Rock Brentwood
Jun21-08, 05:00 AM
2004 February 26
sci.physics.research
http://groups.google.com/group/sci.physics.research/msg/4d403db332920ea1?hl=en&dmode=source
From Danny Ross Lunsford <antimatte...@yahoo.NOSE-PAM.com>:
>Is there a good place online I can find a detailed presentation
>of the Newman-Penrose formalism? Offline?
Better yet, derive your own version. I put the working ingredients in
http://federation.g3z.com/Physics/index.htm#PenroseNewman
There's enough here to generalize this to Riemann-Cartan geometries,
and even further to the subfamily of metric affine geometries where
the covariant derivative of the metric may be diagonal (i.e., Riemann-
Cartan connection + Weyl field).
In fact, Penrose-Newman is most naturally suited in the Riemann-Cartan
setting.
This is slated for future expansion. The points of interest are those
relating to the decompositions:
4 = 2_R x 2_L
of the frame basis 4: (l, m, m*, n) into the right-helicity spinor
basis 2_R: (o, i) and left-helicity basis 2_L: (o*, i*):
l = o x o* = o* x o, m = o x i* = i* x o
m* = i x o* = o* x i, n = i x i* = i* x i.
In relation to a standard orthonormal basis (e_0, e_1, e_2, e_3) with
metric signature (+,-,-,-), one has
l = (e_3 + 1/c e_0)/root 2
m = (e_1 + i e_2)/root 2
m* = (e_1 - i e_2)/root 2
n = (e_3 - 1/c e_0)/root 2
The metric, referred to the (l,m,m*,n) basis therefore has the form
g_{ln} = 1 = g_{nl}, g_{mm*} = -1 = g_{m*m}
all other components 0.
The metric, itself, is
g = LxN - MxM* - M*xM + NxL
where (L,M,M*,N) is the dual basis. Using a similar decomposition
L = OxO* = O*xO, M = OxI* = I*xO,
M* = IxO* = O*xI, N = IxI* = I*xI
one gets (deleting the tensor product "x" for brevity)
g = OO*II* - OI*IO* - IO*OI* + II*OO*
= OIO*I* - OII*O* - IOO*I* + IOI*O*
= (OI - IO) (O*I* - I*O*)
= epsilon x epsilon*
which introduces the right-helicity metric
epsilon = O ^ I = O x I - I x O
and left-helcity metric
epsilon* = O* ^ I* = O* x I* - I* x O*.
The Penrose-Newman coefficients are most easily introduced in terms of
the connection 1-forms corresponding to the (l,m,m*,n) basis.
For a metrical connection (Riemann-Cartan spacetime) one has anti-
symmetry in the frame indices of the connection 1-form omega^{ab} = -
omega^{ba}. When written with 1 upper and 1 lower index, this becomes
omega^l_n = omega^n_l = omega^m_{m*} = omega^{m*}_m = 0
omega^l_l + omega^n_n = 0
omega^m_m + omega^{m*}_{m*} = 0
omega^l_m = omega^{m*}_n, omega^n_m = omega^{m*}_l
omega^m_l = omega^n_{m*}, omega^m_n = omega^l_{m*}.
Thus, one can define the coefficients by
(omega^l_l + omega^m_m)/2
= epsilon L + beta M + alpha M* + gamma N,
omega^n_m
= -(kappa L + sigma M + rho M* + tau N),
omega^m_n = pi L + mu M + lambda M* + nu N.
The conjugates are
(omega^l_l + omega^{m*}_{m*})/2
= epsilon* L + alpha* M + beta* M* + gamma* N,
omega^n_{m*}
= -(kappa* L + rho* M + sigma* M* + tau* N),
omega^{m*}_n = pi* L + lambda* M + mu* M* + nu* N.
The Penrose-Newman paper writes down the "commutators"; i.e., the Lie
brackets of the frame basis (l,m,m*,n). In the more general setting of
a Riemann-Cartan spacetime, these are independent coefficients
[e_a, e_b] = f^c_{ab} e_c.
And are related to the differentials of the dual frame (L,M,M*,N) by
the 1st Cartan structure equation
de^a = -1/2 f^a_{bc} e^b ^ e^c
de^a + omega^a_b ^ e^b = T^a
where T^a is the torsion 2-form. If T^a, one can directly write the
Lie brackets and differentials of the dual frame in terms of the
Penrose-Newman coefficients.
Finally, the decompsotion 4 = 2_R x 2_L ramifies as follows:
* Anti-symmetric tensor products:
4 ^ 4 = (3_R x 1_L) + (1_R x 3_L)
* Symmetric tensor products
4 v 4 = (3_R x 3_L) + (1_R x 1_L).
* Symmetric combinations of (4^4) (relevant for the Weyl and Riemann
tensor)
(4^4)v(4^4) =
(3_R v 3_R) x 1_L + 1_R x (3_L v 3_L)
+ 1_R x (3_L x 3_R) x 1_L + 1_L x (3_R x 3_L) x 1_R.
and (not as well-known):
* Decompositions for 3-forms 4 ^ 4 ^ 4
* Decompositions for 4-forms 4 ^ 4 ^ 4 ^ 4
The symmetric combinations 3 v 3 reduce as:
3 v 3 = 5 + 1.
The bases are
3_R: (E1+, E10, E1-)
E1+ = oo, E10 = (oi+io)/root 2, E1- = ii
5_R: (E2++, E2+, E20, E2-, E2--)
E2++ = oooo,
E2+ = (oooi+ooio+oioo+iooo)/2
E20 = (ooii + oioi + oiio + iooi + ioio + iioo)/root 6
E2- = (oiii + ioii + iioi + iiio)/2
E2-- = iiii/2
Applied to the Riemann tensor (and ignoring the torsion), the 5_R x
1_L and 1_R x 5_L parts give you the 10 components of the Weyl tensor.
The 2 1_R x 1_L parts that result from the decompositions of (3_R v
3_R) x 1_L and 1_R x (3_L v 3_L) give you the scalar part (the Ricci
scalar R) and pseudo-scalar part (epsilon^{mnrs} R_{mnrs}, which is 0
when the torsion is 0).
The remaining 2 parts give you the Ricci tensor.
The derivation of the Riemann tensor from the connection coefficients
follows the Cartan structure equation
d omega^a_b + omega^a_c ^ omega^c_b = Omega^a_b
where
Omega^a_b = 1/2 R^a_{bcd} e^c ^ e^d
is the curvature 2-form.
The Einstein equations, themselves, can be directly obtained by
varying the Lagrangian 4-form
L = a epsilon_{abcd} Omega^{ab} ^ e^c ^ e^d
+ c epsilon_{abcd} e^a ^ e^b ^ e^c ^ e^d
where a and c are, respectively, related to G and the cosmological
coefficient.
To make this interesting, you can also add the other 4 algebraic
combinations of the frame 1-form, curvature 2-form and torsion 2-form:
L = ... + b Omega_{ab} ^ e^a ^ e^b
+ l epsilon_{abcd} Omega^{ab} ^ Omega^{cd}
+ m Omega_{ab} ^ Omega^{ab}
+ n T^a ^ T_a.
I'm fairly sure that, in any case, the field laws can be DIRECTLY
derived by carrying out the variational treating the structure
coefficients and Penrose-Newman coefficients are independent
variables. Otherwise, one will have to resort to directly converting
the Einstein equation.
sci.physics.research
http://groups.google.com/group/sci.physics.research/msg/4d403db332920ea1?hl=en&dmode=source
From Danny Ross Lunsford <antimatte...@yahoo.NOSE-PAM.com>:
>Is there a good place online I can find a detailed presentation
>of the Newman-Penrose formalism? Offline?
Better yet, derive your own version. I put the working ingredients in
http://federation.g3z.com/Physics/index.htm#PenroseNewman
There's enough here to generalize this to Riemann-Cartan geometries,
and even further to the subfamily of metric affine geometries where
the covariant derivative of the metric may be diagonal (i.e., Riemann-
Cartan connection + Weyl field).
In fact, Penrose-Newman is most naturally suited in the Riemann-Cartan
setting.
This is slated for future expansion. The points of interest are those
relating to the decompositions:
4 = 2_R x 2_L
of the frame basis 4: (l, m, m*, n) into the right-helicity spinor
basis 2_R: (o, i) and left-helicity basis 2_L: (o*, i*):
l = o x o* = o* x o, m = o x i* = i* x o
m* = i x o* = o* x i, n = i x i* = i* x i.
In relation to a standard orthonormal basis (e_0, e_1, e_2, e_3) with
metric signature (+,-,-,-), one has
l = (e_3 + 1/c e_0)/root 2
m = (e_1 + i e_2)/root 2
m* = (e_1 - i e_2)/root 2
n = (e_3 - 1/c e_0)/root 2
The metric, referred to the (l,m,m*,n) basis therefore has the form
g_{ln} = 1 = g_{nl}, g_{mm*} = -1 = g_{m*m}
all other components 0.
The metric, itself, is
g = LxN - MxM* - M*xM + NxL
where (L,M,M*,N) is the dual basis. Using a similar decomposition
L = OxO* = O*xO, M = OxI* = I*xO,
M* = IxO* = O*xI, N = IxI* = I*xI
one gets (deleting the tensor product "x" for brevity)
g = OO*II* - OI*IO* - IO*OI* + II*OO*
= OIO*I* - OII*O* - IOO*I* + IOI*O*
= (OI - IO) (O*I* - I*O*)
= epsilon x epsilon*
which introduces the right-helicity metric
epsilon = O ^ I = O x I - I x O
and left-helcity metric
epsilon* = O* ^ I* = O* x I* - I* x O*.
The Penrose-Newman coefficients are most easily introduced in terms of
the connection 1-forms corresponding to the (l,m,m*,n) basis.
For a metrical connection (Riemann-Cartan spacetime) one has anti-
symmetry in the frame indices of the connection 1-form omega^{ab} = -
omega^{ba}. When written with 1 upper and 1 lower index, this becomes
omega^l_n = omega^n_l = omega^m_{m*} = omega^{m*}_m = 0
omega^l_l + omega^n_n = 0
omega^m_m + omega^{m*}_{m*} = 0
omega^l_m = omega^{m*}_n, omega^n_m = omega^{m*}_l
omega^m_l = omega^n_{m*}, omega^m_n = omega^l_{m*}.
Thus, one can define the coefficients by
(omega^l_l + omega^m_m)/2
= epsilon L + beta M + alpha M* + gamma N,
omega^n_m
= -(kappa L + sigma M + rho M* + tau N),
omega^m_n = pi L + mu M + lambda M* + nu N.
The conjugates are
(omega^l_l + omega^{m*}_{m*})/2
= epsilon* L + alpha* M + beta* M* + gamma* N,
omega^n_{m*}
= -(kappa* L + rho* M + sigma* M* + tau* N),
omega^{m*}_n = pi* L + lambda* M + mu* M* + nu* N.
The Penrose-Newman paper writes down the "commutators"; i.e., the Lie
brackets of the frame basis (l,m,m*,n). In the more general setting of
a Riemann-Cartan spacetime, these are independent coefficients
[e_a, e_b] = f^c_{ab} e_c.
And are related to the differentials of the dual frame (L,M,M*,N) by
the 1st Cartan structure equation
de^a = -1/2 f^a_{bc} e^b ^ e^c
de^a + omega^a_b ^ e^b = T^a
where T^a is the torsion 2-form. If T^a, one can directly write the
Lie brackets and differentials of the dual frame in terms of the
Penrose-Newman coefficients.
Finally, the decompsotion 4 = 2_R x 2_L ramifies as follows:
* Anti-symmetric tensor products:
4 ^ 4 = (3_R x 1_L) + (1_R x 3_L)
* Symmetric tensor products
4 v 4 = (3_R x 3_L) + (1_R x 1_L).
* Symmetric combinations of (4^4) (relevant for the Weyl and Riemann
tensor)
(4^4)v(4^4) =
(3_R v 3_R) x 1_L + 1_R x (3_L v 3_L)
+ 1_R x (3_L x 3_R) x 1_L + 1_L x (3_R x 3_L) x 1_R.
and (not as well-known):
* Decompositions for 3-forms 4 ^ 4 ^ 4
* Decompositions for 4-forms 4 ^ 4 ^ 4 ^ 4
The symmetric combinations 3 v 3 reduce as:
3 v 3 = 5 + 1.
The bases are
3_R: (E1+, E10, E1-)
E1+ = oo, E10 = (oi+io)/root 2, E1- = ii
5_R: (E2++, E2+, E20, E2-, E2--)
E2++ = oooo,
E2+ = (oooi+ooio+oioo+iooo)/2
E20 = (ooii + oioi + oiio + iooi + ioio + iioo)/root 6
E2- = (oiii + ioii + iioi + iiio)/2
E2-- = iiii/2
Applied to the Riemann tensor (and ignoring the torsion), the 5_R x
1_L and 1_R x 5_L parts give you the 10 components of the Weyl tensor.
The 2 1_R x 1_L parts that result from the decompositions of (3_R v
3_R) x 1_L and 1_R x (3_L v 3_L) give you the scalar part (the Ricci
scalar R) and pseudo-scalar part (epsilon^{mnrs} R_{mnrs}, which is 0
when the torsion is 0).
The remaining 2 parts give you the Ricci tensor.
The derivation of the Riemann tensor from the connection coefficients
follows the Cartan structure equation
d omega^a_b + omega^a_c ^ omega^c_b = Omega^a_b
where
Omega^a_b = 1/2 R^a_{bcd} e^c ^ e^d
is the curvature 2-form.
The Einstein equations, themselves, can be directly obtained by
varying the Lagrangian 4-form
L = a epsilon_{abcd} Omega^{ab} ^ e^c ^ e^d
+ c epsilon_{abcd} e^a ^ e^b ^ e^c ^ e^d
where a and c are, respectively, related to G and the cosmological
coefficient.
To make this interesting, you can also add the other 4 algebraic
combinations of the frame 1-form, curvature 2-form and torsion 2-form:
L = ... + b Omega_{ab} ^ e^a ^ e^b
+ l epsilon_{abcd} Omega^{ab} ^ Omega^{cd}
+ m Omega_{ab} ^ Omega^{ab}
+ n T^a ^ T_a.
I'm fairly sure that, in any case, the field laws can be DIRECTLY
derived by carrying out the variational treating the structure
coefficients and Penrose-Newman coefficients are independent
variables. Otherwise, one will have to resort to directly converting
the Einstein equation.