Light bending computation, Einstein vs Newton

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marten
Posts: 134
Joined: Thu Aug 30, 2007 10:21 am

Light bending computation, Einstein vs Newton

Postby marten » Wed Dec 12, 2007 5:37 pm

So I was laying awake last night trying to figure this out, and I only keep coming up with more questions. I hope someone can point me towards a book, or help answer my questions. First I was thinking about how to calculate how far a particle (with mass) is deflected from its course by a massive object. Then I started thinking about how relativity predicts twice the deflection for light then Newton's theory. And that is where my questions come:

1) How does Newton's theory predict the bending of light in the first place? Light doesn't have mass, so it can't be gravity. Is it other effects like diffraction?

2) I can think of numerical methods to calculate the deflection of a particle in a gravitational field. But how would one analytically solve the problem? I'm imagining a small particle (maybe a satellite) traveling at a certain velocity that passes near a large massive object, (like a planet). I'm thinking in terms of momentum conservation, angular momentum, energy conservation, and I can't see a way to setup the problem...

3) so then how about light (as above?) ?

Thanks!

surjective
Posts: 39
Joined: Thu Jan 17, 2008 1:16 am

Postby surjective » Thu Jan 17, 2008 1:24 am

1) How does Newton's theory predict the bending of light in the first place? Light doesn't have mass, so it can't be gravity. Is it other effects like diffraction?

In newtonian theory, the acceleration of a test body in a gravitational field is independent of it's mass (same with G.R. - this is the equivalence principal) :
f = m*a = G*m*M/r^2, where M is the source mass (ie: the body the photon passes by) and m is the body mass (in the photon case, zero).

Cancelling out the m's, you get a = G*M/r^2

So you see the photon accelerates in the presence of a gravitational field, even in newtonian theory.

2) I can think of numerical methods to calculate the deflection of a particle in a gravitational field. But how would one analytically solve the problem? I'm imagining a small particle (maybe a satellite) traveling at a certain velocity that passes near a large massive object, (like a planet). I'm thinking in terms of momentum conservation, angular momentum, energy conservation, and I can't see a way to setup the problem...

In G.R., you can just solve the geodesic equation in the appropriate metric (which is likely the schwarschild metric if the planet in question is spherically symmetric, uncharged, and has small angular momentum).

In newtonian theory, it's simpler; you just integrate along the particle path.

3) so then how about light (as above?) ?

Same as for massive body (the mass doesn't matter).

marten
Posts: 134
Joined: Thu Aug 30, 2007 10:21 am

Postby marten » Thu Jan 17, 2008 1:37 pm

Thanks for the reply, surjective. Helpful and informative.

Marten

Bean
Posts: 20
Joined: Thu Jul 29, 2010 7:35 pm

Re:

Postby Bean » Sat Jul 31, 2010 4:13 pm

surjective wrote:1) How does Newton's theory predict the bending of light in the first place? Light doesn't have mass, so it can't be gravity. Is it other effects like diffraction?

In newtonian theory, the acceleration of a test body in a gravitational field is independent of it's mass (same with G.R. - this is the equivalence principal) :
f = m*a = G*m*M/r^2, where M is the source mass (ie: the body the photon passes by) and m is the body mass (in the photon case, zero).

Cancelling out the m's, you get a = G*M/r^2

So you see the photon accelerates in the presence of a gravitational field, even in newtonian theory.

2) I can think of numerical methods to calculate the deflection of a particle in a gravitational field. But how would one analytically solve the problem? I'm imagining a small particle (maybe a satellite) traveling at a certain velocity that passes near a large massive object, (like a planet). I'm thinking in terms of momentum conservation, angular momentum, energy conservation, and I can't see a way to setup the problem...

In G.R., you can just solve the geodesic equation in the appropriate metric (which is likely the schwarschild metric if the planet in question is spherically symmetric, uncharged, and has small angular momentum).

In newtonian theory, it's simpler; you just integrate along the particle path.

3) so then how about light (as above?) ?

Same as for massive body (the mass doesn't matter).


There is an initial mathematical step you forgot in 1): due to the photon being massless, when one divides the Newtonian force by the mass, one has to use L'Hopitale's (bleeping spelling, not to mention that I can't see a single letter I'm typing :x ) rule resulting in (if use Newton's 2nd law) acceleration+((d/dm)acceleration)m->acceleration. While it makes little to no difference in your end result it is still required to avoid mathematicians calling you up at 3 or 4 in the morning and ranting at you ( :) ). And, inconsequentially, I would like to point out that while Newtonian can do this, GR makes more physical sense when setting up this problem as there you can just think of it as moving coordinates (the photon I mean) as opposed to the Newtonian requirement that it be "something" (mass/particle/matter/antimatter(with assumptions)/what-not).
Last edited by Bean on Sat Jul 31, 2010 9:10 pm, edited 1 time in total.

CarlBrannen
Posts: 381
Joined: Mon May 24, 2010 11:34 pm

Re: Light bending computation, Einstein vs Newton

Postby CarlBrannen » Sat Jul 31, 2010 6:43 pm

It might be useful to see how Einstein's gravity looks in Newtonian form. That way you can see exactly which term gives the bending and how it differs: http://arxiv.org/abs/0907.0660

Bean
Posts: 20
Joined: Thu Jul 29, 2010 7:35 pm

Re: Light bending computation, Einstein vs Newton

Postby Bean » Sat Jul 31, 2010 10:03 pm

CarlBrannen wrote:It might be useful to see how Einstein's gravity looks in Newtonian form. That way you can see exactly which term gives the bending and how it differs: http://arxiv.org/abs/0907.0660

Response: A) *muttering* show-off
B) interesting
:o




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