In an earlier notebook, I validated the (up-down) calculus against Carlip’s electrodynamical aberration. In this notebook, I turn to the second and motivating problem in Carlip’s paper, Aberration and the Speed of Gravity: gravitational aberration, through the gravitational analogue of the Liénard-Wiechert potential. The calculus is a collection of rules and operators for both unambiguous input and readable output of tensor index notations. It’s becoming an embedded domain-specific language (eDSL). The electrodynamical case required only special relativity, and was thus a bit smaller than the gravitational case. Here, we use the same to calculate Christoffel connections and the Riemann curvature tensor in a weak-field approximation. These yield a similar aberration effect in gravitation to that seen in electrodynamics: the velocity of a moving source causes the apparent direction of gravitational influence to point approximately to the instantaneous position of the source rather than to its retarded position. In the electrodynamical case, the approximation is linear in the Lorentz-invariant 4-velocity. In the gravitational case, the approximation is better: it’s quadratic in the velocity and exact if the source is accelerating but not jerking. At first glance, aberration seems absurd. How can a particle “know” where the source is “now” when “now” can’t even be defined invariantly, that is, without choosing a coordinate system? But, in retrospect, it has to be this way. The retarded 3-position of a moving source—where it was when the gravitational influence was emitted—is not Lorentz-invariant. Thus it also depends on choice of coordinates. But no choice of coordinate can affect the physics. A test particle cannot “know where the source was” any more than it can know where it is “now.” Instead, the source sees a sum of Lorentz-invariant quantities involving the invariant retarded distance and its derivatives: velocity and acceleration, etc. This sum resembles “knowledge,” but it is just an extrapolation of Lorentz-invariant quantities, however, resolving the conundrum. This notebook, like its predecessor, is more about computation in than it is about physics. To keep this notebook short, we have only cursory, philosophical remarks about physical interpretation. We refer you to the original paper and its references for more crisp discussion of the physics.
Introduction
Introduction
In general relativity, the “force” is not a vector field but a curvature of spacetime, encoded in the Riemann Tensor . To investigate aberration, we must derive this curvature for a moving source, then collect terms in common powers of , where is the invariant retarded distance. The peeling property of the curvature tensor allows that final analysis.
ρ
R
σμν
1/r
r
Standard manual derivations of Riemann are notoriously error-prone due to the proliferation of indices. Here, we apply ’s fourGrad operator to derive indexed objects symbolically from their definitions. The resulting formal verification easily justifies the effort of developing , as checking and double checking a manual calculation would have taken much longer.
The Implementation
The Implementation
Here, we “paste” the implementation without commentary, referring you to the earlier notebook for walkthrough. was developed for the easier electrodynamical case, but ended up working well here, without modification.
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This section contains mostly general-purpose rules for , but there are special rules in the mix, such as those for invariant retarded distance and some rules for fourGrad. A clean separation of general from special awaits more meticulous engineering of .
Display Rules
Display Rules
In[]:=
ClearAll[dpyRules,d,s,λ,σ,η,dλds];dpyRules={(*Specificrulesaheadofgeneralrules*)dλds[[ν_]]:>(FractionBox[dSubscript[λ,ν],ds]//DisplayForm),dλds[[ν_]]:>(FractionBox[dSuperscript[λ,ν],ds]//DisplayForm),(*GeneralRules*)ε_[[τ_]]:>Superscript[ε,τ],ε_[[τ_]]:>Subscript[ε,τ],ε_[[μ_],[ν_]]:>Subscript[ε,Row[{μ,ν}]],ε_[[μ_],[ν_]]:>Superscript[Subscript[ε,μ],ν],ε_[[μ_],[ν_]]:>Superscript[ε,Row[{μ,ν}]],ε_[[μ_],[ν_]]:>Subscript[Superscript[ε,μ],ν]};ClearAll[echo];(*fordebugging*)echo[τ_]:=Module[{},Print[τ/.dpyRules];Echo[τ]];
Tensor Rules
Tensor Rules
In[]:=
ClearAll[tensorRules];tensorRules=(*SpecialRulesFirst*)(*Definitionofσ*)x[ud_]-z[s][ud_]:>σ[ud],(*NullInterval*)σ[[α_]]σ[[α_]]:>0,σ[[α_]]σ[[α_]]:>0,(*Definitionofλ*)z'[s][ud_]:>λ[ud],(*UnitVelocity*)λ[[α_]]λ[[α_]]:>1,λ[[α_]]λ[[α_]]:>1,(*MixedMetricactsas*)η[[μ_],[ρ_]]:>δ[[μ],[ρ]],(*MetricSymmetry,avoidingunboundedrecursionviaOrderedQ*)η[[α_],[β_]]/;OrderedQ[{α,β}]:>η[[β],[α]],(*GeneralRules*)(*CoalesceTermsintoDown-UpOrder*)τ_[[μ_]]υ_[[μ]]/;OrderedQ[{τ,υ}]:>υ[[μ]]τ[[μ]],(*RaisingandLowering:e.g.->*)η[[μ_],[ν_]]τ_[[ν_]]:>τ[[μ]],η[[μ_],[ν_]]τ_[[μ_]]:>τ[[ν]],η[[μ_],[ν_]](τ_+υ_):>η[[μ],[ν]]τ+η[[μ],[ν]]υ,η[[μ_],[ν_]](τ_[[ν_]]υ_[[μ_]]):>τ[[μ]]υ[[μ]],(*unravelingrecursiveindices*)τ_[(h1:|)[(h2:|)[ud_]]]/;h1===h2:>τ[h1[ud]];
ρ
η
μ
ρ
δ
μ
η
μα
α
λ
λ
μ
Invariant Retarded Distance r
Invariant Retarded Distance
r
In[]:=
ClearAll[rExpr];(rExpr=σ[[ρ]]λ[[ρ]])/.dpyRules
Out[]=
σ
ρ
ρ
λ
Custom 4-grad Operator
Custom 4-grad Operator
Basic Rules from Calculus
Basic Rules from Calculus
Linearity, Product Rule, Power Rule
In[]:=
fourGrad[a_+b_,ud_]:=fourGrad[a,ud]+fourGrad[b,ud];fourGrad[a_*b_,ud_]:=bfourGrad[a,ud]+afourGrad[b,ud];fourGrad[,ud_]:=nfourGrad[a,ud];
n_
a_
n-1
a
Constants and k
e
In[]:=
fourGrad[e,_]:=0;fourGrad[k,_]:=0;
Numeric Constants
In[]:=
fourGrad[n_?NumericQ,_]:=0;
Gradient of the Separation Vector σ
Gradient of the Separation Vector σ
In[]:=
fourGrad[σ[udν_],udμ_]:=η[udμ,udν]-λ[udν];
σ[udμ]
r
In[]:=
fourGrad[σ[[ν]],[μ]]/.dpyRules
Out[]=
η
μν
λ
ν
σ
μ
r
Gradient of the Velocity λ
Gradient of the Velocity λ
In[]:=
fourGrad[λ[udν_],udμ_]:=dλds[udν];
σ[udμ]
r
In[]:=
fourGrad[λ[[ν]],[μ]]/.dpyRules
Out[]=
d
λ
ν
ds
σ
μ
r
∂μr: Gradient of r
∂
μ
r
In[]:=
(gradRRaw=fourGrad[rExpr,[μ]])/.dpyRules
Out[]=
d
ρ
λ
ds
σ
μ
σ
ρ
r
η
μρ
λ
ρ
σ
μ
r
ρ
λ
In[]:=
(fourGrad[r,μ_]=Simplify[gradRRaw//.tensorRules]//.tensorRules)/.dpyRules
Out[]=
λ
μ
σ
μ
d
ρ
λ
ds
σ
ρ
r
The Physics
The Physics
The Metric Perturbation hμν
The Metric Perturbation
h
μν
In the weak-field approximation (linearized gravity), the metric is the Minkowski metric plus a small perturbation :
g
μν
η
μν
h
μν
g
μν
η
μν
h
μν
(
1
)For a point mass moving with 4-velocity , the metric perturbation is the direct gravitational analog of the electrodynamical Liénard-Wiechert potential.
M
λ
μ
h
μν
4GM
r
λ
μ
λ
ν
(
2
)◼
Note this is not Carlip’s Equation 2.1, which is the Kinnersley metric for the photon rocket-problem. That problem has similar outcome to the outcome of this paper, as we may show in a future notebook.
In , we capture the constant factor as a symbolic constant without a value, and write
4GM≡k
k
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This kind of collision will be fixed by “alpha renaming” when we write a proper compiler for .
Intuitions
Intuitions
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Aside: Gravitation shares this idea with quantum fields, where force bosons come from the connections of gauge fields that leave Lagrangians invariant. This similarity is so alluring that physicists strongly suspect an underlying unified theory. However, unification has resisted all attempts. Various attempts make predictions refuted by experiment. Other attempts fail renormalization, and so on. It is an open question whether the unification even exists or whether Nature gives us two different but similar-looking kinds of force laws (space-time and gauge), laughing at our petty struggles to understand.
To implement this efficiently in , we define a helper function to “instantiate” metricGradient with specific indices.
Intuitions
Intuitions
Define a second-derivative helper, doubleGradH, and differentiate metricGradient again.
Intuitions
Intuitions
In a manual calculation, a physicist would usually discard higher-order terms and exploit symmetries in intermediate steps to keep the expressions manageable. While elegant, that process is perilous; a single dropped index or misplaced sign propagates errors that are extremely expensive to trace or redo. By contrast, Wolfram’s symbolic engine is syntactically rigid. It does not simplify until told to do so, and it does not make algebraic mistakes. The “wall of math” is assurance that every interaction—every chain rule and every suitable contraction—is applied.
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I have first-hand experience with this, leading me to cherish an ancestor of Mathematica. My first job out of school in the early 80’s was to check the math for relativistic clock corrections in GPS signals, long before such was routine and automated. I recall doing page after page of algebra, only to fail a sign or index check at the end, then failing to find the bug on backtracking, and ending up starting over from scratch, multiple times. This very project caused me to seek out Stephen Wolfram, then at Caltech Physics, and get a copy of SMP from him to check those calculations. Without SMP, I would have had little confidence in my final results and would have felt irresponsible. Of course, SMP eventually became Mathematica, which became Wolfram, and here we are some 45 years later. I have used it every day for all kinds of tasks, including designing C code, since long before it was a commercial product.
We can trust this result not because the output is pretty or ugly or anything else, but because the process is robust. The same fourGrad and tensorRules that successfully reproduced the Liénard-Wiechert fields in electrodynamics apply here without modification.
That said, the big-picture structure is exactly what a relativist hopes to see:
Verification: The Component Inspector
Verification: The Component Inspector
Finally, we apply the following “Component Inspector” logic. We check the Newtonian limit (tidal tensor) and the radiation term.
Intuitions
Intuitions
The Newtonian Tidal Limit contains two distinct scalings:
Conclusion
Conclusion
We have extended the (up-down) calculus from electrodynamics to general relativity. By formalizing index valence and implementing differentiation rules for retarded quantities, we have overlaid a tensorial symbolic engine on top of Wolfram. We showed how to derive a linearized Riemann tensor from first principles in .
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
General relativity in the calculus
by Brian Beckman
Wolfram Community, STAFF PICKS, January 6, 2026
https://community.wolfram.com/groups/-/m/t/3601899
by Brian Beckman
Wolfram Community, STAFF PICKS, January 6, 2026
https://community.wolfram.com/groups/-/m/t/3601899