Mass-Energy Equivalence [Math]

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From: Tom Roberts on 7 Apr 2010 18:37

Tony M wrote:
> On Apr 6, 10:29 am, Tom Roberts <tjroberts...(a)sbcglobal.net> wrote:
>> Normally we apply the word "mass" to OBJECTS, not to collections of unbound
>> objects. In the initial state the electron and the positron EACH have a mass,
>> and the total mass is 1.022 MeV/c^2. In the final state the gammas each have
>> zero mass, and the total mass is 0. The total energies of the initial and final
>> states are the same. The total kinetic energy of the initial state is 0, but the
>> total kinetic energy of the final state is 1.022 MeV. These units have c=1, and
>> it is obvious that mass was indeed converted to kinetic energy.
>>
>> This is relativity; the kinetic energy of an object is its
>> total energy minus its mass (c=1).
>
> Sue, see what you did? Exactly what I was afraid of, you made Tom
> think one can convert mass to energy! Just kidding, he probably
> believed that all along.

You are supposed to READ what I write. I was quite careful to say mass is
converted to KINETIC energy, not to "energy". The distinction is important.

For this annihilation one has:

State: Initial Final
------- -----
Total mass (MeV/c^2) 1.022 0
Kinetic energy (MeV) 0 1.022
Total energy (MeV) 1.022 1.022
Total momentum (MeV/c) 0 0

A simple glance shows that kinetic energy increased by the exact same amount
that mass decreased, and we say "mass was converted to kinetic energy"; the
total energy and total momentum are unchanged. This _IS_ how these words are used.

Please remember that when I say "mass", I use the word in the current sense of
modern physics, in which it is invariant.

Most of your confusion is due to your use of the word "mass"
where today the correct phrase is "relativistic mass". Note
also that the latter is not really a mass, it is an ENERGY
(that's the primary reason it is an anachronism).

Some of your confusion is between objects and systems. When we say "the mass of
a system", we don't mean the norm of the system's 4-momentum, we mean the sum of
the norms of its constituents' individual 4-momenta. This is a holdover from
Newtonian physics, and can be confusing to novices -- to assist the reader,
careful authors say "total mass" here.

Note also that in SR and GR it is rare to mention the total
mass of a system, as it is really only useful when comparing
to Newtonian mechanics. In relativity that quantity has no
special meaning or useful purpose. In contrast, the mass of
an object does have special meaning and is essential; this can
apply to systems that are considered as objects (which
requires they be bound). Like any other specialized field,
physics has a technical vocabulary all its own, and one must
learn it before one can be conversant in the subject.

> Guys, if you accept the following two statements then everything else
> will just fall into place:
> 1. The Law of Conservation of Energy is valid.
> 2. E=mc2 is valid (in all situations, not just at one�s own
> convenience).

Neither is of universal validity. One should not "accept" them, one should
UNDERSTAND them. And their limitations.

The correct way to discuss this is to apply the theory, not just grab a few
sound bites and think they are all one needs. They aren't.

> If Tom admits that energy is conserved, then by E=mc2 => mass is
> conserved as well! Conservation of energy implies conservation of
> mass, there�s no way around it!

This is just plain not true. The energy of an object is the time component of
its 4-momentum, and its mass is the norm of its 4-momentum. When you say "energy
conservation you mean of a SYSTEM OF OBJECTS, not just a single object.

You make several mistakes here, confusing an object with
a system, and confusing "mass" with "relativistic mass".

Let me translate your statement above into a statement using standard meanings:

If Tom admits that energy and momentum are conserved for a system,
then the norm of its 4-momentum is conserved as well. Conservation
of 4-momentum implies conservation of 4-momentum, there's no way
around it! [your E=mc2 is untranslatable.]

> If Tom admits that photons have energy, then by E=mc2 => photons have
> mass as well!

Again this is just plain not true. Energy and mass are different.

Note also that the equation "E=mc2" cannot be applied to
photons, independent of your later statement of what your
"m" means.

But in a later post you said:
> In my equation by m I meant the
> relativistic mass m=gamma*m0, where m0 would be the rest or invariable
> mass.
When you use words with nonstandard meanings you cannot expect other people to
understand what you say.

This is NOT what Einstein meant in his original equation. He was discussing the
REST ENERGY of an object. The new notion that relativity introduced was that an
object at rest has nonzero total energy related to its mass, and it was
essential to include this in the equations of the theory.

Many popular writers have bastardized this and confused the issue no end, in
part because the technical vocabulary has morphed since Einstein wrote.

Note that the "relativistic mass" of an object is THE SAME
THING as its total energy. These phrases both refer to the
time component of its 4-momentum. Note that it is acceptable
to omit the first word from "total energy", but it is not
acceptable to omit the first word of "relativistic mass";
this is so because the first does not affect anything, the
second does.

> [quoting wikipedia]
> �In as much as energy is conserved in closed systems in relativity,
> the mass of a system is also a quantity which is conserved:

This is wrong. For the mass of the system to be conserved requires that its
momentum also be conserved. That is, 4-momentum conservation implies "mass"
conservation, but energy conservation alone does not do so. And this also
requires the rather nonstandard use of "mass" of a system to refer to the norm
of its total 4-momentum.

For systems like the two diverging gammas in the above final state, we normally
don't apply the term "mass" TO THE SYSTEM, because it is essentially useless
(the mass of each gamma is of course zero, and this is standard usage). For
"mass" to be useful, it must be measurable, and that requires a single object or
a bound system, not a collection of unbound objects.

> One must keep in mind that there is ONLY ONE MASS.

This is just plain not true. There are MANY aspects of an object to which the
term "mass" could be applied:

1. (NM) the m in F=ma (inertial mass 1)
2. (NM) the m in F=GMm/r^2 (passive gravitational mass)
3. (NM) the M in F=GMm/r^2 (active gravitational mass)
4. (NM) the m in p=mv [3-vectors] (inertial mass 2)
5. (SR) the m in E=mc^2 (rest mass)
6. (SR) the m in E=\gamma m c^2 (rest mass)
7. (SR) the m in P=mV [4-vectors] (inertial mass 3)
8. (**) the m in E=mc^2 ("relativistic mass")
9. (**) the m in F=ma ("transverse mass")
10.(**) the m in F=\gamma^2 m a ("longitudinal mass")
** is nonstandard usage in SR.

A defining characteristic of "mass" is that it is INTRINSIC to an object.
Traditionally it measured "how much stuff is present". Being intrinsic, it
cannot possibly depend on observer (coordinates), and must therefore be
invariant. In relativity this implies that only 5, 6, and 7 apply.

Purists generally take 4 to be the definition of mass in NM,
and the corresponding 7 to be the definition in SR.

But now, with this understanding in SR, there is indeed only one mass (5, 6, and
7 are all the same). But it is NOT what you think it is.

Tim Little wrote:
> E^2 = m^2 c^4 + p^2 c^2 is the *correct* equation in relativity, with
> p representing momentum. The greatly publicized E = m c2 is a
> simplification, in the case where p = 0.

Yes.

Tom Roberts

From: BURT on 7 Apr 2010 19:52

On Apr 7, 12:08 pm, spudnik <Space...(a)hotmail.com> wrote:
> NB: the equation, E=mcc, may not be in the best form,
> regarding "dimensional analysis;" but,
> that's just algebra.
>
> thus:
> well, you can't rely on every thing that you read in a Wiki,
> either. that is just silly;
> when two quanta of hydrogen are fuzed to make one
> of helium, some of the matter is "converted"
> into energy. when one combines a quantum of hydrogen and
> one of antihydrogen, ideally, all of the matter converts.
>
> aether is just mental fluff around electromagnetism!
>
> > "The equation E = mc2 indicates that energy always exhibits mass in
> > whatever form the energy takes. It does not imply that mass may be
> > converted to energy, for modern theory holds that neither mass nor
> > energy may be destroyed, but only moved from one location to another."
>
> --Light: A History!http://wlym.comhttp://21stcenturysciencetech.comhttp://white-smoke.wetpaint.com

First energy was infinitely small point particle of neutron that
decayed and combined to form the first Hydrogen.

Mitch Raemsch

From: mpc755 on 7 Apr 2010 22:35

On Apr 7, 3:05 pm, spudnik <Space...(a)hotmail.com> wrote:
> well, you can't rely on every thing that you read in a Wiki,
> either. that is just silly;
> when two quanta of hydrogen are fuzed to make one
> of helium, some of the matter is "converted"
> into energy. when one combines a quantum of hydrogen and
> one of antihydrogen, ideally, all of the matter converts.
>
> aether is just mental fluff around electromagnetism!
>
> >http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence
> > "The equation E = mc2 indicates that energy always exhibits mass in
> > whatever form the energy takes. It does not imply that mass may be
> > converted to energy, for modern theory holds that neither mass nor
> > energy may be destroyed, but only moved from one location to another."
>
> --Light: A History!http://wlym.comhttp://21stcenturysciencetech.comhttp://white-smoke.wetpaint.com

http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence

"According to the theory of relativity, mass and energy as commonly
understood, are two names for the same thing, and neither one is
changed or transformed into the other. Rather, neither one appears
without the other. Rather than mass being changed into energy, the
view of relativity is that rest mass has been changed to a more mobile
form of mass, but remains mass. In this process, neither the amount of
mass nor the amount of energy changes. Thus, if energy changes type
and leaves a system, it simply takes its mass with it. If either mass
or energy disappears from a system, it will always be found that both
have simply moved off to another place."

What is being described in the paragraph above is matter transitioning
to aether.

From: mpc755 on 7 Apr 2010 22:35

On Apr 7, 3:08 pm, spudnik <Space...(a)hotmail.com> wrote:
> NB: the equation, E=mcc, may not be in the best form,
> regarding "dimensional analysis;" but,
> that's just algebra.
>
> thus:
> well, you can't rely on every thing that you read in a Wiki,
> either. that is just silly;
> when two quanta of hydrogen are fuzed to make one
> of helium, some of the matter is "converted"
> into energy. when one combines a quantum of hydrogen and
> one of antihydrogen, ideally, all of the matter converts.
>
> aether is just mental fluff around electromagnetism!
>
> > "The equation E = mc2 indicates that energy always exhibits mass in
> > whatever form the energy takes. It does not imply that mass may be
> > converted to energy, for modern theory holds that neither mass nor
> > energy may be destroyed, but only moved from one location to another."
>
> --Light: A History!http://wlym.comhttp://21stcenturysciencetech.comhttp://white-smoke.wetpaint.com

http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence

"According to the theory of relativity, mass and energy as commonly
understood, are two names for the same thing, and neither one is
changed or transformed into the other. Rather, neither one appears
without the other. Rather than mass being changed into energy, the
view of relativity is that rest mass has been changed to a more mobile
form of mass, but remains mass. In this process, neither the amount of
mass nor the amount of energy changes. Thus, if energy changes type
and leaves a system, it simply takes its mass with it. If either mass
or energy disappears from a system, it will always be found that both
have simply moved off to another place."

What is being described in the paragraph above is matter transitioning
to aether.

From: mpc755 on 7 Apr 2010 22:36

On Apr 7, 4:12Â pm, BURT <macromi...(a)yahoo.com> wrote:
> On Apr 6, 9:04Â pm, mpc755 <mpc...(a)gmail.com> wrote:
>
>
>
> > On Apr 6, 9:31Â pm, BURT <macromi...(a)yahoo.com> wrote:
>
> > > On Apr 6, 3:23Â pm, mpc755 <mpc...(a)gmail.com> wrote:
>
> > > > On Apr 6, 6:08Â pm, BURT <macromi...(a)yahoo.com> wrote:
>
> > > > > On Apr 6, 2:34Â pm, mpc755 <mpc...(a)gmail.com> wrote:
>
> > > > > >http://cfa-www.harvard.edu/Walsworth/pdf/PT_Romalis0704.pdf
>
> > > > > > "A possible candidate for dark energy that avoids some of the fine-
> > > > > > tuning problems associated with the cosmological is quintessence, a
> > > > > > very low-energy field with a wavelength comparable to the size of the
> > > > > > known universe. In addition to its effect on the expansion of the
> > > > > > universe, quintessence might also manifest itself through its possible
> > > > > > interactions with matter and radiation."
>
> > > > > >http://dictionary.reference.com/browse/quintessence
>
> > > > > > "quinÂ·tesÂ·sence
> > > > > > â â/kwÉªnËtÉsÉns/ Show Spelled[kwin-tes-uhns] Show IPA
> > > > > > ânoun
> > > > > > 1. the pure and concentrated essence of a substance.
> > > > > > 2. the most perfect embodiment of something.
> > > > > > 3. (in ancient and medieval philosophy) the fifth essence or element,
> > > > > > ether, supposed to be the constituent matter of the heavenly bodies"
>
> > > > > > A low-energy field with a wavelength comparable to the size of the
> > > > > > known universe is aether as a one something.
>
> > > > > > Aether and matter are different states of the same material.
>
> > > > > > Matter is compressed aether and aether is uncompressed matter.
>
> > > > > > Aether is the pure essence of matter.
>
> > > > > > "quintessence might also manifest itself through its possible
> > > > > > interactions with matter"
>
> > > > > > Aether interacts with matter by being displaced by matter.
>
> > > > > > The pressure exerted by aether displaced by matter manifests itself as
> > > > > > gravity.
>
> > > > > > The physical effects the expansion of matter transitioning to aether
> > > > > > has on the neighboring matter and aether manifests itself as energy.
>
> > > > > > 'DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT? By A.
> > > > > > EINSTEIN'http://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf
>
> > > > > > "If a body gives off the energy L in the form of radiation, its mass
> > > > > > diminishes by L/c2."
>
> > > > > > The mass of the body does diminish, but the matter which no longer
> > > > > > exists as part of the body has not vanished. It still exists, as
> > > > > > aether. As the matter transitions to aether it expands in three
> > > > > > dimensions. The effect this transition has on the surrounding aether
> > > > > > and matter is energy.
>
> > > > > Why would the universal speed limit squared define the fundamental
> > > > > energy in mass?
>
> > > > > Mitch Raemsch
>
> > > > It could be the relationship of matter to aether. The equation may be
> > > > written as:
>
> > > > A=Mc^2, where A is Aether and M is matter.- Hide quoted text -
>
> > > > - Show quoted text -
>
> > > Abusing that mathematical truth is important for you to not get away
> > > with. You do so at your own peril.
>
> > > Mitch Raemsch
>
> > I am simply pointing out there is a relationship to matter and aether
> > and what we define as 'energy' is the physical effects matter
> > transitioning to aether has on the neighboring matter and aether. In
> > this regard:
>
> > A=Mc^2 where A is aether and M is matter.- Hide quoted text -
>
> > - Show quoted text -
>
> And why do you think this?
>

Because matter and aether are different states of the same material.