Mike wrote:
Tom Roberts wrote:
In SR and GR, the mass of an object is the
norm of its 4-momentum, and is thus an invariant.

Not in GR, it is impossible to define mass that way since 4-vectors are
found in different tangent spaces.

Yes in GR. For a given pointlike object, its 4-momentum is transported
along its trajectory, and the norm of its 4-momentum is the same in ANY
of the tangent spaces for the points the object itself traverses. This
is trivially true for a geodesic path, but it is true in general because
P.A=g(P,A)=0, where g(.,.) is the metric, P is the object's 4-momentum,
and A is the object's 4-acceleration. Note that all 3 quantities are
evaluated at the same point in the manifold (_any_ given point along the
object's trajectory), so the 3 tensors all "live" in the same
(co-)tangent space.


I suggest GRists try to find a better definition of mass that is
convincing.

There is nothing wrong with this one. Indeed, it is probably the only
one that satisfies the basic requirements of what we mean by "mass" and
has the proper local limit for inertial frames. Certainly it is the
usual definition.


Tom Roberts

Tom Roberts wrote:
Mike wrote:
Tom Roberts wrote:
In SR and GR, the mass of an object is the
norm of its 4-momentum, and is thus an invariant.

Not in GR, it is impossible to define mass that way since 4-vectors are
found in different tangent spaces.

Yes in GR. For a given pointlike object, [shrug]

How convenient to define mass just for pointlike objects. But real life
is not poinlike, don't you think so?

I mean, some people should understand that physics is not a pure
extension of geometry.


I suggest GRists try to find a better definition of mass that is
convincing.

There is nothing wrong with this one. Indeed, it is probably the only
one that satisfies the basic requirements of what we mean by "mass" and
has the proper local limit for inertial frames. Certainly it is the
usual definition.

For your so called "pointlike" objects. But not for any other kind of
the usual objects you encounter in your favirite "local" experiments.

GR is to "pointlike" in every sense. Is there anything elese besides
"pointlike" physics?

Mike





Tom Roberts

Pete wrote:
For a proof please see -
http://www.geocities.com/physics_wor...ork_energy.htm

Pete

Hi Pete,

I went through your proof line by line. I did find a few minor errors
or typos - not sure which.

The result I found is that potential energy is not part of relativistic
mass. Kinetic energy is part of relativistic mass. Potential energy +
Kinetic Energy(which includes relativistic energy) = Total energy
which is constant.

I like doing this type of physics and would like to work with you on
your papers.

wrote in message
ps.com...

Pete wrote:
For a proof please see -
http://www.geocities.com/physics_wor...ork_energy.htm

Pete

Hi Pete,

I went through your proof line by line. I did find a few minor errors
or typos - not sure which.

The result I found is that potential energy is not part of relativistic
mass. Kinetic energy is part of relativistic mass. Potential energy +
Kinetic Energy(which includes relativistic energy) = Total energy
which is constant.

I wasn't speaking about the potential energy contained with the body itself.
I was speaking about the potential energy of the object as a function of its
location. I thought you understood this. I appologize if my writing was
unclear.

The potential energy which you speak of above is related to E as

E = K + E_0

where E = total energy (in absense of external field), K = kinetic energy
and E_0 = rest energy

E_0 is a function of the total internal kinetic energy of the object when
the object is at rest as well as the total internal kinetic energy of the
body also when the body is at rest.

I like doing this type of physics and would like to work with you on
your papers.

Sure. Where would you like to start? Try starting by reading a paper on mass
that I wrote
http://www.geocities.com/physics_world/mass_paper.pdf

Try understanding this too -
http://www.geocities.com/physics_wor...rd_paradox.htm

The root of understanding that is to understand this first
http://www.geocities.com/physics_wor...gy_vs_mass.htm

I look forward to working with you. Best wishes

Pete

ps - my e-mail address is peter102560 [at] comcast [dot] net

Pete wrote:

Sure. Where would you like to start? Pete

ps - my e-mail address is peter102560 [at] comcast [dot] net

The papers that I create tend to confuse people. I can sit down with
someone and
explain in words, but when I write it down it doesn't work. So that is
why I started
following these newsgroups - to sharpen my communication skills.

In http://www.geocities.com/physics_wor...ork_energy.htm, which I
will refer to as
the work-energy paper, in equation (3), c squared appears as if by
magic. Then in equation
(4), it disappears and is replace by the number 2.

"Pete" wrote in message
. ..

wrote in message
ps.com...

Pete wrote:
For a proof please see -
http://www.geocities.com/physics_wor...ork_energy.htm

Pete

Hi Pete,

I went through your proof line by line. I did find a few minor errors
or typos - not sure which.

The result I found is that potential energy is not part of relativistic
mass. Kinetic energy is part of relativistic mass. Potential energy +
Kinetic Energy(which includes relativistic energy) = Total energy
which is constant.

I wasn't speaking about the potential energy contained with the body
itself. I was speaking about the potential energy of the object as a
function of its location. I thought you understood this. I appologize if
my writing was unclear.

The potential energy which you speak of above is related to E as

E = K + E_0

where E = total energy (in absense of external field), K = kinetic energy
and E_0 = rest energy

E_0 is a function of the total internal kinetic energy of the object when
the object is at rest as well as the total internal kinetic energy of the
body also when the body is at rest.

I like doing this type of physics and would like to work with you on
your papers.

Sure. Where would you like to start? Try starting by reading a paper on
mass that I wrote
http://www.geocities.com/physics_world/mass_paper.pdf

Try understanding this too -
http://www.geocities.com/physics_wor...rd_paradox.htm

The root of understanding that is to understand this first
http://www.geocities.com/physics_wor...gy_vs_mass.htm

I look forward to working with you. Best wishes

Pete

ps - my e-mail address is peter102560 [at] comcast [dot] net

Pete nice to have you back!
I plan to read the above with much interest. If my understanding is right,
potential energy has to add to the mass of a system simply because inside a
system potential energy can be converted into kinetic energy, while the
system must (should) conserve momentum.

Regards,
Harald

wrote:
Tom Roberts wrote:
In general, potential energy is not well defined in relativity;
Someone answered my question in the following way:

The mass of a nucleus is more than the mass of the constituent parts.
This communicates to me that assembling such a structure somehow
increases the mass of the constituent parts which sounds very interesting.

Hmmm. The mass of most structures is LESS than the sum of the masses of
their parts, including nuclei (when considered to be constructed out of
protons and neutrons).

1 amu = 931.5 MeV/c^2, but the mass of a proton is
938.3 MeV/c^2, and a neutron's mass is 939.5 MeV/c^2.
Remember 12 amu is the mass of a C^12 atom with 6 p
and 6 n plus 12 electrons. Each electron has a mass of
0.5 MeV/c^2, so that C^12 nucleus has a mass of
12*931.0 MeV/c^2, significantly smaller than 6*938.3+6*939.5.
The electron binding energies are 0.0 to this accuracy.

Nomenclatu generally when we say "the potential energy of an object"
we mean _external_ to the object. When one models a compound object as
having parts that are held together, we call the corresponding energy
its "binding energy".

In assembling a compound object (that remains together as a single
object), necessarily some energy must be released -- this is the binding
energy of the object. If this did not happen, the object could, and
would, fly apart on its own and we would not call it an object. In order
to separate the object into its parts, that binding energy must be
supplied to pull it apart.

When totaling the energy of an object, one adds m*c^2 for each
component, the internal kinetic energy of each component (in the
object's rest frame), and _subtracts_ the binding energy of each
component. Many/most authors consider binding energy negative, in which
case it simply adds to the rest and kinetic energies of the components.

So when assembling such a structure, the mass of the result can be less
than the sum of the masses of its constituent parts, as long as they
have small kinetic energies within the compound object. For nuclei
assembled from protons and neutrons, the binding energy per nucleon
exceeds their kinetic energy, and the nucleus has smaller mass than the
sum of the nucleons. For a proton assembled from a set of quarks and
gluons, the kinetic energies exceed the binding energy, and the mass of
the proton is larger than the sum of the masses of its quarks and gluons
[#].

[#] this is a special case, because quantum effects are large.
What I said is for the expectation values of the kinetic
energy, binding energy, and quark/gluon counts; but if one
tries to pull a quark or gluon out of the proton, the binding
energy will increase so that it cannot be done -- either the
attempt to remove it will fail, or a pair of quarks/gluons will
be created and you no longer have the system you started with.

All of this is well known, and for a compound object the mass is simply
E/c^2 for the total energy described above (in the object's rest frame).
This is solidly established experimentally, for nuclei, atoms,
molecules, and larger objects such as the moon.

One aspect of this is if one starts with a macroscopic object and heats
it (without destroying it), its mass will increase. Unfortunately the
effect is so small it is not practical to measure it in the laboratory.


Tom Roberts

Mike wrote:
Tom Roberts wrote:
Mike wrote:
Tom Roberts wrote:
In SR and GR, the mass of an object is the
norm of its 4-momentum, and is thus an invariant.
Not in GR, it is impossible to define mass that way since 4-vectors are
found in different tangent spaces.
Yes in GR. For a given pointlike object, [shrug]

How convenient to define mass just for pointlike objects. But real life
is not poinlike, don't you think so?

Actually, in the context of GR, in everyday life all objects of interest
can indeed be considered pointlike. That is, all objects we use daily
are vastly smaller than the earth or the sun.

This is physics, and approximations are an inherent part of the process.
The approximation that my automobile is pointlike compared to the earth
is a very good approximation. Indeed, for the purpose of computing its
trajectory, even the moon can be considered pointlike to high accuracy.


I mean, some people should understand that physics is not a pure
extension of geometry.

Yes. YOU need to learn that. The approximations involved are essential.
Physics is not math. shrug


I suggest GRists try to find a better definition of mass that is
convincing.
There is nothing wrong with this one. Indeed, it is probably the only
one that satisfies the basic requirements of what we mean by "mass" and
has the proper local limit for inertial frames. Certainly it is the
usual definition.

For your so called "pointlike" objects. But not for any other kind of
the usual objects you encounter in your favirite "local" experiments.

In GR, for objects of non-negligible size, there is no obvious value
corresponding to mass. Of course in principle it is wrong to consider
such an object, because it is inherently made up of tiny objects (atoms)
that interact with each other in complicated ways; the correct way to
handle such objects is as an enormous number of (pointlike) atoms,
accounting for their inter-atomic interactions as well as the geometry
of spacetime. This is clearly far beyond our capabilities, and is
unecessary for virtually all purposes, so some sort of approximation is
used....

I discuss pointlike objects and local experiments, because they are
SIMPLE, and because actual results are known. For the general case of
two or more objects of non-negligible size and mass no exact solutions
of the field equation are known.


Tom Roberts

Tom Roberts wrote:
Mike wrote:

I mean, some people should understand that physics is not a pure
extension of geometry.

Yes. YOU need to learn that. The approximations involved are essential.
Physics is not math. shrug

Without math, physics is BS. shrug

Witness the Lorentz transform where one of its two properties (the
principle of Relativity) violates scientific experimentations and
everyday experiences. Based on this same mathematics, there seem to be
an infinite ways to interpret it into different piles of BS.

For your so called "pointlike" objects. But not for any other kind of
the usual objects you encounter in your favirite "local" experiments.

In GR, for objects of non-negligible size, there is no obvious value
corresponding to mass. Of course in principle it is wrong to consider
such an object, because it is inherently made up of tiny objects (atoms)
that interact with each other in complicated ways; the correct way to
handle such objects is as an enormous number of (pointlike) atoms,
accounting for their inter-atomic interactions as well as the geometry
of spacetime. This is clearly far beyond our capabilities, and is
unecessary for virtually all purposes, so some sort of approximation is
used....

I discuss pointlike objects and local experiments, because they are
SIMPLE, and because actual results are known. For the general case of
two or more objects of non-negligible size and mass no exact solutions
of the field equation are known.

You must also develop ways including mathematical models to explain the
existence and its behavior of a solid object made out of these
pointlike objects.

"Koobee Wublee" wrote in message ups.com...
Tom Roberts wrote:
Mike wrote:

I mean, some people should understand that physics is not a pure
extension of geometry.

Yes. YOU need to learn that. The approximations involved are essential.
Physics is not math. shrug

Without math, physics is BS. shrug

Witness the Lorentz transform

That you so miserably fail to understand:
http://users.telenet.be/vdmoortel/di...rentzTale.html
;-)

Dirk Vdm