Geodesics equations via variational principle

Question

I would like to recover the (timelike) geodesics equations via the variational principle of the following action:

$$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$

Using an arbitrary auxiliary parameter $\lambda$, then one is able to rewrite the action and obtains: $$ d\tau = \sqrt{-g_{\mu\nu}\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$ $$ -m \int d\tau = -m \int (d\tau / d\lambda) d\lambda = -m\int\sqrt{-g_{\mu\nu}\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$

Now we can vary the paths $x^{\mu}(\tau) \rightarrow x^{\mu}(\tau) + \delta x^{\mu}(\tau)$.

I need the following equation to be true, so i get the geodesics equations, but I don't know how one can say the following: $$ \int d\tau \,\delta g_{\mu\nu}\frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} = \int d\tau\, g_{\mu\nu, \rho} \frac{dx^{\mu}}{d\tau} \frac{dx^{\nu}}{d\tau} \delta x^{\rho} $$ I know there is a simpler action, which leads to the same equation of motion, but there one has to do the same manipulation, which I sadly don't understand.

Thanks for helping me!

Answer 1

Your action is: $$ S[x] = -m\int_{\lambda_0}^{\lambda_1}\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} d\lambda $$ and you have to impose $\delta S=0$ with the constraints $\delta x(\lambda_0) =\delta x(\lambda_1) =0$, that means that the considered curves in the domain of $S$ have fixed endpoints.

To compute $\delta S$ you have to replace $x$ for $x+ \epsilon \delta x$ (so $\frac{dx}{d\lambda}$ must be replaced for $\frac{dx}{d\lambda} + \epsilon\frac{d \delta x}{d\lambda}$ ) and finally to compute the derivative respect to $\epsilon$ for $\epsilon=0$.

$$\delta S[x] = \frac{d}{d\epsilon}|_{\epsilon=0} S[x+ \epsilon \delta x]\:.$$

The computation leads to (assuming that $g$ and the curves are $C^1$, these curves defined on the compact $[\lambda_0,\lambda_2]$ one can safely swap the symbol of integral with that of $\epsilon$ derivative, essentially by a known theorem by Lebesgue) $$ \delta S[x] = -\frac{m}{2}\int_{\lambda_0}^{\lambda_1}\frac{- \frac{\partial g_{\alpha \beta}}{\partial x^\delta} \delta x^\delta \tfrac{dx^{\alpha}}{d\lambda}\tfrac{dx^{\beta}}{d\lambda} - 2g_{\alpha\beta} \frac{d \delta x^\alpha}{d\lambda}\frac{d x^\beta}{d\lambda}}{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} d\lambda\:. $$ Notice that $x$ appears in $g_{\mu\nu}=g_{\mu\nu}(x)$, too, and it gives rise to the contribution $\frac{\partial g_{\mu\nu}(x)}{\partial x^\sigma}\delta x^\sigma$ you mentioned in your question.

The denominator in the integral does not vanish as we are varying our curve in the class of timelike curves joining the two fixed endpoints.

Integrating by parts, one gets: $$ \frac{2}{m}\delta S[x] = \int_{\lambda_0}^{\lambda_1}\delta x^\delta\frac{ \frac{\partial g_{\alpha \beta}}{\partial x^\delta} \tfrac{dx^{\alpha}}{d\lambda}\tfrac{dx^{\beta}}{d\lambda} }{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} d\lambda -\int_{\lambda_0}^{\lambda_1} \delta x^\alpha\frac{d}{d \lambda}\frac{2g_{\alpha\beta} \frac{d x^\beta}{d\lambda} }{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} d\lambda + [...]\delta x^\alpha(\lambda_1)-[...]\delta x^\alpha(\lambda_0)\:. $$ The last two terms can be dropped as they vanish by hypothesis. Changing the name of some summed indices we end up with:

$$ \frac{2}{m}\delta S[x] = \int_{\lambda_0}^{\lambda_1}\delta x^\delta\left[\frac{ \frac{\partial g_{\alpha \beta}}{\partial x^\delta} \tfrac{dx^{\alpha}}{d\lambda}\tfrac{dx^{\beta}}{d\lambda} }{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} -\frac{d}{d \lambda}\frac{2g_{\delta\beta} \frac{d x^\beta}{d\lambda} }{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} \right]d\lambda\:. $$ Since the LHS vanishes for every choice of the variation $\delta x^\delta(\lambda)$, we conclude that $\delta S[x]=0$ on a curve $x=x(\lambda)$ is equivalent to the requirement that the said curve verifies: $$\frac{ \frac{\partial g_{\alpha \beta}}{\partial x^\delta} \tfrac{dx^{\alpha}}{d\lambda}\tfrac{dx^{\beta}}{d\lambda} }{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} -\frac{d}{d \lambda}\frac{2g_{\delta\beta} \frac{d x^\beta}{d\lambda} }{\sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}}} =0\:.\quad (1)$$ We can change parameter and use the proper time $d\tau$ so that: $$d\lambda \sqrt{-g_{\mu\nu}(x(\lambda))\, \tfrac{dx^{\mu}}{d\lambda}\tfrac{dx^{\mu}}{d\lambda}} = d\tau$$ and (1) becomes: $$\frac{1}{2}\frac{\partial g_{\alpha \beta}}{\partial x^\delta} \frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau} -\frac{d}{d \tau}g_{\delta\beta} \frac{d x^\beta}{d\tau} =0\:.\quad (2)\:.$$ Expanding the last derivative changing the name of $\beta$ to $\mu$ in the last term: $$\frac{1}{2}\frac{\partial g_{\alpha \beta}}{\partial x^\delta} \frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau} - \frac{\partial g_{\delta\beta}} {\partial x^\sigma} \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} -g_{\delta\mu} \frac{d^2 x^\mu}{d\tau^2} =0\:.\quad \:.$$ In other words: $$\frac{d^2 x^\mu}{d\tau^2} - g^{\delta\mu} \frac{1}{2}\frac{\partial g_{\alpha \beta}}{\partial x^\delta} \frac{dx^{\alpha}}{d\tau}\frac{dx^{\beta}}{d\tau} + g^{\delta\mu} \frac{\partial g_{\delta\beta}} {\partial x^\sigma} \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} =0\:.$$ Renaming some indices: $$\frac{d^2 x^\mu}{d\tau^2} + \frac{1}{2}g^{\mu\delta}\left(2\frac{\partial g_{\delta \beta}}{\partial x^\sigma} - \frac{\partial g_{\sigma\beta}} {\partial x^\delta}\right) \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} =0\:.$$ Eventually, exploiting $g_{\delta \beta}= g_{\beta\delta}$: $$\frac{d^2 x^\mu}{d\tau^2} + \frac{1}{2}g^{\mu\delta}\left(\frac{\partial g_{\delta \beta}}{\partial x^\sigma} + \frac{\partial g_{\beta \delta}}{\partial x^\sigma}- \frac{\partial g_{\sigma\beta}} {\partial x^\delta}\right) \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} =0\:.$$ Now notice that: $$\frac{\partial g_{\delta \beta}}{\partial x^\sigma} \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} = \frac{\partial g_{\delta \sigma}}{\partial x^\beta} \frac{d x^\beta}{d\tau}\frac{d x^\sigma}{d\tau}=\frac{\partial g_{\delta \sigma}}{\partial x^\beta} \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau}$$ so the found identity can be re-written as: $$\frac{d^2 x^\mu}{d\tau^2} + \frac{1}{2}g^{\mu\delta}\left(\frac{\partial g_{\delta \sigma}}{\partial x^\beta} + \frac{\partial g_{\beta \delta}}{\partial x^\sigma}- \frac{\partial g_{\sigma\beta}} {\partial x^\delta}\right) \frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} =0\:.$$ We have found: $$\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\sigma_\beta}\frac{d x^\sigma}{d\tau}\frac{d x^\beta}{d\tau} =0\:,\tag{GE}$$ as wished.

COMMENTS.

The discussion above proves several facts.

(1) If a curve $x= x(\lambda)$ with $\lambda \in [\lambda_0,\lambda_1]$ exists which is a stationary point of the said functional in the set of timelike curves, then it can be re-parmetrized in order to satisfy the geodesic equation (GE) referred to the affine parameter given by the proper time.

(2) There are infinitely many timelike curves which sationarize the functional $S[x]$ for fixed $[\lambda_0,\lambda_1]$ and fixed $x(\lambda_0)=:x_0$ and $x(\lambda_1)=:x_1$ provided $x_0$ and $x_1$ are timelike related and sufficiently close to each other.

It is sufficient to consider a normal neighborhood centered on $x_0$ and the unique geodesic segment parametrized with the proper time and joining $x_0$ and $x_1$: $x=x(\tau)$, $\tau\in [0,T= length(x)]$. This geodesic segment exists by the properties of a normal neighborhood. A stationary path is then obtained as $x(\tau(\lambda))$ where $$[\lambda_0, \lambda_1] \ni \lambda \mapsto\tau (\lambda)\in [0,T]$$ is an arbitrary smooth map with $\tau'(\lambda)>0$ everywhere. Every such function defines a different stationary path for the same functional and domain of it.

(3) The Cauchy problem based on the QDE arising form the Euler-Lagrange equations arising from $S[x]$ and standard initial conditions is not well posed for the lack of uniqueness (if a solution exists).

In fact, as it easily arises from the discussion above different choices of $\tau=\tau(\lambda)$ (with fixed initial derivative) determine different solutions of the same Cauchy problem instead of a unique solution. (The mathematical obstruction is here the failure of the local Lipschitz condition of the EL differental equations when written in normal form.)

(4) The variational principle can be exploited to determine the equations of timelike and spacelike geodesics, but it cannot be used for light-like geodesics. That is because one cannot pass to Eq.(GE) from the EL equation because the denominator in the parameter change is zero.

Answer 2

There are two definitions of geodesics here.

1. locally distance minimizing curves You minimize the action, as you did, \begin{equation} S(\gamma) = \int_a^b \sqrt{ g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} } dt = \int_a^b L dt \end{equation} The Euler Lagrangian equation associated with this action is \begin{equation} \frac{d}{dt}(\frac{\partial L}{\partial \dot{x}^{\mu}}) - \frac{\partial L}{\partial x^\mu}= 0 \end{equation}

2. curves on which tangent vector is parallel transported. You minimize the action, \begin{equation} E(\gamma) = \int_a^b \frac{1}{2}g_{\mu\nu} \dot{x}^{\mu} \dot{x}^{\nu} dt = \int_a^b \frac{1}{2}L^2 dt \end{equation} through Euler Lagrangian equation \begin{equation} \frac{d}{dt}(\frac{\partial L}{\partial \dot{x}^{\mu}}) - \frac{\partial L}{\partial x^\mu} = -\frac{1}{L}\frac{\partial L}{\partial \dot{x}^{\mu}} \frac{d}{dt}L \end{equation} and get the solution(after some algebra) \begin{equation} \ddot{x}^{\lambda} + \Gamma^{\lambda}_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = 0 \end{equation} This curve parallel transports the tangent vector.

We notice \begin{eqnarray} \frac{d}{dt}(\frac{\partial L}{\partial \dot{x}^{\mu}}) - \frac{\partial L}{\partial x^\mu} = 0\\ \frac{d}{dt}L = 0 \end{eqnarray} solves Euler-Lagrangian equations in both case. $\frac{d}{dt}L$ just fixes the parameterization. For a Riemannian manifold, the parameter differs the length of the curve by an affine transformation \begin{equation} S(\gamma_{sol}(t)) = \int_a^t L(\gamma_{sol}(\tau) d\tau = (t-a) L(\gamma_{sol}(t=0) ) \end{equation} that's why some textbook directly the length(or proper time) as the parameter in the first place.

The second definition is sometimes called affine geodesics. While in GR, in most cases we are talking about affine geodesics, which is a locally minimizing curve plus an affine parameterization.