1 Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 1999 Gravitational Lensing from Hamiltonian Dynamics c�1999 Edmund Bertschinger and Rennan Barkana.Introduction The deflection of light by massive bodies is an old problem having few pedagogical treat-ments. The full machinery of general relativity seems like a sledge hammer when applied to weak gravitational fields. On the other hand, photons are relativistic particles and their propagation over cosmological distances demands more than Newtonian dynamics. In fact, for weak gravitational fields or for small perturbations of a simple cosmologi-cal model, it is possible to discuss gravitational lensing in a weak-field limit similar to Newtonian dynamics, albeit with light being deflected twice as much by gravity as a nonrelativistic particle. The most common formalism for deriving the equations of gravitational lensing is based on Fermat’s principle: light follows paths that minimize the time of arrival (Schei-der et al. 1992). As we will show, light is deflected by weak static gravitational fields as though it travels in a medium with variable index of refraction n = 1 − 2φ where φ is the dimensionless gravitational potential. With the framework of Hamiltonian dynamics given in the notes Hamiltonian Dy-namics of Particle Motion, here we present a synopsis of the theory of gravitational lensing. The Hamiltonian formulation begins with general relativity and makes clear the approximations which are made at each step. It allows us to derive Fermat’s least time principle in a weak gravitational field and to calculate the relative time delay when lensing produces multiple images. It is easily applied to lensing in cosmology, including a correct treatment of the inhomogeneity along the line of sight, by taking advantage of the standard formalism for perturbed cosmological models. Portions of these notes are based on a chapter in the PhD thesis of Barkana (1997). 1� � � 2 Hamiltonian Dynamics of Light Starting from the notes Hamiltonian Dynamics of Particle Motion (Bertschinger 1999), we recall that geodesic motion of a particle of mass m in a metric gµν is equivalent to Hamiltonian motion in 3 + 1 spacetime with Hamiltonian 0ig piH(pi, xj , t) = −p0 = g pi +(gij pipj + m2)+ � 0i�2 �1/2 . (1) g00 −g00 g00 2This Hamiltonian is obtained by solving gµν pµpν = −m for p0. The spacetime coor-dinates xµ = (t, xi) are arbitrary aside from the requirement that g00 < 0 so that t is timelike and is therefore a good parameter for timelike and null curves. The canonical momenta are the spatial components of the 4-momentum one-form pµ. The inverse met-ric components gµν are, in general, functions of xi and t. With this Hamiltonian, the exact spacetime geodesics are given by the solutions of Hamilton’s equations dxi ∂H dpi ∂H = = . (2)dt ∂pi , dt − ∂xi Our next step is to determine the Hamiltonian for the problem at hand, which requires specifying a metric. Because we haven’t yet derived the Einstein field equations, all we can do is to pick an ad hoc metric. In order to obtain useful results, we will choose a physical metric representing a realistic cosmological model, an expanding Big Bang cosmology (a Robertson-Walker spacetime) superposed with small-amplitude spacetime curvature fluctuations arising from spatial variations in the matter density. For now, the reader will have to accept the exact form of the metric without proof. The line element for our metric is 2ds2 = a (t) −(1 + 2φ)dt2 + (1 − 2φ)γij dxidxj . (3) In the literature, t is called �conformal� time and xi are �comoving� spatial coordinates. The cosmic expansion scale factor is a(t) and is related to the redshift of light emitted at time t by a(t) = 1/(1 + z). To get the non-cosmological limit (weak gravitational fields in Minkowski spacetime), one simply sets a = 1. The Newtonian gravitational potential φ(xi, t) obeys (to a good approximation) the Poisson equation. (In cosmology, the source for φ is not ρ but rather ρ − ρ¯ where ¯ρ is the mean mass density; we will show this in more detail later in the course.) We assume |φ| � 1 which is consistent with cosmological observations implying φ ∼ 10−5 . In equation (3) we write γij (xk ) as the 3-metric of spatial hypersurfaces in the unper-turbed Robertson-Walker space. For a flat space (the most popular model with theorists, and consistent with observations to date), we could adopt Cartesian coordinates for 2which γij = δij . However, to allow for easy generalization to nonflat spaces as well as non-Cartesian coordinates in flat space we shall leave γij unspecified for the moment. Substituting the metric implied by equation (3) into equation (1) with m = 0 yields the Hamiltonian for a photon: H(pi, xj , t) = p(1 + 2φ) , p ≡ � γij pipj �1/2 . (4) gWe have neglected all terms of higher order than linear in φ. Not surprisingly, in a perturbed spacetime the Hamiltonian equals the momentum plus a small correction for gravity. However, it differs from the proper energy measured by a stationary observer, E = −V µpµ, because the 4-velocity of such an observer is V µ = (a(1 − φ), 0, 0, 0) (since µν V µV ν = −1) so that E = a−1p(1 + φ). The latter expression is easy to understand because a−1 converts comoving to proper energy (the cosmological redshift) and in the Newtonian limit φ is the gravitational energy per unit mass (energy). Why is the Hamiltonian not equal to the energy? The answer is because it is conjugate to the time coordinate t which does not measure proper time. The job of the Hamiltonian is to provide the equations of motion and not to equal the energy. The factor of 2 in equation (4) is important � it is responsible for the fact that light is deflected twice as much as nonrelativistic particles in a gravitational field. To first order in φ, Hamilton’s equations applied to equation (4) yield dxiidpi i dt = n (1 + 2φ) , dt = −2p�iφ + γkij pk nj (1 + 2φ) , n ≡ γij ppj . (5) We will drop terms O(φ2) throughout. We have defined a unit three-vector ni in the photon’s direction of motion (normalized so that γij ninj = 1). The symbol γkij = 1 γkl(∂iγjl + ∂j γil − ∂lγij ) is a