From: annis@hale.ifa.hawaii.edu (James Annis)
Newsgroups: sci.astro
Subject: Re: grav lensing (long)
Date: 28 Jul 92 09:40:31 GMT
Organization: Institute for Astronomy, Hawaii


I'll let Patrick Mock argue the high road, and I'll stick to the
pedrestrian routes.

Tom Van Flandern writes:

>     My argument is simple and easy for anyone to grasp.  The counter-
> arguments you have quoted require considerable work to grasp even the
> essentials, and then they seem so skirt the basic point I have made here: 
> the *shape* of the image of a point source being gravitationally lensed 
> will be a ring-arc in general, or a full ring if axial alignment is quite 
> close. Moreover, that is what is observed in the only cases which everyone 
> here agrees are true gravitational lenses, namely galactic clusters lensing
> background blue objects.

But Tom, Fermat's principle isn't hard. The name is just obscure.
And its application to gravitational lensing is straightforward.

I found a chapter in Feynman's 1st year physics book:
	Feynman. Lectures in Physics I, 
	Chapter 26: Optics, the Principal of Least Time.

Feynman introduces the ideas of reflection and refraction, and notes the 
relevent law (Snell's law), then tries to find a way of thinking about how 
light behaves.  Feynmann notes that light does not follow the path of least 
distance, in general, because in the particular case of refraction it fails 
to do so. Feynman then introduces a way of thinking about light that does 
explain its behavior in general:

	"Fermat's principle. His idea is this: that out of all possible
paths that it might take to get from one point to another, light takes
the path which requires the -shortest time-."  (p26-3)

(I like Feynman. He makes things easy.) He goes on to say that it
has predictive power, for instance that using it one can predict that 
the speed of light in water is slower than in air. Later he gives 
a more precise statement of Fermat's principle:

	"The correct statement is the following: a ray going in a 
certain particular path has the property that if we make a small 
change in the ray in any manner whatever, (say in the location at 
which it comes to the mirror, or the shape of the curve, or anything)
there will be -no- first order change in the time; there will be only 
a -second- order change in the time. In other words, the principle
is that light takes a path such that there are many other paths nearby
which take almost exactly the -same- time."

There is no reason not to apply Fermat's principle to gravitational 
lenses.  So, to interpret gravitational lensing, we must understand 
the time it takes light to take each path around the lensing galaxy.  
There are two ways a light pulse from a distant object is delayed when 
it passes through the gravitational field of an intervening object. 

    1) the extra path length light must travel because its path was bent. 
    2) the local gravitational time delay due to the presence of mass.

("familiar from the radar-ranging experiments carried out in the solar 
system", to quote Blandford and Narayan; this is just the (GR) effect 
that a clock runs faster out of a potential well than inside of one).

These have different operational dependicies. In the limit of small angles
from the lensing mass, they have the following form:

   1)		t(path) = (xi - xs)**2
   2)		t(grav) = xi 		          (Blandford and Narayan)
where 
	= means proportional to
	xi is the position of the mirage relative to the lensing mass
	xs is the position of the source object relative to the mass, 
				i.e., where the image would appear
					if there were no lensing.

If one calculates t = t(path) + t(grav) for every point in a grid
in front of the lensing mass, one finds a strange, contorted surface.
It has extrema in 3,5,7... places, in general. Fermat's principle
says that the images will show up at the locations where the time 
delay surface's derivative is zero (an extrema).

The surface has an extrema in the shape of a ring -only- for the
case of xs=0 i.e., when the souce is exactly behind the lens mass.
Because t(path) goes as the square of the deviation from exact, the 
ring shaped extrema very rapidly breaks into multiple images.

(How rapid is rapid? ... if this is the only point we are arguing 
about, then one merely has to do the simulation to settle it. Blandford 
and coworkers claim to have done it, and they say rapid is nearly 
instananeous for a point source. Untill I see numbers to the contrary,
I'll believe them.)

Fermat's principle predicts multiple images of a gravitationally lensed
point source. A point very close to the point source might be lensed
into an image very close to that of the point source, thereby creating 
arc shapes.  But a point source is a point source; there is not light
from "a point nearby".

For quasars (which seem to be very good approximations to point sources
from observations which have nothing to do with gravitational lensing), 
the chances of lying exactly on xs=0 is negligiable, and so only mutiply 
imaged quasars are found. For extended radio lobes, the chances are fairly 
good, and rings are found. For galaxies behind the centers of clusters, 
the chances are fair, and giant arcs are found.  -But-, these are not arcs 
because they are misaligned, but because the potential in the bottoms of 
clusters is complicated, and the ring gets butchered. 

The little arclets are a different story. They are too far from the center 
to be multiply imaged, and are just gravitationally distorted. Every
paper analysing these arclets that I've seen says they have to have
unlensed angular sizes greater than 1", or we wouldn't see them as
arc-like.  One of the fundamental observations about quasars is that they
have angular diameters <<1".

Fermat's principle allows a prediction of multiple images of a point 
source, from very close to first principles. What more could one ask?
--

James Annis   			annis@galileo.ifa.hawaii.edu     
Newsgroups: sci.astro