The world function
Despite being one of the most important function in general relativity, little care is paid to the world function, most of it being proven in Synge[1] and in Visser. I thought it might be of interest to give it a nice going over.
Definition
While quite not exactly that, the world function is fundamentally a distance function on a spacetime. Synge and Visser both give it a slightly different definition, so let's include them here. Visser defines what he calls the geodetic interval, where, for $p, q \in M$ and some geodesic $\gamma$ from $p$ to $q$, we define the "norm"
\begin{equation} \| \dot{\gamma} \| = \sqrt{\textrm{sgn}(g(\dot{\gamma}, \dot{\gamma})) g(\dot{\gamma}, \dot{\gamma})} \end{equation}with the sign being here to avoid any negative square roots, being simply positive for spacelike vectors and negative for timelike vectors. The geodetic interval is then defined as
\begin{equation} \sigma_{\gamma}(p,q) = \int_{\lambda_1}^{\lambda_2} \| \dot{\gamma} \| d\lambda \end{equation}In other words, if $\gamma$ is timelike, it measures the proper time, if it is spacelike, it measures the distance, and if it is null, it is simply zero. This definition could technically apply to any curve, but overall, only the case for geodesics will be considered here. In particular, if there exists only one geodesic between $p$ and $q$, we have a unique function $\sigma(p,q)$. In the case of a totally normal neighbourhood $U$, we have a well-defined function
\begin{eqnarray} \sigma : U \times U &\to& \mathbb{R}\\ (p,q)&\mapsto&\sigma(p,q) \end{eqnarray}which does not depend on any geodesic as an input.
On the other hand, Synge defined the world function as
\begin{equation} \Omega_\gamma(p,q) = \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} g(\dot{\gamma}, \dot{\gamma}) d\lambda \end{equation}In both cases, as we are considering geodesics, for which $\| \dot{\gamma} \|$ is constant along the curve, we can simply take them out of the equation. If we call the tangent vector $\dot{\gamma}(\tau_1) = u$,
\begin{eqnarray} \sigma_{\gamma}(p,q) &=& (\lambda_2 - \lambda_1) \| u \|\\ \Omega_\gamma(p,q) &=& \frac{1}{2} (\lambda_2 - \lambda_1)^2 \| u \|^2 \end{eqnarray}meaning that we have for geodesics $\sigma^2_\gamma(p, q) = \Omega_\gamma(p,q) / 2$.
How do these functions act under reparametrization? If we consider a curve $\gamma : I \to M$ and a reparametrization $h : I \to I'$, with $I, I'$ intervals of $\mathbb{R}$, we have the reparametrized curve $\gamma' = \gamma \circ h$, giving it a new parameter $\lambda'$, such that $\lambda' = h(\lambda)$. Its tangent
\begin{eqnarray} \frac{d}{d\lambda} \gamma(h(\lambda')) = \dot{h}(\lambda') \frac{d}{d\lambda'} \gamma(\lambda') \end{eqnarray}The norm transforms as
\begin{eqnarray} \| \dot{\gamma}(\lambda) \| &=& \| \dot{\gamma}(h(\lambda')) \|\\ &=& |h(\lambda)| \| \dot{\gamma}(\lambda') \| \end{eqnarray} [check the use of lambda and lambda prime]so that $\dot{\gamma}(\lambda) = \dot{h}(\lambda') \dot{\gamma}(\lambda')$. We therefore get, by substitution,
\begin{eqnarray} \sigma_{\gamma}(p,q) &=& \int_I \| \dot{\gamma}(\lambda) \| d\lambda\\ &=& \int_{I'} \| \dot{\gamma}(h(\lambda)) \| d\lambda \end{eqnarray}For proper time/length parametrization, where the tangent vector remains at a constant $\|\dot{\gamma}\| = 1$, we a constant integral :
\begin{eqnarray} \sigma_{\gamma}(p,q) &=& \int_{\lambda_1}^{\lambda_2} d\lambda\\ &=& (\lambda_2 - \lambda_1)\\ \Omega_\gamma(p,q) &=& \frac{1}{2} (\lambda_2 - \lambda_1) \int_{\lambda_1}^{\lambda_2} g(\dot{\gamma}, \dot{\gamma}) d\lambda\\ &=& \frac{(\lambda_1 - \lambda_2)^2}{2} \end{eqnarray}which does confirm the notion that they indeed measure proper time/length.
If we consider the vector $u$ at $p$ for which the geodesic of initial conditions $(p, u)$ arrives at $q$ for $\lambda = 1$, we can show that Synge's world function is equal to :
\begin{eqnarray} \Omega(p,q) &=& \frac{1}{2} g(\exp_p^{-1}(q), \exp_p^{-1}(q))\\ &=& \frac{1}{2} (1 - 0) g(u, u)\\ &=& \frac{1}{2} (1 - 0) g(u, u)\\ &=& \frac{1}{2} \int_{0}^1 g(\dot{\gamma}(\lambda), \dot{\gamma}(\lambda)) d\lambda \end{eqnarray}and, up to reparametrization, we have that $\Omega$ can be expressed by that map. Therefore, we also have that for geodesics in a totally geodesic neighbourhood, $\sigma(p,q) = \sqrt{g(\exp_p^{-1}(q), \exp_p^{-1}(q))} / 2$.
The exponential map
As we can define the world function from the exponential map, it will be useful to write down a few important properties of it.
Given a point $p \in M$, there is a neighbourhood $U_p$ such that, for every $q \in U_p$, there is a unique geodesic connecting $p$ to $q$, noted $\vec{pq}$. This geodesic will still have some degrees of freedom, up to some translations and rescalings, so we will consider specifically the geodesic such that
- $\vec{pq}(0) = p$
- $\vec{pq}(1) = q$
- $\frac{d}{d\lambda}[\vec{pq}(\lambda)]|_{\lambda = 0} = v$
where $v$ is the tangent vector at the origin. The exponential map is our map from the tangent space $T_pM$ to our neighbourhood $U_p$, such that
\begin{equation} \exp_p(v) = q \end{equation}By using the properties of the rescaling of geodesics, we also that that
\begin{equation} \exp_p(\lambda v) = \vec{pq}(\lambda) \end{equation}Since given our conditions on the geodesics involved, the tangent vector is unique, we can also define the inverse function,
\begin{equation} \exp_p^{-1}(q) = v \end{equation}For simplicity, let's consider a geodesically convex neighbourhood $U$, such that for every point $p, q \in U$, there exists a unique geodesic joining the two, If
\begin{equation} \vec{pq}(\lambda) = \exp_p (\lambda \exp_p^{-1}(q)) \end{equation}The derivatives we can get out of the exponential functions are the following :
- As $\exp_p^{-1}(q)$ is a vector, any of the usual vector derivative can be applied to it.
Bitensors
Our world functions are bitensors, ie for two vector bundles $\mathcal{V}, \mathcal{W}$ of typical fiber $V, W$, they are sections of the vector bundle $\pi : \mathcal{V} \boxtimes \mathcal{W} \to M \times M$, with typical fiber $V \otimes W$. For more details on the topic of the external tensor product, see this article.
In our case, $\sigma$ and $\Omega$ are bitensors of the trivial line bundle, forming the bundle $\pi : \mathbb{R} \boxtimes \mathbb{R} \to M \times M$. As the tensor product of $\mathbb{R}$ with itself is $\mathbb{R}$ again, this is simply the trivial line bundle again : this is the space of scalar functions over $M \times M$. We could simply have defined everything with $\mathbb{R}$ from the start, but the exterior tensor product will come up when we consider the derivatives of bitensors.
Bitensor bundles are equipped with two projection operators. $\mathrm{pr}_1$ projects on the first half of the base space, $\mathrm{pr}_2$ on the second, so that $\mathrm{pr}_1(p,q) = p$. We therefore have $\mathrm{pr}_1 \circ \pi (s(p,q)) = p$ and similarly for $q$.
The Synge bracket is a map from bitensors to the tensor product bundle, via
\begin{eqnarray} [] : \Gamma(V \boxtimes W) &\to& \Gamma(V \otimes W)\\ s(p,q) &\mapsto& [s] = \lim_{p \to q} s(p,q) \end{eqnarray}if our limit process is unique (generally taken by taking the limit on a unique geodesic between $p$ and $q$.)
More interesting are the derivatives of bitensors. If we simply consider our manifold $M \times M$ as having the trivial product metric $g \otimes g$, take the tensor fields $X_1, Y_1$ and $X_2, Y_2$ over $M$, such that we construct their bitensor fields via $\mathrm{pr}_1$
...
Due to the symmetry of the world function, we only really need to consider the derivative with respect to one variable to generalize it. Let's consider the derivative of the inverse exponential map for constant $p$, which is simply a vector field $V(q) = \exp_p^{-1}(q)$, such that $\exp_p(V(q)) = q$. What is exactly the vector field $V$?
\begin{eqnarray} V[f] &=& \exp_p^{-1}(q) [f] \end{eqnarray} \begin{eqnarray} \nabla_X^i \exp_p^{-1}(q) &=& \end{eqnarray}...
There are two connections we need to define for our bitensor, simply enough the connection associated with each parameter of the function. In a coordinate-free definition, this is the pullback of the connection along a projector,
\begin{eqnarray} \mathrm{pr}_{i}^* \nabla_X \left[ \mathrm{pr}_{i}^* \sigma(p,q)\right] &=& \mathrm{pr}_{i}^* \nabla_{\left(d\mathrm{pr}_{i}\right) (X)} \sigma(p,q)\\ &=& \mathrm{pr}_{i}^* \nabla_{\left(d\mathrm{pr}_{i}\right) (X)} \sigma(p,q)\\ \end{eqnarray}...
The world function is just the composition of the metric and inverse exponential map, so via the chain rule,
\begin{eqnarray} \nabla_X^i g(\exp_p^{-1}(q), \exp_p^{-1}(q)) &=& (\nabla_X^i g)(\exp_p^{-1}(q), \exp_p^{-1}(q)) + g(\nabla_X^i \exp_p^{-1}(q), \nabla_X^i \exp_p^{-1}(q))\\ &=& g(\nabla_X^i \exp_p^{-1}(q), \nabla_X^i \exp_p^{-1}(q)) \end{eqnarray}Bibliography
Last updated : 2022-07-13 10:50:28