Grav lensing: multiple images as opposed to a circle, how is that possible?

Yes, it's time to ask some inane layperson's questions again... :-)

I'm now wondering how gravitational lensing can conspire to (apparently)
_not_ be unidirectional.

E.g., in the case of "Einstein's cross" four images of a distant quasar
appears, singling out those four directions, as opposed to a smeared-out
circle of light around the object doing the lensing.

--
A: Because it messes up the order in which people normally read text.
Q: Why is it such a bad thing?
A: Top-posting.
Q: What is the most annoying thing on usenet and in e-mail?

(Alf P. Steinbach) wrote

I'm now wondering how gravitational lensing can conspire to (apparently)
_not_ be unidirectional.

E.g., in the case of "Einstein's cross" four images of a distant quasar
appears, singling out those four directions, as opposed to a smeared-out
circle of light around the object doing the lensing.

There are (at least) two factors which lead to asymmetric lensed
images. First, the background source, lens, and observer may not be
aligned perfectly. If the lens is off to the right of the true position
of the background source, for example, like this:

S L

^ ^
true pos | | lens position


then the lensed images will be brighter on the left-hand side of
the lens, sort of like this:


* L .

^ ^ ^
bright img | faint image
|
| lens position


In addition, if the lensing mass is not spherically symmetric,
it can produce asymmetric images even if the alignment between
source, lens and observer is perfect.

Perhaps it might help to look at some lectures describing
the topic:

http://spiff.rit.edu/classes/phys240...grav_lens.html

There are links to a few movies, plus links to references for
further reading.

Michael Richmond

* Michael Richmond:
(Alf P. Steinbach) wrote

In addition, if the lensing mass is not spherically symmetric,
it can produce asymmetric images even if the alignment between
source, lens and observer is perfect.

Assymetric, yes.

It's not difficult to see that some assymetric distortion (like a
transformation plus smearing of an image) would result. It's not even
difficult to see that two images can appear, if the source is to the
right of the lens then apparently to the left and to the right of the
lens. But not identical images.

But what about multiple clear and apparently nearly identical images as
in "Einstein's Cross"? To get multiple images with ordinary optics (I'm
drawing an analogy here) one needs more than more than one lens, or
mirrors. From that analogy it seems one needs a lens that has a mass
distribution so that it effectively acts as more than one lens,
scattered, or, way out perhaps, two lenses in series?


Perhaps it might help to look at some lectures describing
the topic:

http://spiff.rit.edu/classes/phys240...grav_lens.html

There are links to a few movies, plus links to references for
further reading.

Thank you.

It does however not explain multiple images. Instead, it just asserts
that they can occur "if [the background source and the lens] line up
well enough that the true position of the background source falls within
the Einstein ring radius of the lens". I don't understand that.

--
A: Because it messes up the order in which people normally read text.
Q: Why is it such a bad thing?
A: Top-posting.
Q: What is the most annoying thing on usenet and in e-mail?

It does however not explain multiple images. Instead, it just asserts
that they can occur "if [the background source and the lens] line up
well enough that the true position of the background source falls within
the Einstein ring radius of the lens". I don't understand that.

There is no way around the math; you'll have to read the technical
literature on gravitational lensing if you want to understand. One
early paper you might read is by Bourassa and Kantowski. The ADS
entry for it

http://adsabs.harvard.edu/cgi-bin/np...f6510b0d827235

includes a scanned version of the entire paper. Somewhat more
complicated models are included in another early paper in the field,
by Padmanabhan and Subramanian:

http://adsabs.harvard.edu/cgi-bin/np...f6510b0d829377

Good luck.

Michael Richmond

* Michael Richmond:

It does however not explain multiple images. Instead, it just asserts
that they can occur "if [the background source and the lens] line up
well enough that the true position of the background source falls within
the Einstein ring radius of the lens". I don't understand that.

There is no way around the math; you'll have to read the technical
literature on gravitational lensing if you want to understand. One
early paper you might read is by Bourassa and Kantowski. The ADS
entry for it

http://adsabs.harvard.edu/cgi-bin/np...f6510b0d827235

includes a scanned version of the entire paper. Somewhat more
complicated models are included in another early paper in the field,
by Padmanabhan and Subramanian:

http://adsabs.harvard.edu/cgi-bin/np...f6510b0d829377

Good luck.

Yes, thanks, it seems luck is the key word in dechiphering that first
mentioned paper, even the presumably clarifying figures... ;-)

Here is then my limited understanding, disregarding for now (until last
paragraph) the impenetrable paper.

Consider three points in space: source, point lens, receiver, not in a
straight line. These three points define a plane P. Now the source can
emit a photon that goes somewhat to the side of the lens, wrt. to the
direction from receiver to lens, and we can vary the distance out from
the lens, and for at most one distance for a particular side (assuming
the photon doesn't go around the lens one or more times, which I in my
naivete think is impossible) it can strike the receiver.

Since there are only two sides of the lens in plane P, photons in P can
at most define two images of the source. That's what I meant when I
wrote that it's easy to visualize _two_ images. But not identical.

If the source emits a photon that is not in plane P then the photon path
plus lens defines a new plane Q, and the receiver is not in that new
plane Q unless the source, lens and receiver are in a straight line. So
assuming first they're not in a straight line. In order for that photon
to strike the receiver the lens will then have to deflect the photon in
a direction normal to the plane Q defined by the original path + lens.

I haven't heard of gravity having any sideways effect before.

Assuming next that source, lens and receiver are in a straight line.
Then the photon can strike the receiver among any of an infinite number
of paths, namely those that pass through a cirle or oval around the
lens. Which gives the "Einstein ring" effect.

Now the only simplifying assumption I see in that is the one about point
source, point lens and point receiver. But as far as I can decipher the
paper you linked to it claims that -- possibly with spatial extension
in the picture -- five or even seven distinct images can appear. I
can see that spatial extension can give an "Einstein ring" even when the
line-up isn't perfect, but I fail to see how photons can arrive at the
receiver when passing through five or seven points around the lens (or
in the case of "Einstein's cross", four points), and _not others_.

--
A: Because it messes up the order in which people normally read text.
Q: Why is it such a bad thing?
A: Top-posting.
Q: What is the most annoying thing on usenet and in e-mail?

(This is also of interest to sci.physics.research readers; hence the
crosspost.)

Alf P. Steinbach wrote:

Here is then my limited understanding, disregarding for now (until last
paragraph) the impenetrable paper.

Unfortunately, I don't have time to say very much, although if I could
draw some pictures on a whiteboard I think I could clear up the problem!
Since I can't do that in ASCII, let me recommend some excellent resources:

Probably my all-time four star scientific visualization site is this:

http://www.iam.ubc.ca/~newbury/lenses/lenses.html

You won't learn the math from this site (but there's some good
supplemental discussion there), but simply playing with the model may help
dispell some misconceptions--- plus it's a heck of a lot of fun.

To understand what you'll find at Newbury's site, probably all you need
can be found in the -third edition- of the excellent textbook by Stephani

author = {Hans Stephani},
title = {General Relativity: An Introduction of the Theory of the
Gravitational Field},
publisher = {Cambridge University Press},
note = {translated by {J}ohn {S}tewart and {M}artin {P}ollock},
year = 1990}

The current (third) edition has a nice concise discussion of the basics of
gravitational lensing, which should clear up your question. (Don't get
the second or first editions---if you can even find them--- since they
lack this section!)

For more detail, the Living Reviews article by Wambsganns is excellent:

http://www.emis.math.ca/EMIS/journal...-12/index.html

Both of these stick to pretty simple models, and are oriented toward gtr
students. That's probably fine for your purposes, but for a much more
complete discussion of possible models, aimed at serious astronomers, see
for example:

author = {P. Schneider and J. Ehlers and E.E. Falco},
title = {Gravitational Lenses},
series = {Astronomy and Astrophysics Library},
volume = 14,
publisher = {Springer-Verlag},
year = 1992}

"T. Essel" (hiding somewhere in cyberspace)

On Thu, 1 Jul 2004, Alf P. Steinbach wrote:
Yes, thanks, it seems luck is the key word in dechiphering that first
mentioned paper, even the presumably clarifying figures... ;-)
Sketch some pictures as you read this. It's a bit beyond my ascii-art
abilities...

On way to think of this is to consider that we and our telescopes are
much smaller than the source, lens and distances involved. So much so that
on a drawing of that the rays you might trace in geometric optics, we are
tiny even compared to the lines! So you can consider those lines as
pencils of light larger than the earth. Light within them can be
considered parallel rays similar those from a distant object, so if we
point a telescope along that pencil we can image the object like any
normal object. --The gravitaion lens doesn't form images, it just
redirects the light so we can form them, like we do with our eyes and a
(flat) mirror.
A glass or plastic lens that behaves like a gravitational lens does not
have a surface that looks like the surface of a section of a sphere, like
a standard lens, it looks like a 1/x curve**. For a standard lens the
normal to the surface of the lens gets close to the normal to the plane of
the lens as you move towards the center. For a gavitaional lens it gets
further away, so it is impossible for a gravitaional lens to focus in the
usual sense --it can't even form a virtual image like a concave lense
does. But it can cause rays of light that pass by at a given radius to
cross at a particular distance, making the ring associated w/ gravitaional
lenses. Now if you blocked out (or dispersed) most of the light except for
a small bit of the arc, you could image the source, the problem being that
normally there are many ovelapping images in the ring.
Note that even in the case of a radially symetric lens you could see
more than a simple ring: If an source is off center is will produce a
diplaced ring, so say we had a black hole with 3 stars arranged in a
triangle behind it: A red one, a blue one and a yellow one. There would be
3 rings, a red, a blue and a yellow one interlocking or partially
overlapping. Look really cool as a special effect in a movie (hey! Ya
gotta put my name in the credits if ya use it!) You might even be able to
deconvolve an image out of a full ring. (More commonly small/point sources
& distortions in the images are used to try to get info about the shape of
the lensing object.)
Now with aberration, the pencils of light may converge before they get
to the earth, then diverge, or diverge at the outset. It turn out that
there is alway one path where the light remains relatively parallel (that
corresponds to the direct path) allowing light to reach the earth (but
which may be physically blocked by the lensing body). And if things are
radially symmetric (and not to strange, like a toroidal glalaxy
cluster...) there will be some radius that will form a ring -but- there
can be an aberation caused by an over or under mass in part of the lensing
object that causes part of the ring to spread it's pencils out, diluting
their light so we see nothing, on the other side of the ring this
over/under mass may cause a convergance in the pencils before they reach
the earth, after that point they diverge and again we see nothing, again
-but- at some point along what would have been the ring, between converges
too much and diverges is "just right" --since it's along a ring, there are
two points, so we add two images. And thus, images are always added in
pairs.
A more detailed consideration using Huygen's principle show that, in
fact my fingers never left my hands during that hand waving: there is
always a minimum time path for the light to travel. If I add a new local
minima there must be a new local maxima added too and images show up in
set of one, three, five, etc.

And that's about 10% more than I know :-)

3ch

*arguably, in this sense, only the final lens in a telescope makes the
image, all the others do is "Bend the light". In fact, the main purpose of
the large mirror in most telescopes is to gather light from a large area
to a small one --the camera or eyepiece. The same field of view and
magmification could be acheived w/ a much smaller telescope.

**It is not (or at least I don't mean to imply) it -is- a 1/x curve, only
that it looks like one in the sense x^4 "looks" like a parabola. --Some
people have plastic "gravitational lenses". Fun and informative to play
with a bit if you can find one.