Is there a simple equation for the trajectory of light in a gravitational field?

[lovetruth]

The light should accelerate towards the earth, the same way as massive objects are attracted towards the earth[General relativity]. We may not find path of the light to be curved because of the enormous velocity of light, the curvature of light is too subtle for our senses.

Consider a thought experiment: There are two vertical mirrors on the earth. If a horizontal light pulse is shone on one mirror, the light pulse will repeatedly bounce between the two mirrors. Since the gravity is acting on the light pulse, the light pulse will began to move downwards with the passage of time. The vertical position of the light pulse can be given by:
s = ut +0.5g(t^2)

Is there any flaw in my thought experiment? If not, is this done experimentally by anyone yet?

 

[ctxyz]

The vertical position of the light pulse can be given by:
s = ut +0.5g(t^2)

No, it won't.

Is there any flaw in my thought experiment? If not, is this done experimentally by anyone yet?

Quite a few, the biggest one being that you are trying to apply Newton's dynamics to massless particles (photons). If you really want to know how light moves in a gravitational field, I could recommend a few books.

 

[bcrowell]

If you really want to know how light moves in a gravitational field, I could recommend a few books.

Yep.

Lovetruth, I have suggested previously to you that it is not a good idea to keep posting speculations about relativity and expecting other people to debug them, and that you should start by building a foundation of knowledge of the basics of special relativity, including essential subjects like the Lorentz transformation. A good place to start would be Spacetime Physics, by Taylor and Wheeler. The present topic relates to general relativity, and in my opinion you still need to fill in your rudimentary knowledge of SR before attempting GR. PF is not a university, nor is it a substitute for reading books. You cannot get a coherent introduction to relativity by posting speculations on PF for others to debug.

 

[Bill_K]

The vertical position of the light pulse can be given by: s = ut +0.5g(t^2)

No, it won't.

Yes it will. To do otherwise would violate the principle of equivalence.

 

[lovetruth]

Yes it will. To do otherwise would violate the principle of equivalence.

Thanks Bill_k. You understood my question very clearly.

I don't know the exact maths of the GR and can not solve tensors equations. But I have the physical insight or mental picture to grasp the problem at hand which you have perfectly recognized.

So is my thought experiment correct. And most importantly, is there any similar experiment done yet.

 

[bcrowell]

Yes it will. To do otherwise would violate the principle of equivalence.

Only for small t. An easy way to see that it can't apply at all t is that if it did, the ray of light would eventually have a vertical velocity greater than c.

If you want to have a discussion of this, we can, but it's not going to be a discussion at a level that lovetruth is ready for. Unless s/he has really hit the books in the last week or so, s/he doesn't know basics like the Lorentz transformation.

Essentially the reason that the argument based on the e.p. fails is that GR doesn't have any spacetime that satisfactorily embodies the notion of a uniform gravitational field. You can use a flat spacetime described in Rindler coordinates http://en.wikipedia.org/wiki/Rindler_coordinates , but the proper acceleration is nonuniform. There is something called the Petrov metric http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.4 , but it has some quirky properties like CTCs.

 

[bcrowell]

I don't know the exact maths of the GR and can not solve tensors equations.

Nobody expects you to master tensors and GR overnight. But you do need to master basics like the Lorentz transformation. Otherwise we'll have a repeat of the kind of situation we've had in threads like this one https://www.physicsforums.com/showthread.php?t=508275 . In that thread, you got several other people to lead you through a long, complex calculation. Then in this thread https://www.physicsforums.com/showthread.php?t=511548 it became evident that you didn't know what a Lorentz transformation was or what it was used for, so that clearly you had no idea what was being done for you in the earlier thread.

 

[Dale]

Only for small t.

And also only for a non-inertial reference frame which is accelerating at g wrt some local inertial frame.

The same is also true for flat spacetime so it actually has little to do with gravity and more to do with coordinates.

 

[lovetruth]

Nobody expects you to master tensors and GR overnight. But you do need to master basics like the Lorentz transformation. Otherwise we'll have a repeat of the kind of situation we've had in threads like this one https://www.physicsforums.com/showthread.php?t=508275 . In that thread, you got several other people to lead you through a long, complex calculation. Then in this thread https://www.physicsforums.com/showthread.php?t=511548 it became evident that you didn't know what a Lorentz transformation was or what it was used for, so that clearly you had no idea what was being done for you in the earlier thread.

I have read Lorentzian transformation from wikipedia after you encouraged me to. I will try to be more aware of the subjects before posting.

 

[atyy]

The light should accelerate towards the earth, the same way as massive objects are attracted towards the earth[General relativity]. We may not find path of the light to be curved because of the enormous velocity of light, the curvature of light is too subtle for our senses.

Consider a thought experiment: There are two vertical mirrors on the earth. If a horizontal light pulse is shone on one mirror, the light pulse will repeatedly bounce between the two mirrors. Since the gravity is acting on the light pulse, the light pulse will began to move downwards with the passage of time. The vertical position of the light pulse can be given by:
s = ut +0.5g(t^2)

Is there any flaw in my thought experiment? If not, is this done experimentally by anyone yet?

If you send out regular pulses of light at interval To from the origin. You seem to imagine that each pulse of light accelerates under the gravitational field. At a height h away from the origin, the pulses of light are received. What do you predict for the interval Tf between the arrival times of the light pulses?

 

[Bill_K]

Ben, I'm all for higher mathematics, and you may be right that the poster is short in that area. However I also recognize situations that do not call for math. I didn't say "valid for all t". And we don't need a solution that fills all of space. This is just in the neighborhood of a single point.

This is a classic elevator experiment. The basic assumption of general relativity is that at any point in any sufficiently small region, there is a set of local inertial reference frames in which the laws of special relativity are obeyed. In such a frame light rays travel along straight lines. At the Earth's surface an inertial frame falls with acceleration g, and an observer at rest will see the light ray falling with the frame, curving downward in a parabola.

Of course it's an incredibly small effect. Grace Hopper reminds us that light takes a nanosecond to travel one foot. If the two mirrors are one foot apart, the light ray will have time to fall about 10³(10^-9)² = 10^-15 cm, which is about a hundredth the diameter of a proton. But in principle at least, the mirrors would need to be tilted ever so slightly upward, to keep the ray bouncing back and forth on the same arcing path, like badminton players hitting a shuttlecock to each other.

Besides the smallness, there's another reason the effect has not (and probably cannot) be observed. We depend on light as the most accurate way of aligning things. It would be doubly difficult to align the mirrors with sufficient accuracy except by using light rays.

 

[lovetruth]

Only for small t. An easy way to see that it can't apply at all t is that if it did, the ray of light would eventually have a vertical velocity greater than c.

If you want to have a discussion of this, we can, but it's not going to be a discussion at a level that lovetruth is ready for. Unless s/he has really hit the books in the last week or so, s/he doesn't know basics like the Lorentz transformation.

Essentially the reason that the argument based on the e.p. fails is that GR doesn't have any spacetime that satisfactorily embodies the notion of a uniform gravitational field. You can use a flat spacetime described in Rindler coordinates http://en.wikipedia.org/wiki/Rindler_coordinates , but the proper acceleration is nonuniform. There is something called the Petrov metric http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.4 , but it has some quirky properties like CTCs.

So do you agree that light falls with an acceleration which is same as that experienced by a massive object towards the earth?

 

[ctxyz]

Yes it will. To do otherwise would violate the principle of equivalence.

Radial motion for light in a gravitational field does not follow the Newtonian equations, so the answer "no, it won't" stands. See for example Rindler, "Relativity:Special, General and Cosmological" (chapter 11.10) or MTW.

 

[lovetruth]

Ben, I'm all for higher mathematics, and you may be right that the poster is short in that area. However I also recognize situations that do not call for math. I didn't say "valid for all t". And we don't need a solution that fills all of space. This is just in the neighborhood of a single point.

This is a classic elevator experiment. The basic assumption of general relativity is that at any point in any sufficiently small region, there is a set of local inertial reference frames in which the laws of special relativity are obeyed. In such a frame light rays travel along straight lines. At the Earth's surface an inertial frame falls with acceleration g, and an observer at rest will see the light ray falling with the frame, curving downward in a parabola.

Of course it's an incredibly small effect. Grace Hopper reminds us that light takes a nanosecond to travel one foot. If the two mirrors are one foot apart, the light ray will have time to fall about 10³(10^-9)² = 10^-15 cm, which is about a hundredth the diameter of a proton. But in principle at least, the mirrors would need to be tilted ever so slightly upward, to keep the ray bouncing back and forth on the same arcing path, like badminton players hitting a shuttlecock to each other.

Besides the smallness, there's another reason the effect has not (and probably cannot) be observed. We depend on light as the most accurate way of aligning things. It would be doubly difficult to align the mirrors with sufficient accuracy except by using light rays.

Thanks again Bill_k for supporting my argument.

You are right about the very small vertical distance that a horizontal light will travel between two mirrors. What I have done ingeniously in my thought experiment is that the small vertical distances are amplified over time by repeated reflections. Thus, light and any massive object should take same time to drop from a certain height (Very much similar to Galileo experiment of dropping objects from tower of pisa).

 

[atyy]

What I have done ingeniously in my thought experiment is that the small vertical distances are amplified over time by repeated reflections. Thus, light and any massive object should take same time to drop from a certain height (Very much similar to Galileo experiment of dropping objects from tower of pisa).

Bill K answered your question for a small distance.
http://www.einstein-online.info/spotlights/equivalence_deflection

The experiment has been done over large distances. The answer over a large distance is "no".
http://www.theory.caltech.edu/people/patricia/lclens.html

I am still interested in what you calculate for my question in post #9.

 

[bcrowell]

However I also recognize situations that do not call for math. I didn't say "valid for all t". And we don't need a solution that fills all of space. This is just in the neighborhood of a single point.

The OP did not make such a qualification, and you did not make such a qualification in #4. That qualifiction doesn't suffice, either. The OP has an initial vertical velocity u, and u<<c is also required.

So do you agree that light falls with an acceleration which is same as that experienced by a massive object towards the earth?

No. It's not true for your original setup with the two mirrors, without the additional restriction to small t and u<<c.

As atyy pointed out with the link in #15, it also doesn't immediately generalize to cases other than your two-mirror setup. This is why the deflection of light by the sun was considered a crucial test of general relativity. If your statement that "light falls with an acceleration which is same as that experienced by a massive object towards the earth" had been true generically, then the same would have been true in the sun's field, Newton's laws would have given the correct result for deflection of light by the sun, and that famous test of GR would not have been considered a test of GR at all.

To understand why the restriction to vertical velocities much less than c is required, consider the case where a ray of light is sent straight up or straight down.

 

[atyy]

Bill K answered your question for a small distance.
http://www.einstein-online.info/spotlights/equivalence_deflection

The experiment has been done over large distances. The answer over a large distance is "no".
http://www.theory.caltech.edu/people/patricia/lclens.html

I am still interested in what you calculate for my question in post #9.

Just read the two links about the distinction between local and global light deflection.

Ignore what I wrote in post #9 (it's misleading).

 

[lugita15]

Essentially the reason that the argument based on the e.p. fails is that GR doesn't have any spacetime that satisfactorily embodies the notion of a uniform gravitational field. You can use a flat spacetime described in Rindler coordinates http://en.wikipedia.org/wiki/Rindler_coordinates , but the proper acceleration is nonuniform. There is something called the Petrov metric http://www.lightandmatter.com/html_books/genrel/ch07/ch07.html#Section7.4 , but it has some quirky properties like CTCs.

Is it a theorem of general relativity that no spacetime can have a uniform gravitational field, or is it just that no such solution to the Einstein equations has been found so far?

 

[yuiop]

To understand why the restriction to vertical velocities much less than c is required, consider the case where a ray of light is sent straight up or straight down.

Just for lovetruth's benefit I thought I would flesh this out a bit. A vertically falling photon has a velocity of c at all times as measured locally. In Schwarzschild coordinate terms the vertically falling photon has a coordinate velocity of c*(1-2GM/(rc^2)) and so in coordinate terms the photon is decelerating as it falls. Hence the complications if u or t is not very small.

It is also instructive to consider the case of particles orbiting a massive gravitational body such as black hole. The orbital velocity of a particle for a given radial coordinate can be predicted from the GR equations and this predicts that the orbital velocity of a particle at r=3GM/c^2 would be c as measured locally. (See http://en.wikipedia.org/wiki/Photon_sphere.) However a particle with non zero rest mass obviously cannot travel at the speed of light, but a photon can orbit at this radius so basically a photons "falls" in just the same way as a massive particle would, if it had that velocity, i.e the falling rate is independent of rest mass (for small masses).

GR predicts that the deflection of light by a massive gravitational body is twice that predicted by Newtonian physics, so presumably the deflection of massive particles at relativistic speeds is also twice that predicted by Newtonian calculations.

 

[bcrowell]

Is it a theorem of general relativity that no spacetime can have a uniform gravitational field, or is it just that no such solution to the Einstein equations has been found so far?

It depends on what list of properties you want it to satisfy. My own take on it is given in this book http://www.lightandmatter.com/genrel/ , in ch. 5, exercise 5 and section 7.4.

 

[lovetruth]

What is the trajectory of the light in gravitational field using the simple co-ordinate system for an observer on earth. Also, are laws of reflection obeyed in gravitational field.

 

[bcrowell]

What is the trajectory of the light in gravitational field using the simple co-ordinate system for an observer on earth. Also, are laws of reflection obeyed in gravitational field.

There is no simple, exact equation for this, although there are various approximations that could be used. If you really want to understand this topic, first you need to read a good book on special relativity. You can't understand general relativity without understanding SR first, and from what you've said recently, your current knowledge of SR sounds pretty sketchy (e.g., you say your source of information about the Lorentz transformation is the WP article). I don't recall whether you say you've had calculus. Assuming that you have, a good next step, if you're interested in trajectories of light rays in a spherically symmetric gravitational field, would be to read Exploring Black Holes by Taylor and Wheeler. Physics Forums and Wikipedia are not substitutes for reading books.