Thinking: General Relativity, 20

4 min readMay 17, 2024

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Geodesics & Euler-Lagrange Equations [1]

Starting with the geodesic equation as we known from the previous articles A “straight line on a curved surface” is called a geodesic, which minimizes the distance between 2 points.

The physical meaning of the above equations, tells us that It’s the equation of motion of freely-falling particles in a curved space-time. Particles traveling on a geodesic feel no forces.

then the particle has zero acceleration,

and it is moving in an inertial frame where gravity is absent.

represents a force due to gravity, which is eventually curving the path of the particle through space-time.

different functions.

Now as we know, [3]

We can think of this equation as expressing the action of a particle moving from A to B along curve C.
Then the rule for calculating the geodesic is to “extremelize” that quantity, or more accurately to make it stationary.
The equation that tells us how to minimize a quantity like S in the equation is called the Euler–Lagrange equation.
When we go from the principle of least action to the Euler–Lagrange equation, the principle of least action turns into a differential equation involving a Lagrangian.
Explicitly, the Euler–Lagrange equation becomes an equation of the type F = ma.

This is the expression that we will want to minimize.
A time-like geodesic maximizes proper time (it is one of the explanations of the twin paradox).
The definition of action is proportional to the minus of the proper time:

The integrand in the equation (where we put m back beneath the integral sign) is the Lagrangian.
It is the quantity that, in the calculation of a geodesic, plays the same role as the Lagrangian when we apply the principle of least action to calculate the trajectory of a particle in classical non-relativistic physics.
Action, by definition, is equal to the integral of the Lagrangian, which is itself a function of velocities and positions:

recalling the Euler–Lagrange equations that the Lagrangian must satisfy:
On the left-hand side, for each of these partial derivatives, we take the derivative with respect to time and equate it to the partial derivative of with respect to the corresponding component of position. Therefore the Euler–Lagrange equations are the three equations: