1 Geodesics in Spacetime

Classical theories of physics involve two different objects: particles and fields. The fields tell the particles how to move, and the particles tell the fields how to sway. For each of these, we need a set of equations.

In the theory of electromagnetism, the swaying of the fields is governed by the Maxwell equations, while the motion of test particles is dictated by the Lorentz force law. Similarly, for gravity we have two different sets of equations. The swaying of the fields is governed by the Einstein equations, which describe the bending and curving of spacetime. We will need to develop some mathematical machinery before we can describe these equations; we will finally see them in Section 4.

Our goal in this section is to develop the analog of the Lorentz force law for gravity. As we will see, this is the question of how test particles move in a fixed, curved spacetime. Along the way, we will start to develop some language to describe curved spacetime. This will sow some intuition which we will then make mathematically precise in later sections.

The Principle of Least Action

Our tool of choice throughout these lectures is the action. The advantage of the action is that it makes various symmetries manifest. And, as we shall see, there are some deep symmetries in the theory of general relativity that must be maintained. This greatly limits the kinds of equations which we can consider and, ultimately, will lead us inexorably to the Einstein equations.

We start here with a lightening review of the principle of least action. (A more detailed discussion can be found in the lectures on Classical Dynamics.) We describe the position of a particle by coordinates $x^i$ where, for now, we take $i=1,2,3$ for a particle moving in three-dimensional space. Importantly, there is no need to identify the coordinates $x^i$ with the $(x,y,z)$ axes of Euclidean space; they could be any coordinate system of your choice.

We want a way to describe how the particle moves between fixed initial and final positions,

$$x^i(t_1)=x_{\mathrm{initial}}^i\mathrm{and}{x}^i(t_2)=x_{\mathrm{final}}^i\text{\hspace{1em}(1.1)}$$

To do this, we consider all possible paths $x^i(t)$ , subject to the boundary conditions above. To each of these paths, we assign a number called the action S. This is defined

as

$$S[x^i(t)]={\int }_{t_1}^{t_2}dtL(x^i(t),{\dot{x}}^i(t))$$

where the function $L(x^i,{\dot{x}}^i)$ is the Lagrangian which specifies the dynamics of the system. The action is a functional; this means that you hand it an entire function worth of information, $x^i(t)$ , and it spits back only a single number.

The principle of least action is the statement that the true path taken by the particle is an extremum of S. Although this is a statement about the path as a whole, it is entirely equivalent to a set of differential equations which govern the dynamics. These are known as the Euler-Lagrange equations.

To derive the Euler-Lagrange equations, we think about how the action changes if we take a given path and vary it slightly,

$$x^i(t)\to x^i(t)+\delta x^i(t)$$

We need to keep the end points of the path fixed, so we demand that $\delta x^i(t_1)=\delta x^i(t_2)=$ 0. The change in the action is then

$$\begin{array}{c}\delta S={\int }_{t_1}^{t_2}dt\delta L={\int }_{t_1}^{t_2}dt\left(\frac{\partial L}{\partial x^i}\delta x^i+\frac{\partial L}{\partial {\dot{x}}^i}\delta {\dot{x}}^i\right)\\ ={\int }_{t_1}^{t_2}dt\left(\frac{\partial L}{\partial x^i}-\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{x}}^i}\right)\right)\delta x^i+{\left[\frac{\partial L}{\partial {\dot{x}}^i}\delta x^i\right]}_{t_1}^{t_2}\end{array}$$

where we have integrated by parts to go to the second line. The final term vanishes because we have fixed the end points of the path. A path $x^i(t)$ is an extremum of the action if and only if $\delta S=0$ for all variations $\delta x^i(t)$ . We see that this is equivalent to the Euler-Lagrange equations

$$\frac{\partial L}{\partial x^i}-\frac{d}{dt}\left(\frac{\partial L}{\partial {\dot{x}}^i}\right)=0\text{\hspace{1em}(1.2)}$$

Our goal in this section is to write down the Lagrangian and action which govern particles moving in curved space and, ultimately, curved spacetime.

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