Redshift and contact forms (Q1685874)

In this note the authors prove that over a globally hyperbolic Lorentzian manifold the photon redshift factor in general relativity arises as the ratio of two appropriate Penrose-Low contact structures on the corresponding space of null curves. More precisely, let \((X,\langle\cdot,\cdot\rangle)\) be a connected time-oriented Lorentzian manifold with \(\dim X\geqq 2\) and consider two infinitesimal observers in \(X\), i.e., two future-pointing unit timelike vectors \(n_E\) and \(n_R\) at the events \(E, R\in X\) connected by a null geodesic \(\gamma\). Then the \textit{photon redshift} \(z=z(n_E,n_R,\gamma)\in{\mathbb R}\) from \(n_E\) (``emitter'') to \(n_R\) (``receiver'') along \(\gamma\) is defined by \[ 1+z:=\frac{\langle n_E,\dot{\gamma}(E)\rangle}{\langle n_R,\dot{\gamma}(R)\rangle}. \] Assume that \((X,\langle\cdot,\cdot\rangle)\) is moreover globally hyperbolic and pick one Cauchy surface \(M\subset X\). Consider the set \({\mathfrak N}_X\) of future-directed inextendible null geodesics along \(X\) up to orientation-preserving reparametrisations. For every \(M\) this space can be endowed with a contact structure as follows. There is an obvious embedding \(i_M:{\mathfrak N}_X\rightarrow T^*M\) via \(\gamma\mapsto \langle \dot{\gamma} (x),\cdot\rangle|_M/\langle \dot{\gamma} (x),n_M(x)\rangle\) where \(x =\gamma\cap M\) is the unique intersection point and \(n_M\) is the future-pointing unit normal vector field along \(M\). Then a \textit{contact structure} \(\alpha_M\) on \({\mathfrak N}_X\) is defined via \(\alpha_M:=i^*_M(\lambda_{\mathrm{can}})\) where \(\lambda_{\mathrm{can}}\) is the canonical contact form on \(T^*M\) (locally given by \(\sum_kq^k\mathrm{d}p^k\)). The main result of the article is that for two Cauchy surfaces \(M,M'\subset X\), the equality \[ \frac{\alpha_{M'}(v)}{\alpha_M(v)}= 1+z(n_M(x),n_{M'}(x'),\gamma ) \] holds with any tangent vector \(v\) at \(\gamma\in {\mathfrak N}_X\) (see Theorem 1.2 and its proof in Section 2). This result is then applied to make a comparison between the Liouville volume forms on the contact manifolds \(({\mathfrak N}_X, \alpha_M)\) and the Riemannian volume forms along \((X,\langle\cdot,\cdot\rangle)|_M\) (see Section 3 in the article).