Re: Theory explains where the extra mass comes from when an object increases in speed
Bob_S
> It has been over one hundred years since Einstein introduced his Theory
> of Relativity. The theory states when an object increases in speed, its
> mass will also increase. This has been verified, but no answer has
> revealed the source from which the extra mass arises. We traditionally
> believe mass is located inside the matter that constitutes the object.
> Since kinetic energy consists of no solid matter, it would be illogical
> to assert that mass arises from nothing.
> The answer should be based on physics, not on mathematics.
If you have a mass, say one specific proton, with a total energy of 980
GeV (like at Fermilab) some people will say it has a mass or a total
mass of 980 GeV. They should say total energy, because no additional
mass has been added. Anyone in its rest frame will say its mass is just
938.3 MeV. The extra "mass" is just an external, not intrinsic property
of that proton. The answer is based on physics.
Bob S
--
Bob_S
Rich L.
There is actually a somewhat deeper answer to this question. In
Relativity theory, energy and momentum are different aspects of the
same thing, and mass is just one form of energy. When we say that a
particle's mass increases at near light speeds, what we really mean is
that if you apply a force to that particle and observe how it is
accelerated by that force, it is accelerated as if it were more
massive than when the particle was nearly at rest in the laboratory.
The only mass that is truely intrinsic to a particle is the rest mass.
If you work with the energy-momentum 4 vector, it becomes apparent
that energy and momentum are really just different aspects of the same
thing. In different rest frames the energy and momentum transform
into each other. I think it is very interesting to realize that if
you represent the rest mass as a frequency, using f=mc^2/h, where h is
Planks constant, and then consider how this frequency would be
perceived by a moving observer, one obtains a new frequency that
corresponds to the total energy of the particle in that moving frame,
and a wavelength (a projection of the frequency along the observers
spatial axis) that corresponds to the momentum of the particle. In
other words, what we call the total energy of the particle is the
projection of the rest mass "frequency" onto our (the observers) time
axis, and what we call "momentum" is the projection of the particles
rest mass "frequency" onto our space axies.
Rich L.
=========== Moderator's note ========================================
Mass is not just one form of energy. As the representation theory of
the proper orthochronous Poincare group teaches us, the mass is a Casimir
operator, characterizing the representation of the Poincare group, while
energy and momentum together combine to a four-vector. Energy is the
time-like component of this four-vector, mass a Lorentz scalar.
This is quite different in the ordinary representation of the Galilei
group in non-relativistic quantum mechanics, where mass is a central
charge. The subtle difference is that in the usual formulation of
non-relativistic quantum mechanics does not permit the superposition of
states with different mass, while this is no problem in relativistic
quantum theory.
jw...@basicisp.net
I've shortened the full quote to a more appropriate length.
===========================================================================
An even simpler way of looking at this, avoiding assumptions of group
theory or QM, is just to look at a dimensional analysis of the equation,
E^2 = (pc)^2 + (mc^2)^2.
Here, mc^2 is rest energy, and pc is energy depending on velocity in
whatever inertial frame (rest frame) has been chosen for the
analysis.
pc further can be expressed as mvGc, where G is the Einstein
gamma = (1 - v^2/c^2)^(-1/2) and is dimensionless.
Therefore, mvc has dimensions of energy; therefore m does not have
dimensions of energy.
So, "mass increases" is not accurate: The equation above
shows that mvG, not m, increases to make E increase with
velocity -- equivalent to the Moderator's explanation.
harry
news:Bob_S....@physicsbanter.com...
Note: in many textbooks (incl. Feynman's) "mass" is defined using p=mv
("relativistic mass") and not limited to observations in an object's rest
frame - just as "time" does not necessarily mean "proper time"; probably
that's what the OP referred to. That definition differs from the one used in
the early days of relativity. In recent years there has been a strong push
(notably from particle physicists) to always mean with "mass", "rest mass"
(or "proper mass"), which in turn has been opposed by a persistent minority
(mostly physics teachers). It has deteriorated in a fight over names and
words, similar to that about planets -> not worthy of discussion in this
thread.
Back to the "physical" question of the OP. Using your definition of mass,
Alonso and Finn pointed out in their textbook on electromagnetism
(Fundamental Physics Part 2) that the mass (or inertia) of the electron
corresponds to its field energies - the energy stored in its magnetic field
corresponds at least approximatively to its kinetic energy (and its rest
energy to its electric field energy). In approximation this was already
known before the inception of relativity theory (e.g Thomson 1881, I can
send you the PDF). Thus, from that perspective mass has been added.
Moreover, the increase of mass corresponds to increase of magnetic field
energy, which is observed as self induction. That also provides a physical
explanation of why its resistance against acceleration is direction
dependent at high speed. What special relativity did was to predict the
exact increase of the electron's inertia and resistance against acceleration
with increasing speed (Lorentz 1904, Einstein 1905).
Current theory is less clear for protons and atoms (IMHO, some insight is
still lacking here). Logically, a physical explanation should be sought
along the same lines as that for the electron. As you seem to suggest, mass
and energy have physical meaning; they are indeed more than merely
accounting tools.
Harald
========= Moderator's note =======================
That's indeed the only weak point about the Feynman lectures I've found: It's
using an outdated notion of mass, which one shouldn't use anymore. I'm quite
puzzled why Feynman did this!
harry
"harry" <harald.NOTT...@epfl.ch> wrote in message
news:1256024...@sicinfo3.epfl.ch...
[..]
[..]
> ========= Moderator's note =======================
>
> That's indeed the only weak point about the Feynman lectures I've found:
> It's
> using an outdated notion of mass, which one shouldn't use anymore. I'm
> quite
> puzzled why Feynman did this!
Reply to the moderator: one should not tell teachers what they should
prefer! ;-)
Alonso and Finn as well as Feynman apparently preferred that definition of
mass, which was anyway the most popular one at the time (the eighties).
There are advantages and disadvantages to both definitions of mass; the
choice to either include a gamma term inside a symbol or not merely affects
the way the physics is taught and not the physics itself.
Harald
Hendrik van Hees
> Reply to the moderator: one should not tell teachers what they should
> prefer! ;-)
> Alonso and Finn as well as Feynman apparently preferred that
> definition of mass, which was anyway the most popular one at the time
> (the eighties). There are advantages and disadvantages to both
> definitions of mass; the choice to either include a gamma term inside
> a symbol or not merely affects the way the physics is taught and not
> the physics itself.
It is always advantageous to use as simple a definition as possible.
Physics is difficult enough by itself, one shouldn't obscure it by
using bad math!
Let's look at the most simple case, namely classical (i.e.,
non-relativistic) relativistic mechanics of a point particle. It is
covariantly described by a trajectory in the space-time manifold:
x^{\mu}(lambda), where lambda is a scalar parameter such that
dt/d(lambda)>0 (*)
in one (and thus in all) inertial frames.
A natural choice is the proper time of the particle, i.e., the time
measured in a set of co-moving locally inertial reference frames, where
the particle is at rest:
d tau=sqrt[dx^{mu}/d(lambda) dx_{mu}/d(lambda)] d(lambda)>0
which by definition is a scalar too.
Then we define the four-velocity and four-acceleration of the particle
by
u^mu=dx^mu/d(tau); a^{mu}=d^2 x^mu/d(tau),
which are both four vectors as the space-time vector.
With this construct covariant equations of motion
m a^{mu}=K^{mu}.
Here, m is the (invariant) mass of the particle, which is a scalar, and
K^{mu} is the four-force acting on the particle.
This four-force on the rhs. of this equation has to fulfill the
constraint
m u_{mu} a^{mu}=u_{mu} K^{mu}=0.
which follows from
u_{mu} u^{mu}=1=const => d/d(tau) [u_{mu} u^{mu}]= a_{mu} u^{mu}=0.
It immediately follows that with the four-momentum
p^{mu}=m u^{mu} => m^2=p_{mu} p^{mu},
and due to (*) one can also use the time of an arbitrary inertial frame,
and then the three-momentum reads
\vec{p}=m d \vec{x}/d(tau)=gamma m d\vec{x}/dt,
where
gamma=dt/d(tau)=1/sqrt[1-(d\vec{x}/dt)^2]
and then the eom. reads in old-fashioned 3D notation:
d/dt [gamma m d\vec{x}/dt]=\vec{K} dtau/dt=\vec{K}/gamma=\vec{F},
where all the beauty of the covariant formulation is gone.
The only practical advantage in solving the eom. is, that one has not to
bother anymore about the constraint. Provided the original relativistic
eom. is consistent in this respect, everything is relativistically
correct, but it's very hidden in the non-covariant 3D notation for the
eom.
So I prefer to teach the logically straight-forward way based on
Minkowski's ingenious analysis of the structure of space-time!
--
Hendrik van Hees Institut f�r Theoretische Physik
Phone: +49 641 99-33342 Justus-Liebig-Universit�t Gie�en
Fax: +49 641 99-33309 D-35392 Gie�en
http://theory.gsi.de/~vanhees/faq/
harry
message news:hbns9t$7pl$1...@news2.open-news-network.org...
> harry wrote:
>
>> Reply to the moderator: one should not tell teachers what they should
>> prefer! ;-)
>> Alonso and Finn as well as Feynman apparently preferred that
>> definition of mass, which was anyway the most popular one at the time
>> (the eighties). There are advantages and disadvantages to both
>> definitions of mass; the choice to either include a gamma term inside
>> a symbol or not merely affects the way the physics is taught and not
>> the physics itself.
>
> It is always advantageous to use as simple a definition as possible.
> Physics is difficult enough by itself, one shouldn't obscure it by
> using bad math!
For sure we all agree on that; however, no erroneous math is promoted in any
texbook that I know. Probably you mean with "bad" math that you don't fancy
it as it obscures what you like to highlight.
> Let's look at the most simple case, namely classical (i.e.,
> non-relativistic) relativistic mechanics of a point particle. It is
> covariantly described by a trajectory in the space-time manifold:
> x^{\mu}(lambda), where lambda is a scalar parameter such that
>
> dt/d(lambda)>0 (*)
>
> in one (and thus in all) inertial frames.
>
> A natural choice is the proper time of the particle, i.e., the time
> measured in a set of co-moving locally inertial reference frames, where
> the particle is at rest:
>
> d tau=sqrt[dx^{mu}/d(lambda) dx_{mu}/d(lambda)] d(lambda)>0
>
> which by definition is a scalar too.
>
> Then we define the four-velocity and four-acceleration of the particle
> by
>
> u^mu=dx^mu/d(tau); a^{mu}=d^2 x^mu/d(tau),
>
> which are both four vectors as the space-time vector.
[..]
> It immediately follows that with the four-momentum
>
> p^{mu}=m u^{mu} => m^2=p_{mu} p^{mu},
>
> and due to (*) one can also use the time of an arbitrary inertial frame,
> and then the three-momentum reads
>
> \vec{p}=m d \vec{x}/d(tau)=gamma m d\vec{x}/dt,
>
> where
>
> gamma=dt/d(tau)=1/sqrt[1-(d\vec{x}/dt)^2]
>
> and then the eom. reads in old-fashioned 3D notation:
>
> d/dt [gamma m d\vec{x}/dt]=\vec{K} dtau/dt=\vec{K}/gamma=\vec{F},
> where all the beauty of the covariant formulation is gone.
>
> The only practical advantage in solving the eom. is, that one has not to
> bother anymore about the constraint. Provided the original relativistic
> eom. is consistent in this respect, everything is relativistically
> correct, but it's very hidden in the non-covariant 3D notation for the
> eom.
> So I prefer to teach the logically straight-forward way based on
> Minkowski's ingenious analysis of the structure of space-time!
To me it looks more "natural" to choose the time as defined in a standard
coordinate system (such as "Universal Time" to which our clocks are
referenced). But I won't linger on such issues as the OP's question is not
addressed in your commentary, as far as I can see. In the context of this
thread, the OP's question to you becomes:
If one prefers Minkowski's take on SRT, "where does the extra mass/energy
come from", or more precisely, how does one explain (physically, not
mathematically) the inertia increase of an object that increases in speed?
Harald
army...@foo.invalid
>
> Mass is not just one form of energy. As the representation theory of the
> proper orthochronous Poincare group teaches us, the mass is a Casimir
> operator, characterizing the representation of the Poincare group, while
> energy and momentum together combine to a four-vector. Energy is the
> time-like component of this four-vector, mass a Lorentz scalar.
That's the rest mass, but Bob_S by "mass" was obviously meaning the
relativistic mass, which is essentially the energy divided by c^2.