When one speaks of the relativistic mass of an object, should one
include the potential energy of the object?

wrote:
When one speaks of the relativistic mass of an object, should one
include the potential energy of the object?

First, the term "relativistic mass" is an anachronism, and has been
found to be more confusing than it is worth, especially to newbies. In
particular, it does not appear in any fundamental theory of physics
(which are all based on SR). In SR and GR, the mass of an object is the
norm of its 4-momentum, and is thus an invariant. "Relativistic mass",
on the other hand is misnamed (being really energy, not mass), and is
dependent on reference frame -- a characteristic that robs it of most
utility.


In general, potential energy is not well defined in relativity;
certainly it cannot be used with the ubiquity it is used in Newtonian
mechanics. For instance, neither electrostatics nor gravity can be
described in terms of a "potential energy" -- the closest analogy to
those Newtonian potentials in relativity are the 4-potential of
electrodynamics, and the metric tensor (whose properties determine the
geometrical relationships commonly known as gravity).


Tom Roberts

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.

wrote in message
ups.com...
When one speaks of the relativistic mass of an object, should one
include the potential energy of the object?

If the object has internal degrees of freedom in which the proper mass can
vary (e.g. an atom can transition between states) then its possible for the
internal potential energy to change along with a corresponding change in
mass. If you're speaking about a charged particle in an EM field then it has
the same meaning as it does in electrodynamics since EM is relativistically
correct. I.e. the total energy E = K + E_o (K = kinetic energy, E_o = rest
energy) is related to relativistic energy T and potential energy V as

E = T + V

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

(Note: I screwed up Eq. 18 so ignore that. It will be corrected sometime in
the future.)

The role of E in 4-vectors is that E is proportional to the time component
of the time component of the canonical momentum 1-form. The potential, V is
the time component of the 4-potential A^u. T is the proportional to the time
component of the 4-momentum.

Note: My notation is not standard but is consitent with Goldstein's
mechanics text.

Pete

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

Thanks, Pete

That's just what I was looking for. I will work through it and get back
to you.

By the way, would you please look at my concurrent thread "interesting
idea".

I want to see for myself if there is anything to that idea.

wrote:
When one speaks of the relativistic mass of an object, should one
include the potential energy of the object?

Absolutely so. The following equation is derived from the Lagrangian
associated with the geodesics in Schwarzschild spacetime.

E = m c^2 sqrt(1 - 2 U) / sqrt(1 - B^2)

Where

** E = Observed energy
** m = Mass at (B = U = 0) or rest mass
** U = G M / c^2 / r
** B^2 c^2 = (dr/dt)^2 / (1 - 2 U) + r^2 (cosH)^2 (dO/dt)^2 / (1 -
2 U) + r^2 (dH/dt)^2 / (1 - 2 U)

Also, the observed mass or the relativistic mass can be described as
follows.

m' = m sqrt(1 - 2 U) / sqrt(1 - B^2)

Notice at low speed and low gravitation, the observed energy becomes

E = m c^2 (1 - U + B^2 / 2)

In which you understand the following as

** m B^2 c^2 / 2 = Kinetic energy
** m U c^2 = Potential energy
** E - m c^2 = Total energy

However, if you accept this very simple equation, it is going to stir
up problems within the foundation of GR. That is why you are not going
to find any GR religious folks to accept that mass being merely an
observed quantity. Instead, they will talk about mass being rest mass
period.

wrote in message
oups.com...

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

Thanks, Pete

See also - http://www.geocities.com/physics_wor...ear_energy.htm

Its a worked out example of how potential energy fits into the calculations.
I wanted something concrete to refer to so I made this example.

That's just what I was looking for. I will work through it and get back
to you.

By the way, would you please look at my concurrent thread "interesting
idea".

I'll try to.

Pete

wrote in message
oups.com...

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

Thanks, Pete

That's just what I was looking for. I will work through it and get back
to you.

There is also a method of calculating it by using the Lagrangian method. See

http://www.geocities.com/physics_wor...tic_energy.htm

Pete

"Koobee Wublee" wrote in message oups.com...
wrote:
When one speaks of the relativistic mass of an object, should one
include the potential energy of the object?

Absolutely so. The following equation is derived from the Lagrangian
associated with the geodesics in Schwarzschild spacetime.

You better stay away from that until you understand the meaning
of the variables in elementary special relativity:
http://users.telenet.be/vdmoortel/di...rentzTale.html

Dirk Vdm

Tom Roberts wrote:
wrote:
When one speaks of the relativistic mass of an object, should one
include the potential energy of the object?

First, the term "relativistic mass" is an anachronism, and has been
found to be more confusing than it is worth, especially to newbies. In
particular, it does not appear in any fundamental theory of physics
(which are all based on SR). In SR and GR, the mass of an object is the
norm of its 4-momentum, and is thus an invariant. " [snip]
Tom Roberts

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

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

Mike