# Can Fermat's principle be applied to gravitational lensing?

1. Jun 27, 2007

### 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?

2. Jun 27, 2007

### robphy

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

Last edited by a moderator: May 2, 2017
3. Jun 27, 2007

### MeJennifer

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

Last edited: Jun 27, 2007
4. Jun 27, 2007

### pervect

Staff Emeritus
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.

5. Jun 27, 2007

### 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.

6. Jun 27, 2007

### SpitfireAce

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

7. Jun 27, 2007

### 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

8. Jun 27, 2007

### 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

9. Jun 28, 2007

### MeJennifer

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

Last edited by a moderator: Jun 28, 2007
10. Jun 28, 2007

### pervect

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

One also occasionally sees "extremal action".

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

11. Jun 29, 2007

### Cusp

Two excellent papers on this;

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

A new formulation of gravitational lens theory, time-delay, and Fermat's principle
Schneider