Rock Brentwood
Feb2-09, 06:00 AM
On Jan 31, 5:25 am, Rock Brentwood <markw...@yahoo.com> wrote:
> The term "Maxwell Equations" is used loosely here. The analogy is with
> the equations
> dA = F; dF = 0
> which describe the field "kinematics" and
> dG = J; dJ = 0
> which describe the field "dynamics".
The description laid out in the last part for the electrodynamic field
laws is actually more general than what goes under the name Maxwell-
Lorentz theory, since it assumed no specific Lagrangian. So, the
missing element to the "dynamics" is the "constitutive law" -- what
actually comprises the conjugate fields. This is determined by the
functional form of the Lagrangian.
Interestingly, you can actually go a long way with the (generalized)
Maxwell dynamics without saying much of anything at all about the
functional form. In general, one may only require that the Lagrangian
be explicitly Lorentz-invariant, in which case it reduces to a
function of the form
L = L(I,J)
where
I = (E^2 - B^2 c^2)/2, J = E.B
are the two Lorentz invariants of the field. More generally, one could
pose the condition that it be *quasi-invariant*. In that case, there
are two more invariants that arise, which are explicitly dependent on
the gauge potentials. But in the former case, this is already a lot to
say. Given the condition, one can already define two coefficients:
epsilon = dL/dI, theta = dL/dJ
and then reduce the variational
D(L) = epsilon D(I) + theta D(J)
with
D(I) = E.D(E) - c^2 B.D(B)
D(J) = E.D(B) + B.D(E)
giving you the result
D(L) = (epsilon E + theta B).D(E) - (epsilon c^2 B - theta E).D(B)
and the conjugate fields
D = epsilon E + theta B
H = epsilon c^2 B - theta E,
where epsilon and theta are functions of I and J.
The two coefficients reflect the appearance of two invariants (this
situation generalizes to non-Abelian gauge theory, except that there
are then also cubic invariants). One of the two invariants is a total
differential, J. Therefore, the coefficient theta is only define up to
an additive constant and can always be taken to be 0 for the null
fields I = 0, J = 0. For null fields, epsilon(I, J) = epsilon(0,0) =
epsilon_0, and the constitutive law reduces to the Lorentz relations,
describing the linear propagation of waves.
More generally, equating the Lagrangian density to L = epsilon I +
theta J - Delta/4, and setting n = epsilon/epsilon_0, this is enough
to write out the stress tensor as the usual one seen in
electrodynamics, multiplied by n, and added onto by a trace part =
Delta/4.
(A footnote: The trace T^{mu}_{mu} is, in fact, the kernel of the
dilation current. Though absent macroscopically, it's most certainly
present at the microscopic level, since the breaking of dilation
invariance is the very underpinning to all matters related to
renormalization. Here, we see that the phenomenon -- mistakenly
attributed to as a feature specific to quantum theory -- is actually
*classical* in grounding, simply as a deviation of the functional form
of L away from a 2nd degree homogeneity, and the deviation of the
sources from Coulomb at short range. This concept is not new -- nor
its employment as a means to resolve the field infinity problem. It
originates with Maxwell and Faraday themselves, who employed it just
for that purpose. Part of the reason for the exercise below and to
follow is to bring out the analogous constitutive coefficients for
gravity.)
In here, and the following, I'll use D(...) no denote the variational
of a quantity. This is not the exterior derivative operator, so it
acts on differential forms by the ordinary Leibnitz rule
D(a ^ b) = D(a) ^ b + a ^ D(b).
The analogue is now played by the fields dual to the one defined
previously:
> *x* = (theta^1,theta^2,theta^3), t = theta^0
> *X* = (Theta^1,Theta^2,Theta^3), T = Theta^0
> *sigma* = (omega^2_3,omega^3_1,omega^1_2)
> *alpha* = (omega^1_0,omega^2_0,omega^3_0)
> *Sigma* = (Omega^2_3,Omega^3_1,Omega^1_2)
> *Alpha* = (Omega^1_0,Omega^2_0,Omega^3_0).
The simplest motivation for them is to start from the classical flat-
space case: to find a local transport law for momentum AND angular
momentum which generalizes to curvilinear coordinates.
The conservation of momentum, in the language of fluid dynamics, can
be cast in the language of differential forms as
dP_a = 0
where P is the DENSITY of momentum, represented as a current 3-vector:
P = P^{mu}_a d^3x_{mu}.
There is a 3-form for each component (a = 0, 1, 2, 3) of momentum. The
index (a) is no longer treated as a local coordinate index, but as a
global frame index. The components P^{mu}_a are components of a tensor
DENSITY, not merely a tensor.
A similar process for the angular momentum immediately reveals a
problem. One starts out defining the angular momentum current as a 3-
form
J_{ab} = S_{ab} + x_a ^ P_b - x_b ^ P_a,
where S_{ab} is the intrinsic part of the angular momentum (the spin
density) and writes out the conservation law as the following
continuity equation
dJ_{ab} = 0.
That's the transport law stated in usual form. But this makes explicit
use of Cartesian coordinates.
However, there is a general prescription for eliminating such
dependence in ALL first-order moments! This makes specific use of the
momentum conservation law.
The method is to "borrow" a derivative from P and put it on the "x".
The justification for this is that since dP_a = 0, then P_a can,
itself, be locally expressed as the exterior differential of a
"kinetic potential" 2-form, p_a:
P_a = dp_a.
So, making use of this, we write (omitting the wedge for brevity)
x_a ^ P_b - x_b ^ P_a = x_a dp_b - x_b dp_a = d(x_a p_b - x_b p_a)
+ dx_b ^ p_a - dx_a ^ p_b.
It doesn't matter in the transport law dJ_{ab} = 0 doesn't matter if
the total differential is present or not, so assume it's been absorbed
in a redefinition of J_ab and S_ab and replace it with:
J_{ab} = S_{ab} + theta_b ^ p_a - theta_a ^ p_b.
In flat space, dx_a = theta_a = eta_{ab} theta^b. So, this is the
basis of the generalization to curvilinear coordinates in flat space.
As was the case with P, since dJ_{ab} = 0, we can assume that the
angular momentum current also arises from a "kinetic potential" 2-form
s_{ab},
J_{ab} = ds_{ab}.
This results in the equation
S_{ab} = ds_{ab} + theta_a ^ p_b - theta_b ^ p_a.
It's assumed that J_{ab}, S_{ab} and s_{ab} are all anti-symmetric in
their two frame indices.
When going over to curvilinear coordinates and curved spacetimes, this
is further generalized with the introduction of the connection 1-
forms. The exterior differentials are replaced by exterior *covariant*
differentials:
ds_{ab} --> ds_{ab} - omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac}
dp_a --> dp_a - omega^c_a ^ p_c.
Thus, the final result is that we obtain the following field
equations:
S_{ab} = ds_{ab} - omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac} +
theta_a ^ p_b - theta_b ^ p_a
P_a = dp_a - omega^c_a ^ p_c.
Amazingly, these can be derived from a Lagrangian as its Euler-
Lagrange equations -- but with one important condition.
To find the actual form of the Lagrangian, we'll start out by assuming
the relation between the spin 3-current and connection is given in
terms of a prospective Lagrangian 4-form L by
D(L) = ... + 1/2 D(omega^{ab}) ^ S_{ab} + ...
To make this work then requires that we also have, in this variation,
the following
* -1/2 D(omega^{ab}) ^ ds_{ab}
* 1/2 D(omega^{ab}) ^ (omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac})
* -1/2 D(omega^{ab}) ^ (theta_a ^ p_b - theta_b ^ p_a).
The first term integrates by parts to (note the signs)
* 1/2 d(D(omega^{ab} ^ s_{ab}) - 1/2 D(d(omega^{ab})) ^ s_{ab}.
The last two terms can be rewritten as
* -D(omega^{ab}) ^ omega^c_b ^ s_{ca}
* D(omega^a_b) ^ theta^b ^ p_a.
By virtue of anti-symmetry, we also have
-D(omega^{ab}) ^ omega^c_b ^ s_{ca} = 1/2 D(omega^{ab} ^ omega^c_b)
^ s_{ac}.
With some index manipulation, this can be rewritten as
1/2 D(omega^{ac} ^ omega^b_c) ^ s_{ab}
= -1/2 D(omega^a_c ^ omega^{cb}) ^ s_{ab}.
Thus, these terms combine to yield
-1/2 D(omega^{ab} + omega^a_c ^ omega^{cb}) ^ s_{ab} + d(...)
= -1/2 D(Omega^{ab}) ^ s_{ab} + d(...).
This shows that the kinetic potential s_{ab} must be paired off with
the CURVATURE 2-form Omega^{ab}.
For the term involving theta, we obtain
D(omega^a_b) ^ theta^b ^ p_a.
= D(omega^a_b ^ theta^b) ^ p_a - omega^a_b ^ D(theta^b) ^ p_a
= D(omega^a_b ^ theta^b) ^ p_a + D(theta^b) ^ omega^a_b ^ p_a.
Thus, the only way to fit in the equation for the momentum 3-current
and its kinetic potential is so as to cancel out the last term. This
requires pairing off
D(L) = ... - D(theta^b) ^ P_b + ...
In all, applying this to the remaining terms of the Euler-Lagrange
equation for P_a, we obtain the following additions
D(L) = ... - D(theta^b) ^ P_b + D(theta^b) ^ dp_b - D(theta^b) ^
omega^c_b ^ p_c.
Integrating the second of these terms by parts, we have (again, note
the signs)
-d(D(theta^b) ^ p_b) + D(d(theta^b)) ^ p_b.
It's only at this point that we reach the proviso. The condition on
which the momentum equation be a Euler-Lagrange equation is seen by
placing all the terms for all the variationals of (theta) together.
This yields the following expression
D(L) = ... + D(omega^a_b ^ theta^b) ^ p_a + D(theta^b) ^ omega^a_b
^ p_a
- D(theta^b) ^ P_b + D(d(theta^a)) ^ p_a - D(theta^b) ^ omega^c_b ^
p_c + d(...)
Thus, we obtain (with some index-renaming)
D(L) = ... + D(d(theta^a + omega^a_b ^ theta^b) ^ p_a - D(theta^a)
^ P_a + d(...).
The kinetic potential p_a must be paired off with the TORSION 2-form,
Theta^a:
D(L) = ... + D(Theta^a) ^ p_a + ...
So, it's finally at this point that we find that the total variation
in the Lagrangian should be:
D(L) = -D(theta^a) ^ P_a + D(Theta^a) ^ p_a + 1/2 D(omega^{ab}) ^ S_
{ab} - 1/2 D(Omega^{ab}) ^ s_{ab}.
In summary, we've derived the "field laws" as the following Euler-
Lagrange equations
ds_{ab} - omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac} = J_{ab} = S_{ab}
- theta_a ^ p_b + theta_b ^ p_a
dp_a - omega^c_a ^ p_c = P_a
This is analogous to the Maxwell field law dG = J. Taking exterior
derivatives gives us the following as the generalization of the flat-
space transport laws
dP_a - omega^c_a ^ P_c = -Omega^c_a ^ p_c
dJ_{ab} - omega^c_a ^ J_{cb} - omega^c_b ^ J_{ca} = -Omega^c_a ^ s_
{cb} - Omega^c_b ^ s_{ac}.
The (Euler-Lagrange) field equations can, thus, be thought of as the
first integrals of the transport equations of the underlying fluid
dynamics. This is entirely analogous to the development of the
Bernoulli law or the Navier-Stokes equation, and that's the role being
played here by the field law.
When the second equation is written out in terms of S_{ab}, instead of
J_{ab} then the following additions appear on the left
theta_a ^ P_b - theta_b ^ P_a
and on the right
Theta_a ^ p_b - Theta_b ^ p_a.
The left-hand term, written in terms of the components of the momentum
3-current, is
theta_a ^ P^{mu}_b d^3x_{mu} - theta_b ^ P^{mu}_a d^3x_{mu}
= (theta_{a mu} P^{mu}_b - theta_{b mu} P^{mu}_a) d^4 x
= (P_{ab} - P_{ba}) d^4 x.
This is the anti-symmetric part of the index-lowered energy-momentum
tensor
P_{ab} = theta_{a mu} P^{mu}_b.
Unlike the stress tensor (which is a multiple of the derivative of the
Lagrangian with respect to the contravariant metric g^{mu}), the
energy-momentum tensor need not be symmetric.
A DISTINCTION is being made, here, between "stress tensor" and "energy-
momentum tensor", both in the names (T_{ab} and P_{ab}, respectively)
and their conceptions. The indices in the latter, P_{ab}, play
completely unsymmetric roles. The a-index is associated with the
volume 3-form, while the b-index is a GLOBAL frame index for the
momentum current.
So ...
>d*Alpha* - *sigma* x *Alpha* - *alpha* x *sigma* = 0
>
> This will be continued in the next part...
continuing on, first note the correction:
d*Alpha* - *sigma* x *Alpha* - *alpha* x *Sigma* = 0
The asterisks are being used here and below to denote vector boldface.
We supplement the definitions
> *x* = (theta^1,theta^2,theta^3), t = theta^0
> *X* = (Theta^1,Theta^2,Theta^3), T = Theta^0
> *sigma* = (omega^2_3,omega^3_1,omega^1_2)
> *alpha* = (omega^1_0,omega^2_0,omega^3_0)
> *Sigma* = (Omega^2_3,Omega^3_1,Omega^1_2)
> *Alpha* = (Omega^1_0,Omega^2_0,Omega^3_0).
with the following definitions:
H = -P_0, *P* = (P_1, P_2, P_3),
h = -p_0, *p* = (p_1, p_2, p_3),
*j* = (s_23, s_31, s_12), *k* = (1/c)^2 (s_01, s_02, s_03),
*J* = (S_23, S_31, S_12), *K* = (1/c)^2 (S_01, S_02, S_03),
resulting in the following equations
d*p* - *sigma* x *p* + (1/c)^2 *alpha* h = *P*
dh + *alpha*.*p* = H
d*k* - *sigma* x *k* + (1/c)^2 *alpha* x *j* = *K* + t *p* - (1/c)
^2 *x* h
d*j* - *sigma* x *j* - *alpha* x *k* = *J* - *x* x *p*
The transport laws are embodied in the continuity equations
d*P* - *sigma* X *P* + (1/c)^2 *alpha* H = -*Sigma* X *p* + (1/c)^2
*Alpha* h
dH + *alpha*.*P* = *Alpha*.*p*.
d*K* - *sigma* x *K* + (1/c)^2 *alpha* x *J* - t *P* + (1/c)^2 *x*
H
= -*Sigma* x *k* + (1/c)^2 *ALpha* x *j* - T *p* + (1/c)^2 *X* h
d*J* - *sigma* x *J* - *alpha* x *K* + *x* x *P*
= -*Sigma* x *j* - *Alpha* x *k* + *X* x *p*.
In the vicinity of a free-fall frame, where we can write *alpha* = 0,
*sigma* = 0, and in a weak field where the curvature and torsion
components are negligible, these equations reduce to more familiar
form as
d*P* = 0
dH = 0
d*K* - t *P* + (1/c)^2 *x* H = 0
d*J* + *x* x *P* = 0
Reverting to flat space coordinates (x^0 = s, (x^1,x^2,x^3) = *r*), we
have
*x* = d*r*, t = ds
and
d(*K* - s *P* + (1/c)^2 *r* H) = 0
d(*J* + *r* x *P*) = 0
which are the original form posed for the transport laws for the
angular momentum, where the form-valued 3-vector *J* here represents
the 3-current density of the intrinsic spin.
> The term "Maxwell Equations" is used loosely here. The analogy is with
> the equations
> dA = F; dF = 0
> which describe the field "kinematics" and
> dG = J; dJ = 0
> which describe the field "dynamics".
The description laid out in the last part for the electrodynamic field
laws is actually more general than what goes under the name Maxwell-
Lorentz theory, since it assumed no specific Lagrangian. So, the
missing element to the "dynamics" is the "constitutive law" -- what
actually comprises the conjugate fields. This is determined by the
functional form of the Lagrangian.
Interestingly, you can actually go a long way with the (generalized)
Maxwell dynamics without saying much of anything at all about the
functional form. In general, one may only require that the Lagrangian
be explicitly Lorentz-invariant, in which case it reduces to a
function of the form
L = L(I,J)
where
I = (E^2 - B^2 c^2)/2, J = E.B
are the two Lorentz invariants of the field. More generally, one could
pose the condition that it be *quasi-invariant*. In that case, there
are two more invariants that arise, which are explicitly dependent on
the gauge potentials. But in the former case, this is already a lot to
say. Given the condition, one can already define two coefficients:
epsilon = dL/dI, theta = dL/dJ
and then reduce the variational
D(L) = epsilon D(I) + theta D(J)
with
D(I) = E.D(E) - c^2 B.D(B)
D(J) = E.D(B) + B.D(E)
giving you the result
D(L) = (epsilon E + theta B).D(E) - (epsilon c^2 B - theta E).D(B)
and the conjugate fields
D = epsilon E + theta B
H = epsilon c^2 B - theta E,
where epsilon and theta are functions of I and J.
The two coefficients reflect the appearance of two invariants (this
situation generalizes to non-Abelian gauge theory, except that there
are then also cubic invariants). One of the two invariants is a total
differential, J. Therefore, the coefficient theta is only define up to
an additive constant and can always be taken to be 0 for the null
fields I = 0, J = 0. For null fields, epsilon(I, J) = epsilon(0,0) =
epsilon_0, and the constitutive law reduces to the Lorentz relations,
describing the linear propagation of waves.
More generally, equating the Lagrangian density to L = epsilon I +
theta J - Delta/4, and setting n = epsilon/epsilon_0, this is enough
to write out the stress tensor as the usual one seen in
electrodynamics, multiplied by n, and added onto by a trace part =
Delta/4.
(A footnote: The trace T^{mu}_{mu} is, in fact, the kernel of the
dilation current. Though absent macroscopically, it's most certainly
present at the microscopic level, since the breaking of dilation
invariance is the very underpinning to all matters related to
renormalization. Here, we see that the phenomenon -- mistakenly
attributed to as a feature specific to quantum theory -- is actually
*classical* in grounding, simply as a deviation of the functional form
of L away from a 2nd degree homogeneity, and the deviation of the
sources from Coulomb at short range. This concept is not new -- nor
its employment as a means to resolve the field infinity problem. It
originates with Maxwell and Faraday themselves, who employed it just
for that purpose. Part of the reason for the exercise below and to
follow is to bring out the analogous constitutive coefficients for
gravity.)
In here, and the following, I'll use D(...) no denote the variational
of a quantity. This is not the exterior derivative operator, so it
acts on differential forms by the ordinary Leibnitz rule
D(a ^ b) = D(a) ^ b + a ^ D(b).
The analogue is now played by the fields dual to the one defined
previously:
> *x* = (theta^1,theta^2,theta^3), t = theta^0
> *X* = (Theta^1,Theta^2,Theta^3), T = Theta^0
> *sigma* = (omega^2_3,omega^3_1,omega^1_2)
> *alpha* = (omega^1_0,omega^2_0,omega^3_0)
> *Sigma* = (Omega^2_3,Omega^3_1,Omega^1_2)
> *Alpha* = (Omega^1_0,Omega^2_0,Omega^3_0).
The simplest motivation for them is to start from the classical flat-
space case: to find a local transport law for momentum AND angular
momentum which generalizes to curvilinear coordinates.
The conservation of momentum, in the language of fluid dynamics, can
be cast in the language of differential forms as
dP_a = 0
where P is the DENSITY of momentum, represented as a current 3-vector:
P = P^{mu}_a d^3x_{mu}.
There is a 3-form for each component (a = 0, 1, 2, 3) of momentum. The
index (a) is no longer treated as a local coordinate index, but as a
global frame index. The components P^{mu}_a are components of a tensor
DENSITY, not merely a tensor.
A similar process for the angular momentum immediately reveals a
problem. One starts out defining the angular momentum current as a 3-
form
J_{ab} = S_{ab} + x_a ^ P_b - x_b ^ P_a,
where S_{ab} is the intrinsic part of the angular momentum (the spin
density) and writes out the conservation law as the following
continuity equation
dJ_{ab} = 0.
That's the transport law stated in usual form. But this makes explicit
use of Cartesian coordinates.
However, there is a general prescription for eliminating such
dependence in ALL first-order moments! This makes specific use of the
momentum conservation law.
The method is to "borrow" a derivative from P and put it on the "x".
The justification for this is that since dP_a = 0, then P_a can,
itself, be locally expressed as the exterior differential of a
"kinetic potential" 2-form, p_a:
P_a = dp_a.
So, making use of this, we write (omitting the wedge for brevity)
x_a ^ P_b - x_b ^ P_a = x_a dp_b - x_b dp_a = d(x_a p_b - x_b p_a)
+ dx_b ^ p_a - dx_a ^ p_b.
It doesn't matter in the transport law dJ_{ab} = 0 doesn't matter if
the total differential is present or not, so assume it's been absorbed
in a redefinition of J_ab and S_ab and replace it with:
J_{ab} = S_{ab} + theta_b ^ p_a - theta_a ^ p_b.
In flat space, dx_a = theta_a = eta_{ab} theta^b. So, this is the
basis of the generalization to curvilinear coordinates in flat space.
As was the case with P, since dJ_{ab} = 0, we can assume that the
angular momentum current also arises from a "kinetic potential" 2-form
s_{ab},
J_{ab} = ds_{ab}.
This results in the equation
S_{ab} = ds_{ab} + theta_a ^ p_b - theta_b ^ p_a.
It's assumed that J_{ab}, S_{ab} and s_{ab} are all anti-symmetric in
their two frame indices.
When going over to curvilinear coordinates and curved spacetimes, this
is further generalized with the introduction of the connection 1-
forms. The exterior differentials are replaced by exterior *covariant*
differentials:
ds_{ab} --> ds_{ab} - omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac}
dp_a --> dp_a - omega^c_a ^ p_c.
Thus, the final result is that we obtain the following field
equations:
S_{ab} = ds_{ab} - omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac} +
theta_a ^ p_b - theta_b ^ p_a
P_a = dp_a - omega^c_a ^ p_c.
Amazingly, these can be derived from a Lagrangian as its Euler-
Lagrange equations -- but with one important condition.
To find the actual form of the Lagrangian, we'll start out by assuming
the relation between the spin 3-current and connection is given in
terms of a prospective Lagrangian 4-form L by
D(L) = ... + 1/2 D(omega^{ab}) ^ S_{ab} + ...
To make this work then requires that we also have, in this variation,
the following
* -1/2 D(omega^{ab}) ^ ds_{ab}
* 1/2 D(omega^{ab}) ^ (omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac})
* -1/2 D(omega^{ab}) ^ (theta_a ^ p_b - theta_b ^ p_a).
The first term integrates by parts to (note the signs)
* 1/2 d(D(omega^{ab} ^ s_{ab}) - 1/2 D(d(omega^{ab})) ^ s_{ab}.
The last two terms can be rewritten as
* -D(omega^{ab}) ^ omega^c_b ^ s_{ca}
* D(omega^a_b) ^ theta^b ^ p_a.
By virtue of anti-symmetry, we also have
-D(omega^{ab}) ^ omega^c_b ^ s_{ca} = 1/2 D(omega^{ab} ^ omega^c_b)
^ s_{ac}.
With some index manipulation, this can be rewritten as
1/2 D(omega^{ac} ^ omega^b_c) ^ s_{ab}
= -1/2 D(omega^a_c ^ omega^{cb}) ^ s_{ab}.
Thus, these terms combine to yield
-1/2 D(omega^{ab} + omega^a_c ^ omega^{cb}) ^ s_{ab} + d(...)
= -1/2 D(Omega^{ab}) ^ s_{ab} + d(...).
This shows that the kinetic potential s_{ab} must be paired off with
the CURVATURE 2-form Omega^{ab}.
For the term involving theta, we obtain
D(omega^a_b) ^ theta^b ^ p_a.
= D(omega^a_b ^ theta^b) ^ p_a - omega^a_b ^ D(theta^b) ^ p_a
= D(omega^a_b ^ theta^b) ^ p_a + D(theta^b) ^ omega^a_b ^ p_a.
Thus, the only way to fit in the equation for the momentum 3-current
and its kinetic potential is so as to cancel out the last term. This
requires pairing off
D(L) = ... - D(theta^b) ^ P_b + ...
In all, applying this to the remaining terms of the Euler-Lagrange
equation for P_a, we obtain the following additions
D(L) = ... - D(theta^b) ^ P_b + D(theta^b) ^ dp_b - D(theta^b) ^
omega^c_b ^ p_c.
Integrating the second of these terms by parts, we have (again, note
the signs)
-d(D(theta^b) ^ p_b) + D(d(theta^b)) ^ p_b.
It's only at this point that we reach the proviso. The condition on
which the momentum equation be a Euler-Lagrange equation is seen by
placing all the terms for all the variationals of (theta) together.
This yields the following expression
D(L) = ... + D(omega^a_b ^ theta^b) ^ p_a + D(theta^b) ^ omega^a_b
^ p_a
- D(theta^b) ^ P_b + D(d(theta^a)) ^ p_a - D(theta^b) ^ omega^c_b ^
p_c + d(...)
Thus, we obtain (with some index-renaming)
D(L) = ... + D(d(theta^a + omega^a_b ^ theta^b) ^ p_a - D(theta^a)
^ P_a + d(...).
The kinetic potential p_a must be paired off with the TORSION 2-form,
Theta^a:
D(L) = ... + D(Theta^a) ^ p_a + ...
So, it's finally at this point that we find that the total variation
in the Lagrangian should be:
D(L) = -D(theta^a) ^ P_a + D(Theta^a) ^ p_a + 1/2 D(omega^{ab}) ^ S_
{ab} - 1/2 D(Omega^{ab}) ^ s_{ab}.
In summary, we've derived the "field laws" as the following Euler-
Lagrange equations
ds_{ab} - omega^c_a ^ s_{cb} - omega^c_b ^ s_{ac} = J_{ab} = S_{ab}
- theta_a ^ p_b + theta_b ^ p_a
dp_a - omega^c_a ^ p_c = P_a
This is analogous to the Maxwell field law dG = J. Taking exterior
derivatives gives us the following as the generalization of the flat-
space transport laws
dP_a - omega^c_a ^ P_c = -Omega^c_a ^ p_c
dJ_{ab} - omega^c_a ^ J_{cb} - omega^c_b ^ J_{ca} = -Omega^c_a ^ s_
{cb} - Omega^c_b ^ s_{ac}.
The (Euler-Lagrange) field equations can, thus, be thought of as the
first integrals of the transport equations of the underlying fluid
dynamics. This is entirely analogous to the development of the
Bernoulli law or the Navier-Stokes equation, and that's the role being
played here by the field law.
When the second equation is written out in terms of S_{ab}, instead of
J_{ab} then the following additions appear on the left
theta_a ^ P_b - theta_b ^ P_a
and on the right
Theta_a ^ p_b - Theta_b ^ p_a.
The left-hand term, written in terms of the components of the momentum
3-current, is
theta_a ^ P^{mu}_b d^3x_{mu} - theta_b ^ P^{mu}_a d^3x_{mu}
= (theta_{a mu} P^{mu}_b - theta_{b mu} P^{mu}_a) d^4 x
= (P_{ab} - P_{ba}) d^4 x.
This is the anti-symmetric part of the index-lowered energy-momentum
tensor
P_{ab} = theta_{a mu} P^{mu}_b.
Unlike the stress tensor (which is a multiple of the derivative of the
Lagrangian with respect to the contravariant metric g^{mu}), the
energy-momentum tensor need not be symmetric.
A DISTINCTION is being made, here, between "stress tensor" and "energy-
momentum tensor", both in the names (T_{ab} and P_{ab}, respectively)
and their conceptions. The indices in the latter, P_{ab}, play
completely unsymmetric roles. The a-index is associated with the
volume 3-form, while the b-index is a GLOBAL frame index for the
momentum current.
So ...
>d*Alpha* - *sigma* x *Alpha* - *alpha* x *sigma* = 0
>
> This will be continued in the next part...
continuing on, first note the correction:
d*Alpha* - *sigma* x *Alpha* - *alpha* x *Sigma* = 0
The asterisks are being used here and below to denote vector boldface.
We supplement the definitions
> *x* = (theta^1,theta^2,theta^3), t = theta^0
> *X* = (Theta^1,Theta^2,Theta^3), T = Theta^0
> *sigma* = (omega^2_3,omega^3_1,omega^1_2)
> *alpha* = (omega^1_0,omega^2_0,omega^3_0)
> *Sigma* = (Omega^2_3,Omega^3_1,Omega^1_2)
> *Alpha* = (Omega^1_0,Omega^2_0,Omega^3_0).
with the following definitions:
H = -P_0, *P* = (P_1, P_2, P_3),
h = -p_0, *p* = (p_1, p_2, p_3),
*j* = (s_23, s_31, s_12), *k* = (1/c)^2 (s_01, s_02, s_03),
*J* = (S_23, S_31, S_12), *K* = (1/c)^2 (S_01, S_02, S_03),
resulting in the following equations
d*p* - *sigma* x *p* + (1/c)^2 *alpha* h = *P*
dh + *alpha*.*p* = H
d*k* - *sigma* x *k* + (1/c)^2 *alpha* x *j* = *K* + t *p* - (1/c)
^2 *x* h
d*j* - *sigma* x *j* - *alpha* x *k* = *J* - *x* x *p*
The transport laws are embodied in the continuity equations
d*P* - *sigma* X *P* + (1/c)^2 *alpha* H = -*Sigma* X *p* + (1/c)^2
*Alpha* h
dH + *alpha*.*P* = *Alpha*.*p*.
d*K* - *sigma* x *K* + (1/c)^2 *alpha* x *J* - t *P* + (1/c)^2 *x*
H
= -*Sigma* x *k* + (1/c)^2 *ALpha* x *j* - T *p* + (1/c)^2 *X* h
d*J* - *sigma* x *J* - *alpha* x *K* + *x* x *P*
= -*Sigma* x *j* - *Alpha* x *k* + *X* x *p*.
In the vicinity of a free-fall frame, where we can write *alpha* = 0,
*sigma* = 0, and in a weak field where the curvature and torsion
components are negligible, these equations reduce to more familiar
form as
d*P* = 0
dH = 0
d*K* - t *P* + (1/c)^2 *x* H = 0
d*J* + *x* x *P* = 0
Reverting to flat space coordinates (x^0 = s, (x^1,x^2,x^3) = *r*), we
have
*x* = d*r*, t = ds
and
d(*K* - s *P* + (1/c)^2 *r* H) = 0
d(*J* + *r* x *P*) = 0
which are the original form posed for the transport laws for the
angular momentum, where the form-valued 3-vector *J* here represents
the 3-current density of the intrinsic spin.