Computing Einstein's equations using variational principles

Edgardo S. Cheb-Terrab

Freddy Baudine

One of the biggest challenges for a computer algebra system is to compute Einstein's equations from first principles, by equating to zero the functional derivative of the Action for gravity, without using human-shortcuts or tricks. Developments during 2024, appearing as new in Maple 2025, broke through that barrier and now the LagrangeEquations  Physics command, introduced in 2023, can perform that elusive computation of Einstein's equations, not using tabulated cases, properly handling several (traditional or not) alternative ways of presenting the Lagrangian (the integrand in the Action), taking advantage of the functional differentiation  capabilities of the Physics  package .

The computation can be performed in one call to Physics:-LagrangeEquations, or in steps interactively using the Physics:-Fundiff  and Physics:-Simplify  commands that include newly implemented capabilities for simplifying tensorial expressions in curved spacetimes. This is an exciting breakthrough/milestone in Computer Algebra, also in an area out of reach of neural network AIs.

Einstein's equations are a system of second order nonlinear coupled partial differential equations for the 10 components of the spacetime metric

In the Lagrangian formulation, the coordinates of the problem are the 10 components of the metric,

and the parameters of the variational problem are the spacetime coordinates .

The traditional Lagrangian

The simplest case is that of Einstein's equation in vacuum, for which the Lagrangian density is expressed in terms of the trace of the Ricci  tensor and the determinant of the metric by

What was almost a dream in previous years, Einstein's equations can now be computed in one simple instruction as the Lagrange equations for this Lagrangian, in the traditional compact form, taking the components of the metric tensor as the coordinates (for an interactive, step by step computation, see further below)

The tensorial equation computed is also the definition of the Einstein  tensor

A Lagrangian depending only on first derivatives of the metric

The Lagrangian  used to compute Einstein's equations (4)  contains first and second derivatives of the metric; the latter make the computation significantly more complicated. The second order derivatives, however, can be removed from the formulation. To see that, rewrite L in terms of Christoffel  symbols

Recalling the definition

in  the two terms containing derivatives of Christoffel symbols contain second order derivatives of .

Now, it is always possible to add a total spacetime derivative to  without changing Einstein's equations (assuming the variation of the metric in the corresponding boundary integrals vanishes), and in that way, in this particular case of , obtain a Lagrangian involving only 1st order derivatives. The total derivative, expressed using the inert  command to see it before the differentiation operation is performed, is

Adding this term to , performing the  differentiation operation and simplifying we get

which is a Lagrangian depending only on 1st order derivatives of the metric through Christoffel  symbols. As expected, the equations of motion resulting from this Lagrangian are the same Einstein equations computed in (4); this computation can also now be performed in a single call

Simplification capabilities developed to cover these computations

To illustrate the new Maple 2025 tensorial simplification capabilities note that  is no just  ≡ (6) after discarding its two terms involving derivatives of Christoffel symbols. To verify this, split  into the terms containing or not derivatives of Christoffel

Comparing, the total derivative TD≡ (8) is not just , but

Things like these, , can now be verified directly with the new tensorial simplification capabilities: take the left-hand side minus the right-hand side, evaluate the inert derivative  and simplify to see the equality is true

That said, it is also true that  results in the Lagrangian , and since the equations of movement don't depend on the sign of the Lagrangian, for this Lagrangian  adding the term  happens to be equivalent to just discarding the terms of  involving derivatives of Christoffel symbols.

Derivation step by step using functional differentiation (variational principle)

Also new in Maple 2025, due to the extension of Fundiff  to compute in curved spacetimes, it is now possible to compute Einstein's equations from first principles by constructing the action,

and equating to zero the functional derivative with respect to the metric. To avoid displaying the resulting large expression, end the input line with ":"

Simplifying this result, we get an expression in terms of Christoffel  symbols and its derivatives

In this result, we see derivatives of Christoffel  symbols, expressed using the D_  command for covariant differentiation. Although, such objects have not the geometrical meaning of a covariant derivative, computationally, they here represent what would be a covariant derivative if the Christoffel symbols were a tensor. For example,

With this computational meaning for the derivatives of Christoffel symbols appearing in (19), rewrite EEC≡(19) in terms of the Ricci  and Riemann  tensors. For that, consider the definition

Rewrite the noncovariant derivatives  in terms of derivatives using the computational representation (20), simplify and isolate one of them

Analogously, derive an expression to rewrite derivatives of Christoffel symbols using the Riemann  tensor

Substitute  these two equations, in sequence, into Einstein's equations EEC≡(19)

Simplify  to arrive at the traditional compact form of Einstein's equations