Einstein Prize Talk: Light-Cones in Relativity: Real, Complex and Virtual - with Applications

We present some observations about certain unusual geometric structures that appear in both Minkowski space and asymptotically flat space-times. Three different types of light-cones are considered: ordinary real light-cones in Minkowski space, M, complex light-cones in the complexified Minkowski space, M$_{C}$,(Minkowski coordinates x$^{a}$ go to complex z$^{a}$) and third, virtual light-cones in asymptotically flat space-times. All three types are defined at future null infinity, I$^+$, (I$^+$ defined by the endpoints of infinite extensions of future directed null geodesics) via the vanishing of the shear of the null geodesics lying in the null surface. The virtual light-cones appear to converge to points in an auxiliary virtual space, H-space. Cones are labeled by their apex coordinate x$^{a}$ or z$^{a}$. Two applications are discussed. The first begins with asymptotically flat Maxwell fields written as W=E+iB. On each light cone, with apex x$^{a}$, extracting the l=1 harmonic of the Maxwell field determines the complex electromagnetic dipole moment, D$_{E\&M}=$D$_{E}+iD_{M}$. D$_{E\&M}$, a function of x$^{a}$, can be analytically extending into M$_{C}$. Its zero set, points in M$_{C}$ where D$_{E\&M}$(z$^{a}$) vanishes, is a complex curve called the complex center of charge world-line. The second application virtually repeats the Maxwell case but now for asymptotically flat Einstein/Einstein-Maxwell fields. In the asymptotic region of each virtual light-cone, extracting the l=1 harmonics from the asymptotic gravitational field (the Weyl tensor) yields the complex gravitational dipole, D$_{Grav}=$D$_{Mass}+$iD$_{Spin}$. Each cone is labeled by its H-space apex z$^{a}$. D$_{Grav}$(z$^{a}$) is thus a function on H-space. Its zero set determines an H-space curve: the complex center of mass world-line. Interior space-time physical quantities and dynamics, (e.g. center of mass, spin, angular momentum, linear momentum, force, eqs. of motion) are identified at I$^+$ and described in terms of this complex world-line.