The Work Density Four-vector; Revision

[David Rutherford <drutherford@SOFTCOM.NET>]

I've made a few changes to Section 19, "The Work Density Four-vector" in
my paper at

http://www.softcom.net/users/der555/newtransform.pdf

I've also added a new part to my "Applications" titled "Energy Density
Correction" in pdf format at

http://www.softcom.net/users/der555/enerdens.pdf

where I show that the energy density u of the electric field E is twice
the conventional value, that is,

u = e_0 E^2

instead of

u = (1/2) e_0 E^2

where e_0 is the permitivity constant. It's mostly the same material I
posted, here, recently under the thread "Energy density: the correct
one" (but much easier to read than ascii).

Below is the complete Section 19, from my paper, with some alterations
due to the translation from the tex version to ascii. I've used an
exclamation point preceding a quantity to indicate that the quantity
is a four-vector (bold in my paper).


The Work Density Four-vector
----------------------------

We introduce the work density four-vector !W, which is
the work per unit proper volume done on a body by a constant force
density !F over a spacetime displacement !S as the four-vector product

!W = !F!S

where !F is the force density four-vector and !S is the displacement
four-vector from the event P_1(x_1, y_1, z_1, t_1) to the event
P_2(x_2, y_2, z_2, t_2),

!S = S_x e_1 + S_y e_2 + S_z e_3 + S_t e_4

where

S_x = x_2 - x_1, S_y = y_2 - y_1, S_z = z_2 - z_1, S_t = ct_2 - ct_1

Included in the spatial part of !W is the expression !Fx!S, which is
related to the torque on a body, but also present is an expression for
a new quantity !F:!S, which includes the impulse. The time part of !W
contains the expression !F.!S, which includes the conventional,
three-dimensional expression for work as well as the additional term
F_t S_t.

We can write the differential work d!W done by a variable force on a
body, as

d!W = !Fd!s

where d!s = dx e_1 + dy e_2 + dz e_3 + cdt e_4 is the differential
displacement four-vector and !Fd!s is the four-vector product of !F and
d!s (the differential d is not to be confused with the derivative
four-vector !d). To find the work done by this variable force on the
body, we integrate along its worldline from the event P_1 to the event
P_2, to get

!W = \int_{P_1}^{P_2}{!Fd!s}                (1)

In the case of the work done by a variable force on a current
distribution in an electric field, we can write (1), using !F = !J!d!A,
where !J is the current density four-vector and !d!A is the derivative
product of the derivative four-vector !d and the potential four-vector
!A, as

d!W = (!J(!d!A))d!s                           (2)

We can then reduce (2) to

d!W = !Jd!A                                  (3)

where d!A is now the differential vector potential. This implies that we
can write the work density four-vector !W, in this case, as

!W = !J!A

(Technically, due to the product rule, we should then write
d!W = !Jd!A + !Ad!J, but we are considering !J to be a constant, here)

The differential magnitude dW (this is now a scalar quantity) of d!W,
from (3), is simply the magnitude of !Jd!A, which is \rho_e d\phi,
where \rho_e is the charge density on which the work is being done, and
d\phi is the differential electric potential at the location of \rho_e.
Thus, we can write dW as,

dW = \rho_e d\phi                   (4)

We can now find the magnitude of the work per unit proper volume W
done on \rho_e, by integrating (4) from the initial potential \phi_1
to the final potential \phi_2 at the location of \rho_e,

W = \int_{\phi_1}^{\phi_2}{\rho_e d\phi}          (5)

> From (5) we see that the magnitude of the work per unit proper volume
done on \rho_e is independent of the displacement of \rho_e. It depends
only on the difference between the initial and final potentials at the
location of \rho_e.

Evidently, work can be done on a body whether or not it undergoes
a displacement in space. For example, in order to create a
distribution of charged particles, work must be done on each
particle to move it into place against the fields of the particles
already in place (assembled particles). In addition, however, work
must be done on the assembled particles, in order to keep them in
place, against the field of each new particle, as the new particle
is moved into place. This additional work done on the assembled
particles in order to keep them in place must be included, along
with the work initially done on each new particle to move it into
place, in the total work required to create the distribution and
thus in the total energy of the distribution.

--
Dave Rutherford
"New Transformation Equations and the Electric Field Four-vector"
http://www.softcom.net/users/der555/newtransform.pdf

Applications:
"4/3 Problem Resolution"
http://www.softcom.net/users/der555/elecmass.pdf
"Action-reaction Paradox Resolution"
http://www.softcom.net/users/der555/actreact.pdf
"Energy Density Correction"
http://www.softcom.net/users/der555/enerdens.pdf

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.