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The Work Density Four-vector



Recently, I posted a series of messages regarding an extention of the
usual definition of work as the three-dimensional dot product of the
force three-vector and the displacement three-vector. I suggested that
this might be altered to a four-dimensional dot product of the force
four-vector and the displacement four-vector. I've changed my mind,
again, and now think that it is better to describe work (density) as a
four-vector which can be reduced to a very simple form in the case of a
charge density in an electric field.

I've added a section to my paper explaining it at

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

The new stuff is in Section 19. Below is the addition I made to 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 = FS

where F is the force density four-vector and S is the displacement
four-vector from the event P1(x1, y1, z1, t1) to the event
P2(x2, y2, z2, t2),

S = Sx e1 + Sy e2 + Sz e3 + St e4

where

Sx = x2 - x1, Sy = y2 - y1, Sz = z2 - z1, St = ct2 - ct1

Included in the spatial part of W is the expression FxS, 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
FtSt.

We can write the differential work dW done by a variable force on a
body, as

dW = Fds (1)

where ds = dx e1 + dy e2 + dz e3 + cdt e4 is the differential
displacement four-vector and Fds is the four-vector product of F and
ds (The differential d is not to be confused with the derivative
four-vector d (in bold)). To find the work done by this variable force
on the body, we integrate the along its worldline from event the P1 to
the event P2, to get

W = \int_P1^P2{Fds}

The time component dW_4 of (1) can be reduced, in the case of F [defined
as F = JE, where J is the current density four-vector and E is the
electric field four-vector], to

dW_4 = \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.
The time component of the work per unit proper volume W_4 done on
\rho_e, from the initial potential \phi_1 to the final potential \phi_2
at the location of \rho_e, is

W_4 = \int_{\phi_1}^{\phi_2}{\rho_e d\phi} (2)

From (2) we see that the time component 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.

--
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

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