# Fermats P & Gravitational Lensing

Fermat's principle states that light travels the path of stationary time, I've read up on this from several sources and I've come to realize that this means that it will travel paths of extrema, (maxima, minima & saddle points).

For minima, refraction is often given as an example (minizing the action or time of path)

For saddle/inflection points, Elliptical mirrors are given as examples (multiple paths given the same time from one focus to the other)

I understand these two examples... (atleast the first one very well).

My question is about the Maximal timec/ action path which is often applied gravitional lensing. How can there ever be a maximal time path, shouldn't the maximal time be infinite?

Here is a blurb from the following source

http://relativity.livingreviews.org/open?pubNo=lrr-2007-4&key=vuiss07 [Broken]

"4.1.1 Basics of lensing
Light is bent by the action of a gravitational field. In the case where a galaxy lies close to the line of sight to a background quasar, the quasar’s light may travel along several different paths to the observer, resulting in more than one image.

The easiest way to visualise this is to begin with a zero-mass galaxy (which bends no light rays) acting as the lens, and considering all possible light paths from the quasar to the observer which have a bend in the lens plane. From the observer’s point of view, we can connect all paths which take the same time to reach the observer with a contour, which in this case is circular in shape. The image will form at the centre of the diagram, surrounded by circles representing increasing light travel times. This is of course an application of Fermat’s principle; images form at stationary points in the Fermat surface, in this case at the Fermat minimum. Put less technically, the light has taken a straight-line path between the source and observer.

If we now allow the galaxy to have a steadily increasing mass, we introduce an extra time delay (known as the Shapiro delay) along light paths which pass through the lens plane close to the galaxy centre. This makes a distortion in the Fermat surface. At first, its only effect is to displace the Fermat minimum away from the distortion. Eventually, however, the distortion becomes big enough to produce a maximum at the position of the galaxy, together with a saddle point on the other side of the galaxy from the minimum. By Fermat’s principle, two further images will appear at these two stationary points in the Fermat surface. This is the basic three-image lens configuration, although in practice the central image at the Fermat maximum is highly demagnified and not usually seen.

If the lens is significantly elliptical and the lines of sight are well aligned, we can produce five images, consisting of four images around a ring alternating between maxima and saddle points, and a central, highly demagnified Fermat maximum. Both four-image and two-image systems (“quads” and “doubles”) are in fact seen in practice. The major use of lens systems is for determining mass distributions in the lens galaxy, since the positions and brightnesses of the images carry information about the gravitational potential of the lens. Gravitational lensing has the advantage that its effects are independent of whether the matter is light or dark, so in principle the effects of both baryonic and non-baryonic matter can be probed.
"

So gravitational lensing demonstrates this maximal path. I understand how a maxima is an extrema, but I can't reconcile how you could have such a path.

help!!

Thanks

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George Jones
Staff Emeritus
Gold Member
How can there ever be a maximal time path, shouldn't the maximal time be infinite?

I don't understand this. Could you elaborate?

yossell
Gold Member
A wild guess: the authors are working in a metric whose signature makes timelike curves have negative squared separation rather than positive, and the maximum of a set of negative reals is finite.

Taken from WIKIPEDIA

http://en.wikipedia.org/wiki/Fermat's_principle

"The modern version of Fermat's principle states that the optical path length must be stationary, which means that it can be either minimal, maximal or a point of inflection (a saddle point). Minima occur when a wave passes from one medium into another (refraction) and in the reflection of light from a planar mirror. Maxima occur in gravitational lensing. A point of inflection describes the path light takes when it is reflected off an elliptical mirrored surface."

The way I understand it, Fermat's Revised Principle says you will find light taking the following three paths, maxima, minima, inflection points.

When I think of maximal time, Im thinking of whats the longest possible time it takes to get there, so I want to say infinite.

The following Pdf explains the elliptical mirror example with minima and maxima very well

see page 2.

George Jones
Staff Emeritus
Gold Member
When I think of maximal time, Im thinking of whats the longest possible time it takes to get there, so I want to say infinite.

This is what I don't understand. In gravitational lensing, what light takes an infinite amount of time to travel from source to reception?

I don't know thats why I'm asking the question.

Im basically infering that Fermats Revised principle says that light takes paths of extrema.

Sources say (wikipedia & the first that I posted), that gravitational lensing demonstrates fermats principle in the maxima case. I'm asking how could there be a maximal time.

Maybe a local maximal, but a global maxima? Obviously Im not seeing something

George Jones
Staff Emeritus
Gold Member
Sorry that I keep asking the same question. Suppose a quasar is a source of light, and a black hole that is between that is between the quasar and me acts as a gravitational lense. For what paths does the time taken for the light to travel from the quasar to me approach infinity?

If the path is approaching infinity, im assuming you would never see the light.

We apparently see the effects of gravitaional lensing

"If the lens is significantly elliptical and the lines of sight are well aligned, we can produce five images, consisting of four images around a ring alternating between maxima and saddle points, and a central, highly demagnified Fermat maximum. Both four-image and two-image systems (“quads” and “doubles”) are in fact seen in practice."

Not sure anymore
Very confused

atyy
The text says maxima in the "Fermat surface" - this isn't necessarily either proper time or coordinate time.

Edit: Yes, I guess it is local maxima.

George Jones
Staff Emeritus
Gold Member
Maybe you have a look at 3.3 Gravitational Lensing Via Fermat's Principle from

http://arxiv.org/abs/astro-ph/9606001.

All the diagrams are at the end of the article, and study of previous sections may be necessary to understand 3.3.

Stingray
Maybe a local maximal, but a global maxima? Obviously Im not seeing something

Fermat principles guarantee local extrema. It should also be pointed out that gravitational lensing is not particularly special for involving local maxima. The same thing also happens when looking at the paths of ordinary objects freely falling in a gravitational field.

Anyway, the time that is maximized is a proper time. Understanding how this can be maximal requires looking at its definition. If $x^\mu(s)$ is the path of an object (photon, ball, ...) as a function of some parameter $s$, the proper time is
$$\tau(s_i,s_f) = \int_{s_i}^{s_f} \sqrt{ - g_{\mu\nu} (x(s)) \frac{d x^\mu}{d s} \frac{d x^\nu}{d s} } ds.$$
$g_{\mu\nu}$ is the metric. Crucially, this has a negative eigenvector. In flat spacetime, for example, you can find coordinates such that
$$\tau(s_i,s_f) = \int_{s_i}^{s_f} \sqrt{ \dot{t}^2 - \dot{x}^2 - \dot{y}^2 - \dot{z}^2 } ds.$$

Now suppose that you're interested in a ball traveling from $[0,0,0,0]$ to $[T,0,0,0]$. The correct curve is given by $[t(s) = s,x(s) = y(s)= z(s) = 0]$. The proper time elapsed on this path is $T$. Now try wiggling the spatial velocities. It's clear that any way you do this will reduce the proper time. It's slightly more tricky to see this for variations of $t(s)$, but still possible. Similar things happen in curved spacetime, although you also get saddle points when there are multiple extremal paths between two points (i.e. multiple images).

The point here is that Lorentzian metrics are weird. Adding a huge detour actually reduces the time taken by someone moving on that path.

Thanks for clearing that up

I was thinking that curved space time was the culprit here.

Maybe if we lived moving at high speeds, near extremely curved space-time, and with debroglie wavelengths on the order to see QM effects dominate, our brains would evolve to understand & apply these theories easier.