Deflection of light due to gravity

[lancelot]

Two questions:

 

A hand-held torch (on earth) is shone horizontally. What is the path of an emitted photon in earth based coordinates?

 

A photon approaches a black hole, but not head-on. It is deflected and continues past the BH. What is its path?

[ajb]

Light follows what we call null geodesics, they satisfy [math]ds^{2}=0[/math]. On a flat space-time these geodesics are straight lines and on curved geometries this is no longer the case.

 

Now as you are talking about black-holes, I assume we are thinking of the Schwarzschild

solution. This is a well known result, it is discussed nicely in Wald. I also found these notes, they look correct.

 

Armed with this, I guess we should be able to construct some nice pictures.

[lancelot]

ajb/spock, Not what I am lookng for. To clarify, my first question rephrased is :

 

What is the trajectory of the photon as measured at the earth's surface in cartesian terms. For example a rifle bullet has a parabolic path, the resultant of horizontal velocity vector and vertical acceleration vector (g). What is the path of the photon?

 

Second question - please ignore "black hole" and substitute 'heavy dense body in free space" to make it simpler and avoid getting into photon spheres and Schwarzchild radius. So, what is path of a photon as it passes by a dense body in free space, in cartesian terms?

[swansont]

I don't think any of this clarification changes ajb's answer.

[Pete]

ajb/spock, Not what I am lookng for. To clarify, my first question rephrased is :

 

What is the trajectory of the photon as measured at the earth's surface in cartesian terms. For example a rifle bullet has a parabolic path, the resultant of horizontal velocity vector and vertical acceleration vector (g). What is the path of the photon?

 

Second question - please ignore "black hole" and substitute 'heavy dense body in free space" to make it simpler and avoid getting into photon spheres and Schwarzchild radius. So, what is path of a photon as it passes by a dense body in free space, in cartesian terms?

The link that ajb gave was to the solutions for geodesics in a Schwarzchild spacetime, which is the spacetime that you are referring to. The paths that you seek are the spatial portions of the geodesics. Once those are obtained from the notes you can simply change them to Cartesian coordinates.

 

Do you need help in solving these equations explicitly?

 

Pete

[lancelot]

Hi everyone. Maybe there is something odd with my computer or my settings, but I cant see any link appearing in ajb's post. I wondered what he meant by 'these notes'. (?)

 

ajb could you possibly try to post that link again, see if it appears?

 

* Ahah! Have just noticed that 'notes' is a link. It hardly shows up as different from main text on my screen . Will check that out and thanks.

Edited July 31, 2008 by lancelot

[ajb]

Hi everyone. Maybe there is something odd with my computer or my settings, but I cant see any link appearing in ajb's post. I wondered he meant by 'these notes'. (?)

 

ajb could you possibly try to post that link again, see if it appears?

 

The link seems to be working for me.

 

And when you say a "heavy dense body", do you mean a "heavy dense spherically symmetric non-rotating body"? If so, then the reply I have given you is correct.

 

Am I correct in thinking you want to work out how light is deflected around a star/the Sun etc as seen from Earth. If so, this is a well known result, (in the weak field limit for sure).

[Klaynos]

Hi everyone. Maybe there is something odd with my computer or my settings, but I cant see any link appearing in ajb's post. I wondered what he meant by 'these notes'. (?)

 

ajb could you possibly try to post that link again, see if it appears?

 

* Ahah! Have just noticed that 'notes' is a link. It hardly shows up as different from main text on my screen . Will check that out and thanks.

 

It's under the word "notes".

[lancelot]

ajb thanks, the link on 'notes' does work, it wasnt apparent to me before, as normally the blue text is more prominent on messages. That's an interaction between the forum software and my RGB monitor I guess.

 

I have looked at the link - and I see the maths and terminology is quite complex. I will endeavour to work through the maths but it could take me a while!

 

To deal first with a dense body in space question - that seems to be quite a complicated matter in GR. For a massive particle in Newtonian physics it's easy - an elliptical orbit or other variation of a hyperbolic function. But it would seem perhaps there is not simple equivalent in GR so I will just accept that.

 

To deal with a photon emitted from a torch on earth:

A rifle bullet willl have a vertical acceleration vector of g, which defines its trajectory. As any student knows, a = g, because m cancels out in the Newtonian derivation, using f=ma.

My curiosity was aroused by wondering what happens if m is made infinitesimally small (but not zero). Even if m were 10^-10000 eV, or whatever, it would still cancel out and a = g. So i wondered how that path calculated in Newtonian fashion for a 'bullet' of infinitesimally small mass at velocity close to c, would compare with the path of a particle of absolute zero mass when calculated correctly in GR.

i.e what would be the 'rate of fall' of the photon towards the earth surface? And the resultant trajectory?

 

However, I begin to suspect that using GR, the answer may not be a simple one, at least not for me.

Edited July 31, 2008 by lancelot

[Sisyphus]

I think the idea is that the photon won't fall at all because its mass is zero, and hence so is the gravitational force, and so is the acceleration. That's why it's not analogous to a rifle bullet, and why you won't see elliptical orbits like you would with Newtonian equations. It will go in a straight line - through space. The reason it nevertheless does (in a sense) curve is that space itself is not "straight." One of the crazy aspects of GR is that straight lines don't behave like we intuitively expect them to, and don't appear straight from other perspectives. That's how you can get things like black holes, where space is "bent" so much that no straight line is ever going to lead you anywhere but elsewhere inside the black hole.

 

What you're looking for, then, is I guess what the photon's path would look like from Earth. And no, it wouldn't be straight. I'm not going to do the math myself, though, because a)I'm lazy and b)I'm afraid I might embarrass myself.

[ajb]

In Newtonian gravity we have 3 kinds of "orbits"; elliptical (bound), parabolic and hyperbolic (both unbound).

 

In general relativity you can have more than this, even for the Schwartzchild solution.

 

In general, it is going to be impossible to solve for geodesic exactly. We only really stand a chance of doing this for space-times with a lot of symmetries. Otherwise, one may have to resort to numerical methods.

 

I have no idea if you would get something that resembles a null-geodesic of the Schwarzschild metric by taking very small masses in Newtonian gravity. What is true is that general relativity in the (very) weak limit is Newtonian gravity. So, I am confident in saying something like for massive bodies Newtonian gravity is very accurate. You need GR to explain some of the details such as the precession of the perihelion of Mercury. For "near massless" bodies I don't know.

[lancelot]

 

What you're looking for, then, is I guess what the photon's path would look like from Earth.

 

That exactly what I was looking for.

It could be an interesting 'trival pursuit' challenge for someone!

I suppose I am right to assume from ajb that it would not be a para/hyperbolic function, as for a mass particle? Pity, that would have been neat, and within my maths capabilities!

[lancelot]

In Newtonian gravity we have 3 kinds of "orbits"; elliptical (bound), parabolic and hyperbolic (both unbound).

 

In general relativity you can have more than this, even for the Schwartzchild solution.

 

In general, it is going to be impossible to solve for geodesic exactly. We only really stand a chance of doing this for space-times with a lot of symmetries. Otherwise, one may have to resort to numerical methods.

 

I have no idea if you would get something that resembles a null-geodesic of the Schwarzschild metric by taking very small masses in Newtonian gravity. What is true is that general relativity in the (very) weak limit is Newtonian gravity. So, I am confident in saying something like for massive bodies Newtonian gravity is very accurate. You need GR to explain some of the details such as the precession of the perihelion of Mercury. For "near massless" bodies I don't know.

 

 

ajb: It seems that the observation of distant starlight deflected by the sun in 1919 proved Einstein's GR to be correct. If so it must have been possible even at that time to calculate the predicted deflection and/or path, I would assume. (?)

[Janus]

ajb: It seems that the observation of distant starlight deflected by the sun in 1919 proved Einstein's GR to be correct. If so it must have been possible even at that time to calculate the predicted deflection and/or path, I would assume. (?)

 

http://theory.caltech.edu/people/patricia/lclens.html

[ajb]

ajb: It seems that the observation of distant starlight deflected by the sun in 1919 proved Einstein's GR to be correct. If so it must have been possible even at that time to calculate the predicted deflection and/or path, I would assume. (?)

 

Yes.

[Pete]

ajb: It seems that the observation of distant starlight deflected by the sun in 1919 proved Einstein's GR to be correct.

No. That observation confirmed the prediction made by general relativity. Confirm of a prediction does not logically prove that a theory is correct. When many predictions are confirmed then scientists have more confidence in the theory.

 

Pete

[lancelot]

http://theory.caltech.edu/people/patricia/lclens.html

 

Janus - I have just opened this - excellent! Thank you so much.