Geodesics as Paths of Zero Acceleration

1. Four-Acceleration in Curved Spacetime

In special relativity, in an inertial frame, the four-acceleration is simply the derivative of the four-velocity with respect to proper time, $A^{\mu }=d{U}^{\mu }/d\tau$. However, as we've seen, ordinary derivatives of vector components are not tensor components in a general coordinate system. To get a valid tensor expression for acceleration that holds in any coordinate system on a curved manifold, we must replace the ordinary derivative with the covariant derivative.

We define the four-acceleration vector $A^{\mu }$ as the covariant derivative of the four-velocity vector field $U^{\mu }$ along its own worldline:

This definition is fundamental. An observer carrying an accelerometer measures $A^{\mu }$. If the accelerometer reads zero, the observer is in free-fall, following a path dictated purely by the geometry of spacetime.

Let's see this definition expanded out in coordinates. The following video provides a clear and rigorous breakdown.

As the video demonstrates, expanding the covariant definition gives us the components of the acceleration vector:

The first term is the familiar coordinate acceleration, while the second term, involving the connection, accounts for the effects of spacetime curvature and the choice of a non-inertial coordinate system.

2. Geodesics as Paths of Zero Acceleration

With the four-acceleration defined, we can now state the physical principle that determines the motion of free particles. The Equivalence Principle tells us that an observer in a small, freely-falling reference frame will not feel any gravitational forces. Within this local frame, the laws of physics are those of Special Relativity.

A free test particle (one with no non-gravitational forces acting on it) will, according to an observer in this freely-falling frame, move in a straight line at a constant velocity. This means its four-acceleration is zero. Since $A^{\mu }=0$ is a tensor equation, if it holds in one coordinate system (the local inertial frame), it must hold in all coordinate systems.

Therefore, the equation of motion for a freely-falling particle in any gravitational field is simply:

This leads directly to the geodesic equation:

This equation is the mathematical embodiment of a "straightest possible line" in curved spacetime—a path of zero covariant acceleration.

The next video provides a superb physical motivation for this, starting from the idea of parallel transporting the tangent vector.

For a formal, written summary of this derivation, you can refer to the following resources:

• The notes "GRG I, part 3" by Norman Gray (ID: [LINK](https://www.astro.gla.ac.uk/users/norman/lectures/GR/part3-usletter.pdf)), Section 4, "Geodesics" (up to Eq. 3.37).
• The MIT notes "Parallel transport and geodesics" (ID: [LINK](https://dspace.mit.edu/bitstream/handle/1721.1/36859/8-962Spring2002/NR/rdonlyres/Physics/8-962Spring2002/940ABFC4-1A26-483C-992A-DDCD48D1F272/0/gr2_2.pdf)), Section 3.3, "Geodesics."

3. The Affine Parameter

In our derivation, we used proper time $\tau$ as the parameter along the curve. This is a natural choice for massive particles (timelike geodesics). But what about light rays (null geodesics), for which $d{\tau }^2=0$?

The geodesic equation holds its simple form only when parameterized by a special class of parameters called affine parameters.

• An affine parameter $\lambda$ is one for which the geodesic equation takes the form $\frac{d^2{x}^{\mu }}{d{\lambda }^2}+\cdots =0$.
• If $\lambda$ is an affine parameter, then any linear transformation ${\lambda }^{\prime }=a\lambda +b$ (where $a,b$ are constants) also produces an affine parameter. This corresponds to changing the zero point and the scale of your "clock."

For timelike geodesics, proper time $\tau$ is an affine parameter. For null geodesics, we must use a different affine parameter, which can be thought of as tracking the progress of the light ray.

The concept of an affine parameter is crucial for actually solving the geodesic equation. The following video segment clarifies its meaning.

4. Derivation from a Variational Principle

We now turn to a completely different, yet equally powerful, derivation of the geodesic equation. This approach will resonate with your background in classical mechanics, specifically the use of Lagrangians and the principle of stationary action.

In relativity, the path taken by a free particle between two spacetime events (say, P and Q) is the one that extremizes the proper time experienced by the particle. For timelike paths, this turns out to be the path of maximal proper time—a concept known as the "principle of maximal aging."

We can formalize this by defining an "action," $S$, equal to the total proper time along a path:
$S={\int }_P^{Q}d\tau ={\int }_{{\lambda }_P}^{{\lambda }_Q}\sqrt{-g_{\mu \nu }\frac{d{x}^{\mu }}{d\lambda }\frac{d{x}^{\sigma }}{d\lambda }}d\lambda$
The quantity inside the integral acts as our Lagrangian, $L$. To find the path that extremizes this action ($\delta S=0$), we use the Euler-Lagrange equations. This derivation is a beautiful piece of calculus of variations.

The following video walks through this derivation step-by-step. Given your mathematical background, you should find the rigor of this derivation very satisfying.

The fact that two completely different starting points—the geometric idea of a "straightest line" and the physical principle of "extremal aging"—lead to the exact same equation of motion is a profound statement about the consistency and elegance of General Relativity.

An Advanced Aside: Orthogonality of Velocity and Acceleration

Here's an interesting property that demonstrates the consistency of the formalism. In curved spacetime, the four-velocity $U^{\mu }$ and four-acceleration $A^{\mu }$ are always orthogonal, i.e., their inner product is zero: $g_{\mu \nu }{U}^{\mu }{A}^{\nu }=0$.

This is a direct consequence of the fact that the four-velocity is normalized ($g_{\mu \nu }{U}^{\mu }{U}^{\nu }=-1$) and that the connection is metric-compatible (${\nabla }_{\sigma }{g}_{\mu \nu }=0$). The proof is a nice exercise in index manipulation. If you're curious to see it worked out, the 'Physics Unsimplified' video you just watched provides the full proof.

Conclusion

In this lesson, we have thoroughly investigated the geodesic equation, the law of motion in General Relativity.

Key Takeaways:

• The four-acceleration in curved spacetime is defined covariantly as $A^{\mu }=U^{\nu }{\nabla }_{\nu }{U}^{\mu }$.
• The geodesic equation is the statement of zero four-acceleration, $A^{\mu }=0$, which describes the worldline of a freely-falling test particle.
• This equation only takes its simple form when the path is parameterized by an affine parameter, such as proper time $\tau$ for massive particles.
• The geodesic equation can also be derived from a variational principle, where a free particle follows the path of extremal proper time between two events.
• The convergence of these two distinct derivations—one geometric, one action-based—highlights the deep internal consistency of the theory.

We have now established how particles move in a given spacetime geometry. But what determines the geometry itself? And how can we quantify "curvature"? In our next lesson, we will finally answer this by defining the Riemann curvature tensor, which arises from considering the path-dependence of parallel transport. This will be the key to understanding how mass and energy curve spacetime.