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xPert: Computer algebra for metric perturbation theory
David Brizuela, Jose M. Martin-Garcia, Guillermo A. Mena Marugan
TL;DR
High-order metric perturbation theory produces large, combinatorially difficult equations, motivating specialized computer algebra. xPert combines explicit nonrecursive expansion formulas with efficient tensor canonicalization in Mathematica, enabling calculations through order 10 in practical timings.
Problem
Higher-order perturbation calculations in general relativity generate huge expressions and require specialized computer algebra because their complexity is intrinsically combinatorial and grows exponentially with perturbative order.
Method
xPert combines explicit nonrecursive formulas for perturbative expansions with efficient tensor canonicalization and tools for standard metric-perturbation schemes.
Results
xPert handles perturbative orders n = 4 or n = 5 within seconds and n = 10 with canonicalization times of approximately 1 hour, subject to available RAM.
Takeaways & Limitations
xPert provides a practical system for constructing and manipulating high-order metric-perturbation equations around arbitrary backgrounds.
Abstract
from arXiv · showhide
We present the tensor computer algebra package xPert for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. It is based on the combination of explicit combinatorial formulas for the n-th order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct, for Mathematica. We give examples of use and show the efficiency of the system with timings plots: it is possible to handle orders n=4 or n=5 within seconds, or reach n=10 with timings below 1 hour.
I. INTRODUCTION
The introduction motivates extending linear perturbation theory beyond its nonlinear limitations and presents xPert as a high-order symbolic-computation package. It combines explicit perturbative formulas with efficient tensor canonicalization to support calculations up to order 10.
- Linearization studies small deviations around known general-relativistic backgrounds but omits the intrinsic nonlinear interactions among perturbation modes.
- Second-order perturbations have been applied to dust collapse, rotating fluid stars, Kerr black holes, and gravitational radiation from close-limit collisions.
- Arbitrary-order perturbation formalisms for spherical spacetimes motivated specialized tensor computer-algebra tools whose efficiency and usability were subsequently improved.
- xPert combines explicit precomputed curvature-expansion formulas with efficient Mathematica tensor canonicalization, avoiding slow recursive computations.
- The article reviews high-order perturbation formulas, explains their Mathematica implementation, and evaluates efficiency through timing examples.
A. Basic concepts
The basic formalism treats perturbations as derivatives of smooth one-parameter families of manifolds and tensor fields, compared through gauge-dependent point identifications. It defines metric perturbations and emphasizes index placement, product rules, and alternative background-field restrictions.
- Perturbation theory compares tensor fields on parameterized manifolds with background fields through point-to-point identification mappings, or gauges.
- The perturbative operator is a derivative and therefore obeys a generalized Leibniz rule for products of multiple tensors.
- Metric perturbations are denoted {n}h_ab ≡ Δ^n[g_ab], with the perturbative order recorded as a left superscript.
- Nonvanishing metric perturbations require careful index-position tracking, with metric factors and the Leibniz rule used to correct inconvenient positions.
- The background-field method is recovered by setting all covariant metric perturbations of order n ≥ 2 to zero.
B. Perturbations of curvature tensors
xPert constructs high-order curvature perturbations directly from metric perturbations using explicit formulas rather than recursive order-by-order computation. The section develops expansions for the inverse metric, connection, and curvature tensors.
- B. Perturbations of curvature tensors: High-order curvature calculations are long, so xPert implements explicit formulas that directly produce perturbations of the principal tensors in general relativity.
- B. Perturbations of curvature tensors: The inverse-metric expansion is organized by sorted partitions of the perturbative order into positive integers.
- B. Perturbations of curvature tensors: Christoffel-symbol perturbations are constructed from metric perturbations and their first covariant derivatives at all orders.
- B. Perturbations of curvature tensors: Perturbations of the Riemann tensor follow from the generalized Leibniz rule, while Ricci, Ricci-scalar, and Einstein-tensor perturbations are derived from Riemann and metric contractions.
- B. Perturbations of curvature tensors: The Ricci-scalar and Einstein-tensor formulas are left unsimplified because they already provide efficient direct expressions for arbitrary perturbative order.
C. Perturbation of the metric determinant
The metric determinant perturbation is expressed through antisymmetric densities and the generalized Leibniz rule. At every order, it factors into the background determinant times a contracted scalar containing at most N metric perturbations.
- The metric determinant depends on the chosen basis and transforms as a density of weight +2.
- The determinant is defined using upper and lower antisymmetric densities whose components are basis-dependent but remain invariant under perturbation.
- The n-th order determinant perturbation equals the determinant itself multiplied by a scalar formed by contracting metric perturbations.
- The scalar factor contains at most N metric perturbations, and this is the only place where the perturbed manifold’s dimension enters the article.
D. Perturbations of derivatives
The section derives perturbation rules for partial, covariant, and Lie derivatives, emphasizing that covariant derivatives change when the metric is perturbed while the Lie structure does not.
- Partial derivatives commute with the perturbation operator because they are tied to coordinates and remain unchanged under perturbations.
- Perturbing a covariant derivative requires accounting for the metric-dependent Levi-Civita connection and transforming through partial derivatives and Christoffel symbols.
- The covariant-derivative perturbation formula can be rewritten using the Leibnitz rule for arbitrary tensor densities of weight ω.
- The Lie-derivative perturbation formula handles perturbations of both the vector field and tensor while leaving the underlying Lie structure unchanged.
III. THE xPert PACKAGE
xPert implements arbitrary-order metric perturbation expansions in Mathematica by combining explicit nonrecursive formulas with tensor canonicalization and commands for defining, expanding, and manipulating perturbations.
- Specialized tensor computer algebra is needed beyond low perturbative orders because combinatorial expansion terms grow exponentially with n.
- xPert is an xAct module for Mathematica, whose computational-group-theory algorithms canonicalize indexed expressions with arbitrary symmetries.
- The package provides four principal commands: DefMetricPerturbation, Perturbation, ExpandPerturbation, and GaugeChange.
- Perturbation represents the perturbation operator, combines perturbative orders of composed heads, and supports linearity, the Leibnitz rule, and derivative operations.
- xPert combines explicit precomputed curvature expansions with efficient xAct canonicalization to construct perturbations nonrecursively at arbitrary order n.
- ExpandPerturbation replaces arbitrary-order perturbations of known background objects with explicit expansions in metric perturbations and separates metric factors when index positions require it.
- A second-order perturbation of the Einstein tensor is constructed and canonicalized in less than one second.
IV. EXAMPLES
The examples characterize xPert’s computational scaling with perturbative order and show that canonicalization, rather than expansion, is the main practical bottleneck. Despite exponential combinatorial growth, the implementation handles useful orders efficiently on moderate hardware.
- The overall complexity is governed mainly by perturbative order n, which increases both the number of terms exponentially and the number of factors typically linearly.
- xPert’s canonicalization algorithms make expressions with a few dozen indices effectively polynomial in index count, enabling third-order perturbation expressions to be manipulated in 1 second.
- The nth-order Leibniz and Faà di Bruno expansions are fast enough to be negligible relative to canonicalization, with Leibniz expansion faster and less memory-intensive.
- RAM limits practical expressions to roughly 10^5 terms with a few gigabytes, corresponding approximately to n = 10.
- Orders n = 4 or n = 5 can be manipulated within seconds, whereas n = 10 requires canonicalization times of approximately 1 hour.
V. CONCLUSIONS
xPert addresses the large expressions generated by extending general-relativistic perturbation theory beyond linear order. Its explicit formulas and tensor-manipulation algorithms support high-order calculations, while memory and term-count growth remain the principal practical constraints.
- Higher-order perturbations extend linear perturbation theory by incorporating general relativity’s nonlinear character as sources for linearized equations, but these sources are often huge.
- xPert combines explicit nonrecursive formulas with computational-group-theory algorithms to construct and manipulate metric perturbation equations through n = 10.
- Orders n = 4 and n = 5 are achievable in seconds, while n = 10 is reachable with average timings below 1 hour on moderate computer resources.
- The huge number of generated terms is xPert’s main time and memory limitation, although independent terms can be treated by different processors for parallelization.
- As an xAct module, xPert provides additional tensor-manipulation commands, background particularization, and conversion of equations to C or FORTRAN code.
- xPert has been used for high-order spherical-background perturbations, nonlinear radiation transfer, and scalar-field inflation in anisotropic spacetimes.
- The package is free software distributed under the GNU general public license.