Math light bending by gravity

[hover]

Hey all.
I was wondering what is the math behind calculating how much light is bent by gravity/curvature of space. Can someone teach me??

thanks

 

[Paulanddiw]

Are you interested in the the bending of starlight from conservation of energy and angular momentum, or do you want to see the relativistic approach?

 

[yogi]

If your question pertains to the amount of space curvature for a given amount of matter, the excess radius is MG/3c^2 where M is the mass of the object (say earth), G is the gravitational constant and c^2 is the square of the velocity of light. If you put in the numbers, it is seen that the entire mass of the Earth only modifies the effective radius by about 1.5 millimeters

 

[hover]

Are you interested in the the bending of starlight from conservation of energy and angular momentum, or do you want to see the relativistic approach?

I'd prefer the relativistic approach.

 

[hover]

If your question pertains to the amount of space curvature for a given amount of matter, the excess radius is MG/3c^2 where M is the mass of the object (say earth), G is the gravitational constant and c^2 is the square of the velocity of light. If you put in the numbers, it is seen that the entire mass of the Earth only modifies the effective radius by about 1.5 millimeters

I kind of don't know what your saying. What does that equation do? I'm curious now.

Anyway my real question is how can you calculate the bending of light passing a massive object, for example the sun??

 

[cesiumfrog]

Are you interested in the the bending of starlight from conservation of energy and angular momentum, [..]?

Sure! I'd like to know how that gives you the result, could you at least outline it?

 

[pervect]

Hey all.
I was wondering what is the math behind calculating how much light is bent by gravity/curvature of space. Can someone teach me??

thanks

You'll need at least basic differential equations. There's a series of formulas using different variables in for instance, MTW "gravitation" pg 673 for the orbit of light in the Schwarzschild geometry (that is, near a single large mass).

The orbit is determined by a single constant, the "impact parameter" b.

It has various forms, it will look something like the following differential equation which expresses r as a function of $\phi$ in the $\theta=0$ plane. r, $\theta$, and $\phi$ are all Schwarzschild coordinates, which are derived from the familar "spherical coordinates" by the same names.


$$\left( \frac{1}{r^2} \frac{dr}{d\phi} \right)^2 + \frac{1-2M/r}{r^2} = \frac{1}{b^2}$$

 

[Paulanddiw]

Hey, cesiumfrog,
Thanks for asking. Here's how I got the experimentally determined value of bending using angular momentum and energy conservation http://alzasol.spaces.live.com/blog/?&_c02_owner=1 [Broken]

There's a drawing at the bottom of the text. You'll have to klick on it.

 

[yogi]

I kind of don't know what your saying. What does that equation do? I'm curious now.

Anyway my real question is how can you calculate the bending of light passing a massive object, for example the sun??

There is a lot of math that goes into the derivation of the deviation of a light beam passing by the surface of a massive object - during the development of GR, Einstein had first reached a wrong conclusion about the deviation - later he realized his error and published the correct deviation which has been verified by subsequent experiments - the correct value is twice that which would be calculated using Newtonian approximations - in a recent thread, I think it was Space Tiger that posted a clever Newtonian derivation using the notion of an impulse function which is a short cut way of arriving at would happen if one calculates the force at each instant using Newtonian gravity. But even these methods require a bit of mathematical maturity to be able to appreciate what is being said -

The bottom line is that the spatial deviations are very small even for massive bodies - for the Earth the change in the spatial curvature to which I referred is very small - and in the case of the Sun the deviation of light grazing its surface is was barely measurable in 1919 when the theory of GR was first put to the test. But according to GR, both space and time are affected by matter - and it is not at all difficult to set up experiments that measure the temporal changes - for example GPS satellites are corrected for altitude before launch - the change in clock rates for even small differences in height are easily measured and routinely correct in order to synchronize the satellite clocks with GPS receivers on the surface of the earth

 

[lightarrow]

Hey, cesiumfrog,
Thanks for asking. Here's how I got the experimentally determined value of bending using angular momentum and energy conservation http://alzasol.spaces.live.com/blog/?&_c02_owner=1 [Broken]
There's a drawing at the bottom of the text. You'll have to klick on it.

I saw that document. There is something I would like to ask you: is it really possible to write c^2 = c_x^2 + c_y^2? If a body has velocity components v_x = 0.9c and v_y = 0.9c, then its speed should be c*Sqrt(2*0.9^2) = 1.27c > c So, it's impossible for a body to have such velocity components?

 

[pervect]

I saw that document. There is something I would like to ask you: is it really possible to write c^2 = c_x^2 + c_y^2? If a body has velocity components v_x = 0.9c and v_y = 0.9c, then its speed should be c*Sqrt(2*0.9^2) = 1.27c > c So, it's impossible for a body to have such velocity components?

Unfortunately, the whole derivation in that document is a mess as far as getting any accurate answers go.

If you consder the energy-momentum 4-vector, it does turn out that there is a conserved energy $p_0$ and, assuming that the orbit is in the $\theta=0$ plane, a conserved anguler momentum $p_\phi$, but the correct relativistic forms of these conserved quantities aren't given by the URL above, which is someones ambitious but unfortunately not technically correct attempt at solving the problem.

One good online source that gives the right answers for the orbits of material bodies http://www.fourmilab.ch/gravitation/orbits/, though it doesn't really go into the correct derviation of the formulas for energy and angular momentum.

Unfortunately it doesn't directly handle the OP's question of the orbit of light, though one can get that orbit by taking the orbit of a material body in the limit as it approaches 'c'.

As far as lightarrow's question about velocities goes, in flat space-time it is true that v^2 = (dx/dt)^2 + (dy/dt)^2, however the space-time for the Schwarzschild geometry isn't flat, and this formula doesn't work right there.

 

[pervect]

http://arxiv.org/abs/gr-qc/0506075" [Broken] is another reference for a correct way of deriving the equations of motion of a particle in the Schwarzschild geometry near a large mass.

Basically, the easiest way to get the right answer is to use a Lagrangian approach, and to find the path that extremizes "proper time" - this has also been called the "principle of maximal aging".

http://www.eftaylor.com/pub/GRtoPLA.pdf is also useful in describing the idea, but rather than find an exact solution it demonstrates that the principle of least action (maximal aging) yields the Newtonian answer in the non-relativistic limit.

There are some more references at http://www.eftaylor.com/leastaction.html#actionsummary

about the general idea of the "least action" approach to physics.

 

[Paulanddiw]

I saw that document. There is something I would like to ask you: is it really possible to write c^2 = c_x^2 + c_y^2? If a body has velocity components v_x = 0.9c and v_y = 0.9c, then its speed should be c*Sqrt(2*0.9^2) = 1.27c > c So, it's impossible for a body to have such velocity components?

Yeah, you are right.

 

[MeJennifer]

Basically, the easiest way to get the right answer is to use a Lagrangian approach, and to find the path that extremizes "proper time" - this has also been called the "principle of maximal aging".

One interesting thing here to ponder on is, since the proper time of a light beam is always zero how would we find a single path that extremizes proper time.

Would you agree that that approach has some issues?

 

[George Jones]

One interesting thing here to ponder on is, since the proper time of a light beam is always zero how would we find a single path that extremizes proper time.

Would you agree that that approach has some issues?

A geodesic can be parametrized by something other than proper time.

 

[MeJennifer]

A geodesic can be parametrized by something other than proper time.

Which, including their physical interpretations, do you consider good parameterization options for null curves?

 

[coalquay404]

Which, including their physical interpretations, do you consider good parameterization options for null curves?

The mistake you're making is an implicit assumption that the action principle for the equations of motion of a particle are unique. This is obviously untrue. Besides the obvious choices for massive particles where one supposes that the value of the action is proportional to the proper length, there are myriad other possibilities. One can, if one wishes to be fancy about it, construct an action principle by demanding that the action be stationary with respect to harmonic functions of compact support defined along the world line. Alternatively, you could just go the old route of introducing an einbein along the world line and demanding that the action be stationary with respect to variations in it; this is covered, for example, in the first string theory lecture every student receives as a motivating example for the Brink-Di Vecchia-Howe action for a bosonic string.

 

[pervect]

Another approach I've seen is to simply take the limit for the path of a massive particle as m->0. For instance, a neutrino has mass according to recent observations, but the path it follows is essentially the same path that light follows.

Using an einbein (sidenote: we've disucssed einbein's in https://www.physicsforums.com/showthread.php?p=1054196 ) has a lot of advantages, too. Even in the case of a massive particle, taking the direct route of minimizing the action (which can be expressed as the square root of a quadratic form, because proper time squared is a quadratic form) gets very messy due to the square root.

However, while it's convenient computationally, einbein's may not be as familiar to people as the approach using the calculus of variations, so it might not be the first choice if one is trying to present the material at an undergraduate college level. Unfortunately, I don't think there is any way to approach the topic using only high-school level math - one really needs calculus!

 

[MeJennifer]

Another approach I've seen is to simply take the limit for the path of a massive particle as m->0. For instance, a neutrino has mass according to recent observations, but the path it follows is essentially the same path that light follows.

Pervect, I particularly have trouble with the word "essentially" in this context. To me it is far from obvious that we can say that. I would be more inclined to say that a lightlike path is essentialy different from a timelike path.

Now if someone would refer me to some paper that proves that a lightlike path is simply the limit of a timelike path on a manifold with a Minkowski like metric, then I would be happy. But honestly, I am not holding my breath.

Actually, while we are at it, I would not be surprised that someone conjured up a proof that the contrary is true!

 

[George Jones]

Which, including their physical interpretations, do you consider good parameterization options for null curves?

Not all good parameterizations have obvious physical interpretations, but for null curves, one that that does is the parameterization that makes the tangent 4-vector to the curve equal to the four-momentum.

 

[pervect]

Now if someone would refer me to some paper that proves that a lightlike path is simply the limit of a timelike path on a manifold with a Minkowski like metric, then I would be happy. But honestly, I am not holding my breath.

I've seen this approach used in MTW's "Gravitation", a standard textbook.

There are probably ways to understand it that are mathematically more precise, but I find MTW's approach of taking the limit as mass goes to zero very intuitive.

The way I see it, the problem is basically this. Suppose you have two points, AB, close together, which are connected by a null geodesic.

You can draw a large number of null geodesics from point B, going in different directions.

But only one of these null geodesics originating at B is the "correct" continuation of AB.

I think that is is probably necessary and sufficient for AB, BC, and AC to all be null intervals for BC to be a "continuation" of AB. But I haven't really thought this through.

[add]There is at least one issue here. C could be coincident with A, for instance. One probably needs to add some requirement that B must be "between" A and C.

 

[MeJennifer]

I've seen this approach used in MTW's "Gravitation", a standard textbook.

There are probably ways to understand it that are mathematically more precise, but I find MTW's approach of taking the limit as mass goes to zero very intuitive.

Do you have a page number for me, I take a look at it.

So, for instance are you saying that the limit, of say the velocity addition formula, where u and v approach c, equals c, rather than being undefined?

 

[pervect]

Do you have a page number for me, I take a look at it.

pg 672, section $25.6 "Orbit of a photon, neutrino, or graviton in Schwarzschild geometry".

Note that this was written back in the days when the neutrino was still thought to be massless.

The concepts of "energy per unit of rest mass" and "angualr momentum per unit of rest mass" make no sense for an object of zero rest mass (photon, ...). However, there is nothing about the motion of such an entity that cannot be discovered by considering the motion of a particle of rest mass u and taking the limit as u->0.

 

[MeJennifer]

pg 672, section $25.6 "Orbit of a photon, neutrino, or graviton in Schwarzschild geometry".

Note that this was written back in the days when the neutrino was still thought to be massless.

Thanks for looking that up!