208
8 Reissner–Nordström Particle
for some real-valued function of u. Consequently
F = −
p
r
˙
G y ϑ
(1)
∧ ϑ
(4)
+
p
r
˙
G x ϑ
(2)
∧ ϑ
(4)
+
g
r 2 ϑ
(1)
∧ ϑ
(2) .
(8.148)
The partial derivative of (8.147) with respect to u gives
˙
G = − ˙
g + 2 g H ,
(8.149)
with H given by (8.6). If we put
˙
G = g w ,
(8.150)
for some w(x, y, u) then (8.149) becomes
w = −g
−1
˙
g + 2 H ,
(8.151)
and (8.148) reads now
F =
g
r 2 ϑ
(1)
∧ ϑ
(2)
+
g p
r
∂w
∂x
ϑ
(2)
∧ ϑ
(4)
−
g p
r
∂w
∂y
ϑ
(1)
∧ ϑ
(4) .
(8.152)
Here (8.151) coincides with the Maxwell equation (8.6) and (8.152) coincides with
the dual field (8.5), with now the electric charge e of the source replaced by the
magnetic monopole moment g in each case. Since the electromagnetic energy
momentum tensor is invariant under the interchange of the Maxwell field and its
dual field, the remaining field equation (8.12), along with (8.10) and (8.11), is
unchanged except for the replacement of e with g. Hence the equations of motion
of a magnetic Reissner–Nordström particle in first or second approximation will be
given again by (8.78) and (8.108) respectively but with e replaced by g.
References
1. I. Robinson, A. Trautman, Proc. R. Soc. A 265, 463 (1962)
2. I. Robinson, P.A. Hogan, Found. Phys. 15, 617 (1985)
3. E.T. Newman, R. Posadas, Phys. Rev. 187, 1784 (1969)
4. P.A. Hogan, M. Imaeda, J. Phys. A 12, 1061 (1979)
5. J.L. Synge, Annali di Matematica Pura ed Applicata (IV) LXXXIV, 33 (1970)
6. J.L. Synge, Relativity: The Special Theory (North-Holland Publishing Company, Amsterdam,
1965)
8 Reissner–Nordström Particle
for some real-valued function of u. Consequently
F = −
p
r
˙
G y ϑ
(1)
∧ ϑ
(4)
+
p
r
˙
G x ϑ
(2)
∧ ϑ
(4)
+
g
r 2 ϑ
(1)
∧ ϑ
(2) .
(8.148)
The partial derivative of (8.147) with respect to u gives
˙
G = − ˙
g + 2 g H ,
(8.149)
with H given by (8.6). If we put
˙
G = g w ,
(8.150)
for some w(x, y, u) then (8.149) becomes
w = −g
−1
˙
g + 2 H ,
(8.151)
and (8.148) reads now
F =
g
r 2 ϑ
(1)
∧ ϑ
(2)
+
g p
r
∂w
∂x
ϑ
(2)
∧ ϑ
(4)
−
g p
r
∂w
∂y
ϑ
(1)
∧ ϑ
(4) .
(8.152)
Here (8.151) coincides with the Maxwell equation (8.6) and (8.152) coincides with
the dual field (8.5), with now the electric charge e of the source replaced by the
magnetic monopole moment g in each case. Since the electromagnetic energy
momentum tensor is invariant under the interchange of the Maxwell field and its
dual field, the remaining field equation (8.12), along with (8.10) and (8.11), is
unchanged except for the replacement of e with g. Hence the equations of motion
of a magnetic Reissner–Nordström particle in first or second approximation will be
given again by (8.78) and (8.108) respectively but with e replaced by g.
References
1. I. Robinson, A. Trautman, Proc. R. Soc. A 265, 463 (1962)
2. I. Robinson, P.A. Hogan, Found. Phys. 15, 617 (1985)
3. E.T. Newman, R. Posadas, Phys. Rev. 187, 1784 (1969)
4. P.A. Hogan, M. Imaeda, J. Phys. A 12, 1061 (1979)
5. J.L. Synge, Annali di Matematica Pura ed Applicata (IV) LXXXIV, 33 (1970)
6. J.L. Synge, Relativity: The Special Theory (North-Holland Publishing Company, Amsterdam,
1965)