Einstein Relatively Easy

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 We have already shown how to derive the geodesic equation directly from the Equivalence Principle in in our article Geodesic equation and Christoffel symbols.

Here our aim is to focus on the second definition of the geodesic (path of longer Proper Time^[1]) to derive the Geodesic Equation from a variationnal approach, using the Principle of least Action. That's actually how Einstein deduced it in his 1916 synthetic paper The Foundation of the General Relativity of Relativity

We can express the proper time along a time-like wordline  (while ignoring the limits) as:

with λ being the affine parameter parametrizing the path.

For a given proper spacetime interval ds² = g_μνdx^μdx^ν, we get:

But according to the calculus of variations, finding the path with maximal proper time is equivalent to find the path with extremal action^[2] (in this case a maximum), with the action S and the integrated Lagrangian L defined as:

where the dot over the x denotes the derivative with respect to λ. 

Passing the above Lagrangian through the Euler-Lagrange equation gives:

For the  four-dimensional space-time, as the indice α can take four values, this equation represent four differential equations.

Looking at the first member and remembering that the metric only depends on x^α (and not on the dot/derivatives coordinates), we get using the usual derivative rules

 We can now remark that from the above definition of τ, we can write dτ as:

Applying the chain rule we finally obtain

Similarly, considering the second term δL/δx^α of the Euler-Lagrange equation:

Putting all the terms together, our initial Euler-Lagrange equation becomes:

Finally, multiplying both sides by dλ/dτ gives:

We get the geodesic equation in its most common form, which is more simple to use than its alternative form with the Christoffel symbol as in this case the metric tensor g_αβ get referenced directly.

For the sake of it, let us make sure that the two expressions are equivalent.

First let us develop the first term of the above equation

Injecting this expression in our original Euler-Lagrange expression gives:

Multiplying now by the inverse metric g^γα we get

As μ and ν are interchangeable indices, one can write:

We then confirm that the two expressions of the geodesic equation are strictly equivalent.

[1] Refer to our article Geodesics as proper time maximization to see how this applies also in Special Relativity.

[2] The path taken by the system between times t1 and t2 is the one for which the action is stationary (no change) to first order, which means δS=0.