Rock Brentwood
Jan31-09, 06:34 AM
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". These are rendered in 3-vector
form with respect to a fixed basis (when in flat-space) by the usual
convention:
A = *A*.d*r* - phi dt
F = *B*.d*S* + *E*.d*r* dt
G = *D*.d*S* - *H*.d*r* dt
J = *J*.d*S* dt - rho dV
where
d*r* = (dx,dy,dz)
d*S* = (dydz,dzdx,dxdy)
dV = dxdydz.
This is in the language of differential forms, and the exterior
product is denoted by juxtaposition (e.g. dxdy = -dydx).
In a general field theory one starts with a Lagrangian 4-form L and
defines the conjugate fields G and J by the variational of L as:
DL = (DA)J - (DF)G.
The variational for (DF)G can be done algebraically by
(DF)G = D(dA)G = d(DA)G = d((DA G) + (DA) dG
thus leading to the result
DL = DA (J - dG) + d(DA G).
Out of the former comes the field law J = dG, and out of the latter
comes the canonical form Theta = DA G from which the Hamiltonian
structure is derived.
For the gravity field, the analogous role of A is played by the frame
1-forms (theta^a: a = 0, 1, 2, 3) and connection 1-forms (omega^a_b:
a,b = 0,1,2,3). It is assumed here that the connection is
"metrical" (i.e. yields 0 covariant derivatives) with respect to the
metric given by
g = delta_{ab} theta^a (x) theta^b - c^2 theta^0 (x) theta^0
(summation convention used; (x) denotes tensor product).
This leads to the result that
0 = dg_{ab} = g_{ac} omega^c_b + g_{bc} omega^c_a
or
omega_{ab} + omega_{ba} = 0
lowering the index with the metric.
The analogous role of F is played by the curvature 2-forms Omega^a_b
and torsion 2-form Theta^a.
Similarly, Omega_{ab} is anti-symmetric in its two (lowered) indices.
The reduction is carried out by writing everything in form-valued
scalar and 3-vector form with the following 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).
The resulting of writing the Cartan equations
d theta^a + omega^a_b theta^b = Theta^a
d omega^a_b + omega^a_c omega^c_b = Omega^a_b
and the Ricci and Bianchi identities
d Theta^a + omega^a_b Theta^b = Omega^a_b theta^b
d Omega^a_b + omega^a_c Omega^c_b = Omega^a_c omega^c_b
is
d*x* - *sigma* x *x* + *alpha* t = *X*
dt + (1/c)^2 *alpha*.*x* = T
d*X* - *sigma* x *X* + *alpha* T = -*Sigma* x *x* + *Alpha* t
dT + (1/c)^2 *alpha*.*X* = (1/c)^2 *Alpha*.*x*
d*sigma* - 1/2 *sigma* x *sigma* + (1/2) (1/c)^2 *alpha* x *alpha*
= *Sigma*
d*alpha* - *sigma* x *alpha* = *Alpha*
d*Sigma* - *sigma* x *Sigma* + (1/c)^2 *alpha* x *Alpha* = 0
d*Alpha* - *sigma* x *Alpha* - *alpha* x *sigma* = 0
This will be continued in the next part...
the equations
dA = F; dF = 0
which describe the field "kinematics" and
dG = J; dJ = 0
which describe the field "dynamics". These are rendered in 3-vector
form with respect to a fixed basis (when in flat-space) by the usual
convention:
A = *A*.d*r* - phi dt
F = *B*.d*S* + *E*.d*r* dt
G = *D*.d*S* - *H*.d*r* dt
J = *J*.d*S* dt - rho dV
where
d*r* = (dx,dy,dz)
d*S* = (dydz,dzdx,dxdy)
dV = dxdydz.
This is in the language of differential forms, and the exterior
product is denoted by juxtaposition (e.g. dxdy = -dydx).
In a general field theory one starts with a Lagrangian 4-form L and
defines the conjugate fields G and J by the variational of L as:
DL = (DA)J - (DF)G.
The variational for (DF)G can be done algebraically by
(DF)G = D(dA)G = d(DA)G = d((DA G) + (DA) dG
thus leading to the result
DL = DA (J - dG) + d(DA G).
Out of the former comes the field law J = dG, and out of the latter
comes the canonical form Theta = DA G from which the Hamiltonian
structure is derived.
For the gravity field, the analogous role of A is played by the frame
1-forms (theta^a: a = 0, 1, 2, 3) and connection 1-forms (omega^a_b:
a,b = 0,1,2,3). It is assumed here that the connection is
"metrical" (i.e. yields 0 covariant derivatives) with respect to the
metric given by
g = delta_{ab} theta^a (x) theta^b - c^2 theta^0 (x) theta^0
(summation convention used; (x) denotes tensor product).
This leads to the result that
0 = dg_{ab} = g_{ac} omega^c_b + g_{bc} omega^c_a
or
omega_{ab} + omega_{ba} = 0
lowering the index with the metric.
The analogous role of F is played by the curvature 2-forms Omega^a_b
and torsion 2-form Theta^a.
Similarly, Omega_{ab} is anti-symmetric in its two (lowered) indices.
The reduction is carried out by writing everything in form-valued
scalar and 3-vector form with the following 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).
The resulting of writing the Cartan equations
d theta^a + omega^a_b theta^b = Theta^a
d omega^a_b + omega^a_c omega^c_b = Omega^a_b
and the Ricci and Bianchi identities
d Theta^a + omega^a_b Theta^b = Omega^a_b theta^b
d Omega^a_b + omega^a_c Omega^c_b = Omega^a_c omega^c_b
is
d*x* - *sigma* x *x* + *alpha* t = *X*
dt + (1/c)^2 *alpha*.*x* = T
d*X* - *sigma* x *X* + *alpha* T = -*Sigma* x *x* + *Alpha* t
dT + (1/c)^2 *alpha*.*X* = (1/c)^2 *Alpha*.*x*
d*sigma* - 1/2 *sigma* x *sigma* + (1/2) (1/c)^2 *alpha* x *alpha*
= *Sigma*
d*alpha* - *sigma* x *alpha* = *Alpha*
d*Sigma* - *sigma* x *Sigma* + (1/c)^2 *alpha* x *Alpha* = 0
d*Alpha* - *sigma* x *Alpha* - *alpha* x *sigma* = 0
This will be continued in the next part...