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Photons of different energy follow different geodesics?

  1. Jan 26, 2012 #1
    a massive body like a star creates a warped spacetime in its vicinty. this warped geometry of space is reflected by the geodesics appropriate to its mass. a photon passing by this massive object is not, as per GR, "attracted" to the star via some "force", but rather simply follows what it sees to be a "straight" line, while in reality that "straight" line is actually curved due to the mass of the star warping the surrounding space. ie, the photon is not accelerated due to a change in its apparent direction, since it thinks it is traveling in a straight line - it "feels" no force acting upon it. (i hope that is all correct...)

    if we start changing the energy of the passing photon, does this also change the geodesic which it follows past the star? ie, will a radio wavelength photon follow the same "straight" line as a gamma ray photon?

    i assume that if we replace "photon" in the question with "massive body" and consider the path of a body of small mass, such as an asteroid, compared to the path of a larger mass body such as a planet sized object, they will follow different paths because the mass of each will both be affecting the overall geodesics of the system. (correct?)

    since photons do not have "mass" do photons of different energy follow the same geodesic path, or does the energy of a photon also affect the overall warping of spacetime, such that different wavelngth photons follow different paths past the star?

    thanks.
     
  2. jcsd
  3. Jan 26, 2012 #2

    tom.stoer

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    photons of different energy (emitted in the same rest frame with different frequency) follow the same (light-like) geodesic; otherwise you would see a "gravitational rainbow effect"
     
  4. Jan 26, 2012 #3

    Bill_K

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    In principle, photons of different energies follow different paths, although it is a very small effect. For a very low energy photon, the size of it will be correspondingly large. Parts of it will be in different places and hence feel different attraction. In fact, part of it will even be on the opposite side of the attracting body.

    On the other hand a very high energy photon will distort the gravitational field. In fact both the photon and the attracting body will orbit a common center of mass.
     
  5. Jan 26, 2012 #4

    tom.stoer

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    Bill, now you are not talking about photons as pointlike particles but about wave pakets.
     
  6. Jan 26, 2012 #5

    PAllen

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    Well, there is a result in GR that two anti-parallel beams of light attract, while parallel, they don't. Clearly, in the former case, the light path would depend on energy (and while both paths would be lightlike, neither would be a geodesic of the background geometry).
     
  7. Jan 26, 2012 #6

    tom.stoer

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    Now you include back-reaction.

    I agree - as you said, it's no longer about geodesics in pure gravity
     
  8. Jan 26, 2012 #7
    yes. I think you are asking about a [theoretical] test particle geodesic:
    Here is how Wikiepdia explains it...

    http://en.wikipedia.org/wiki/Geodesic_(general_relativity)#Approximate_geodesic_motion
     
  9. Jan 27, 2012 #8
    A geodesic is essentially an one-body solution. First you take the start as if it is alone and obtain the spacetime distortion. Once you computed it (e.g. Schwarszchild) you place an object (test body) and assume that spacetime distortion is unchanged.

    This is an excellent approximation when mass (energy) of test body is much smaller than mass of the source body.

    Therefore, the geodesic does not depend of test body properties. Indeed light bending around sun depends only on Sun gravitational 'field'.

    In a more complete model you would solve the two-body problem, and obtain the two-body equation of motion, which of course is not a geodesic.
     
  10. Jan 27, 2012 #9

    pervect

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    Let's recap a bit:

    The null geodesic is independent of the energy of the photon.

    It's really only an approximation that photons follow geodesics - though it's a pretty good one. Sometimes elementary courses forget to mention that it's an approximation - this isn't really a terribly big issue because it's such a good approximation.

    So within the null geodesic approximation, light travels the same path regardless of energy. Because the null geodesic approximation is a very good one, we'd expect that any deviations from this, any gravitational wavelength dependent effects, would be very tiny under most circumstances.

    Let's add a bit of new material.

    The more exact result is the Papapetrou equations for a spinning particle. These can apparently be derived from the dirac equations in curved space-time (with some other assumptions), so I think it's reasonable to apply them to a photon.

    http://arxiv.org/abs/0810.0447 (I didn't study this paper terribly closely).

    As far as I know, the Papapetrou equations would be the best way to go beyond the null-geodesic approximation.

    I'd guess that the effects of a more exact analysis of the departure of photons from null geodesics would break down into some effects that depends on gravitational radiation being emitted by the photon which would probably be energy-dependent, and other effects due to the spin of the photon that probably wouldn't be energy dependent, but I can't really justify this rigorously so take it with a grain of salt.
     
  11. Jan 27, 2012 #10

    Matterwave

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    I don't believe the 2 body problem is solved, in general, for GR, is it?
     
  12. Jan 27, 2012 #11

    PAllen

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    It depends what you mean by solved. Exact solution for a dynamical case, no; accurate numerical solutions even extreme black hole and neutron star scenarios, yes. N-body problems are not analytically solved in Newtonian theory, but they are analyzed, in practice, all the time.
     
  13. Jan 27, 2012 #12

    Matterwave

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    Sure, I was just trying to say that without resort to numerical simulations of this situation, it seems difficult to give a good answer.
     
  14. Jan 27, 2012 #13

    PAllen

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    I found a couple of papers discussing and amplifying the old Tolman, Ehrenfest and Podolsky result I mentioned (anti-parallel light beams attract, parallel don't). The first considers the issue strictly classically. The second addresses quantum and spin issues (though published in a major journal, the second has not been cited yet, and I cannot vouch for its reliability; the first is well cited).

    http://arxiv.org/abs/gr-qc/9811052

    http://arxiv.org/abs/1009.3849
     
  15. Jan 27, 2012 #14

    PAllen

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    The first paper in my last post provides a cute numeric validation of the normal practice of ignoring these theoretical effects. They compare the impact on two anti-parallel laboratory lasers (10 cm apart) over a 3 km course of their self gravitation, compared to the influence on the beam paths from gravitational waves from the Virgo cluster. The former effect is over 80 orders of magnitude smaller!
     
  16. Jan 28, 2012 #15
    GR, as classical electrodynamics, cannot explain the general two-body problem. Both only can deal with two-body problem in certain approximated cases (as when one body is massive).
     
  17. Jan 28, 2012 #16

    PAllen

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    That's a rather pessimistic way of putting it. Equal mass neutron star binaries are regularly dealt with numerically in the last 10 years or so. Do you have to posit some approximate equation of state for the star? Of course. Does this mean the theory can't deal with it? Not as virtually anyone but you would phrase it.

    The literature is filled with references like the following:

    http://arxiv.org/abs/1103.3874


    [EDIT: "can't deal" and "requires approximation" are very different statements. Newtonian gravity requires approximation for the N-body problem. That is very different from saying Newtonian gravity can't deal with the N-body problem.]
     
    Last edited: Jan 28, 2012
  18. Jan 28, 2012 #17

    PAllen

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    Adding to this nice recap, one way (in principle) light deviates from null geodesics independent of gravitational radiation is simply the analog of two body motion. The moon and earth versus a satellite and earth are radically different. Similarly, an enormously energetic photon would be expected to follow a different path near earth than a low energy photon.
     
  19. Jan 28, 2012 #18
    I would not say pessimistic but realistic. GR, as classical electrodynamics, cannot deal with the N-body problem.

    Relativistic theories are essentially one-body theories and only can deal with two-body systems in those trivial cases when the two-body system can be somewhat reduced to an one-body problem. I cited a trivial case, that when one of the bodies is massive and the two-body problem is reduced to a trivial one-body problem (external field).

    The article that you cite is considering two massive bodies but is working in a special regime where the two-body problem is reduced to solving the (their own words) "effective-one-body" problem. Indeed nowhere in the paper they write a single two-body equation of motion... because does not exist such equation in GR.

    This is in contrast with Newtonian gravity, which can deal with the N-body problem. The Newtonian Hamiltonian (or Lagrangian) for a N-body system is well-defined, the N-body equation of motion as well, and the N-body solution can be given in closed form. Of course, although the N-body solution exists in Newtonian gravity, another issue is if you can obtain it in analytical form or if you have a powerful enough computer to obtain a numerical form, but the solution exists by well-established theorems.
     
    Last edited: Jan 28, 2012
  20. Jan 29, 2012 #19

    PAllen

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    I disagree with your interpretation of this paper and papers it references.

    First, note that problems setting up well defined N-body Lagrangian is at least annoying, but not necessarily fundamental for a theory. It may be viewed as the failure of the integrability of some systems covered by the field equations. If you declare this to be a requirement for a theory, it is a big problem; if you don't, it is just an issue.

    Second, the issue of 'equations of motion' is also not relevant, because it is a feature of GR field equations that these are not needed at all (unlike classical EM). The one body equations are derivable as an approximation from the field equations, but are completely inessential to application of the field equations to a physical scenario (in principle).

    Third, the article is comparing numerical evolution from two body initial conditions with analytic one body models as a test on the latter. It doesn't mention two body equations of motion because, indeed, they don't exist but are also unnecessary in GR. Note, for example, this key phrase in the description of the setup of the numerical evolution of the field equations: "As initial data we use quasi-equilibrium binaries generated
    with the multi-domain spectral-method code LORENE developed
    at the Observatoire de Paris-Meudon [38]"

    Is all of this extremely difficult, in practice? Yes. Is it a function of principle (taking a particular classical theory as is, to apply it on its own terms)? No

    So, I repeat, GR can handle N body problems, in principle, but the practice is exceedingly difficult.
     
  21. Jan 30, 2012 #20
    OP here - thanks to you all for an interesting discussion.
     
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