There is a set of 16 polynomial curvature scalars called the Carminati-McLenaghan invariants: https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants . For some fairly broad classes of spacetimes, they are supposed to give enough information to distinguish one spacetime from another. As far as I know, they have been implemented in GRTensorII (a free package that runs on top of the proprietary Maple CAS), but not in Maxima/ctensor, which is free. I thought I would try to implement some of them in Maxima. So far, I've implemented the first four on the list, which are called ##R##, ##R_1##, ##R_2##, ##R_3##. This is a learning experience for me, since I haven't done much actual coding in Maxima before. The following is the code I came up with:(adsbygoogle = window.adsbygoogle || []).push({});

I tried it out on the following four metrics:Code (Text):/*-----------------------------------------------------------------

Carminati-McLenaghan invariants,

https://en.wikipedia.org/wiki/Carminati%E2%80%93McLenaghan_invariants

(c) 2015 B. Crowell, GPL v. 2 licensed

------------------------------------------------------------------*/

/* Make sure the following have been calculated: */

cmetric(false)$

lriemann(false)$

uriemann(false)$

ricci(false)$ /* compute the ricci tensor ric[i,j]=R_ij */

uricci(false)$ /* compute the mixed ricci tensor uric[i,j]=R_i^j */

/* R=R_i^i, the Ricci scalar, scalar curvature */

cm_r():=trigsimp(ratsimp(sum(uric[i,i],i,1,dim)))$

cm_r();

/* trace-free Ricci tensor, cm_s[i,j]=S_ij=R_ij-(1/4)Rg_ij */

for i thru dim do

for j thru dim do

cm_s[i,j] : ric[i,j]-(1/4)*cm_r()*lg[i,j]$

/* mixed version, cm_us[i,j]=S^i_j */

for i thru dim do

for j thru dim do

cm_us[i,j] : sum(ug[i,k]*cm_s[k,j],k,1,dim)$

/* R_1=(1/4)S^i_j S^j_i */

cm_r1():=trigsimp(ratsimp((1/4)*sum(sum(cm_us[i,j]*cm_us[j,i],j,1,dim),i,1,dim)))$

cm_r1();

/* R_2=(-1/8)S^i_j S^j_k S^k_i */

cm_r2():=trigsimp(ratsimp((-1/9)*sum(sum(sum(cm_us[i,j]*cm_us[j,k]*cm_us[k,i],k,1,dim),j,1,dim),i,1,dim)))$

cm_r2();

/* R_3=(1/16)S^i_j S^j_k S^k_l S^l_i */

cm_r3():=trigsimp(ratsimp((1/16)*sum(sum(sum(sum(cm_us[i,j]*cm_us[j,k]*cm_us[k,l]*cm_us[l,i],

l,1,dim),k,1,dim),j,1,dim),i,1,dim)))$

cm_r3();

I. Schwarzschild

II. de Sitter space

III. ##ds^2 =d u d v-a^2(u)d w^2##, with ##a(u)=1+\cos(u)/2## -- see

Schmidt, "Why do all the curvature invariants of a gravitational wave vanish?" http://arxiv.org/abs/gr-qc/9404037[/PLAIN] [Broken]

IV. ##ds^2 = A(t)(dt^2-dx^2)##, with ##A(t)=1/(1+e^t)## -- see https://www.physicsforums.com/threads/curvature-singularity-with-well-behaved-kretschmann-scalar.842614/#post-5290133

I. Schwarzschild. This gave zero for all four of the invariants that I'd implemented. I assume that the Kretschmann invariant is algebraically independent of these, but that if I implement the other 12 invariants, at least one will detect the singularity in the same way the the Kretschmann invariant does.

II. de Sitter space. R=-12, and the other three are zero. It makes sense that they're all constant, since de Sitter space doesn't evolve over time.

III. All four of the invariants I'd implemented vanish, as they should for this solution. (That's the point of the Schmidt paper.)

IV. All four are nonzero and approach a finite limit as ##t\rightarrow\infty##, even though there is geodesic incompleteness. It would be interesting to see if one of the other 12 invariants does blow up as ##t\rightarrow\infty##. If not, then it would support @PAllen 's suspicion that this spacetime could be extended past the singularity and so is not really singular.

It would be interesting to implement the other 12 invariants. Do do that, I would have to figure out how to compute the dual ##^*C## of the Weyl tensor.

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# Fun with Carminati-McLenaghan invariants

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