Ben Crowell, Fullerton College
Contact information: http://www.lightandmatter.com/area4author.html
An implementation of the Carminati-McLenaghan invariants in Maxima, using ctensor. The original paper (paywalled) describing the CM invariants was Carminati and McLenaghan, J. Math. Phys. 32 (1991) 3135. There is a brief article on Wikipedia that describes them, https://en.wikipedia.org/wiki/Carminati%E2%80%93McLenaghan_invariants . The notation used here is the same as the notation in the Wikipedia article.
The package is contained in the file cm_invariants.mac, which can be installed wherever you like. If you wish, you can test your installation by running the test suite, as described below.
The following example shows the use of the package to compute the CM invariants of the Schwarzschild spacetime.
load(ctensor); ct_coords:[t,r,theta,phi]; lg:matrix([(1-2*m/r),0,0,0], [0,-1/(1-2*m/r),0,0], [0,0,-r^2,0], [0,0,0,-r^2*sin(theta)^2])$ load("cm_invariants.mac"); cm_invariants(); /* Calculate all the invariants. */ cm_print_invariants(); /* Print out the ones that are nonzero. */
The output is as follows:
Nonvanishing CM invariants: 2 - 6 W1 = 6 m r 3 - 9 W2 = - 6 m r
After you call cm_invariants(), the invariants are all calculated and stored in the array cm_invariant, stored in the order R, R1, R2, R3, M3, M4, W1, W2, M1, M2, M5. The names ("R", "R1", ...) are stored in the array cm_invariant_name. The array cm_nonvanishing_invariants contains a list of the array indices of the invariants that are nonzero, e.g., [7,8] in the case of the Schwarzschild metric to show that W1 and W2 are nonzero.
Note that the ctensor package, without the cm_invariants package, already has a function scurvature() for calculating the Ricci scalar and rinvariant() for the Kretschmann scalar (which is not algebraically independent of the CM invariants).
Calculating only some of the invariants
The full set of invariants includes polynomials of up to fifth order in the Riemann tensor. In moderately complicated cases, the expressions for the components of the Riemann tensor itself can easily have hundreds or thousands of terms. If the Riemann tensor has, for example, 10^3 terms, then brute-force computation of a fifth-order polynomial in the Riemann tensor will involve something like 10^15 terms, which is obviously infeasible. For this reason, it may be advantageous to compute only some of the invariants, or compute them one at a time in order to see how long it takes. This can be done as in the following example:
load(ctensor)$ ct_coordsys(kerr_newman,all)$ load("cm_invariants.mac")$ showtime:true$ cm_init()$ cm_r(); cm_r1();
(This example works in Maxima 5.37 but not in Maxima 5.32.) The calculation above takes about 20 minutes on my machine to determine that R=0 for the Kerr-Newman spacetime, and also that (after a little more simplification) R1=[e/(a^2 cos^2 theta+r^2)]^4.
Starting over with a new metric
If you've calculated the invariants for one metric and then want to calculate them for some other metric within the same script, then you need to tell ctensor to forget the first metric. To do this, call the function init_ctensor(), then set up your new metric, and call cm_invariants() again.
Eigenvalues of the Weyl tensor
Many of the CM invariants depend on either the Ricci tensor or the Weyl tensor, but not both. An alternative way of expressing the information in these invariants is to find the eigenvalues of these tensors. The cm_invariants package is not needed in order to do this with the Ricci tensor. In the case of the Weyl tensor, it is necessary to reexpress the tensor as a matrix with two bivector indices, and cm_invariants provides this as the matrix cm_c_bivector. The file examples/kerr.mac gives an example of such a calculation, for the Kerr spacetime (rotating, uncharged black hole).
By default we have
cm_trig_simp:true; /* apply trigonometric substitutions? */ cm_rat_simp:true; /* rationally simplify results? */
To disable these simplifications, set these variables to false after calling cm_invariants().
The following functions and variables are the public interface of the package.
Global boolean variable, true by default. If true, then trigonometric substitutions are applied to all results. To turn off this simplification, set this variable to false after loading the package.
Same as cm_trig_simp but for rational simplifications.
Function that computes various tensors from the metric, and then computes all the CM invariants.
Initializes tensors used in computing the CM invariants. This is called automatically by cm_invariants(), so it only needs to be used when calculating some rather than all of the invariants, as in the example above using the Kerr-Newman metric.
Functions that calculate the individual CM invariants. The names follow the notation defined in the Wikipedia article.
cm_s, cm_us, cm_uus
Global arrays holding the trace-free Ricci tensor, cm_s[i,j]=S_ij=R_ij-(1/4)Rg_ij, as well as its fully contravariant and mixed versions cm_uus[i,j]=S^ij and cm_us[i,j]=S^i_j.
A function that returns -1, 0, or +1, depending on whether p is an even or odd permutation away from being sorted. The input p is an array of integers. This routine is meant to be used for small arrays, and will not have good performance on large arrays.
Global array containing the tensorial Levi-Civita tensor, i.e., the parity of its four indices, multiplied by the square root of minus the determinant of the metric.
cm_c, cm_c_mixed, cm_c_upper
Global arrays containing the Weyl tensor in the forms C_ijkl, C_ij^kl, and C^ijkl. The array cm_c is simply a copy of ctensor's weyl with the indices reshuffled to conform to the literature, i.e., cm_c[l,i,j,k]=weyl[i,j,k,l].
cm_c_star, cm_c_star_mixed, cm_c_star_upper
Global arrays containing the left Hudge duals of the various forms of the Weyl tensor, i.e., *C_ijkl, *C_ij^kl, *C^ijkl.
Global matrix containing the representation C_A^B of the Weyl tensor in terms of bivector indices A and B.
Function that calculates all the invariants and stores the results in the array cm_invariant.
Global array containing [cm_r(),cm_r1(),cm_r2(),cm_r3(),cm_m3(),cm_m4(),cm_w1(),cm_w2(),cm_m1(),cm_m2(),cm_m5()].
Global array containing ["R","R1","R2","R3","M3","M4","W1","W2","M1","M2","M5"]. The names follow the notation defined in the Wikipedia article.
Function that returns a list of array indices in cm_invariant_name for which the CM invariant vanishes.
Prints out all the nonvanishing CM invariants, or a message saying that all of them vanish.
The code is distributed along with a test suite in the subdirectory named tests. To test your installation, do "make test", which will run the maxima programs located in that directory, one after another. The following is a brief description of what these tests are. They are somewhat automated, so that many possible bugs will result in an error message. However, some of the output does need to be inspected by a human.
The program parity.mac tests that the cm_parity() function works correctly in a few cases.
The rest of the test programs set up the metric for some spacetime, calculate its CM invariants, print them out, and, in some cases, check that they seem to be correct. Several of these tests (flat.mac, plane_wave.mac, and vsi.mac) are spacetimes for which it is known on theoretical grounds that we should get zero for all curvature invariants that are continuous functions of the Riemann tensor and its derivatives (not just the CM invariants). If any of these fail to be zero, an error is generated.
The programs schwarzschild.mac, closed.mac, and de_sitter.mac do the calculations for the Schwarzschild spactime; a closed, radiation-dominated FLRW cosmology; and an FLRW cosmology containing only a repulsive cosmological constant (de Sitter space). For the Schwarzschild and closed spacetimes, we expect some curvature singularities where certain CM invariants blow up; the test programs check for this and throw an error if this is not the case. In the case of de Sitter space, the output is a constant, which makes sense because of the space's time-translation symmetry.
The program end.mac calculates the invariants for a somewhat pathological spacetime adapted from an example by Geroch, in which timelike geodesics are incomplete as time approaches infinity. All of the CM invariants remain finite, however. Since this spacetime is conformally flat, the only CM invariants that can be nonzero are the R's, and the test script checks whether this is true.
The program godel.mac calculates results for the Godel spacetime, comparing against published results.
The program polynomial.mac calculates the CM invariants for a metric defined in terms of polynomials in the coordinates, comparing against results provided by Ian Anderson.
The program reissner_nordstrom.mac tests against values tabulate by Ron Lenk.
All of the test programs call the function cm_do_tests(). This function checks that the traceless Ricci tensor is indeed traceless, checks the symmetries of the Weyl tensor and its Hodge dual, and checks the tracelessness of the Weyl tensor and its Hodge dual.
Find some known-good expressions for the CM invariants in some spacetimes, either published in the literature or calculated using the GRTensorII package, and check the results against those in some cases where the results are finite.
Make use of symmetries to improve efficiency.
The calculation of the R's can be broken down into a series of matrix multiplications. It would probably be more efficient to do it that way, rather than treating the calculation of each R as a separate computation. However, the computational effort probably grows exponentially with the order of the polynomial, so the improvement in performance might not be very much.