Question

The Riemann normal coordinates are given by \(x^{i}=a^{i} t .\) For each set of \(a^{i}\), one obtains a different set of geodesics. Thus, we can think of \(a^{i}\) as the parameters that distinguish among the geodesics. (a) By keeping all \(a^{i}\) (and \(t\) ) fixed except the \(j\) th one and using the definition of tangent to a curve, show that \(\mathbf{n}_{j}=t \partial_{j}\), where \(\mathbf{n}_{j}\) is (one of) the \(\mathbf{n}(\) 's) appearing in the equation of geodesic deviation. (b) Substitute (a) plus \(u^{i}=\dot{x}^{i}=a^{i}\) in Eq. (37.29) to show that $$ R_{i j k}^{m}+R_{j i k}^{m}=3\left(\Gamma_{i k, j}^{m}+\Gamma_{j k, i}^{m}\right) $$ Substitute for one of the \(\Gamma\) 's on the RHS using Eq. (36.45). (c) Now use the cyclic property of the lower indices of the curvature tensor to show that $$ \Gamma_{i j, k}^{m}=-\frac{1}{3}\left(R_{i j k}^{m}+R_{j i k}^{m}\right) $$.

Short answer

By using the definition of tangent to a curve and substituting \(n_{j}=t \partial_{j}\) for \(n_{j}\), one obtains \(R_{i j k}^{m}+R_{j i k}^{m}=3\left(\Gamma_{i k, j}^{m}+\Gamma_{j k, i}^{m}\right)\). A substitution using Equation (36.45) further simplifies this equation, and by applying the cyclic property of the curvature tensor, it is shown that \(\Gamma_{i j, k}^{m}=-\frac{1}{3}\left(R_{i j k}^{m}+R_{j i k}^{m}\right)\).

Step-by-step solution

Derive the tangent to a curve

In the case of \(x^{i}=a^{i} t\), by keeping all \(a^{i}\) (and \(t\)) fixed except the \(j\) th one, the derivative of the curve with respect to the variable \(a^{j}\) is taken, which will give the tangent (\(n_{j}\)) to the curve. This can be expressed as \(n_{j}=t \partial_{j}\), where \(\partial_{j}\) represents the partial derivative with respect to \(a^{j}\).

Substitute tangent and velocity vector into Equation of Geodesic Deviation

Next, substitute the derived tangent \(n_{j}=t \partial_{j}\) into the equation for geodesic deviation. Simultaneously, consider the velocity vector given by \(u^{i}=\dot{x}^{i}=a^{i}\). Substituting these values into the Equation (37.29) yields a new equation \(R_{i j k}^{m}+R_{j i k}^{m}=3\left(\Gamma_{i k, j}^{m}+\Gamma_{j k, i}^{m}\right)\). This is the equation that somehow relates the Riemann tensor with the Christoffel symbols.

Use Equation (36.45) to substitute one of the \(\Gamma\) 's on the RHS

Now use the provided Equation (36.45) in the exercise to replace one of the Christoffel symbols on the right hand side of the equation obtained in Step 2.

Apply the cyclic property of the curvature tensor

Finally, use the circular property of the lower indices of the curvature tensor to rearrange the equation from step 3 and show that \(\Gamma_{i j, k}^{m}=-\frac{1}{3}\left(R_{i j k}^{m}+R_{j i k}^{m}\right)\), which concludes the exercise.

Key concepts

Geodesics

Imagine you're hiking and need to find the shortest path from one point to another on a hillside - in mathematics, this is similar to what we call a 'geodesic'. Geodesics are the straightest possible paths in a curved space, akin to the concept of a straight line in flat geometry.

In the context of the exercise given, geodesics are represented by the Riemann normal coordinates, \(x^{i}=a^{i} t\). Here, the variables \(a^{i}\) essentially determine the direction and 'steepness' of the geodesic, and by varying these parameters, you explore different geodesic paths. Just as choosing different trails can lead to different viewpoints on your hike, changing the values of \(a^{i}\) leads to different geodesics in the mathematical 'landscape'.

Tangent to a Curve

Think of bike trails in a park: The direction you're heading at any moment is akin to a tangent at that point of the path. Similarly, in mathematics, a 'tangent to a curve' at a point gives you the direction in which the curve is heading. It's a vector that points 'straight ahead' if you were traveling along the curve.

In our exercise, the tangent vector at a point on a geodesic is denoted by \(\mathbf{n}_{j}=t \partial_{j}\). It's derived by tweaking one of the \(a^{i}\) values, specifically the \(j\)th parameter, while keeping the rest fixed, similar to adjusting your bike's handlebars slightly to steer along a different path while riding.

Equation of Geodesic Deviation

The 'equation of geodesic deviation' might sound complex, but it simply measures how geodesics, these 'natural paths', diverge from or converge toward each other over space – much like noticing two parallel roads gradually coming closer or moving apart as you travel.

In the steps outlined, we begin by representing the tangent vector to the geodesic, and subsequently, we combine it with the velocity vector of the curve to delve into this equation. By doing so, we establish a relationship between the changing separation of geodesics and the underlying geometry of the space. This can tell us much about the 'hills and valleys' in the mathematical landscape we are traversing.

Riemann Curvature Tensor

Now, let's add a layer of complexity to our mental image of a hilly terrain: The Riemann curvature tensor is like a tool that measures the 'bumpiness' or curvature of the mathematical surface you are examining. Think of it as checking the gradient and twists of a hill you’re on to determine just how steep or bumpy the path ahead might be.

During the problem-solving steps, we see that the Riemann curvature tensor \(R_{ijk}^{m}\) is manipulated in conjunction with the Christoffel symbols to express the curvature of our space in terms of these symbols. This is crucial for understanding how geodesics behave because it relates the path's curvature directly to the terrain's features.

Christoffel Symbols

Imagine Christoffel symbols as signposts that tell us how to 'correct' our path as we move along our trail, preventing us from straying off course. They are an essential part of the mathematical apparatus that dictates how curves (or 'paths') bend within a given space.

Through the exercise, we unravel a relationship between these Christoffel symbols and the Riemann curvature tensor, which ultimately allows us to describe the curvature of space with a set of simpler instructions - how to turn the handlebars on our bike, so to speak, to follow the geodesic trail without getting lost. By manipulating these symbols as per the exercise, we gain insights into navigating the complex landscape of curved spaces.