formulas of
SR to a photon which is "standing still", the mass we'd find for it
would be zero -- but since they don't stand still, again, it doesn't
mean much.
And by the way, what I said about physicists is true. I've been in
arguments between physicists on exactly this point, and some assert
that it certainly has mass and they even compute a value for it.
Others turn purple in the face and start shouting at that point.
> The mass=energy equation is false, yet you use the equation as proof
> that itself must be correct.
I didn't "prove" anything there. I just pointed out that there's no
trivial contradiction to the fact that the photon carries energy and
has zero rest mass, because in relativity theory, the "relativistic
mass" and the rest mass are not the same thing.
You may think reality doesn't behave that way, of course, and that's
an issue to be settled by experiment.
> Moreover, there is no such empirically observed thing as "rest
> mass."
I would argue that "rest mass" is trivial to measure -- we do it every
day. It's the "relativistic mass" of a moving body that is a tough
nut to crack -- it's hard to weigh something in motion, and it's very
hard to weigh something moving at a good fraction of C. AFAIK any
measurements of "relativistic mass" of moving objects which showed any
difference from the "rest mass" must have been done indirectly (and I
don't know if there have been any such).
From here down things got progressively out of hand, so I'm going to
make a big <snip>. Feel free to re-post any of the cut text to which
you want a response. I want to respond to one more thing in your
note, which I will take somewhat out of context:
[ ... snip ... ]
[SAL:]
>> The mass of an iron nucleus is less than the sum of the masses of
>> the nucleons, if we use hydrogen and helium for our "standard" to
>> figure out what nucleons weigh. Fusion releases energy, and the
>> "ash" which is left behind has less mass. Makes sense to me.
[DT:]
> It doesn't make sense to science, however. If the sum of the
> individual protons and neutrons mass is greater than the combined
> mass, then that means the nucleus has less mass than the sum of the
> individual parts, right?
>
> E=mc^2
>
> If the mass decreases, the energy decreases, right?
>
> Where is the "sense" you refer to? It looks like nonsense to me.
>
> One would have to believe the exact opposite effect occurs when
> nuclei are unbound than when they are bound. If they gave up energy
> when being bound, then they must absorb energy when being split,
> otherwise the protons and neutrons would be short of mass. If
> E=mc^2 is true, and mass is converted to energy during nuclear
> binding, nuclear fission reactions should create a vast cold
> implosion, not a vast hot explosion.
OK. Let's look at some numbers, rounded off a bit, while ignoring the
mass of the electrons, and with the assumption that protons and
neutrons mass the same.
Atomic number of iron is 26. Atomic weight of iron_56 (the most
common isotope): 55.935, nucleon count 56 (26 protons and 30
neutrons). Mass of a nucleon, in an iron_56 nucleus, is 0.9988 units.
Atomic weight of hydrogen: 1.0078
So, a bare proton or neutron weighs 1.0078 units, or about 1% _more_
than the same particle embedded in an iron nucleus. Fusing 56
assorted protons and neutrons results in a mass _loss_ of about 0.5
units -- about half the mass of a proton. And that operation would
release energy, in the form of radiation and/or kinetic energy of any
released particles.
Down to iron, fusion is _exothermic_ -- fusing light elements releases
energy. What's more, _fission_ of any element no heavier than iron is
ENDOTHERMIC -- it _costs_ energy to split an iron nucleus.
BUT let's look at tellurium, the element with twice iron's atomic
number. Its atomic number is 52. Its most common isotope is Te_78,
with an atomic weight of 129.906, and 130 nucleons (52 protons and 78
neutrons). Weight of an average nucleon in a Te_78 nucleus is 0.9993
units, or about 0.04% _more_ than the same nucleon would weigh inside
an iron nucleus.
If we fused two iron_56 nuclei to build a tellurium_78 nucleus (never
mind the neutron count mismatch for the moment!), we'd _GAIN_
about 0.0004 units of mass per nucleon, or a total of about
0.034 atomic mass units total.
And fusion of elements heavier than iron is _ENDOTHERMIC_ -- you need
to put energy in to do it. On the other hand, fission of elements
heavier than iron is exothermic. When uranium splits, the pieces
weigh less than, in aggregate, than the original nucleus, and the
reaction gives off energy.
Fusion bombs use hydrogen; fission bombs use uranium (or other
super-heavy elements). Fission of a heavy element -- heavier than
iron -- releases energy, and fission bombs and reactors _must_ use
heavy elements for their fuel. Fusion bombs, on the other hand,
cannot use heavy elements as their fuel, because fusion
between heavy elements actually costs energy.
I do not see any contradiction between this and the assertion that
mass and energy are being interconverted during fission and fusion
reactions.
