At this point, we have a tool to represent the curvature of a given curved space using the Riemann curvature tensor. Let’s deviate from this for a while. Another natural question to ask: what is the simplest and shortest path for two points in a flat space? Obviously, by simple human intuition, it is a straight line.
To put in a more precise manner, straight lines that are initially parallel to each other would remain parallel when extended. Is this still true on curved space? As you would have imagined, straight lines would not be straight due to the curvature. Initially parallel straight lines would cross path with each other in curved space.
In fact, there is no parallel straight lines in general for curved space as straight lines would “bend” due to the curvature, making them no longer parallel with each other. The definition of a straight-line using parallelism is no longer a useful concept for curved space. What should we do? What is the simplest and straightest pathway for curved space? We can use the concept of parallel transport. In Euclidean space, tangent vectors for straight lines remain unchanged when extended.
We could use this idea and extend it to curved space. Remember, parallel transport is a process where it transports a vector to be as straight as possible along a curve. By combining these two concepts together, we could use parallel transport to transport the tangent vector in its own direction. These are the “straight lines” of curved space, known as geodesics.
Let’s be more precise and mathematical about it. Suppose there is a curve that is parameterized by lambda as shown with its corresponding tangent vector.
From previous section, we have derived the expression of parallel transporting a vector along the basis vector direction as shown below.
Is there a way to have an expression in any curve? Yes, there is. The above expression can be multiplied by the tangent vector to obtain the resulting vector after being parallel-transported through a curve parameterized by lambda. This works because the above expression is defined in terms of per δx. The tangent vector is δx/δλ. Thus, multiplication of tangent vector effectively results in 1/δλ as δx’s cancels each other.
Right now, we have the generalized equation for parallel transporting a vector. Let’s remind ourselves again: in flat or Euclidean space, parallel straight lines remain parallel after extension. Effectively, tangent vectors of parallel lines do not change in direction or magnitude.
For a curved space, this is no longer true. Initially parallel line would not stay parallel after extension due to the curvature of space. The idea of normal “straight line” that we are familiar with is not applicable in curved space.
We must redefine what is “straight line” in the context of curved space. One obvious way is to keep the tangent vector as straight as possible using parallel transport. Thus, parallel transporting the tangent vector in its own direction is exactly what we need to define the “straight lines” in curved space.
The above differential equation is called the geodesic equation. The solutions of this differential equation are the “straight lines” of that particular curved space. The above expression is in compacted version. It is more convenient to solve the differential equation in expanded form as shown.
This is the expanded version of the geodesic equation. As you know, initially parallel lines do not remain parallel when extended. A natural question to ponder: how fast do two initially parallel straight lines converge or diverge from each other after extension? Imagine there are two nearby “straight lines” or geodesics as shown below. Both geodesics are parameterized by λ.
From the figure above, we could connect a vector between points in the first and second geodesic that corresponds to the same λ. This vector is called the separation vector which is the difference between two position vectors of two geodesics at the same value. The value of this separation vector essentially tracks the magnitude or direction of two geodesics from each other.
Before we move on, there are some assumptions that we must establish. These two geodesics are infinitesimally close to each other. By choosing point A to be the center of our analysis, the space surrounding point A is nearly flat. Tangent vectors in point A and A’ are assumed to be parallel for further analysis.
Since the analysis is centered around point A, space at point A is flat. Thus, the metric tensor at point A is the simple Minkowski metric or flat space metric with vanishing Christoffel symbols. We have vanishing Christoffel symbols as we assumed to work in a coordinate system that has constant basis vectors in flat space.
Point A’ is nearby point A. Thus, the space at point A’ is nearly flat. Again, we can assume that the metric tensor for just the Minkowski metric as well. However, Christoffel symbol is no longer vanishing as it is just nearly flat at point A’ i.e. there is some curvature at point A’. We could approximate the Christoffel symbols at point A’ by using the first order Taylor’s approximation.
Now, let us apply the geodesic equation at point A and A’. At point A, the geodesic equation is reduced to a very simple differential equation because it has a vanishing Christoffel symbol.
However, the Christoffel symbol is its first order Taylor approximation at point A’. The differential equation at point A’ is the following expression.
A useful quantity that warrants our attention is the second derivative of the separation vector with respect to λ. This quantity is essentially the “acceleration” of the separation vector. By definition, it is the difference between the second derivative of the position vectors of each geodesic equation.
From the geodesic equation previously, it can be written by the following expression.
However, the above derivative is just the ordinary derivative. A normal derivative does not work in curved space because basis vectors are not constant. A covariant derivative is the most appropriate derivative in this case.
By taking the first covariant derivative of the separation vector, we result in two terms, which are the normal derivative and a term that tracks the changes in basis vector.
We have to perform another covariant derivative to get its acceleration.
We must perform covariant derivative on the first and second term. Let’s focus on the covariant derivative of the first term first. The covariant derivative of the first term is the resulting expression.
To tidy up the expression, we can redefine some terms as shown below.
Thus, the covariant derivative of the first term is the following expression with the convention establishes earlier.
The second term consists of three main variables. Its covariant derivative is just the chain rule for three variables as shown below.
Since it is nearly flat at point A, the first derivative of separation vector is zero as it is almost constant near point A.
The second covariant derivative of separation vector is the following expression after adding the covariant derivative results of the first and second terms.
Any first derivative of the separation vector vanishes due to the initially parallel assumption. The covariant derivative of tangent vector is effectively zero because we are dealing with geodesics. The expression can be simplified as shown below.
The covariant derivative of the Christoffel symbol can be expanded. Any term that has squares of Christoffel symbols is assumed to be small and ignored. This can be done as the value of Christoffel symbol is already infinitesimal small near point A. Thus, squares of Christoffel symbols are effectively zero.
The ordinary second derivative of the separation vector has been previously derived as the following expression.
By substituting this expression, the final second covariant derivative of the separation vector as follows.
Some of the dummy indices can be renamed so that common variables can be factor out to make the expression simpler.
We briefly return to the Riemann curvature tensor derived earlier , which is given by the following expression:
Again, squares of Christoffel symbols can be ignored as it is effectively zero.
The indices in the Riemann curvature tensor are renamed to suit the variables used so far.
We can substitute this expression to obtain our final expression. This is the reason why we go back to Riemann curvature tensor.
This is known as the equation of geodesic deviation. It captures the dynamics of multiple straight lines in a curved space. The rate of convergence or divergence of straight lines largely depends on the curvature of space itself (from Riemann curvature tensor).
Almost Tensorial 