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From GR

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Introduction: General Relativity for Lensing


Metric in the weak field limit


The Cosmological Principle states that the Universe should be homogenous and isotropic (essentially a cosmological extension of the Copernican principle - that we inhabit no special place in the Universe). In 1936 Robertson and Walker showed that for a homogeneous and isotropic Universe the metric g could be parameterised by a number of independent functions


Formula GR1


where r is a time independent comoving distance (a distance defined to remain constant between a source and an observer), θ

and φ are the usual traverse polar coordinates. t is the cosmic time: the proper time measured by an observer at rest to the local matter distribution (or `substratum'). R(t) is the scale factor of the Universe. The function S0k(r) is defined as:




where k encodes the geometric curvature of spacetime: k=0 being a flat spacetime, k=+1 describes a positively curved hyperspherical

spacetime, k=-1 describes a negatively curved hyperbolic spacetime. Another common form of the RW metric uses a different definition of the comoving distance where S0k(r)=r so that:




A dimensionless scale factor $a(t)$ can be defined:




where R0 is the present day scale factor; a=1 at the present day.


The metric encapsulates the evolution of the Universe as a function of time and space, one of the main aims of modern cosmology has been to work out the factors that enter into the metric and thus describe the evolution of our Universe.  For example, one of the most commonly stated questions deals with the curvature, k, and asks if the Universe is closed (k > 0), open (k < 0) or flat (k = 0)?  From current observations, it appears that we live in an approximately flat Universe, although the curvature is so close to zero that it is very unlikely the Universe would spontaneously choose this value


A full treatment of the way light interacts with the matter-energy density of the Universe requires solving the Einstein Field Equations using the fact that light travels along null geodesics.  Solving for a general space-time metric is not possible due to the complexity of the system.  In astronomical applications, however, there are a number of simplifying approximations that can be used. 


The Weak Field Limit 


In the weak field limit, where the gravitational potential is small Φ<<c and the time variation of the potential is small, the metric can be rewritten as 




where Φ is the Newtonian potential. The light deflection angle in the weak field limit can now be derived. 


We start with the relativistic equation of motion or geodesic equation




where Γ is the Christoffel symbol/matrix and u is the 4D velocity. This basically describes the path that a photon will take through a 4D space-time. The space-time is described by the metric which enters equation (GR6) through the Christoffel matrix that can be written in terms of the metric g




If equation (GR7) this is substituted in the geodesic equation (GR6) the we are left with




Looking at the spatial part of equation (GR8), collecting all like terms, and using the fact that g0i=0 (no strong time variation of the metric) the equation of motion can be again rewritten as




 where roman letters have been used for the spatial part of the metric and greek letters for time and space parts. Substituting the metric in the weak field limit equation (GR5) we can finally right down the geodesic equation in the weak field limit




We can now look at some of the limits of this equations. In the slow motion limit, when ui<<c we find ourselves with the familiar Newtonian equation of motion




However in the ulra-relativistic limit, where ui ~ ric where ris the unit vector along the direction of a photons path, and with riri=1, we have 




where the gradient operator perpendicular to r has been defined as 




Now we can investigate where the Newtonian potential gradient originates from.  By tracing things back to equation (GR9) it can be seen that in the Newtonian limit the g00,j/2 term dominates. So the Newtonian result comes from the time part of the metric only i.e. in the Newtonian case the curvature of space is neglected (which we know is indeed the case). 


Now going to the ultra-relativistic limit and comparing equation (GR11) and equation (GR12) we can pick out two main differences. Firstly there is a factor of 2 difference in front of the potential gradient term. Comparing equations (GR10) and (GR9) the origin of this extra factor of 2 can be traced to spatial curvature terms. So ultra-relativistic particle feel an extra "force" due to spatial curvature that slow particles do not. The second difference is that the force parallel to the particles trajectory is cancelled out by spatial curvature terms so that only a transverse deflection is felt by the particle. Again this second difference has an intuitive interpretation; a particle moving at c cannot be accelerated any faster along the line of its trajectory. 


So to conclude the force felt along a photon path due to a lensing mass is zero whilst the force in the transverse direction is doubled relative to the Newtonian limit. 


Finally the deflection angle is simply the integral of the changes in a particles path over time




We have now arrived at the starting point, from General Relativity, for lensing studies.


For example for a point mass the potential can be written as




where ε is the perpendicular distance from the point mass and z is the parallel distance. By substituting this into equation (GR14) and integrating over the path z the deflection for a point mass 




where we have used the definition of the Schwarzchild radius



Five Easy Steps


  1. Derive the weak field metric
  2. Exercise 2
  3. Exercise 3
  4. Exercise 4
  5. Exercise 5


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