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phy006
2007-Aug-20, 06:04 PM
Hi, sorry if this has been asked before; there's only so much of a large forum I'm able to trawl through!

I'm kind of wrestling with a few of the consequences of time dilation in a gravitational field. First, how does the speed of light propagation in a gravity well compare to the local lightspeed for an observer in a nonaccelerating frame? Is the actual propagation speed measurably different, or does the light down the well appear to move at 2.997e8 m/s but is redshifted compared to local light?

As a consequence of that, what effect (if any) does it have on astronomical observation? Time obviously runs at a measurably slower "rate" down here in the well, and the local constancy of lightspeed is also obviously preserved. I was tempted to draw the conclusion that there is a sort of Discworldian light "backlog", and that an observer in orbit would see a significantly older universe than on the ground, but that's silly, and contradicted by evidence. For instance, the planets are pretty much where we expect them to be from ground observation. Is it the case, then, that incoming light appears slightly more blueshifted at sea level than it does from, say, an orbital telescope? Or am I still out to lunch?

And third, does this imply that the planet (or any massy object) is some fraction of its lifespan younger than it would be if gravitational time dilation did not hold, in a kind of twin paradox? I'm not trying to get all YEC or anything, I know there's independant evidence for the age of the earth, but would more time have elapsed for a nonaccelerating observer than for a planetbound one over 4.5 billion years?

Ken G
2007-Aug-20, 11:43 PM
First, how does the speed of light propagation in a gravity well compare to the local lightspeed for an observer in a nonaccelerating frame? Is the actual propagation speed measurably different, or does the light down the well appear to move at 2.997e8 m/s but is redshifted compared to local light? Locally, in a gravitational well, nothing seems at all different, not the speed of light or the frequency of anything that is generated and measured locally. Redshifting and time dilation only appear when two different places in that well try to interpret the same phenomenon, that's what relativity is really all about-- not the phenomenon itself, but it's interpretation in different reference frames. It's about what happens to the information when it propagates between different observers.

Time obviously runs at a measurably slower "rate" down here in the well, and the local constancy of lightspeed is also obviously preserved. The rate of time seems normal to those down in there, they think our time is running fast.


Is it the case, then, that incoming light appears slightly more blueshifted at sea level than it does from, say, an orbital telescope? Yes, but the difference is minute.



And third, does this imply that the planet (or any massy object) is some fraction of its lifespan younger than it would be if gravitational time dilation did not hold, in a kind of twin paradox? Yes, the age of the Earth is less at sea level than it is for distant aliens in free space. It's not a significant effect. Normally we would let an object determine its own age, this is what is meant by "proper time".

phy006
2007-Aug-21, 03:35 AM
Edit for restatement: So, local lightspeed is always 3e8 m/s, roughly, if you have a vacuum available. Is it the case that an observer at a high gravity potential will observe light at a lower gravity potential propagating at a slower speed than light that is local to the observer? As in, if an orbiting observer had a meter-long rod next to her, and a meter-long rod at around sea-level, and an accurate clock, light would actually take longer to get from one end of the low-potential rod to the other than it would for the high-potential one? Or is the lower-potential light merely redshifted for the observer, and the light appears to take the same time for both rods when measured from the high-potential frame?

Ken G
2007-Aug-21, 02:37 PM
Edit for restatement: So, local lightspeed is always 3e8 m/s, roughly, if you have a vacuum available. Is it the case that an observer at a high gravity potential will observe light at a lower gravity potential propagating at a slower speed than light that is local to the observer? Yes, but then you are using what is known as "coordinate speed". Coordinate speed is pretty much completely arbitrary, it's just distance over time in some cockamamie coordinate system that you have cooked up in your mind. Such coordinates are still mathematically useful because you know how to continuosly transform them into your own local concept of distance and time (called proper distance and proper time)-- so coordinates are a way to conceptualize that which is not happening where you are. But coordinates only directly yield a "real" speed if you use distance as it would locally be measured by an observer on the scene, and time the same way, and when you do that, you always get c in every situation. That, by the way, is why we get "speeds" faster than c in cosmology-- because they are just coordinate speeds of the rate of separation between us and some distant galaxy, not a "real" speed of that galaxy relative to its own surroundings.


As in, if an orbiting observer had a meter-long rod next to her, and a meter-long rod at around sea-level, and an accurate clock, light would actually take longer to get from one end of the low-potential rod to the other than it would for the high-potential one? Yes, you can get that if you use coordinates of time and space that do not both locally conform to any observer on the scene. What you are basically doing is math not physics when you do that.


Or is the lower-potential light merely redshifted for the observer, and the light appears to take the same time for both rods when measured from the high-potential frame?These are not questions with "invariant" answers, the answers depend on the chosen coordinatization. One of the most subtle, and important, things to learn in relativity is the difference between that which is invariant (the same in all frames) and therefore physical, and that which is an interpretation (depends on coordinates) and differs from observer to observer.

phy006
2007-Aug-21, 05:33 PM
And local lightspeed is invariant.

Ok, that helps, but I think I'm still hanging up somewhere, and I'm afraid it's on the naive "backlog" thing. Does the time dilation we currently experience relative to free space in any sense matter to our observation and exploration of the solar system? Would it matter if we were above the event horizon of a black hole? Does simple light-lag even matter? Or is it all a case of choosing which coordinates to interpret from?

Apologies if I'm trying to cover a question you've already answered for me and I haven't figured out that you have.

Ken G
2007-Aug-21, 07:03 PM
Does the time dilation we currently experience relative to free space in any sense matter to our observation and exploration of the solar system? In practical terms, no, but it would if we were living on a neutron star, or near a black hole event horizon. We would use our same telescopes and clocks in the usual way, but the speed at which the rest of the universe would appear to evolve would be much accelerated. Processes that we observe in our laboratories would appear to proceed much faster in space, but we would have relativity to tell us why that apparent rate is not an invariant physical property, and we would not be surprised that astronauts we sent out in probes would find that those processes were not "actually" sped up at all, once you get out to where they are happening. It was all just an artifact of the way we were coordinatizing things, the way we were extrapolating our local concept of space and time continuously out into space. Other coordinatizations are possible, that don't show any speeding up-- for example, the "comoving frame" coordinatization that is used in cosmology, where you basically let local observers moving in a way that is "typical" for that location specify the meaning of space and time at that place. When we use those coordinatizations, we get a different interpretation for why things appear speeded up-- such as "gravitational blueshift". This is the key point-- the result of the experiment is invariant, but the interpretation we use to explain it depends on our choice of coordinates.

Let me give you a very mundane and everyday example of this same phenomenon. Imagine motion in a straight line on an air hockey table. What could be simpler than that? We can use a Cartesian coordinate system, say with the motion along the "x axis", and interpret the motion as force-free constant-velocity motion, a la Galileo, and are completely happy this is what is "really going on". But in fact that's just our interpretation, stemming from those coordinates. I can just as correctly use a polar coordinate system, set up so that the puck starts out moving azimuthally (in the "theta" direction), and lo and behold we find it ends up moving entirely in the radial direction after a long time! So in those coordinates, we have a "force" that turns the azimuthal motion into radial motion, and this force has all the trappings of a "ficticious force", or a "coordinate force". In a world where we thought in terms of polar coordinates, such forces would be a default part of all our interpretations of what we see.

Now imagine yet a third coordinate system, in which we use that same polar coordinate system but go into a rotating reference frame such that the azimuthal puck motion is rendered initially stationary. Then we will have yet another interpretation-- we'll see a ficticious force we'd call "centrifugal force" first cause the puck to assume some radial motion, and then we'd see that radial motion bending due to "coriolis forces". The motion would never appear radial, the puck would spiral away from us into the distance, and we'd understand it entirely in terms of those ficticious forces that are a default part of our coordinate system. We know these are not real forces because they don't require that our puck interacts with anything, yet we see their effects in our coordinate system.

In all three of those situations, the motion is just the same in an invariant sense, but we only interpret it as "straight line constant-velocity" motion in one of the three coordinate systems. The physics lives at a deeper level than the interpretations we give to what is happening, and general relativity exposes that deeper level, even as it figures out how to incorporate gravity, which is another matter altogether. The key point about gravity, and gravitational time dilation, is that it appears as one of these "ficticious" or "coordinate" forces, there is no real force there but if we use a coordinate system that is extrapolated from our local surroundings in what seems like a completely natural way, then we have these default coordinate effects Newton called "the force of gravity". We also get the default coordinate effect that time appears to have sped up outside our gravity well, which is an even more subtle coordinate effect but still just a coordinate effect.


Does simple light-lag even matter? Simple light-lag is always corrected for-- relativity is what persists after that correction. But light lag is still involved in some way, for if there were no light lag then c would be infinite and relativity would be very different.

Or is it all a case of choosing which coordinates to interpret from?
Yes, that's the key. I wouldn't say it is all a case of which coordinates you choose, but everything that is not an invariant property of the physics itself is going to be an arbitrary result of that choice. Interpretations and explanations are an example of things we have in physics that are generally not invariant, that's a point I wish more people understood when they start describing what is "really going on" at a level that is not sufficiently sophisticated, i.e., not invariant.

phy006
2007-Aug-21, 07:41 PM
Thanks for your help. I think that's about as clear as it's going to get without me doing some math homework. For now I think I want to do some casual research into how distant orbits are plotted.