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| 29.1 Introduction to ctensor | ||
| 29.2 Definitions for ctensor |
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ctensor is a component tensor manipulation package. To use the ctensor
package, type load(ctensor).
To begin an interactive session with ctensor, type csetup(). You are
first asked to specify the dimension of the manifold. If the dimension
is 2, 3 or 4 then the list of coordinates defaults to [x,y], [x,y,z]
or [x,y,z,t] respectively.
These names may be changed by assigning a new list of coordinates to
the variable ct_coords (described below) and the user is queried about
this. Care must be taken to avoid the coordinate names conflicting
with other object definitions.
Next, the user enters the metric either directly or from a file by
specifying its ordinal position.
The metric is stored in the matrix
lg. Finally, the metric inverse is computed and stored in the matrix
ug. One has the option of carrying out all calculations in a power
series.
A sample protocol is begun below for the static, spherically symmetric
metric (standard coordinates) which will be applied to the problem of
deriving Einstein's vacuum equations (which lead to the Schwarzschild
solution) as an example. Many of the functions in ctensor will be
displayed for the standard metric as examples.
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) csetup();
Enter the dimension of the coordinate system:
4;
Do you wish to change the coordinate names?
n;
Do you want to
1. Enter a new metric?
2. Enter a metric from a file?
3. Approximate a metric with a Taylor series?
1;
Is the matrix 1. Diagonal 2. Symmetric 3. Antisymmetric 4. General
Answer 1, 2, 3 or 4
1;
Row 1 Column 1:
a;
Row 2 Column 2:
x^2;
Row 3 Column 3:
x^2*sin(y)^2;
Row 4 Column 4:
-d;
Matrix entered.
Enter functional dependencies with the DEPENDS function or 'N' if none
depends([a,d],x);
Do you wish to see the metric?
y;
[ a 0 0 0 ]
[ ]
[ 2 ]
[ 0 x 0 0 ]
[ ]
[ 2 2 ]
[ 0 0 x sin (y) 0 ]
[ ]
[ 0 0 0 - d ]
(%o2) done
(%i3) christof(mcs);
a
x
(%t3) mcs = ---
1, 1, 1 2 a
1
(%t4) mcs = -
1, 2, 2 x
1
(%t5) mcs = -
1, 3, 3 x
d
x
(%t6) mcs = ---
1, 4, 4 2 d
x
(%t7) mcs = - -
2, 2, 1 a
cos(y)
(%t8) mcs = ------
2, 3, 3 sin(y)
2
x sin (y)
(%t9) mcs = - ---------
3, 3, 1 a
(%t10) mcs = - cos(y) sin(y)
3, 3, 2
d
x
(%t11) mcs = ---
4, 4, 1 2 a
(%o11) done
|
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A function in the ctensor (component tensor) package
which initializes the package and allows the user to enter a metric
interactively. See ctensor for more details.
A function in the ctensor (component tensor) package
that computes the metric inverse and sets up the package for
further calculations.
If cframe_flag is false, the function computes the inverse metric
ug from the (user-defined) matrix lg. The metric determinant is
also computed and stored in the variable gdet. Furthermore, the
package determines if the metric is diagonal and sets the value
of diagmetric accordingly. If the optional argument dis
is present and not equal to false, the user is prompted to see
the metric inverse.
If cframe_flag is true, the function expects that the values of
fri (the inverse frame matrix) and lfg (the frame metric) are
defined. From these, the frame matrix fr and the inverse frame
metric ufg are computed.
Sets up a predefined coordinate system and metric. The argument coordinate_system can be one of the following symbols:
SYMBOL Dim Coordinates Description/comments -------------------------------------------------------------------------- cartesian2d 2 [x,y] Cartesian 2D coordinate system polar 2 [r,phi] Polar coordinate system elliptic 2 [u,v] Elliptic coordinate system confocalelliptic 2 [u,v] Confocal elliptic coordinates bipolar 2 [u,v] Bipolar coordinate system parabolic 2 [u,v] Parabolic coordinate system cartesian3d 3 [x,y,z] Cartesian 3D coordinate system polarcylindrical 3 [r,theta,z] Polar 2D with cylindrical z ellipticcylindrical 3 [u,v,z] Elliptic 2D with cylindrical z confocalellipsoidal 3 [u,v,w] Confocal ellipsoidal bipolarcylindrical 3 [u,v,z] Bipolar 2D with cylindrical z paraboliccylindrical 3 [u,v,z] Parabolic 2D with cylindrical z paraboloidal 3 [u,v,phi] Paraboloidal coordinates conical 3 [u,v,w] Conical coordinates toroidal 3 [u,v,phi] Toroidal coordinates spherical 3 [r,theta,phi] Spherical coordinate system oblatespheroidal 3 [u,v,phi] Oblate spheroidal coordinates oblatespheroidalsqrt 3 [u,v,phi] prolatespheroidal 3 [u,v,phi] Prolate spheroidal coordinates prolatespheroidalsqrt 3 [u,v,phi] ellipsoidal 3 [r,theta,phi] Ellipsoidal coordinates cartesian4d 4 [x,y,z,t] Cartesian 4D coordinate system spherical4d 4 [r,theta,eta,phi] Spherical 4D coordinate system exteriorschwarzschild 4 [t,r,theta,phi] Schwarzschild metric interiorschwarzschild 4 [t,z,u,v] Interior Schwarzschild metric kerr_newman 4 [t,r,theta,phi] Charged axially symmetric metric |
coordinate_system can also be a list of transformation functions,
followed by a list containing the coordinate variables. For instance,
you can specify a spherical metric as follows:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
r*sin(theta),[r,theta,phi]]);
(%o2) done
(%i3) lg:trigsimp(lg);
[ 1 0 0 ]
[ ]
[ 2 ]
(%o3) [ 0 r 0 ]
[ ]
[ 2 2 ]
[ 0 0 r cos (theta) ]
(%i4) ct_coords;
(%o4) [r, theta, phi]
(%i5) dim;
(%o5) 3
|
Transformation functions can also be used when cframe_flag is true:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) cframe_flag:true;
(%o2) true
(%i3) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
r*sin(theta),[r,theta,phi]]);
(%o3) done
(%i4) fri;
[ cos(phi) cos(theta) - cos(phi) r sin(theta) - sin(phi) r cos(theta) ]
[ ]
(%o4) [ sin(phi) cos(theta) - sin(phi) r sin(theta) cos(phi) r cos(theta) ]
[ ]
[ sin(theta) r cos(theta) 0 ]
(%i5) cmetric();
(%o5) false
(%i6) lg:trigsimp(lg);
[ 1 0 0 ]
[ ]
[ 2 ]
(%o6) [ 0 r 0 ]
[ ]
[ 2 2 ]
[ 0 0 r cos (theta) ]
|
The optional argument extra_arg can be any one of the following:
cylindrical tells ct_coordsys to attach an additional cylindrical coordinate.
minkowski tells ct_coordsys to attach an additional coordinate with negative metric signature.
all tells ct_coordsys to call cmetric and christof(false) after setting up the metric.
If the global variable verbose is set to true, ct_coordsys displays the values of dim, ct_coords, and either lg or lfg and fri, depending on the value of cframe_flag.
Initializes the ctensor package.
The init_ctensor function reinitializes the ctensor package. It removes all arrays and matrices used by ctensor, resets all flags, resets dim to 4, and resets the frame metric to the Lorentz-frame.
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The main purpose of the ctensor package is to compute the tensors
of curved space(time), most notably the tensors used in general
relativity.
When a metric base is used, ctensor can compute the following tensors:
lg -- ug
\ \
lcs -- mcs -- ric -- uric
\ \ \
\ tracer - ein -- lein
\
riem -- lriem -- weyl
\
uriem
|
ctensor can also work using moving frames. When cframe_flag is
set to true, the following tensors can be calculated:
lfg -- ufg
\
fri -- fr -- lcs -- mcs -- lriem -- ric -- uric
\ | \ \ \
lg -- ug | weyl tracer - ein -- lein
|\
| riem
|
\uriem
|
A function in the ctensor (component tensor)
package. It computes the Christoffel symbols of both
kinds. The argument dis determines which results are to be immediately
displayed. The Christoffel symbols of the first and second kinds are
stored in the arrays lcs[i,j,k] and mcs[i,j,k] respectively and
defined to be symmetric in the first two indices. If the argument to
christof is lcs or mcs then the unique non-zero values of lcs[i,j,k]
or mcs[i,j,k], respectively, will be displayed. If the argument is all
then the unique non-zero values of lcs[i,j,k] and mcs[i,j,k] will be
displayed. If the argument is false then the display of the elements
will not occur. The array elements mcs[i,j,k] are defined in such a
manner that the final index is contravariant.
A function in the ctensor (component tensor)
package. ricci computes the covariant (symmetric)
components ric[i,j] of the Ricci tensor. If the argument dis is true,
then the non-zero components are displayed.
This function first computes the
covariant components ric[i,j] of the Ricci tensor.
Then the mixed Ricci tensor is computed using the
contravariant metric tensor. If the value of the argument dis
is true, then these mixed components, uric[i,j] (the
index i is covariant and the index j is contravariant), will be displayed
directly. Otherwise, ricci(false) will simply compute the entries
of the array uric[i,j] without displaying the results.
Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.
A function in the ctensor (component tensor)
package. einstein computes the mixed Einstein tensor
after the Christoffel symbols and Ricci tensor have been obtained
(with the functions christof and ricci). If the argument dis is
true, then the non-zero values of the mixed Einstein tensor ein[i,j]
will be displayed where j is the contravariant index.
The variable rateinstein will cause the rational simplification on
these components. If ratfac is true then the components will
also be factored.
Covariant Einstein-tensor. leinstein stores the values of the covariant Einstein tensor in the array lein. The covariant Einstein-tensor is computed from the mixed Einstein tensor ein by multiplying it with the metric tensor. If the argument dis is true, then the non-zero values of the covariant Einstein tensor are displayed.
A function in the ctensor (component tensor)
package. riemann computes the Riemann curvature tensor
from the given metric and the corresponding Christoffel symbols. The following
index conventions are used:
l _l _l _l _m _l _m
R[i,j,k,l] = R = | - | + | | - | |
ijk ij,k ik,j mk ij mj ik
|
This notation is consistent with the notation used by the itensor
package and its icurvature function.
If the optional argument dis is true,
the non-zero components riem[i,j,k,l] will be displayed.
As with the Einstein tensor, various switches set by the user
control the simplification of the components of the Riemann tensor.
If ratriemann is true, then
rational simplification will be done. If ratfac
is true then
each of the components will also be factored.
If the variable cframe_flag is false, the Riemann tensor is
computed directly from the Christoffel-symbols. If cframe_flag is
false, the covariant Riemann-tensor is computed first from the
frame field coefficients.
Covariant Riemann-tensor (lriem[]).
Computes the covariant Riemann-tensor as the array lriem. If the
argument dis is true, unique nonzero values are displayed.
If the variable cframe_flag is true, the covariant Riemann
tensor is computed directly from the frame field coefficients. Otherwise,
the (3,1) Riemann tensor is computed first.
For information on index ordering, see riemann.
Computes the contravariant components of the Riemann
curvature tensor as array elements uriem[i,j,k,l]. These are displayed
if dis is true.
Forms the Kretchmann-invariant (kinvariant) obtained by
contracting the tensors
lriem[i,j,k,l]*uriem[i,j,k,l]. |
This object is not automatically simplified since it can be very large.
Computes the Weyl conformal tensor. If the argument dis is
true, the non-zero components weyl[i,j,k,l] will be displayed to the
user. Otherwise, these components will simply be computed and stored.
If the switch ratweyl is set to true, then the components will be
rationally simplified; if ratfac is true then the results will be
factored as well.
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The ctensor package has the ability to truncate results by assuming
that they are Taylor-series approximations. This behavior is controlled by
the ctayswitch variable; when set to true, ctensor makes use
internally of the function ctaylor when simplifying results.
The ctaylor function is invoked by the following ctensor functions:
Function Comments
---------------------------------
christof() For mcs only
ricci()
uricci()
einstein()
riemann()
weyl()
checkdiv()
|
The ctaylor function truncates its argument by converting
it to a Taylor-series using taylor, and then calling
ratdisrep. This has the combined effect of dropping terms
higher order in the expansion variable ctayvar. The order
of terms that should be dropped is defined by ctaypov; the
point around which the series expansion is carried out is specified
in ctaypt.
As an example, consider a simple metric that is a perturbation of the Minkowski metric. Without further restrictions, even a diagonal metric produces expressions for the Einstein tensor that are far too complex:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) ratfac:true;
(%o2) true
(%i3) derivabbrev:true;
(%o3) true
(%i4) ct_coords:[t,r,theta,phi];
(%o4) [t, r, theta, phi]
(%i5) lg:matrix([-1,0,0,0],[0,1,0,0],[0,0,r^2,0],[0,0,0,r^2*sin(theta)^2]);
[ - 1 0 0 0 ]
[ ]
[ 0 1 0 0 ]
[ ]
(%o5) [ 2 ]
[ 0 0 r 0 ]
[ ]
[ 2 2 ]
[ 0 0 0 r sin (theta) ]
(%i6) h:matrix([h11,0,0,0],[0,h22,0,0],[0,0,h33,0],[0,0,0,h44]);
[ h11 0 0 0 ]
[ ]
[ 0 h22 0 0 ]
(%o6) [ ]
[ 0 0 h33 0 ]
[ ]
[ 0 0 0 h44 ]
(%i7) depends(l,r);
(%o7) [l(r)]
(%i8) lg:lg+l*h;
[ h11 l - 1 0 0 0 ]
[ ]
[ 0 h22 l + 1 0 0 ]
[ ]
(%o8) [ 2 ]
[ 0 0 r + h33 l 0 ]
[ ]
[ 2 2 ]
[ 0 0 0 r sin (theta) + h44 l ]
(%i9) cmetric(false);
(%o9) done
(%i10) einstein(false);
(%o10) done
(%i11) ntermst(ein);
[[1, 1], 62]
[[1, 2], 0]
[[1, 3], 0]
[[1, 4], 0]
[[2, 1], 0]
[[2, 2], 24]
[[2, 3], 0]
[[2, 4], 0]
[[3, 1], 0]
[[3, 2], 0]
[[3, 3], 46]
[[3, 4], 0]
[[4, 1], 0]
[[4, 2], 0]
[[4, 3], 0]
[[4, 4], 46]
(%o12) done
|
However, if we recompute this example as an approximation that is
linear in the variable l, we get much simpler expressions:
(%i14) ctayswitch:true;
(%o14) true
(%i15) ctayvar:l;
(%o15) l
(%i16) ctaypov:1;
(%o16) 1
(%i17) ctaypt:0;
(%o17) 0
(%i18) christof(false);
(%o18) done
(%i19) ricci(false);
(%o19) done
(%i20) einstein(false);
(%o20) done
(%i21) ntermst(ein);
[[1, 1], 6]
[[1, 2], 0]
[[1, 3], 0]
[[1, 4], 0]
[[2, 1], 0]
[[2, 2], 13]
[[2, 3], 2]
[[2, 4], 0]
[[3, 1], 0]
[[3, 2], 2]
[[3, 3], 9]
[[3, 4], 0]
[[4, 1], 0]
[[4, 2], 0]
[[4, 3], 0]
[[4, 4], 9]
(%o21) done
(%i22) ratsimp(ein[1,1]);
2 2 4 2 2
(%o22) - (((h11 h22 - h11 ) (l ) r - 2 h33 l r ) sin (theta)
r r r
2 2 4 2
- 2 h44 l r - h33 h44 (l ) )/(4 r sin (theta))
r r r
|
This capability can be useful, for instance, when working in the weak field limit far from a gravitational source.
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When the variable cframe_flag is set to true, the ctensor package
performs its calculations using a moving frame.
The frame bracket (fb[]).
Computes the frame bracket according to the following definition:
c c c d e ifb = ( ifri - ifri ) ifr ifr ab d,e e,d a b |
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A new feature (as of November, 2004) of ctensor is its ability to
compute the Petrov classification of a 4-dimensional spacetime metric.
For a demonstration of this capability, see the file
share/tensor/petrov.dem.
Computes a Newman-Penrose null tetrad (np) and its raised-index
counterpart (npi). See petrov for an example.
The null tetrad is constructed on the assumption that a four-diemensional orthonormal frame metric with metric signature (-,+,+,+) is being used. The components of the null tetrad are related to the inverse frame matrix as follows:
np = (fri + fri ) / sqrt(2) 1 1 2 np = (fri - fri ) / sqrt(2) 2 1 2 np = (fri + %i fri ) / sqrt(2) 3 3 4 np = (fri - %i fri ) / sqrt(2) 4 3 4 |
Computes the five Newman-Penrose coefficients psi[0]...psi[4].
If psi is set to true, the coefficients are displayed.
See petrov for an example.
These coefficients are computed from the Weyl-tensor in a coordinate base.
If a frame base is used, the Weyl-tensor is first converted to a coordinate
base, which can be a computationally expensive procedure. For this reason,
in some cases it may be more advantageous to use a coordinate base in the
first place before the Weyl tensor is computed. Note however, that
constructing a Newman-Penrose null tetrad requires a frame base. Therefore,
a meaningful computation sequence may begin with a frame base, which
is then used to compute lg (computed automatically by cmetric
and then ug. At this point, you can switch back to a coordinate base
by setting cframe_flag to false before beginning to compute the
Christoffel symbols. Changing to a frame base at a later stage could yield
inconsistent results, as you may end up with a mixed bag of tensors, some
computed in a frame base, some in a coordinate base, with no means to
distinguish between the two.
Computes the Petrov classification of the metric characterized by psi[0]...psi[4].
For example, the following demonstrates how to obtain the Petrov-classification of the Kerr metric:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) ug:invert(lg)$
(%i5) weyl(false);
(%o5) done
(%i6) nptetrad(true);
(%t6) np =
[ sqrt(r - 2 m) sqrt(r) ]
[ --------------- --------------------- 0 0 ]
[ sqrt(2) sqrt(r) sqrt(2) sqrt(r - 2 m) ]
[ ]
[ sqrt(r - 2 m) sqrt(r) ]
[ --------------- - --------------------- 0 0 ]
[ sqrt(2) sqrt(r) sqrt(2) sqrt(r - 2 m) ]
[ ]
[ r %i r sin(theta) ]
[ 0 0 ------- --------------- ]
[ sqrt(2) sqrt(2) ]
[ ]
[ r %i r sin(theta) ]
[ 0 0 ------- - --------------- ]
[ sqrt(2) sqrt(2) ]
sqrt(r) sqrt(r - 2 m)
(%t7) npi = matrix([- ---------------------, ---------------, 0, 0],
sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)
sqrt(r) sqrt(r - 2 m)
[- ---------------------, - ---------------, 0, 0],
sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)
1 %i
[0, 0, ---------, --------------------],
sqrt(2) r sqrt(2) r sin(theta)
1 %i
[0, 0, ---------, - --------------------])
sqrt(2) r sqrt(2) r sin(theta)
(%o7) done
(%i7) psi(true);
(%t8) psi = 0
0
(%t9) psi = 0
1
m
(%t10) psi = --
2 3
r
(%t11) psi = 0
3
(%t12) psi = 0
4
(%o12) done
(%i12) petrov();
(%o12) D
|
The Petrov classification function is based on the algorithm published in "Classifying geometries in general relativity: III Classification in practice" by Pollney, Skea, and d'Inverno, Class. Quant. Grav. 17 2885-2902 (2000). Except for some simple test cases, the implementation is untested as of December 19, 2004, and is likely to contain errors.
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ctensor has the ability to compute and include torsion and nonmetricity
coefficients in the connection coefficients.
The torsion coefficients are calculated from a user-supplied tensor
tr, which should be a rank (2,1) tensor. From this, the torsion
coefficients kt are computed according to the following formulae:
m m m
- g tr - g tr - tr g
im kj jm ki ij km
kt = -------------------------------
ijk 2
k km
kt = g kt
ij ijm
|
Note that only the mixed-index tensor is calculated and stored in the
array kt.
The nonmetricity coefficients are calculated from the user-supplied
nonmetricity vector nm. From this, the nonmetricity coefficients
nmc are computed as follows:
k k km
-nm D - D nm + g nm g
k i j i j m ij
nmc = ------------------------------
ij 2
|
where D stands for the Kronecker-delta.
When ctorsion_flag is set to true, the values of kt
are substracted from the mixed-indexed connection coefficients computed by
christof and stored in mcs. Similarly, if cnonmet_flag
is set to true, the values of nmc are substracted from the
mixed-indexed connection coefficients.
If necessary, christof calls the functions contortion and
nonmetricity in order to compute kt and nm.
Computes the (2,1) contortion coefficients from the torsion tensor tr.
Computes the (2,1) nonmetricity coefficients from the nonmetricity vector nm.
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A function in the ctensor (component tensor)
package which will perform a coordinate transformation
upon an arbitrary square symmetric matrix M. The user must input the
functions which define the transformation. (Formerly called transform.)
returns a list of the unique differential equations (expressions)
corresponding to the elements of the n dimensional square
array A. Presently, n may be 2 or 3. deindex is a global list
containing the indices of A corresponding to these unique
differential equations. For the Einstein tensor (ein), which
is a two dimensional array, if computed for the metric in the example
below, findde gives the following independent differential equations:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2) true
(%i3) dim:4;
(%o3) 4
(%i4) lg:matrix([a,0,0,0],[0,x^2,0,0],[0,0,x^2*sin(y)^2,0],[0,0,0,-d]);
[ a 0 0 0 ]
[ ]
[ 2 ]
[ 0 x 0 0 ]
(%o4) [ ]
[ 2 2 ]
[ 0 0 x sin (y) 0 ]
[ ]
[ 0 0 0 - d ]
(%i5) depends([a,d],x);
(%o5) [a(x), d(x)]
(%i6) ct_coords:[x,y,z,t];
(%o6) [x, y, z, t]
(%i7) cmetric();
(%o7) done
(%i8) einstein(false);
(%o8) done
(%i9) findde(ein,2);
2
(%o9) [d x - a d + d, 2 a d d x - a (d ) x - a d d x + 2 a d d
x x x x x x x
2 2
- 2 a d , a x + a - a]
x x
(%i10) deindex;
(%o10) [[1, 1], [2, 2], [4, 4]]
|
Computes the covariant gradient of a scalar function allowing the
user to choose the corresponding vector name as the example under
contragrad illustrates.
Computes the contravariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example below for the Schwarzschild metric illustrates:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) depends(f,r);
(%o4) [f(r)]
(%i5) cograd(f,g1);
(%o5) done
(%i6) listarray(g1);
(%o6) [0, f , 0, 0]
r
(%i7) contragrad(f,g2);
(%o7) done
(%i8) listarray(g2);
f r - 2 f m
r r
(%o8) [0, -------------, 0, 0]
r
|
computes the tensor d'Alembertian of the scalar function once dependencies have been declared upon the function. For example:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) derivabbrev:true;
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) depends(p,r);
(%o4) [p(r)]
(%i5) factor(dscalar(p));
2
p r - 2 m p r + 2 p r - 2 m p
r r r r r r
(%o5) --------------------------------------
2
r
|
computes the covariant divergence of the mixed second rank tensor
(whose first index must be covariant) by printing the
corresponding n components of the vector field (the divergence) where
n = dim. If the argument to the function is g then the
divergence of the Einstein tensor will be formed and must be zero.
In addition, the divergence (vector) is given the array name div.
A function in the ctensor (component tensor)
package. cgeodesic computes the geodesic equations of
motion for a given metric. They are stored in the array geod[i]. If
the argument dis is true then these equations are displayed.
generates the covariant components of the vacuum field equations of
the Brans- Dicke gravitational theory. The scalar field is specified
by the argument f, which should be a (quoted) function name
with functional dependencies, e.g., 'p(x).
The components of the second rank covariant field tensor are
represented by the array bd.
generates the mixed Euler- Lagrange tensor (field equations) for the
invariant density of R^2. The field equations are the components of an
array named inv1.
*** NOT YET IMPLEMENTED ***
generates the mixed Euler- Lagrange tensor (field equations) for the
invariant density of ric[i,j]*uriem[i,j]. The field equations are the
components of an array named inv2.
*** NOT YET IMPLEMENTED ***
generates the field equations of Rosen's bimetric theory. The field
equations are the components of an array named rosen.
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Returns true if M is a diagonal matrix or (2D) array.
Returns true if M is a symmetric matrix or (2D) array.
gives the user a quick picture of the "size" of the doubly subscripted tensor (array) f. It prints two element lists where the second element corresponds to NTERMS of the components specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification.
displays all the elements of the tensor ten, as represented by
a multidimensional array. Tensors of rank 0 and 1, as well as other types
of variables, are displayed as with ldisplay. Tensors of rank 2 are
displayed as 2-dimensional matrices, while tensors of higher rank are displayed
as a list of 2-dimensional matrices. For instance, the Riemann-tensor of
the Schwarzschild metric can be viewed as:
(%i1) load(ctensor);
(%o1) /share/tensor/ctensor.mac
(%i2) ratfac:true;
(%o2) true
(%i3) ct_coordsys(exteriorschwarzschild,all);
(%o3) done
(%i4) riemann(false);
(%o4) done
(%i5) cdisplay(riem);
[ 0 0 0 0 ]
[ ]
[ 2 ]
[ 3 m (r - 2 m) m 2 m ]
[ 0 - ------------- + -- - ---- 0 0 ]
[ 4 3 4 ]
[ r r r ]
[ ]
riem = [ m (r - 2 m) ]
1, 1 [ 0 0 ----------- 0 ]
[ 4 ]
[ r ]
[ ]
[ m (r - 2 m) ]
[ 0 0 0 ----------- ]
[ 4 ]
[ r ]
[ 2 m (r - 2 m) ]
[ 0 ------------- 0 0 ]
[ 4 ]
[ r ]
riem = [ ]
1, 2 [ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ m (r - 2 m) ]
[ 0 0 - ----------- 0 ]
[ 4 ]
[ r ]
riem = [ ]
1, 3 [ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ m (r - 2 m) ]
[ 0 0 0 - ----------- ]
[ 4 ]
[ r ]
riem = [ ]
1, 4 [ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 2 m ]
[ - ------------ 0 0 0 ]
riem = [ 2 ]
2, 1 [ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 2 m ]
[ ------------ 0 0 0 ]
[ 2 ]
[ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ m ]
2, 2 [ 0 0 - ------------ 0 ]
[ 2 ]
[ r (r - 2 m) ]
[ ]
[ m ]
[ 0 0 0 - ------------ ]
[ 2 ]
[ r (r - 2 m) ]
[ 0 0 0 0 ]
[ ]
[ m ]
[ 0 0 ------------ 0 ]
riem = [ 2 ]
2, 3 [ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ m ]
[ 0 0 0 ------------ ]
riem = [ 2 ]
2, 4 [ r (r - 2 m) ]
[ ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ m ]
3, 1 [ - 0 0 0 ]
[ r ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ m ]
3, 2 [ 0 - 0 0 ]
[ r ]
[ ]
[ 0 0 0 0 ]
[ m ]
[ - - 0 0 0 ]
[ r ]
[ ]
[ m ]
[ 0 - - 0 0 ]
riem = [ r ]
3, 3 [ ]
[ 0 0 0 0 ]
[ ]
[ 2 m - r ]
[ 0 0 0 ------- + 1 ]
[ r ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 2 m ]
3, 4 [ 0 0 0 - --- ]
[ r ]
[ ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 0 0 0 0 ]
4, 1 [ ]
[ 2 ]
[ m sin (theta) ]
[ ------------- 0 0 0 ]
[ r ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 0 0 0 0 ]
4, 2 [ ]
[ 2 ]
[ m sin (theta) ]
[ 0 ------------- 0 0 ]
[ r ]
[ 0 0 0 0 ]
[ ]
[ 0 0 0 0 ]
[ ]
riem = [ 0 0 0 0 ]
4, 3 [ ]
[ 2 ]
[ 2 m sin (theta) ]
[ 0 0 - --------------- 0 ]
[ r ]
[ 2 ]
[ m sin (theta) ]
[ - ------------- 0 0 0 ]
[ r ]
[ ]
[ 2 ]
[ m sin (theta) ]
riem = [ 0 - ------------- 0 0 ]
4, 4 [ r ]
[ ]
[ 2 ]
[ 2 m sin (theta) ]
[ 0 0 --------------- 0 ]
[ r ]
[ ]
[ 0 0 0 0 ]
(%o5) done
|
Returns a new list consisting of L with the n'th element deleted.
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ctensor Default value: 4
An option in the ctensor (component tensor)
package. dim is the dimension of the manifold with the
default 4. The command dim: n will reset the dimension to any other
value n.
Default value: false
An option in the ctensor (component tensor)
package. If diagmetric is true special routines compute
all geometrical objects (which contain the metric tensor explicitly)
by taking into consideration the diagonality of the metric. Reduced
run times will, of course, result. Note: this option is set
automatically by csetup if a diagonal metric is specified.
Causes trigonometric simplifications to be used when tensors are computed. Presently,
ctrgsimp affects only computations involving a moving frame.
Causes computations to be performed relative to a moving frame as opposed to
a holonomic metric. The frame is defined by the inverse frame array fri
and the frame metric lfg. For computations using a Cartesian frame,
lfg should be the unit matrix of the appropriate dimension; for
computations in a Lorentz frame, lfg should have the appropriate
signature.
Causes the contortion tensor to be included in the computation of the
connection coefficients. The contortion tensor itself is computed by
contortion from the user-supplied tensor tr.
Causes the nonmetricity coefficients to be included in the computation of
the connection coefficients. The nonmetricity coefficients are computed
from the user-supplied nonmetricity vector nm by the function
nonmetricity.
If set to true, causes some ctensor computations to be carried out using
Taylor-series expansions. Presently, christof, ricci,
uricci, einstein, and weyl take into account this
setting.
Variable used for Taylor-series expansion if ctayswitch is set to
true.
Maximum power used in Taylor-series expansion when ctayswitch is
set to true.
Point around which Taylor-series expansion is carried out when
ctayswitch is set to true.
The determinant of the metric tensor lg. Computed by cmetric when
cframe_flag is set to false.
Causes rational simplification to be applied by christof.
Default value: true
If true rational simplification will be
performed on the non-zero components of Einstein tensors; if
ratfac is true then the components will also be factored.
Default value: true
One of the switches which controls
simplification of Riemann tensors; if true, then rational
simplification will be done; if ratfac is true then each of the
components will also be factored.
Default value: true
If true, this switch causes the weyl function
to apply rational simplification to the values of the Weyl tensor. If
ratfac is true, then the components will also be factored.
The covariant frame metric. By default, it is initialized to the 4-dimensional Lorentz frame with signature (+,+,+,-). Used when cframe_flag is true.
The inverse frame metric. Computed from lfg when cmetric is called while cframe_flag is set to true.
The (3,1) Riemann tensor. Computed when the function riemann is invoked. For information about index ordering, see the description of riemann.
if cframe_flag is true, riem is computed from the covariant Riemann-tensor lriem.
The covariant Riemann tensor. Computed by lriemann.
The contravariant Riemann tensor. Computed by uriemann.
The mixed Ricci-tensor. Computed by ricci.
The contravariant Ricci-tensor. Computed by uricci.
The metric tensor. This tensor must be specified (as a dim by dim matrix)
before other computations can be performed.
The inverse of the metric tensor. Computed by cmetric.
The Weyl tensor. Computed by weyl.
Frame bracket coefficients, as computed by frame_bracket.
The Kretchmann invariant. Computed by rinvariant.
A Newman-Penrose null tetrad. Computed by nptetrad.
The raised-index Newman-Penrose null tetrad. Computed by nptetrad.
Defined as ug.np. The product np.transpose(npi) is constant:
(%i39) trigsimp(np.transpose(npi));
[ 0 - 1 0 0 ]
[ ]
[ - 1 0 0 0 ]
(%o39) [ ]
[ 0 0 0 1 ]
[ ]
[ 0 0 1 0 ]
|
User-supplied rank-3 tensor representing torsion. Used by contortion.
The contortion tensor, computed from tr by contortion.
User-supplied nonmetricity vector. Used by nonmetricity.
The nonmetricity coefficients, computed from nm by nonmetricity.
Variable indicating if the tensor package has been initialized. Set and used by
csetup, reset by init_ctensor.
Default value: []
An option in the ctensor (component tensor)
package. ct_coords contains a list of coordinates.
While normally defined when the function csetup is called,
one may redefine the coordinates with the assignment
ct_coords: [j1, j2, ..., jn] where the j's are the new coordinate names.
See also csetup.
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The following names are used internally by the ctensor package and
should not be redefined:
Name Description --------------------------------------- _lg() Evaluates to lfg if frame metric used, lg otherwise _ug() Evaluates to ufg if frame metric used, ug otherwise cleanup() Removes items drom the deindex list contract4() Used by psi() filemet() Used by csetup() when reading the metric from a file findde1() Used by findde() findde2() Used by findde() findde3() Used by findde() kdelt() Kronecker-delta (not generalized) newmet() Used by csetup() for setting up a metric interactively setflags() Used by init_ctensor() readvalue() resimp() sermet() Used by csetup() for entering a metric as Taylor-series txyzsum() tmetric() Frame metric, used by cmetric() when cframe_flag:true triemann() Riemann-tensor in frame base, used when cframe_flag:true tricci() Ricci-tensor in frame base, used when cframe_flag:true trrc() Ricci rotation coefficients, used by christof() yesp() |
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In November, 2004, the ctensor package was extensively rewritten.
Many functions and variables have been renamed in order to make the
package compatible with the commercial version of Macsyma.
New Name Old Name Description -------------------------------------------------------------------------- ctaylor() DLGTAYLOR() Taylor-series expansion of an expression lgeod[] EM Geodesic equations ein[] G[] Mixed Einstein-tensor ric[] LR[] Mixed Ricci-tensor ricci() LRICCICOM() Compute the mixed Ricci-tensor ctaypov MINP Maximum power in Taylor-series expansion cgeodesic() MOTION Compute geodesic equations ct_coords OMEGA Metric coordinates ctayvar PARAM Taylor-series expansion variable lriem[] R[] Covariant Riemann-tensor uriemann() RAISERIEMANN() Compute the contravariant Riemann-tensor ratriemann RATRIEMAN Rational simplification of the Riemann-tensor uric[] RICCI[] Contravariant Ricci-tensor uricci() RICCICOM() Compute the contravariant Ricci-tensor cmetric() SETMETRIC() Set up the metric ctaypt TAYPT Point for Taylor-series expansion ctayswitch TAYSWITCH Taylor-series setting switch csetup() TSETUP() Start interactive setup session ctransform() TTRANSFORM() Interactive coordinate transformation uriem[] UR[] Contravariant Riemann-tensor weyl[] W[] (3,1) Weyl-tensor |
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