Will anti-parallel light beams between two stars stay parallel?

[KurtLudwig]

TL;DR

In our galaxy, two stars, A and B, move about 20 km/s in slightly different directions to each other. The stars are about 20 light years apart. Will light beams emitted from star A travelling towards light beams emitted from star B stay anti-parallel to each other or will they eventually diverge?

In our galaxy, two stars, A and B, move about 20 km/s in slightly different directions to each other. The stars are about 15 light years apart. Initially, the light beams are exactly anti-parallel. Will light beams emitted from star A traveling towards light beams emitted from star B stay anti-parallel to each other or will they eventually diverge?
Will these light beams move along a common geodesic line?

 

[PeterDonis]

Will light beams emitted from star A traveling towards light beams emitted from star B stay anti-parallel to each other or will they eventually diverge?

Do you mean, diverge due to the gravity of the stars themselves? Or due to something else?

Will these light beams move along a common geodesic line?

Not in spacetime, no, since they are moving in opposite directions.

 

[pervect]

Light from stars isn't focused into a beam - there is usually diffraction. The approximation that ignores diffraction (but doesn't include gravity) is geometric optics. The approximation that also include the effects of gravity would be the "null dust" approximation. The presence of the stars complicates the calculation. How important are the stars to what you are trying to ask?

If your question is more related to anti-parallel beams in flat space-time with nothing else (so no need to include the stars), and if you also don't mind making the geometric optics/ null dust approximations, it is generally true that anti-parallel beams do attract, while parallel beams do not.

One way of understanding this odd behavior is that there is a coulomb-like force between the beams and a magnetic-like force between them. In the case of parallel beams, the two force components cancel, in the case of the anti-parallel case they add together. This is a bit oversimplified, but IIRC the basic idea was published in a peer-reviewed paper I read, though I can't quote it offhand. One reason this is oversimplifed is that it treats gravity as a force, so some of the more subtle elements of the theory get lost. However, it's probably more understandable expressed in the "force" paradigm.

Gravitoelectromagnetism <wiki link>, a formal analogy between Maxwell's equations and linearized, low-speed gravity, is IIRC the context for how we can make an analogy between the coluomb forces between charges, and the 1/r^2 force between masses. In this anaology, we have an equivalent gravitational effect to the "magnetic force". In electromagnetism, the magnetic force occurs between moving charges (i.e. currents), in GEM the gravitomagentic force occurs between moving masses, i.e. "mass currents".

A concern of mine, which I remember having when I read the paper in question, is that GEM requies low speeds as well as linearized gravity, and light is of course not a "slow speed". I don't recall exactly how well the paper addressed this point - I think at the time I was annoyed by the omission, but I don't have the link to the paper to refreresh my fallible memory.

 

[PeterDonis]

The approximation that ignores diffraction (but doesn't include gravity) is geometric optics.

I think by "gravity" here you mean "spacetime curvature due to the stress-energy of the light beam". It is perfectly possible to use the geometric optics approximation in a curved spacetime--for example, in the standard analysis of bending of light by the Sun. It just treats the light beam itself as a "test object" that has no effect on the spacetime geometry.

The approximation that also include the effects of gravity would be the "null dust" approximation.

More precisely, the approximation that also includes the effects of the stress-energy of the light itself.

 

[pervect]

I was thinking about this probelm a bit more, and probably the closest thing to the original question that could actulally calculated analytically is to consider a case with one star, and an idealized laser. We ask how the star(s) deflects the laser. We can then compare a dark star, not emitting radiation, to a bright star, that is, and compare the trajectories of the laser.
When we compapre the path of our test laser beam between the dark star and the radiating star, we find that it takes a slightly different path. Under most circumstances, I'd expect that the there would be only a very very tiny effect due to the radiation. I haven't run any numbers, though.

Technically, for the dark star we use a Schwarzschild metric (an idealized non-rotating star), and for the bright star emitting radiation we use the Vaidya metric. To compute the path of the idealized light beams, we'd just calculate the null geodesics in each space-time geometry.

 

[KurtLudwig]

Thank you all for your responses. (I have learned again how little I know about physics. I am reading in blogs in Physics Forums, articles in Wikipedia and I am reading chapters in college textbooks.)
Let me restate the question: An observer near star A directs a laser towards an observer near to star B. The light from the laser and star light from star A that is parallel to the laser are directed towards star B. (Stars emit light in all directions, but in this case just consider the light that is parallel to the laser beam.) The observer at star B does the same towards star A. (I realize that there is a problem with what is meant by simultaneous.) These two stars are in our galaxy, but they are moving in slightly different orbits and at slightly different velocities. The observers will continue see each others stars, but will they see each others lasers? That is, will they see that star light which was parallel to the emitted laser light? (Providing such an intense laser can be built and a telescope powerful enough is available to the observers.)

 

[PeterDonis]

An observer near star A directs a laser towards an observer near to star B.

How does the observer at star A determine in which direction to point his laser?

The observer at star B does the same towards star A.

Same question as above for the observer at star B.

The observers will continue see each others stars, but will they see each others lasers?

This question can't be answered until you answer the two questions above. Without those answers your scenario is incompletely specified.

 

[pervect]

The following diagram might help. It illustrates only one aspect of your problem, gravitational lensing.

Star A, at the bottom of the page, emits light towards star B. We use the geometric optics approximation to treat the path the light takes.

The gravity of star B "bends" the light beams. (Feel free to say that the space-time curvature of star B bends the light beams if you prefer. I think saying that gravity bends the light beams is a better choice for the level of this thread). Beams that approach star B more closely bend more. The diagram greatly exaggerates the bending efffect ans was not computed in any way, it's very qualitiative.

I've only drawing a few possible light beams. There are actually an infinite number of them, of course, so if you pick a particular observer, there will generally be a light beam passing through the observer, that light beam will show what the observer sees. Exceptions exist of course, one of the simplest example is that beams that are emitted from star A directly towards star B will hit the surface of star B and stop. THis means that an observer "behind" star B can't see star A, because star B is in the way.

Exactly where you observers should be on this diagram is unclear to me. In general, though, I'd say that the laser beam would have to be tilted to be non-parallel to the light from the star to be visible. This is true even in the absence of gravitational lensing.

If you draw a similar picture, and use the rule that "closer light beams are bent more", you can possibly answer the part of your question that relates to gravitational lensing. To draw such a diagram you would need to have a basic familiarity with geometric optics (though you might be familiar without being familiar with what it's called). I can only hazard a vague guess as to what you might or might not be familiar with, based on what questions you ask. The keywords might serve as a good guide for further research on your part, though.

There is another effect, though, due to the motion of the stars, called stellar aberration. I'm not going to go into that effect, though, it would be too much to cover. But it's potentially relevant to your question.

You asked about the effect of the radiation from star B. The basic effect of the radiation is that it would increase the amount of gravitational lensing as compared to a star that did not radiate. The effect should be very minor and may be unmeasurable in practice.

 

[KurtLudwig]

Please let me study up on gravitational lensing and stellar aberration before I reply.
I can see a problem with the idea of directing a laser from star B towards star A, since an intelligent observer is needed at star B, which is not possible. Also, how to establish when to fire the lasers at stars A and B. What is simultaneity on stars 20 light years apart?
I will have to restate the problem.
Geometric optics assumes that light moves in a perfectly straight line, as taught in an introductory physics course, with lenses and mirrors.

 

[PeterDonis]

What is simultaneity on stars 20 light years apart?

Simultaneity is a convention. Since the stars are in relative motion, there is no natural convention for simultaneity in the problem you give.

Geometric optics assumes that light moves in a perfectly straight line

In flat spacetime, yes. But your scenario is not set in flat spacetime. In curved spacetime, geometric optics assumes that light moves on null geodesics of the spacetime geometry.

 

[KurtLudwig]

Thank you. I think I need to understand special relativity and general relativity. Can you recommend an introductory book on these subjects?

 

[PeterDonis]

I think I need to understand special relativity and general relativity. Can you recommend an introductory book on these subjects?

You might try Sean Carroll's online lecture notes:

https://arxiv.org/abs/gr-qc/9712019

They are more focused on GR, but cover SR in the early parts as well.

 

[KurtLudwig]

Thank you again

 

[PeroK]

Thank you. I think I need to understand special relativity and general relativity. Can you recommend an introductory book on these subjects?

Morin has the first chapter of his book on SR online:

https://scholar.harvard.edu/files/david-morin/files/cmchap11.pdf

 

[pervect]

Please let me study up on gravitational lensing and stellar aberration before I reply.
I can see a problem with the idea of directing a laser from star B towards star A, since an intelligent observer is needed at star B, which is not possible. Also, how to establish when to fire the lasers at stars A and B. What is simultaneity on stars 20 light years apart?
I will have to restate the problem.
Geometric optics assumes that light moves in a perfectly straight line, as taught in an introductory physics course, with lenses and mirrors.

For some purposes, you can treat gravitational lensing just like regular lensing, i.e. you can treat space in a particular coordinate system as if it had a diffractive index.

I don't generally recommend that approach, really, as I am in favor of coordinate independent methods. It could still be useful in this particular case, though I am concerned that it might end up causing confusion in the end.

The lensing model invites one to think of the space as obeying Euclidean geometry, with the light being defelected by "lensing effects" - and one also usually ignores the entire question of the behavior of time. That's not really what happens. If one is careful not to ask questions that involve the underlying spatial geometry, it can give correct answers to finding the paths of light beams. It invites one to get incorrect answers about distances and angles, however - it can misleads one on the correct answers to those sort of questions. It also won't address the answers to questions that involve time.

Using this method of gravitational lensing in it's domain of applicability, it's fine though. The point is that the lensing effect due to a single dark mass is different from the lensing effect of a single mass that is emitting radiation - though the difference is usually minor.

Furthermore, if one has two masses, regardless of whether they are emitting light or not, there is some lensing model that describes the behavior of light around the pair of masses . But the calculational details are difficult. What one needs is the metric associated with the pair of light emitting bodies. The full non-linear calculation is very hard, and even the linearized approximation isn't easy.

As far as special relativity goes - my favorite introductory approach is Bondi's "Relativity and Common Sense", which is a very good popularization that has some actual mathematical content. However, that treatment won't really teach you what you need to know to lead into GR. The approach that would most naturally lead into GR is the treatment given in Taylor & Wheeler's "Space Time Physics". In particular, their "Parable of the Surveyor" talks about one of the fundamental points, the idea that we should talk about space-time in a unified manner , rather than as two separate concepts.

GR is a very advanced topic, though. Don't be too dissapointed if you don't get very far with it. There aren't many popular level GR books I could really recommend. Geroch's "General Relativity from A to B" is OK as far as a popularization goes, but I can't say that reading it will allow you to answer your question or help you understand anything in this thread.

I am partial to MTW's "Gravitation" for a serious treatment of GR. But it's not a popularization at all - it's a graduate level textbook. It does have a chatty style, and you might be able to get something from the chatty portion of it, but understanding the meat of it still requires a graduate level background. There are also free alternatives, such as Caroll's online lecture notes that have been previously mentioned. Caroll isn't chatty at all, but it may illustrate the difficulty of the topic.

 

[KurtLudwig]

I did start to read "Gravitation" at a local university library, until it closed to due the corona virus pandemic. (A big black book by Misner, Thorne and Wheeler.) This is absolutely the most difficult book that I have ever read. Among my many deficiencies is the level of mathematics required to understand the mathematics presented within this book. (You must be three orders of magnitude above me in mathematics.)
Also, I did not find anything about quantum gravity in "Gravitation". Granted, gravity and space-time are fully described by the theory of General Relativity. GR has been experimentally verified many times. (Still, I tend to believe, that Newton's Law of Universal Gravitation may not complete, as observed by the rotational velocities of outlying stars. The ideal gas law was not complete until Van der Waals forces were taken into account.)
Bondi's "Relativity and Common Sense" may a better entrance level for me.
Thank you for you advise on books.

 

[PeterDonis]

(A big black book by Misner, Thorne and Wheeler.)

The old joke used to be that its size and weight was an attempt to provide experimental evidence for GR by having the book undergo gravitational collapse and become a black hole.

This is absolutely the most difficult book that I have ever read.

I think you will find this to be a common opinion among its readers. (I don't know that I'd say its the most difficult book I've ever read, but it's certainly near the top.)

 

[PeterDonis]

I did not find anything about quantum gravity in "Gravitation".

There are some brief discussions which are relevant to quantum gravity (I give some examples below), but the book is not a QG textbook so no, it doesn't really try to teach that subject.

Some examples of brief discussions in MTW relevant to QG:

Route #5 of the six routes to the Einstein Field Equation described in Box 17.2. This is the "spin-2 field on flat spacetime" model that was investigated in the 1960s and early 1970s as an attempt to find a quantum field theory for the graviton, by analogy with the known QFT of a spin-1 field, the photon. This theory is valid mathematically, but is not considered a good candidate for a fundamental QG theory.

Route #6 in Box 17.2 describes another, different early attempt to base gravity on some form of quantum theory: the idea that spacetime itself might be modeled similarly to an elastic medium, with something more fundamental arising from particle physics underlying it. AFAIK it has not made any significant headway since, although IIRC you will find some posts by @Demystifier in the Quantum Physics forum that refer to similar models.

Chapter 43 on Superspace, which is an alternative way of formulating GR that leads to a straightforward equation for the quantum dynamics of a 3-geometry, the Wheeler-DeWitt equation, that can be thought of as an analogue to the ordinary Schrödinger Equation of QM for something like the entire universe. I believe there is still a small community of cosmologists working on ideas that came from this, but I don't know if their work has much impact overall.

 

[pervect]

I did start to read "Gravitation" at a local university library, until it closed to due the corona virus pandemic. (A big black book by Misner, Thorne and Wheeler.) This is absolutely the most difficult book that I have ever read. Among my many deficiencies is the level of mathematics required to understand the mathematics presented within this book. (You must be three orders of magnitude above me in mathematics.)
Also, I did not find anything about quantum gravity in "Gravitation". Granted, gravity and space-time are fully described by the theory of General Relativity. GR has been experimentally verified many times. (Still, I tend to believe, that Newton's Law of Universal Gravitation may not complete, as observed by the rotational velocities of outlying stars. The ideal gas law was not complete until Van der Waals forces were taken into account.)
Bondi's "Relativity and Common Sense" may a better entrance level for me.
Thank you for you advise on books.

Graduate level books _are_ difficult, and "Gravitation" is a graduate level book. I have no better warnings for difficulty than to say that a book is graduate level, and hope the person knows what I mean by that. Having looked, you now have some idea of what that entails.

As difficult as "Gravitation" is to fully understand, it also has some good insights at a lower level of difficulty, what I called the "chatty" sections. For instance, the first section of the first chapter is "The Parable of the Apple".

Space-time physics is a lot less difficult than "Gravitation" - though it will only cover special relativity. And it's not the easiest introduction to SR, but it is a treatment of SR that is more useful for learning GR than other easier approaches to SR.

And yes, "Gravitation" does not cover quantum gravity at all, just the classical theory.