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Light Clock in a Gravity well ?

  1. Sep 6, 2009 #1
    Light Clock in a Gravity well ????

    Hi Thinking of the effect on a light clock taken down to regions of greater gravitational dilation . it occured to me that the only way there could be a decreased frequency was if light itself was moving slower. That the decrease in local speed would have to be equivalent to the degree of dilation , ultimately approaching zero speed entering a black hole?
    Is this correct?
    It also seemed like there might be spatial contraction of the clock itself according to the EP
    Is this considered to be the case???
    If this what is expected to be the case , is it a decrease in speed or somehow an increase in distance due to extreme curvature? Or is that a meaningless question??

    thanks
     
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  3. Sep 6, 2009 #2

    JesseM

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    Re: Light Clock in a Gravity well ????

    Only in a locally inertial coordinate system in an infinitesimal region of curved spacetime is the speed of light guaranteed to be c. A global coordinate system in curved spacetime is not inertial and thus there is no reason the speed of light should necessarily be c in such a system, just as the speed of light can be different than c in non-inertial coordinate systems in flat SR spacetime. In GR you are free to use any global coordinate system you like in curved spacetime, the Einstein field equations work in all of them so they are all equally valid physically, and they can say different things about the speed of light. For example, in the curved spacetime around a nonrotating black hole, if you use Schwarzschild coordinates the coordinate velocity of light decreases as you approach the horizon and actually reaches zero on it, whereas if you use Kruskal-Szekeres coordinates in the same spacetime, the coordinate speed of light is the same everywhere (see this page for a discussion of different coordinate systems that can be used in a spacetime containing a nonrotating black hole).
     
  4. Sep 6, 2009 #3
    Re: Light Clock in a Gravity well ????

    Thanks for the link, unfortunately it didn't work but I looked up Kruskal Szekeres
    but was unable to really follow the math. I got that light speed was uniform going into or away from a nonrotating BH but remained unsure if this also meant a uniform time metric throughout the gradient or whether this coordinate system was applicable under less extreme conditions.
    Isn t there any inherent relationship between light and gravitational spacetime in the fundamental field equations themselves? How was the gravitational deviation of light paths due to solar or cosmic masses derived??
    I assume that if a light clock was equiped with a light emmision device that pulsed per tick that this would be observable from an inertial frame at rest wrt the clock so the dilation would be empirically detectable regardless of coordinate system ? SO in that frame it would appear that light speed was slowed correct?? Or Not??
    Thanks
     
  5. Sep 6, 2009 #4

    JesseM

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    Re: Light Clock in a Gravity well ????

    Looks like the website I linked to was down for a bit, it seems to be back up again now. I can also give you some additional info on Kruskal-Szekeres coordinates here, a while ago I scanned some diagrams from Gravitation by Misner/Thorne/Wheeler for part of another discussion. You don't really need to know too much about the math (I don't) to get a basic conceptual understanding of the meaning of the diagrams.

    First of all, it helps to understand some of the weaknesses of Schwarzschild coordinates which are "fixed" by Kruskal-Szekeres coordinates. The first is that in Schwarzschild coordinates it takes an infinite coordinate time for anything to cross the horizon, even though physically it only takes a finite proper time for a falling object to cross the horizon. The second is that inside the horizon, Schwarzschild coordinates reverse the role of time and space--the radial coordinate in Schwarzschild coordinates is physically spacelike outside the horizon but timelike inside, while the time coordinate in Schwarzschild coordinates is physically timelike outside the horizon but spacelike inside. In Kruskal-Szekeres coordinates, in contrast, objects crossing the horizon will cross it in a finite coordinate time, and the Kruskal-Szekeres time coordinate is always timelike while its radial coordinate is always spacelike. And light rays in Kruskal-Szekeres coordinates always look like straight diagonal lines at 45 degree angles, while the timelike worldlines of massive objects always have a slope that's closer to vertical than 45 degrees.

    Here's one of the diagrams from Gravitation, showing the surface of a collapsing star (the black line bounding the gray area which represents the inside of the star) in both Schwarzschild coordinates and Kruskal-Szekeres coordinates. They've also drawn in bits of light cones from events alone the worldline of the surface, and the event horizon is shown as a vertical dotted line in the Schwarzschild diagram on the left, while it's shown as the line labeled r=2M at 45 degrees in the Kruskal-Szekeres diagram on the right (the sawtoothed line in that diagram represents the singularity).

    realisticBHkruskalsmall.jpg

    Here's another diagram from p. 835 of Gravitation, showing schematically how various paths through spacetime--timelike, lightlike, and spacelike--would look when plotted in both Schwarzschild and Kruskal coordinates. The Kruskal-Szekeres diagram here represents the "maximally extended" version of the Schwarzschild spacetime, which in addition to the exterior region and the black hole interior region also contains a white hole interior region at the bottom and an exterior region in "another universe" at the left--there was a thread discussing the meaning of this maximally extended solution, along with Kruskal-Szekeres diagrams in general, here.

    p835Gravitation.jpg

    Finally, here's a diagram from p. 834 which shows how lines of constant radius and constant time in Schwarzschild coordinates look when plotted in Kruskal Szekeres coordinates--you can see that lines of constant Schwarzschild radius look like hyperbolas, while lines of constant Schwarzschild time look like straight lines at different angles. Note that the Kruskal diagram is divided into four regions bounded by event horizons at 45 degrees, regions I (our universe, outside the horizon), II (inside the horizon, black hole region), III (another universe, outside the horizon) and (IV) (inside the horizon, white hole region), this having to do with the "maximally extended solution" I mentioned above.

    p834Gravitation.jpg

    Not sure what you mean by "uniform time metric throughout the gradient". The Kruskal-Szekeres system is just a different coordinate system on the same physical spacetime, so all physical predictions about coordinate-invariant things will be the same as if you used Schwarzschild coordinates. For example, if you have two clocks hovering at constant Schwarzschild radius above the horizon, then regardless of what coordinate system you use, you should get the prediction that the clock at greater radius sees the clock at smaller radius ticking more slowly than itself (and if one member of a pair of twins on the farther clock takes a journey to the closer clock, stays there a while, then rejoins his brother on the farther clock, he will now be younger--gravitational time dilation).
    What do you mean by "inherent relationship"? Can you give an example of a specific physical phenomena which expresses something you'd call an "inherent relationship between light and gravitational spacetime"?
    Not sure of the technical details, but here as always you have to distinguish between coordinate-invariant physical statements and statements which depend on your choice of coordinate system. For example, the way that gravity influences what an observer at a particular location will see in terms of the position of various stars is a coordinate-invariant fact, but the deflection angle of a particular light beam is coordinate-dependent--you should always be able to find a weird coordinate system where a particular light beam had a perfectly straight path in terms of the coordinate chart, for example.
    If you're talking about local measurements, then GR effects shouldn't be important here--if the clock is in freefall, then a freefalling observer measuring it locally would get the same sort of observations as an inertial observer measuring an inertial clock in flat spacetime, whereas if the clock is not in freefall (if it's hovering at constant Schwarzschild radius for example), then a freefalling observer measuring it locally would get the same sort of observations as an inertial observer measuring an accelerating clock in flat spacetime. This is just a special case of the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html [Broken] which work for all local observations.
     
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  6. Sep 7, 2009 #5
    Re: Light Clock in a Gravity well ????

    Originally Posted by Austin0
    I got that light speed was uniform going into or away from a nonrotating BH but remained unsure if this also meant a uniform time metric throughout the gradient or whether this coordinate system was applicable under less extreme conditions.

    Not sure what you mean by "uniform time metric throughout the gradient". The Kruskal-Szekeres system is just a different coordinate system on the same physical spacetime, so all physical predictions about coordinate-invariant things will be the same as if you used Schwarzschild coordinates. For example, if you have two clocks hovering at constant Schwarzschild radius above the horizon, then regardless of what coordinate system you use, you should get the prediction that the clock at greater radius sees the clock at smaller radius ticking more slowly than itself
    Yes this is what I was asking.

    ( Originally Posted by Austin0
    Isn t there any inherent relationship between light and gravitational spacetime in the fundamental field equations themselves?

    If the equations predict a specific deviation of path from a specific coordinate viewpoint and a specific but different deviation from another viewpoint I would assume that this would indicate a certain degree of inherent prediction of deviation that was then coordinate dependant for actual specific magnitudes.


    Originally Posted by Austin0
    I assume that if a light clock was equiped with a light emmision device that pulsed per tick that this would be observable from an inertial frame at rest wrt the clock so the dilation would be empirically detectable regardless of coordinate system ? SO in that frame it would appear that light speed was slowed correct?? Or Not??

    I was talking about the original scenario where the clock is deep in a well and the inertial observer is not in freefall but is outside the well at rest wrt the clock. Say a non-rotating gravitational source. Wouldn't the interpretation of the reduced periodicity of the clock at the small radius be that the light was reduced in speed???
    Or what other interpretation would there be??
    Thanks for the diagrams. Pretty heady stuff. Other universes and white holes.
    I am going to have to take a while to really make sense of it.
     
  7. Sep 7, 2009 #6

    JesseM

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    Re: Light Clock in a Gravity well ????

    By "deviation of path", are you talking about something like the bending of starlight when passing near a source of gravity? If so, then to the extent the shape of the path is described in a coordinate-dependent way, the deviation is coordinate-dependent too--don't know if that answers your question. Like I said, physical predictions about what a given observer sees when the light from different stars reaches her eyes will be the same in all coordinate systems.
    In GR, the only observers that have their own local inertial rest frame are freefalling ones--all those who aren't in freefall will be seen as accelerating in locally inertial frames (they will experience G-forces which tell them their local observations can't be equivalent to those of an inertial observer in flat spacetime, for example). So you can consider an observer not in freefall, but it doesn't make sense to call him "inertial". Also, in GR there is no coordinate-independent way to define the relative velocity of two objects that aren't in the same local region of curved spacetime, so you need to define what you mean by "at rest wrt the clock"--for example, do you just want the observer and clock to both be at constant Schwarzschild radius (both at rest in Schwarzschild coordinates), with the observer at a greater radius than the clock?
    In Schwarzshild coordinates light does actually move slower at a smaller radius, but this has nothing to do with the percieved slowdown of ticks as seen by an observer at greater radius (if you want the observer to be at fixed Schwarzschild radius), since each successive signal from the clock takes the same amount of coordinate time to travel from the clock to the observer. The slowdown would be explained in terms of gravitational time dilation, which says the clock at smaller radius will tick slower than the observer's own clock, relative to Schwarzschild coordinate time.
    Keep in mind that regions III and IV (the other universe and white hole) only appear in an idealized eternal black hole--a more realistic black hole that forms at some finite time, like the one shown in the first of the three diagrams, won't have these regions. As I mentioned on the other thread, an eternal Schwarzschild black hole is really more like a "gray hole" from the outside, in that an observer hovering outside the horizon can see particles coming out towards him from the horizon as well as other particles falling in (though in Schwarzschild coordinates the infalling particles won't actually reach the horizon until a coordinate time of +infinity, and the outgoing particles have been moving away from the horizon since a coordinate time of -infinity). The outgoing particles are the ones coming from the internal "white hole" region (which can also emit particles into the other universe), and the infalling particles enter the internal "black hole" region after crossing the horizon (where they can meet particles that fell into the black hole region from the other universe). Definitely take a look at that other thread if you're interested in understanding this stuff better, but if you just want to understand how the Kruskal-Szekeres diagram works for a realistic nonrotating black hole, focus on the one in the first diagram and ignore regions III and IV of the next two diagrams.
     
  8. Sep 8, 2009 #7
    Re: Light Clock in a Gravity well ????

     
    Last edited: Sep 8, 2009
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