# arXiv Jan 4th-8th 2020

## 2012.14984 xPPN: An implementation of the parametrized post-Newtonian formalism using xAct for Mathematica

Many in the physics community are familiar with the `xAct` tensor manipulation suite for `Mathematica`, and those who aren't probably should be. In recent years, `xAct` has consolidated its place as the go-to computer algebra tool in gravitational theory, outperforming various predecessors with its powerful 'canonicalise' function, and attracting a flurry of applied package development. Some of these packages float and others sink: I'm curious to see how Manuel Hohmann's new `xPPN` package will fare.

Most modifications to Einstein's theory can be cast in the parametrized post-Newtonian (PPN) formalism. PPN allows the theory to be compared, as a ten-parameter modification of the Newtonian theory, against the gold standard set by GR. The implementation of PPN is usually extremely complicated, and varies substantially from theory to theory. Thus xPPN, a general implementation of the PPN formalism, is potentially very exciting.

At the heart of the implementation seems to be the definition of two bespoke manifolds for the \(3+1\) decomposition. Assuming that the covariant theory lives in some sense on \(M_4\), a spacelike foliation \(S_3\) and timelike threading \(T_1\) are introduced as separate `xTensor` manifolds, where \(M_4\cong T_1\times S_3\). This cuts right through the pre-existing notion of ADM decomposition in `xAct`, which I've wrestled with recently in the context of the Hamiltonian analysis. In the end, I kept a single Minkowski manifold \(M_4\), and defined a set of projections accompanied by very many rules. This is not ideal, but the relevant parts of `xAct` (such as `xCoba`) have quite patchy documentation, which makes life less than easy! I won't go into the guts of the higher functions of the package, but suffice to say the post-Newtonian potentials are all defined, along with certain 'utility functions' which facilitate the human-assisted expansion. A nice walk-through is provided for a simple Brans-Dicke-like theory, but since I've not tried it out, I can't offer further comment.

My main concern is the 'theory-scope' of the package. This is of course the hardest part to implement, since you can never tell quite in what terms the next theory will be cast. The Brans-Dicke theory obviously inherits much of the machinery of GR, but with an extra scalar \(\psi\) - I expect e.g. the variations on mimetic gravity could be similarly tackled. This scalar naturally has to be defined when using the package, and presumably one may extend to higher-spin fields also. Accordingly I've already recommended `xPPN` to a colleague who is working on the new relativistic completion of MOND.
However, it is often interesting to build theories out of a gauge-covariant derivative. Accordingly, the `xPPN` package defines three connections: Levi-Civita, teleparallel and symmetric-teleparallel. I'm very happy about the last of these, which follows on from the non-metricity theories of Jiménez, Heisenberg and Koivisto, and is something I'd like to work on at some point. However, in the short term I'm interested in a free connection, in the context of torsion theories. This is likely workable in the `xPPN` setup, but might take some `xAct`-jitsu. Again, I'd have to try.

Overall, this paper introduces software rather than physics, and is structured accordingly. Nonetheless, I am happy to have come across `xPPN`, and am looking forward to trying it out in the near future.