Can Fermat's principle be applied to gravitational lensing?

[SpitfireAce]

Light having to travel through a gravitational field deflects towards the mass and thus increases the length and duration of its journey (traveling through more curved space-time takes more proper time than traveling through less curved space-time) I understand that unlike in refraction, light's path in a gravitational field is, in some sense, predetermined by space-time. Is there a way to describe general relativity in terms of actions?

 

[robphy]

Possibly useful starting point:
http://wwwitp.physik.tu-berlin.de/hellwig/vB/homepage/perlick.htm

 

[MeJennifer]

traveling through more curved space-time takes more proper time than traveling through less curved space-time

I think that you misunderstand something about spacetime.
It does not take time to travel through spacetime, that idea does not make any sense.

 

[pervect]

Light having to travel through a gravitational field deflects towards the mass and thus increases the length and duration of its journey (traveling through more curved space-time takes more proper time than traveling through less curved space-time) I understand that unlike in refraction, light's path in a gravitational field is, in some sense, predetermined by space-time. Is there a way to describe general relativity in terms of actions?

You might also want to look at http://www.eftaylor.com/leastaction.html

Light follows geodesic paths through space-time, so the path of light is indeed determined by an action principle.

The action principle for matter is very simple - a geodesic path is a path that extremizes (generally maximizes) proper time.

Light does not have "proper time", so unfortunately one cannot use the above action princple directly. But while light does not have proper time, it does have an affine parameterization. I believe that one way to describe the action principle satisfied by light would be to use a monochromatic laser beam for the light, and to count the number of wavelengths. Rather than maximizing proper time, I think one can say that light minimizes the number of wavelengths. While I think this is correct, I couldn't find a reference to confirm it.

Note that due to gravitational time dilation, wavelengths do not cover the same distance far away and near to a massive body. To cover the maximum distance with a fixed number of wavelengths, an optimum path avoids approaching a massive body too closely.

 

[SpitfireAce]

I'm afraid the link doesn't work... I read that the fact that a particle takes more proper time to move from point A to B when there is a gravitational field present as opposed to in the absence of one, is proof for the curved space-time construct... the point is that light's trip takes more time because of its deflection... Newtonian gravity can be fully described without differential equations, using the principle of least action, yet it appears that general relativity cannot because Fermat's principle would predict that light deflect away from mass if anything towards less curved space-time and thus save time.

 

[SpitfireAce]

1st link that is, sorry I didn't see you're post pervect

 

[SpitfireAce]

"Note that due to gravitational time dilation, wavelengths do not cover the same distance far away and near to a massive body"
This is probably due to length contraction as perceived by an observer outside the field, but I thought action principle was based on proper time and length.

"To cover the maximum distance with a fixed number of wavelengths, an optimum path avoids approaching a massive body too closely."
But light doesn't seem to avoid massive bodies at all, quite the opposite

still reading website...

 

[SpitfireAce]

"In general relativity a particle moves along the worldline of maximal proper time (maximal aging). In the limit of small spacetime curvature and low velocity this reduces to the principle of least action"

http://www.eftaylor.com/pub/GRtoPLA.pdf

so I guess in GR objects follow paths of most action, I imagine the derivation would be interesting if it wasn't completely beyond me =(

* thanks for the link pervect

btw, can anyone recommend a very introductory mathematics text (Calculus1+)with a lot of physics context

 

[MeJennifer]

so I guess in GR objects follow paths of most action...

I don't think that's right, don't objects in GR take paths of minimal action?

 

[pervect]

I don't think that's right, don't objects in GR take paths of minimal action?

The most technically correct term is probably the principle of stationary action. See for instance http://www.eftaylor.com/pub/call_action.html.

The action for an actual path may be a saddle point rather than a minimum. So the most general, but rather esoteric, term for our basic law is the principle of stationary action.

One also occasionally sees "extremal action".

These are very minor points, the O.P. has basically got the right idea.

 

[Cusp]

Two excellent papers on this;

Fermat's principle, caustics, and the classification of gravitational lens images
Blandford and Narayan

http://ads.grangenet.net/abs/1986ApJ...310..568B

A new formulation of gravitational lens theory, time-delay, and Fermat's principle
Schneider
http://ads.grangenet.net/abs/1985A%26A...143..413S

 

[Ron2d2]

I know this link is very old, but I thought I might chime in (despite the already progressed aging of this topic). You can in fact apply Fermat's principle to gravitational lensing. What you do is you start off with your general null-geodesic equation (which is the path light takes) parametrized with some arbitrary term, but make your parametrization in terms of time t and, assuming a weak-gravitational field with a static potential (by static I mean not moving anywhere), derive the effective speed of light. It should be something like v=1-2*|Phi| (which is 1 "minus" 2 "times" the absolute value of potential Phi). Which you can use the definition of the index refraction to get the index of refraction induced by your static potential. From here its pretty much a lensing problem. You can also prove light is deflected towards the source by applying Fermat's Principle in terms of your index of refraction (basically its the first variation of the integral from your source to observer along index of refraction n dl). Parametrize this in terms of your arbitrary parameter and you get the Lagrangian, and using the Euler-Lagrange equation you get that the infinitesimal change of the path light takes from its initial trajectory is just the perpendicular gradient of the natural log of the index of refraction... or approximating this it is just -2*"perpendicular gradient of"Phi. Integrating this (by the so-called Born Approximation you can just integrate from negative infinity to infinity) from you source to observer results in -4G/b*c^2 in the b-hat direction (this is in vector form). So, the direction light is deflected is toward the mass. I got a negative sign because I parametrized my unperturbed path of the light to be the x-direction and the mass to be the origin. Also, this is just a rough format at approaching the problem... a more proper way would be to compare the background geodesic, extended from the source to the observer, to that of a perturbed geodesic (Carroll's Intro. to GR does this)... but this formal process does not incorporate Fermat's Principle.

Basically, its good to remember that in Einstein's GR equation the energy-momentum tensor is in fact related to curvature of spacetime. Gravity corresponds to changes in the properties of space and time. It alters the straightest possible, or shortest, paths that objects naturally follow (including light!).