Expression Growth, Branches, and Independent Checks

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Symbolic output is trustworthy only on the domain for which every rewrite is valid. Large tensor calculations amplify this problem: an early square-root or logarithm simplification can contaminate thousands of later terms while still producing an elegant final formula. Validation therefore treats assumptions, branches, expression growth, and independent representations as part of the computation rather than as cleanup after it.

Required background. Connections and Covariant Derivatives supplies the derivative expressions being simplified, while Curvature supplies invariant relations for the geometric benchmark.

Helpful background. Formulations of General Relativity gives the continuum identities that may be used legitimately; Ordinary Differential Equations connects symbolic expressions to numerical evaluation; and Spacetime Symmetries and Killing Fields clarifies which reductions follow from imposed symmetry.

Assumptions define the expression

Over the real numbers,

\sqrt{x^2}=\lvert x\rvert,

not $x$ unless $x\ge0$ is known. Over the complex numbers, logarithms, powers, and roots require branch choices. The principal logarithm has a cut and cannot obey $\log(zw)=\log z+\log w$ globally. The principal-value definitions and their domains are fixed in NIST DLMF, § 4.2(i), Eqs. 4.2.1–4.2.7. A computer-algebra assumption such as positive is therefore mathematical input, not a performance hint.

Every expression record should include the base field, reality and sign assumptions, excluded zeros, coordinate domain, and branch conventions. A simplifier may return a conditional expression rather than one formula. That is often the correct answer. Here $M$ is the physical mass, so $GM$ is its geometrized length in the inherited $c=1$ units. Replacing $\sqrt{(r-2GM)^2}$ by $r-2GM$ , for example, is valid in the Schwarzschild exterior but changes sign inside the horizon.

Control expression growth before expansion

Canonicalization, factorization, common-subexpression elimination, and delayed evaluation can prevent intermediate swell. The best representation depends on the next operation: factored expressions expose singular factors, while expanded polynomials may canonicalize quickly. Record both the final form and a serialization or command record of the unsimplified construction so the route can be replayed; an accompanying hash can verify integrity but cannot reproduce the construction by itself.

A useful strategy partitions a calculation into named stages. After each derivative, contraction, or substitution, canonicalize only the symmetries needed at that stage; test a cheap identity; then checkpoint. Time and peak memory belong in the environment record because a mathematically correct algorithm that grows beyond available resources has a practical failure mode. Comparing alternative orders of contraction and simplification reveals whether a result depends on an accidental heuristic.

Exact, sampled, and independent checks

Validation should combine three levels. First, subtract two expressions and reduce the difference exactly under declared assumptions. Second, evaluate at reproducibly generated generic points that respect all domains, using exact rationals or increasing precision. Third, derive the same quantity through a structurally independent representation. Random samples can find counterexamples quickly, but no finite sample proves a symbolic identity.

Branch-sensitive tests must approach the cut from both sides. For the principal logarithm, values above and below the negative real axis differ by the expected discontinuity; see NIST DLMF, § 4.2(iv), especially the branch-cut discussion. Sampling only positive real numbers would never reveal an invalid complex logarithm rewrite.

For a test away from the cut itself, choose $z=w=e^{3\pi i/4}$ . Then the principal logarithms satisfy

\operatorname{Log}z+\operatorname{Log}w=\frac{3\pi i}{2}, \qquad \operatorname{Log}(zw)=\operatorname{Log}(-i)=-\frac{\pi i}{2}.

The two expressions differ by $2\pi i$ . Both inputs and their product are ordinary nonzero points, so avoiding the cut during point selection does not make the product identity globally valid. A valid simplification needs an argument-range condition, not only nonzero factors.

For tensor expressions, an independent component evaluation is especially valuable. Abstractly canonicalize a contraction, then build the same component from a coordinate metric without importing the first result. The separation between abstract-index and component systems is surveyed in MacCallum 2018, §§ 4 and 6.3.4. Agreement between two print formats from one internal expression is not independent evidence.

Schwarzschild and branch benchmarks

Use the Schwarzschild metric with assumptions $G\gt 0$ , $M\gt 0$ , and exterior domain $r\gt 2GM$ . The geometric benchmark requires

R_{ab}=0, \qquad R_{abcd}R^{abcd}=\frac{48(GM)^2}{r^6}.

Run the contraction through at least two simplification routes—early factorization and late factorization—and require identical exact results. Recompute one Riemann component directly from the connection. Then relax the assumption to $r\gt 0$ in an ingoing chart: invariant curvature remains meaningful at $r=2GM$ , while any exterior-only replacement involving $\lvert r-2GM\rvert$ must no longer be used without cases.

A compact algebra benchmark evaluates $\sqrt{x^2}$ for negative, zero, and positive real $x$ , then for generic complex $x$ . Another checks logarithm identities on points near the principal cut. These tests isolate the simplifier before it is embedded in a tensor pipeline. The accepted output includes conditional or absolute-value forms whenever required.

Reproducible rerun and rejection rules

Pin the computer-algebra system, tensor package, assumptions engine, exact-arithmetic mode, simplification functions, random seed, sample generator, working-precision sequence, and resource limits. The rerun sequence loads a clean environment; executes the square-root and logarithm tests; constructs Schwarzschild in a declared chart; computes the Ricci and Kretschmann residuals by two routes; evaluates generic rational points at 30, 60, and 120 digits; and exports inputs, checkpoints, timing, memory, branches, and discrepancies.

Convergence means that numerical discrepancies decrease as precision increases and exact residuals remain zero under route changes. Reject a result if it is valid only at a special point, changes under harmless dummy-index renaming, crosses an excluded denominator, or depends on an undocumented assumption. Also reject silent timeout recovery that substitutes a numerical approximation for an exact step. A useful limitation statement names the verified domain and leaves other branches undecided.

Check your understanding

A simplifier turns $\sqrt{(r-2GM)^2}$ into $r-2GM$ and then obtains a compact curvature expression. Which single condition makes that rewrite valid, and what should happen if the calculation crosses the horizon?

Solution

The rewrite is valid on a real domain with $r-2GM\ge0$ , such as the declared exterior. Without that condition the correct real form is $\lvert r-2GM\rvert$ . A horizon-crossing calculation must retain the absolute value, split into cases, or use a representation that avoids this square root; it may not carry the exterior rewrite silently into the interior.

References

MacCallum, M. A. H. (2018), “Computer Algebra in Gravity Research,” Living Reviews in Relativity 21, 6. DOI.
National Institute of Standards and Technology (n.d.), Digital Library of Mathematical Functions, § 4.2, “Definitions: Logarithm, Exponential, Powers.” Stable URL.