The speed of light in a gravitational field

[Passionflower]

I don't think it's really right to say that using rulers and proper times makes something "coordinate free". After all, inertial coordinate systems are typically defined in terms of a set of rulers and synchronized clocks...anything that allows you to assign a position and a time to any arbitrary event can be treated as defining a coordinate system. (and if two people describe definitions of distances and times using the exact same system of rulers and clocks, but the first uses the words says that the readings on the rulers and clocks define the 'coordinates' of events while the second just talks about 'distances' and 'times', then surely this mere difference in wording does not mean the first is giving a coordinate-dependent description of events while the second is giving a coordinate-independent description)

So you are claiming that ruler distance and radar distance as measured by a clock for static observers depend on the chosen coordinate chart in a Schwarzschild solution?

This is getting weirder by the minute. Anything goes to 'prove' me wrong I suppose.

 

[JesseM]

So you are claiming that ruler distance and radar distance as measured by a clock for static observers depend on the chosen coordinate chart in a Schwarzschild solution?

No, I just say that any definition of distances and times in terms of rulers and clocks can itself be taken as a definition of a coordinate system; in fact, the only physical way to define any coordinate system is in terms of some set of physical rulers (or other physical distance measures like radar distance) and physical clocks. Do you agree that textbooks (following the example of Einstein's original 1905 paper) typically define inertial coordinate systems in such a physical way too, by picturing a network of rigid rulers and synchronized clocks like the one illustrated here? If so, would you say this somehow means that statements about "velocity" in such a system of rulers and clocks are "coordinate independent"? I would say the only coordinate-independent statements about such a system are purely local ones like "object A was next to marking x=5 on the ruler when the clock at that marking read t=13", any attempt to turn such local facts into statements about distances or velocities means you are making coordinate-dependent statements.

Anything goes to 'prove' me wrong I suppose.

Why do you have to make things personal like this? I would have said the same if someone else was making your argument about coordinate-independence, and it's not like there haven't been plenty of times I've agreed with you about stuff. This is a discussion board, everyone's opinions get challenged periodically, no need to make bitter remarks about it and create bad feelings for no good reason.

 

[PAllen]

I've seen a number of derivations of the idea that an observer away from an event horizon 'sees light as going slower' closer to the event horizon. I would guess that, properly defined, there is little dispute about this (e.g. MTW and Sean Carroll both have such derivations). I think it is also accepted that any sufficiently local measurement of lightspeed will be c (almost all posters here, plus a couple of GR texts say this). The seemingly open question here is whether there is a physically meaningful, preferred, way to talk about a non-local measurement of lightspeed in the radial direction by a (non-inertial) static observer at some fixed position above the event horizon. If there is well defined answer to this, it would not surprise me that this comes out different from c.

I have another thought on this. Ich has described of a 'maximally minkowski frame' that can be defined for highly non-inertial observers. He said, for example, that for a uniformly accelerting observer in SR it ends up with the Rindler coordinates and metric. It was based on building coordinates out from a 'prime observer' world line. As shown by the Rindler case, 'paths of simultaneity' are not necessarily (spacelike) 4-geodesics of the spacetime; and a simultaneity hypersurface is not necessarily locally Euclidean. Just as with the SR case, the existence of Euclidean hypersurfaces in the Schwarzschild metric has nothing to do with the way this procdedure would choose hypersurfaces of simultaneity in reference to a chosen prime observer.

Perhaps such a frame provides a meaningful way to talk about a static observer near the event horizon making a non-local measurement c in the radial direction (away and back from the center). I recall Ich's description did involve light signals, but perhaps it isn't circular because he only used it define 'comoving' chains of observers that show no red/blue shift relative to each other. Hopefully, there is no hidden speed assumption.

(I have no idea how to perform these operations, or what the results would be).

 

[Passionflower]

Why do you have to make things personal like this? I would have said the same if someone else was making your argument about coordinate-independence, and it's not like there haven't been plenty of times I've agreed with you about stuff. This is a discussion board, everyone's opinions get challenged periodically, no need to make bitter remarks about it and create bad feelings for no good reason.

I apologize JesseM.

There are a few individuals (not you) who pretend to know everything about relativity and at the same time have the urge of telling others how little they understand. They never show a formula or do a calculation, when their statements are challenged and supported by mathematics and graphs they simply ignore those challenges. That is very frustrating at times.

The seemingly open question here is whether there is a physically meaningful, preferred, way to talk about a non-local measurement of lightspeed in the radial direction by a (non-inertial) static observer at some fixed position above the event horizon. If there is well defined answer to this, it would not surprise me that this comes out different from c.

So just one posting before you posted this I showed a graph plotting the ruler distances between two observers divided by the radar time as measured by a clock at the observer closest to the EH for different distance. Clearly you can see that light takes more time for pairs closer to the EH. But you simply seem to re-raise the same question as if this posting never happened? Am I wrong in this?

Out of curiosity do you understand the plot and did you make any conclusions yourself? Or perhaps you disagree with the result, perhaps you think I miscalculated something? Or did you decide to simply ignore what I wrote?

Well - if you've done the calculation, you should be able to answer Pallen's questions, of how you define simultaneity, and along what particular curve you integrate the distance.

Just coming up with an answer isn't very convicing, if you can't show your work and define what it is that you're calculating when asked for details.

Good points, so I showed my work here: https://www.physicsforums.com/showpost.php?p=2980825&postcount=17.

Now since you asked for them it would be very polite to comment on them right? And if you think that there is anything wrong with it back it up with mathematics? Does that sound reasonable to you?

 

[PAllen]

So just one posting before you posted this I showed a graph plotting the ruler distance between two observers divided by the radar time as measured by a clock at the observer closest to the EH for different distance. Clearly you can see that light takes more time for pairs closer to the EH. But you raise the question as if this posting never happened.

Out of curiosity do you understand the plot and did you make any conclusions yourself? Or perhaps you disagree with the result, perhaps you think I miscalculated something? Or did you decide to simply ignore what I wrote?

I see the plot but don't know it's meaning because I haven't seen definitions of key terms like ruler distance.

I am not the only one who seems unsure of the physical applicability of this graph to a reasonable measurement process that might be used by a static observer near an event horizon.
-

l
Good points, so I showed my work here: https://www.physicsforums.com/showpost.php?p=2980825&postcount=17.

Now since you asked for them it would be very polite to comment on them right?
So you think the calculations are they right or wrong?

Here you simply give a formula for ruler distance, without defining its basis. A simple question I've asked and never gotten any answer to is how this ruler distance is defined. Further, I have made a claim which you may or may not agree with (others here have agreed with it): to define a 'ruler' you must define a spacelike path considered to be 'simultaneous' by some observer; then the interval along this path may be taken to be its length. I have asked whether you agreed with this and, if so, what is the path and rationale for it in your ruler distance formula (I believe Pervect also asked this).

 

[Passionflower]

Here you simply give a formula for ruler distance, without defining its basis.

You wrote you are very familiar with GR but now you say you do not know how to obtain the ruler distance between two stationary observers at two different r coordinates in a Schwarzschild solution.

This is the formula:
<br /> \sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) }-\sqrt {r_{{1}} \left( r_{<br /> {1}}-2\,M \right) }+2\,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{<br /> {2}}-2\,M}}{\sqrt {r_{{1}}}+\sqrt {r_{{1}}-2\,M}}} \right) <br />

You obtain this formula by integration of:

<br /> \left({1-{\frac {2M}{r}}\right)^{-1/2}<br />

I and others went over this many times in several postings on this forum, did you read any of these?

 

[PAllen]

You wrote you are very familiar with GR but now you say you do not know how to obtain the ruler distance between two stationary observers at two different r coordinates in a Schwarzschild solution.

This is the formula:
<br /> \sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) }-\sqrt {r_{{1}} \left( r_{<br /> {1}}-2\,M \right) }+2\,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{<br /> {2}}-2\,M}}{\sqrt {r_{{1}}}+\sqrt {r_{{1}}-2\,M}}} \right) <br />

You obtain this formula by integration of:

<br /> \left({1-{\frac {2M}{r}}\right)^{-1/2}<br />

I and others went over this many times in several postings on this forum, did you read any of these?

I have never claimed to be any kind of expert in GR. I have, in a few posts, described my background (which has peaks and valley's of understanding, and rusty computational skills, and no access to math and graphing software). I see no problem in these forums posing questions that may be fuzzy and that I don't know how to solve. People may discuss/answer/ignore as they see fit.

This definition doesn't relate to what I am asking. It looks like it is just using simultaneity as defined by the Schwarzschild t coordinate (could be wrong here, tell me if so). I am asking for a spacelike simultaneity path from the world line of a static oserver at e.g. 2R to the world line of a static observer at 3R, that comes from application of some reasonable operational definition of simultaneity applied by the non-inertial observer at 2R . The complication being that since we want to independently measure light speed, it is at least dicey to use a light based definition of simultaneity. Anything to do with Schwartzschild t coordinate does not seem relevant to this.

 

[Passionflower]

Well I gave you the formula for the ruler distance, if you decide to ignore it then so be it.

Perhaps some of the people who you seem to think so highly of can give you the formula for whatever it is you are looking for but I would not hold my breath. Why don't you ask them to write down the formula? It would be fun for me to see what they will come up with.

They will either admit that the formula I gave is the correct one or circumvent admitting this by acknowledging your statement without backing it up with any mathematics or formulas instead they will tell you it is all very difficult or so. And if all that fails they perhaps will give you some vague formula in tensorial form, that will shut up 99.9% of the respondents here and then they can keep claiming they know it all and others simply don't understand.

Yes why don't you do that ask for this formula?

 

[Mentz114]

It seems to me that the ruler-distance formula given by PassionFlower
<br /> s = \int \sqrt{g_{rr}}dr<br />
can be interpreted in two ways

1. an observer using an infinitesimal ruler placed end-to-end ( following what path ?)
2. a number of peaks in a monchromatic light ray traveling between the two shell observers.
The ruler's proper length or the wavelength measured in the starting frame are used in calibration.

Are these the same thing ?

(PassionFlower, please don't exercised about this, I also use this formula, but it seems good time to analyse what it means operationally).

 

[PAllen]

Well I gave you the formula for the ruler distance, if you decide to ignore it then so be it.

Perhaps some of the people who you seem to think so highly of can give you the formula for whatever it is you are looking for but I would not hold my breath. I suspect they will acknowledge your statement without backing it up with any mathematics or formulas instead they will tell you it is all very difficult or so. And if all that fails they perhaps will give you some vague formula in tensorial form, that will shut up 99.9% of the respondents here and then they can keep claiming they know it all and others simply don't understand.

Let me ask you this:

Consider 3 observers momentarily adjacent (but with different relative velocities) at 2R. One is free falling, the other is orbiting (unstably, since we 2R is well inside of the closest stable orbit), the last is head static by a rocket. Do you think they will agree on simultaneity? Do you think there is any reason any of their perceptions of simultaneity will be the same as the Schwarzschild t coordinate? I'm pretty sure the answer to these questions are no and no. In which case, each would have a different concept of ruler distance.

 

[PAllen]

It seems to me that the ruler-distance formula given by PassionFlower
<br /> s = \int \sqrt{g_{rr}}dr<br />
can be interpreted in two ways

1. an observer using an infinitesimal ruler placed end-to-end ( following what path ?)
2. a number of peaks in a monchromatic light ray traveling between the two shell observers.
The ruler's proper length or the wavelength measured in the starting frame are used in calibration.

Are these the same thing ?

(PassionFlower, please don't exercised about this, I also use this formula, but it seems good time to analyse what it means operationally).

And for whom is it meaningful? See my post #41. It seems to me any observers with different perceptions of simultaneity will want to use different ruler defintions. What is a ruler but a spacelike path perceived as simultaneous by *some observer*.

 

[Passionflower]

Let me ask you this:

Consider 3 observers momentarily adjacent (but with different relative velocities) at 2R. One is free falling, the other is orbiting (unstably, since we 2R is well inside of the closest stable orbit), the last is head static by a rocket. Do you think they will agree on simultaneity? Do you think there is any reason any of their perceptions of simultaneity will be the same as the Schwarzschild t coordinate? I'm pretty sure the answer to these questions are no and no. In which case, each would have a different concept of ruler distance.

First of all the problem statement you gave concerned stationary observers now you want to talk about observer who are not stationary. Would you agree that it is a lot better to get the first problem resolved first without complicating the matter by introducing non stationary observers? As discussed a few weeks ago on this forum there are several notions of distance for non stationary observers in a Schwarzschild solution.

 

[Mentz114]

And for whom is it meaningful? See my post #41. It seems to me any observers with different perceptions of simultaneity will want to use different ruler defintions. What is a ruler but a spacelike path perceived as simultaneous by *some observer*.

Seems obvious 'for whom it is meaningful'. Both definitions are counts of events along a worldline and therefore agreed by all observers.

An observer at any given time only has one 'perception of simultaneity'.

"What is a ruler but a spacelike path perceived as simultaneous by *some observer*".

That's what you'd like it to be. So give a rigorous definition before you wear out your wrists with all the handwaving.

 

[bcrowell]

Below is a plot of light speeds between pairs of static observers (o1, o2) separated a fixed ruler distance of 1 with the radar distance as measured by a clock at observer o1. In the plot you can see the ruler distance (which is 1 for each pair) divided by the radar distance, this ratio is larger for pairs closer to the EH. This is a coordinate free plot as only the ruler distance and proper time is used.

The proper time and the thing you're calling ruler distance are both coordinates.

 

[PAllen]

First of all the problem statement you gave concerned stationary observers now you want to talk about observer who are not stationary. Would you agree that it is a lot better to get the first problem resolved first without complicating the matter by introducing non stationary observers? As discussed a few weeks ago on this forum there are several notions of distance for non stationary observers in a Schwarzschild solution.

The only reason I brought up other observers was to accentuate the issue of observer dependence of simultaneity. I am only intersted in the static observer. However, do you know of any reason this non-inertial observer will perceive of Schwarzschild t coordinate as their simultaneity? It seems implausible to me, but I could be convinced by a demonstration or argument.

 

[PAllen]

Seems obvious 'for whom it is meaningful'. Both definitions are counts of events along a worldline and therefore agreed by all observers.

An observer at any given time only has one 'perception of simultaneity'.

"What is a ruler but a spacelike path perceived as simultaneous by *some observer*".

That's what you'd like it to be. So give a rigorous definition before you wear out your wrists with all the handwaving.

Everyone may agree this is the invariant length of this world line (actually, it isn't a world line, as it is a spacelike path along r at fixed t). However, observers with different states of motion will disagree on whether it is functionally a ruler.

Consider SR. For one observer, the path (t,x) : (0,0),(0,1) is a ruler. For a different observer, the invariant length of this path won't change, but they will not remotely view it as a ruler because its ends are not simultaneous. So I am wondering why an arbitrary (non-inertial) static observer sees a path defined by constant Schwarzschild t as meaningful ruler. I don't know that it is not, but I haven't seen the question addressed.

 

[Mentz114]

Everyone may agree this is the invariant length of this world line. However, observers with different states of motion will disagree on whether it is functionally a ruler.

I agree, but a ruler is not required to have global significance, it is defined by some observer.

path defined by constant Schwarzschild t as meaningful ruler. I don't know that it is not, but I haven't seen the question addressed.

Am I right that we can find a observer where t and \tau coincide ? For such an observer this objection would disappear.

Maybe this ruler is a chimera. There seems to be no satisfactory operational definition on a large scale.
It's perhaps not surprising there's disagreement over the finer points.

 

[Passionflower]

I am worried at attempts to move the goalposts.

Are we still talking about the distance between two stationary observers in a Schwarzschild solution where the EH=R and the observers are located at 2R and 3R in Schwarzschild coordinates?

I gave a description on how to calculate such a distance, it has been met by non-responses and 'This definition doesn't relate to what I am asking'.

That is where we are at right?

 

[PAllen]

I am worried at attempts to move the goalposts.

Are we still talking about the distance between two stationary observers in a Schwarzschild solution where the EH=R and the observers are located at 2R and 3R in Schwarzschild coordinates?

I gave a description on how to calculate such a distance, it has been met by non-responses and 'This definition doesn't relate to what I am asking'.

That is where we are at right?

Yes. The fact that the observer is static doesn't address what their reasonable definition of simultaneity is. If we weren't trying to set up a framework for non-circular measurement of speed of light, I would trivially define simultaneity of different events for the static observer using the standard radar convention. But for the current purpose it seems absurd - if we use light to set up a coordinate system, how do we independently measure the speed of light?

 

[yuiop]

I've seen a number of derivations of the idea that an observer away from an event horizon 'sees light as going slower' closer to the event horizon. I would guess that, properly defined, there is little dispute about this (e.g. MTW and Sean Carroll both have such derivations). I think it is also accepted that any sufficiently local measurement of lightspeed will be c (almost all posters here, plus a couple of GR texts say this). The seemingly open question here is whether there is a physically meaningful, preferred, way to talk about a non-local measurement of lightspeed in the radial direction by a (non-inertial) static observer at some fixed position above the event horizon. If there is well defined answer to this, it would not surprise me that this comes out different from c.

In Schwarzschild coordinates the Schwarzschild observer "at infinity" claims the vertical speed of light is c*(1-2m/r) while a stationary local observer at r claims the speed of light is simply c. Now while we are generally used to observers in relativity having different points of view of the same set of events it is easy to see that the points of view are physically and conceptually in contradiction at the event horizon. The local measurement of the speed of light is normally taken to the physically "real" measurement and this implies the speed of light is c at the event horizon (but we assume that it not possible to have a stationary local observer exactly the event horizon) while the measurement of the speed of light by the Schwarzschild observer at infinity implies that the speed of light at the event horizon is exactly zero. The observation by the observer at infinity implies light cannot pass through the event horizon while the observation by the local observer concludes that the light passes through the event horizon without any difficulty. These are physically different conclusions and one must be "right" and the other must be "wrong", but which? Normally it is concluded that the conclusions of the observer at infinity are the "wrong" conclusions because they are the conclusions of a distant observer who just a "bookeeper" (and no one like accountants, right? ) and coordinate measurements are just an abstraction without physical meaning. The event horizon is said to be a "coordinate horizon" without physical significance and the coordinate speed in Schwarzschild coordinates is just as arbitrary as plotting the velocity of vehicles on the surface of the Earth in terms of degrees latitude or longitude per hour.

Yes. The fact that the observer is static doesn't address what their reasonable definition of simultaneity is. If we weren't trying to set up a framework for non-circular measurement of speed of light, I would trivially define simultaneity of different events for the static observer using the standard radar convention. I doubt that would match coordinate t. But for the current purpose it seems absurd - if we use light to set up a coordinate system, how do we independently measure the speed of light?

To try and shed some light on this situation, I would like to try and present "a reasonable definition of simultaneity" in the gravitational field of a non rotation gravitational mass that seems to give some reality to coordinate measurements and see how these arguments are countered. First we should consider a "gravitational twin experiment". A pair of twins are located at R1. One sibling is dropped and freefalls to R2. At some later time (say 50 years as measured at R1) the second sibling freefalls to R2 and comes to rest with their twin at R2. The ages of the twins when they are once again alongside each other, differ by an amount that exactly agrees with the gravitational time dilation measured by the Schwarzschild observer at infinity. Therefore we can conclude that the coordinates measurements of the observer at infinity do have physical significance and time really does slow down lower down in the gravitational field and the speed of light really does slow down lower down in the gravitational field (with the implication that light stops at the event horizon).

To set up a "reasonable definition of simultaneity" in this situation, we can speed up the stationary clocks lower down in the field by the gravitational gamma factor and stationary local observers using the coordinated clocks would measure the local speed of light to be slower than in flat space. Before this synchronisation procedure, observers lower down say that clocks higher up appear to be running fast and observers higher up say that clocks lower down appear to be running slow. (Note that the red or blue shift is not reciprocal as in SR). After the synchronisation procedure, observers at any radius agree that clocks at any other radius are running at exactly the same rate. We would now seem to have a reasonable definition of simultaneity, and having defined simultaneity this way, we would seem to able to conclude that light (and any other physical process) really does slow down lower down in a physically meaningful way. Any thoughts?

 

[bcrowell]

I agree with yuiop that the definition of simultaneity is not the main issue here. The Schwarzschild spacetime is static, so it has a preferred notion of simultaneity. That doesn't mean that other definitions of simultaneity are impossible, just that they aren't as natural and useful.

 

[PAllen]

In Schwarzschild coordinates the Schwarzschild observer "at infinity" claims the vertical speed of light is c*(1-2m/r) while a stationary local observer at r claims the speed of light is simply c. Now while we are generally used to observers in relativity having different points of view of the same set of events it is easy to see that the points of view are physically and conceptually in contradiction at the event horizon. The local measurement of the speed of light is normally taken to the physically "real" measurement and this implies the speed of light is c at the event horizon (but we assume that it not possible to have a stationary local observer exactly the event horizon) while the measurement of the speed of light by the Schwarzschild observer at infinity implies that the speed of light at the event horizon is exactly zero. The observation by the observer at infinity implies light cannot pass through the event horizon while the observation by the local observer concludes that the light passes through the event horizon without any difficulty. These are physically different conclusions and one must be "right" and the other must be "wrong", but which? Normally it is concluded that the conclusions of the observer at infinity are the "wrong" conclusions because they are the conclusions of a distant observer who just a "bookeeper" (and no one like accountants, right? ) and coordinate measurements are just an abstraction without physical meaning. The event horizon is said to be a "coordinate horizon" without physical significance and the coordinate speed in Schwarzschild coordinates is just as arbitrary as plotting the velocity of vehicles on the surface of the Earth in terms of degrees latitude or longitude per hour.



To try and shed some light on this situation, I would like to try and present "a reasonable definition of simultaneity" in the gravitational field of a non rotation gravitational mass that seems to give some reality to coordinate measurements and see how these arguments are countered. First we should consider a "gravitational twin experiment". A pair of twins are located at R1. One sibling is dropped and freefalls to R2. At some later time (say 50 years as measured at R1) the second sibling freefalls to R2 and comes to rest with their twin at R2. The ages of the twins when they are once again alongside each other, differ by an amount that exactly agrees with the gravitational time dilation measured by the Schwarzschild observer at infinity. Therefore we can conclude that the coordinates measurements of the observer at infinity do have physical significance and time really does slow down lower down in the gravitational field and the speed of light really does slow down lower down in the gravitational field (with the implication that light stops at the event horizon).

To set up a "reasonable definition of simultaneity" in this situation, we can speed up the stationary clocks lower down in the field by the gravitational gamma factor and stationary local observers using the coordinated clocks would measure the local speed of light to be slower than in flat space. Before this synchronisation procedure, observers lower down say that clocks higher up appear to be running fast and observers higher up say that clocks lower down appear to be running slow. (Note that the red or blue shift is not reciprocal as in SR). After the synchronisation procedure, observers at any radius agree that clocks at any other radius are running at exactly the same rate. We would now seem to have a reasonable definition of simultaneity, and having defined simultaneity this way, we would seem to able to conclude that light (and any other physical process) really does slow down lower down in a physically meaningful way. Any thoughts?

Great post, thanks. I will think more about a few details, but this seems like real progress.

 

[PAllen]

yuiop's analysis convinces me that using simultaneity based on Schwarzschild coordinate time is meaningful for static observers. I also thought of another argument for the same conclusion. Using light signals to define simultaneity actually seems like it can be done without any assumptions about speed (of course, you are still assuming there is something fundamental about light speed, just not its value). If this is done, you also conclude that points with same t on 2R and 3R worldlines are simultaneous.

This finally justifies Passionflowers calculations in post #17. However, when I do them, I get a slightly different ruler length expression. One of us made some arithmetic mistake, not sure which. I get exactly the same proper time expressions, but for proper distance I get (sorry, no latex):

R(sqrt(6) - sqrt(2) + ln((sqrt(2)+1)/(sqrt(3)+sqrt(2)))

This is actually extremely close, in that one sign difference accounts for the discrepancy (leading to division of sqrt expressions inside the log versus multiplication).

For the sake of argument, I use mine, and I compute that we actually expect an observer at 2R, measuring lightspeed to 3R, will come up with about .6435 c.

Hopefully, passionflower and I can discuss issues in the future without rancor.

 

[PAllen]

A follow up question is what a static observer at 2R would measure for lightspeed perpendicular to the radial direction? This calculation seems more messy. I would guess that it comes out different. If true, we predict that an abserver on the surface of neutron star would measure speed of light noticeably different radially versus tangentially.

 

[pervect]

If the static observer at 2R uses his local clocks and rulers, the speed of light will always be equal to 'c' in all directions, radial or otherwise.

So I assume that you're using some sort of "coordinate clock" rather than a local clock, based on the fact that you don't know the answer immediately by inspection. But in that case I'm not sure what you're using for your meter (maybe you're not either?), though personally I think the logical choice to go along with coordinate clocks would be to measure the coordinate angle phi and express your radial velocity in radians / second.

The fact that one can (and has to) tweak one's actual clocks to make them match up to a coordinate system isn't really a particularly good argument for assuming the coordinate clocks are "more real". I'd argue that the clocks that keep proper time are "more real", because they're actual, untweaked clocks, and that's what you measure time with. But any time a discussion gets down to what's "more real", it's mostly an exercise in philosophy. But you do need to specify what measurement procedures you're going to use to get any agreement on what results you should expect.

As I mentioned before, if you were going strictly by the modern SI standards for your measurement, you'd be using cesium time sources for your time, and the same cesium source for your distance (but counting interference fringes). Hence the speed of light would be defined as a constant, and you'd need something else to measure if you were to measure anything, presumably you'd be carrying along an old standard "meter bar" to compare it's length to the SI standard meter.

You can certainly "correct" your cesium time sources to keep coordinate time if you specify that you want to, though I'm not sure why you'd want to turn a physical, coordinate-independent measurement into a coordinate-dependent one.

 

[PAllen]

If the static observer at 2R uses his local clocks and rulers, the speed of light will always be equal to 'c' in all directions, radial or otherwise.

So I assume that you're using some sort of "coordinate clock" rather than a local clock, based on the fact that you don't know the answer immediately by inspection. But in that case I'm not sure what you're using for your meter (maybe you're not either?). Or perhaps I guessed wrong about which clock you're using.

The fact that one can (and has to) tweak one's actual clocks to make them match up to a coordinate system isn't really a particularly good argument for assuming the coordinate clocks are "more real". I'd argue that the clocks that keep proper time are "more real", because they're actual, untweaked clocks, and that's what you measure time with.

No, the calculation that Passionflower did and I re-did for myself, is as follows:

1) compute proper time along 2R world line for a null geodesic to reach 3R and back.

2) Having finally reached consensus that coordinate time constant defines the most reasonable hypersurface of simultaneity for any static observer, compute proper distance between simultaneous events at 2R and 3R.

3) compute speed of light as 2x(distance compute in (2)) / proper time computed in (1).

As far as I believe, coming late to the party, this seems to model as well as possible what a real observer at 2R would measure for lightspeed to 3R and back.

The only thing I see subject to argument is that the distance computed in (2) is not really the distance the observer at 2R would observer. Can you explain why it wouldn't? (since finally Passionflower, yuiop, bcrowell, Mentz114, and myself have all come to agree that coordinate time defines the most reasonable definition of simultaneity for static observers in this geometry).

 

[PAllen]

One other comment is that this is a very non-local measurement. I believe if did what you (pervect) had suggested and computed things for 2R and 2R+epsilon, I would get c (or maybe c - O(epsilon)). I don't think there is any expectation that a non-local measurement of c in a non-inertial frame must come out the same as an inertial observer. These static observers are definitely not inertial observers.

Of course if one uses definitions of units that use light, you always find lightspeed = c. That's why obsessed for so many posts about making sure we didn't use circular definitions if we really wanted to model an independent measurement of lightspeed.

 

[JesseM]

I apologize JesseM.

There are a few individuals (not you) who pretend to know everything about relativity and at the same time have the urge of telling others how little they understand. They never show a formula or do a calculation, when their statements are challenged and supported by mathematics and graphs they simply ignore those challenges. That is very frustrating at times.

No problem, I known it can be frustrating to give a quantitative argument and have it dismissed in a non-quantitative way; our own disagreement was just about terminology, so it's obviously more subjective and there isn't really a totally clear-cut "physically correct" answer to what it means for a particular measurement to be coordinate-dependent or coordinate-independent.

 

[Passionflower]

This finally justifies Passionflowers calculations in post #17. However, when I do them, I get a slightly different ruler length expression. One of us made some arithmetic mistake, not sure which. I get exactly the same proper time expressions, but for proper distance I get:
<br /> R(\sqrt{6} - \sqrt{2} + ln\left({\sqrt{2}+1 \over \sqrt{3}+\sqrt{2}}\right)<br />

No R inside the ln does not seem to be correct.
How did you derive this?

Hopefully, passionflower and I can discuss issues in the future without rancor.

Yes I hope that too.

If the static observer at 2R uses his local clocks and rulers, the speed of light will always be equal to 'c' in all directions, radial or otherwise.

So I assume that you're using some sort of "coordinate clock" rather than a local clock, based on the fact that you don't know the answer immediately by inspection. But in that case I'm not sure what you're using for your meter (maybe you're not either?), though personally I think the logical choice to go along with coordinate clocks would be to measure the coordinate angle phi and express your radial velocity in radians / second.

Pervect are you at all following this discussion? I get the impression that you are not as your coordinate clock comment comes straight out of the blue. Perhaps you really should assume a little less and start reading what people are actually writing about.

 

[pervect]

No R inside the ln does not seem to be correct.
Pervect are you at all following this discussion? I get the impression that you are not as your coordinate clock comment comes straight out of the blue. Perhaps you really should assume a little less and start reading what people are actually writing about.

The amount of time I have to devote to PF is limited, but I do the best I can to follow the posts (mostly the interesting looking ones) with the time I have available.

Occasionally I miss things - either for a lack of time, or because posts "snuck in" on me and I missed them outright (I tend to start reading from my last post, but sometimes posts were started before mine and hence appear in spots where I'm not looking for them).

Other times, posts weren't particularly well written (it does a lot of time and effort to write a detailed post with _all_ the necessary information in it, especially when people have different backgrounds and approaches to the problems.)

Also, the details of a calculation aren't especially interesting to me if the underlying assumptions behind the calculations are not clear.