The speed of light in a gravitational field

[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:
$$\sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) }-\sqrt {r_{{1}} \left( r_{ {1}}-2\,M \right) }+2\,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{ {2}}-2\,M}}{\sqrt {r_{{1}}}+\sqrt {r_{{1}}-2\,M}}} \right)$$

You obtain this formula by integration of:

$$\left({1-{\frac {2M}{r}}\right)^{-1/2}$$

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:
$$\sqrt {r_{{2}} \left( r_{{2}}-2\,M \right) }-\sqrt {r_{{1}} \left( r_{ {1}}-2\,M \right) }+2\,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{ {2}}-2\,M}}{\sqrt {r_{{1}}}+\sqrt {r_{{1}}-2\,M}}} \right)$$

You obtain this formula by integration of:

$$\left({1-{\frac {2M}{r}}\right)^{-1/2}$$

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
$$s = \int \sqrt{g_{rr}}dr$$
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
$$s = \int \sqrt{g_{rr}}dr$$
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", becuase 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:
$$R(\sqrt{6} - \sqrt{2} + ln\left({\sqrt{2}+1 \over \sqrt{3}+\sqrt{2}}\right)$$

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 intersting to me if the underlying assumptions behind the calculations are not clear.

 

[yuiop]

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

Thanks

I should however add a word of caution in line with the rules of the forum. If you use coordinate clocks as I described earlier and conclude that time and the speed of light (and all other physical processes) slow down as we get closer to the event horizon, then you reach the opposite conclusion to the accepted wisdom or textbook conclusions. The textbook position is that the coordinate measurements of the Schwarzschild observer are non physical and that the local measurements are the "real" physically meaningful measurements and that light continues at the speed of of light right through the event horizon. However, even the local measurements have their limitations, because the the local measurements imply that you cannot have a static observer exactly at the event horizon so any measurements by a static observer at the event horizon are invalid or at the very least indeterminate. This is usually supported by calculations of what a free falling observer measures as they fall through the event horizon, using their proper time and rulers. The coordinate observer would counter that even the proper time of the free falling observer is subject to time dilation (gravitational and velocity related) and that the clock of the free falling observer stops at the event horizon would stop and such an observer would not be in a position to make any measurements. For example the velocity of anything measured by a static or free falling observer at the event horizon would effectively be d/t = 0/0. In brief, the conclusion that time or the speed of light slows down lower down in a gravitational field is *NOT* the official position. To me personally, there are two contradictory, but apparently (to me) equally valid physical interpretations of what happens at the event horizon and below and maybe in the future a quantum theory on gravity might shed more light on the physical interpretation. Until then all we can do is predict what a given observer would actually measure and GR gives an unambiguous answer to this, while the "reality" is left open to interpretation.

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.

The horizontal speed of light is fairly easy to calculate. Just set the proper time dtau and dr to zero in the Schwarzschild metric and solve for the coordinate angular velocity. The result is that the coordinate horizontal speed of light is c*sqrt(1-2m/r), while the coordinate vertical speed of light is c*(1-2m/r).

There appears to be a directional asymmetry in the coordinate speed of light, but this may or not show up in corrected local measurements depending on how local distance is operationally defined. One practical way to measure this would be to construct a vertical MMX type apparatus in flat space far from the gravitational source. The device is set up so that so that the arms are of equal length (say one meter) as confirmed by a interferometer at the centre of the device. This device is carefully transported closer to the gravitational source. Now one difficulty with vertical rulers is that they may be stretched or compressed by tidal forces so we cannot be sure that their proper length has not changed making them useless as measuring devices. To compensate for this we adjust the vertical arm so that the MMX device gives a null result and rotate the device by 90 degrees, so the horizontal arm becomes the vertical arm and the vertical arm becomes the horizontal arm. When we are satisfied that that the device is supported and tuned in such a way that it gives a null result an any orientation then we may have a practical ruler. Now if we use a local clock that is synchronised with coordinate time, then the locally measured speed of light will be c*sqrt(1-2m/r) in any direction over a short distance. In a nutshell, the speed of light measured using coordinate time and local rulers is isotropic, but height dependent.

If we carry out radar speed of light measurements over extended distances from a given coordinate r, the speed of light above r will appear to be faster and the speed of light below r will appear to be slower than the local speed of light, whether we use short rulers laid end to end to define distance, or use coordinate radius difference (1/2*circumference/pi) to define distance, but the magnitudes will be different. This is true even if we use the un-tweeked proper time of a single stationary natural clock located at r. The coordinate difference distance is shorter than the ruler distance because the local rulers are subject to gravitational length contraction (but that is a slightly controversial way of putting things, but it works as a convenient mental convenience for me and the maths works out).

Since there are many ways to measure the speed of light it might be less confusing to specify a particular observer and a particular measurement method and we should be able to tell you exactly what that observer will measure, but different people may differ in the physical interpretation of what that measurement "really" means physically.

Finally, I should perhaps mention that the integrated ruler distance can appear in forms that look very different, but yield identical numerical results, so it is good to check results numerically if there appears to be a contradiction. We have had this confusion in the past in different threads.

 

[PAllen]

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

I derived it the same way you did. Look at your post #17 vs. your post #36. #17 multiplied inside of the the nat.log instead of dividing. My expression is like your #36 except I get the r1 expression over the r2 expression inside the nat.log. I have checked carefully, and believe I am correct.

Then, as to no R in the quotient inside the nat.log in my explicit formula for 2R and 3R, that is simply a matter dividing numerator and denominator by R, to get everything in rations, and plugging in the actual numbers.

Thanks for latexing my expression. Note the latex version is missing the final parentheses.

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.

Passionflower, this is how tone starts to get out of control. Pervect is simply thrown off by yuiop's post. This discussed observers choosing to use a different clock rate than their proper time. I took this as a justification for a common simultaneity convention; that they wouldn't actually use this adjusted time for measurements. yuiop did leave the impression that this adjusted time might be used for measurements, thowing pervect off track.

 

[PAllen]

I should however add a word of caution in line with the rules of the forum. If you use coordinate clocks as I described earlier and conclude that time and the speed of light (and all other physical processes) slow down as we get closer to the event horizon, then you reach the opposite conclusion to the accepted wisdom or textbook conclusions. The textbook position is that the coordinate measurements of the Schwarzschild observer are non physical and that the local measurements are the "real" physically meaningful measurements and that light continues at the speed of of light right through the event horizon. However, even the local measurements have their limitations, because the the local measurements imply that you cannot have a static observer exactly at the event horizon so any measurements by a static observer at the event horizon are invalid or at the very least indeterminate. This is usually supported by calculations of what a free falling observer measures as they fall through the event horizon, using their proper time and rulers. The coordinate observer would counter that even the proper time of the free falling observer is subject to time dilation (gravitational and velocity related) and that the clock of the free falling observer stops at the event horizon would stop and such an observer would not be in a position to make any measurements. For example the velocity of anything measured by a static or free falling observer at the event horizon would effectively be d/t = 0/0. In brief, the conclusion that time or the speed of light slows down lower down in a gravitational field is *NOT* the official position. To me personally, there are two contradictory, but apparently (to me) equally valid physical interpretations of what happens at the event horizon and below and maybe in the future a quantum theory on gravity might shed more light on the physical interpretation. Until then all we can do is predict what a given observer would actually measure and GR gives an unambiguous answer to this, while the "reality" is left open to interpretation.

Passionflower and I are not using coordinate values directly. We are using proper time and distance within a model of a specific actual measurement. The part I liked about your post was the idea of adjusted clocks purely for the purpose of establishing a natural simultaneity convention. The calculations we did do not actually use such adjusted clocks.

The horizontal speed of light is fairly easy to calculate. Just set the proper time dtau and dr to zero in the Schwarzschild metric and solve for the coordinate angular velocity. The result is that the coordinate horizontal speed of light is c*sqrt(1-2m/r), while the coordinate vertical speed of light is c*(1-2m/r).

This isn't what I meant at all. I only want to model actual measurements using invariant intervals. So the analog of my radial measurement (which Passionflower first calculate) would be find the actual light path between (r, theta1) and (r,theta2), and compute the proper time for either of these observers elapsed as the light follows this path. Then compute proper distance along the spacelike geodesic between these points. I see exactly how to do this, but is rather messy.

Since there are many ways to measure the speed of light it might be less confusing to specify a particular observer and a particular measurement method and we should be able to tell you exactly what that observer will measure, but different people may differ in the physical interpretation of what that measurement "really" means physically.

That is exactly what I did, and fortunately passionflower understood what I was proposing. With a lot of heat, our difference boiled down to my insistence on understanding the basis of his calculation, specifically what he was assuming about simultaneity and why that was justified, and his not communicating this in a way that was convincing for me.

 

[PAllen]

Picking up the main remaining issue in this thread: how 'real' is the claim that an observer actually measures the radial speed computed by Passionflower and myself? I responded to pervect that it didn't seem implausible because it was a non-local measurement by a non-inertial observer. I suggested if a sufficiently local computation was done you would get c as expected.

So now I did this computation, and lo and behold, it does come out 1 (in c=1 units). That is the exact method leading to .6435 for c as a non-local measurement, when done in terms of r to r+dr, gives c=1. I view this as confirming that the use of invariants has properly compensated for coordinate anisotropy of the Schwarzschild coordinates, and that the predicted non-local measurement is a real prediction.

 

[Passionflower]

This isn't what I meant at all. I only want to model actual measurements using invariant intervals. So the analog of my radial measurement (which Passionflower first calculate) would be find the actual light path between (r, theta1) and (r,theta2), and compute the proper time for either of these observers elapsed as the light follows this path. Then compute proper distance along the spacelike geodesic between these points. I see exactly how to do this, but is rather messy.

In my production of the plot I went a step further and eliminated the usage of r completely. The plot shows pairs of stationary observers (o1 and o2 removed a physical distance of 1, so NOT a coordinate difference of 1) with descending physical distances from the EH divided by the proper time of light from o1 to o2 on o1's clock.

In flat spacetime this obviously gives 1 for each pair. In a Schwarzschild solution this gives 1 at infinity since the solution is asymptotically flat, but for decreasing physical distances to the EH the value increases. Clearly this implies the speed of light over this distance slows down for decreasing distances to the EH.

Clearly there is more to tell than just repeating the mantra that the speed of light measured locally is always c. It is a little bit like telling a person who is driving a car on the highway that locally his car does not use any gasoline at all.

As to the formula, I used:

$$\sqrt {{r_{{2}}}^{2}-2\,r_{{2}}M}-\sqrt {{r_{{1}}}^{2}-2\,r_{{1}}M}+2 \,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{{2}}-2\,M}}{\sqrt {r_ {{1}}}+\sqrt {r_{{1}}-2\,M}}} \right)$$

And then applied 2M=R, r₁ = 2R and r₂ = 3R. From this I obtained:

$$\sqrt {6}\sqrt {{R}^{2}}-\sqrt {2}\sqrt {{R}^{2}}+R\ln \left( {\frac {\sqrt {3}\sqrt {R}+\sqrt {2}\sqrt {R}}{\sqrt {2}\sqrt {R}+\sqrt {R}}} \right)$$

After simplification this becomes:

$$R \left( \sqrt {2}\sqrt {3}-\sqrt {2}+\ln \left( -\sqrt {3}+\sqrt {2} \sqrt {3}-\sqrt {2}+2 \right) \right)$$

An interesting exercise would be to do the same thing for an free falling observer at escape velocity with a (non inertial!) observer tagged along in his 'tail' a constant physical distance of 1 removed. If I am not mistaken we can drive this example all the way through passed the EH. Any takers?

 

[Mentz114]

An interesting exercise would be to do the same thing for an free falling observer at escape velocity with a (non inertial!) observer tagged along in his 'tail' a constant physical distance of 1 removed. If I am not mistaken we can drive this example all the way through passed the EH. Any takers?

Sounds like fun. If the trailing oberver must have the same velocity as the leader ( which begs a question* ) then the trailing observer must reach c before the EH, which can't happen, obviously.
I'm assuming G-P comoving faller for the leading observer here.

(*the same velocity when, and from what perspective ?)

I'm not at all sure if this correct, just a quick analysis.

 

[PAllen]

In my production of the plot I went a step further and eliminated the usage of r completely. The plot shows pairs of stationary observers (o1 and o2 removed a physical distance of 1, so NOT a coordinate difference of 1) with descending physical distances from the EH divided by the proper time of light from o1 to o2 on o1's clock.

But this is not at all the calculation I was saying is messy. All your formulas below are strictly radial. I believe, non-local measurements of the speed of light by a static observer in this geometry are anisotropic. Thus if you actually did what I suggested, you would find a completely different result.

In flat spacetime this obviously gives 1 for each pair. In a Schwarzschild solution this gives 1 at infinity since the solution is asymptotically flat, but for decreasing physical distances to the EH the value increases. Clearly this implies the speed of light over this distance slows down for decreasing distances to the EH.

Clearly there is more to tell than just repeating the mantra that the speed of light measured locally is always c. It is a little bit like telling a person who is driving a car on the highway that locally his car does not use any gasoline at all.

As to the formula, I used:

$$\sqrt {{r_{{2}}}^{2}-2\,r_{{2}}M}-\sqrt {{r_{{1}}}^{2}-2\,r_{{1}}M}+2 \,M\ln \left( {\frac {\sqrt {r_{{2}}}+\sqrt {r_{{2}}-2\,M}}{\sqrt {r_ {{1}}}+\sqrt {r_{{1}}-2\,M}}} \right)$$

I believe this formula is wrong. Inside the nat.log, the numerator and denominator should be reversed.

And then applied 2M=R, r₁ = 2R and r₂ = 3R. From this I obtained:

$$\sqrt {6}\sqrt {{R}^{2}}-\sqrt {2}\sqrt {{R}^{2}}+R\ln \left( {\frac {\sqrt {3}\sqrt {R}+\sqrt {2}\sqrt {R}}{\sqrt {2}\sqrt {R}+\sqrt {R}}} \right)$$

After simplification this becomes:

$$R \left( \sqrt {2}\sqrt {3}-\sqrt {2}+\ln \left( -\sqrt {3}+\sqrt {2} \sqrt {3}-\sqrt {2}+2 \right) \right)$$

I think you've made a mistake in your simplification. I believe my explicit formula for this case is the correct one.

An interesting exercise would be to do the same thing for an free falling observer at escape velocity with a (non inertial!) observer tagged along in his 'tail' a constant physical distance of 1 removed. If I am not mistaken we can drive this example all the way through passed the EH. Any takers?

For this, the question of what line of simultaneity to use for calculating proper distance would rear its head with a vengeance. None of the arguments that coordinate t=constant apply. Thus, before calculating anything you would have to arrive at some physically convincing model of the the path of simultaneity is for one of these observers. It would be different for each one (head versus tail). Thus, if head thinks tail is fixed 1 meter away, tail will disagree and find distance varying because of a different simultaneity. And no, I don't know what the actual answer is except that it will be different for head and tail, and neither will see coordinate time as the basis of simultaneity.

 

[Passionflower]

But this is not at all the calculation I was saying is messy. All your formulas below are strictly radial. I believe, non-local measurements of the speed of light by a static observer in this geometry are anisotropic. Thus if you actually did what I suggested, you would find a completely different result.

I wrote 'I went a step further'. Do you understand what the plot represents? Do you understand that by using this we can express the measured speed of light as ruler or proper distance divided by the time it takes light to get there?

All I did was to express things with physical distances instead of coordinate values mainly because there are a few individuals here who seems to think that everybody else knows next to nothing and that they are the only ones who know that r2-r1 is not a physical distance.

I believe this formula is wrong. Inside the nat.log, the numerator and denominator should be reversed.

I think you've made a mistake in your simplification. I believe my explicit formula for this case is the correct one.

I think my formulas are right, the derivation was done by Maple, perhaps you want to argue that Maple has a bug?

For this, the question of what line of simultaneity to use for calculating proper distance would rear its head with a vengeance. None of the arguments that coordinate t=constant apply. Thus, before calculating anything you would have to arrive at some physically convincing model of the the path of simultaneity is for one of these observers. It would be different for each one (head versus tail). Thus, if head thinks tail is fixed 1 meter away, tail will disagree and find distance varying because of a different simultaneity. And no, I don't know what the actual answer is except that it will be different for head and tail, and neither will see coordinate time as the basis of simultaneity.

We would measure the distance from the head as we use the clock on the head to measure the elapsed time of light. We can use a distance based on Lorentz factoring the integrand of a stationary observer or we can use a distance in Fermi coordinates. Point is we can calculate it and it is a good exercise doing it.

All it takes is a positive attitude, if I make a mistake then help correcting it instead of throwing 'doom' at it.

There is nothing 'expert' or positive about an attitude that it is all messy, too hard, not defined, meaningless etc. There is nothing 'expert' or positive about implying that people are too uninformed to even attempt to do it.
There is nothing 'expert' about giving only 'baby talk' by repeating that the the speed of light is c locally when clearly the discussion goes beyond that.

Contributing means sticking out one's neck and provide things we can calculate, so what if a mistake is made, I make many. But I do not sit back and say 'no, no, no, wrong, you don't understand' without actually doing anything positive except for implying I know it all and the other knows nothing.

Sounds like fun. If the trailing oberver must have the same velocity as the leader ( which begs a question* ) then the trailing observer must reach c before the EH, which can't happen, obviously.
I'm assuming G-P comoving faller for the leading observer here.

(*the same velocity when, and from what perspective ?)

I'm not at all sure if this correct, just a quick analysis.

Indeed, we can perhaps collectively attempt to calculate it?

And yes GP coordinates would work, would be a nice change from Schw. coordinates.

Can we find the formulas for the 'raindrop'?

e.g.

- Distance to the EH (both in terms of Lorentz factoring the integrand used for a stationary observers and in Fermi coordinates (and if the 'doomsayers' there are many more, please come up with them, a few formulas more or less will not break it) ) for a given r value.
- Time it takes light to go a given physical distance away from a 'raindrop' for a given r value?

The trailing part moves obviously non inertially (a raindrop with a little rocket engine :) ) but I do not see how that matters for calculations at the head, but of course we have to watch for the acceleration at the tail to go to infinity.

Seems that is all we need, then we can start to drive this towards the EH and see what happens.

 

[pervect]

For this, the question of what line of simultaneity to use for calculating proper distance would rear its head with a vengeance. None of the arguments that coordinate t=constant apply. Thus, before calculating anything you would have to arrive at some physically convincing model of the the path of simultaneity is for one of these observers. It would be different for each one (head versus tail). Thus, if head thinks tail is fixed 1 meter away, tail will disagree and find distance varying because of a different simultaneity. And no, I don't know what the actual answer is except that it will be different for head and tail, and neither will see coordinate time as the basis of simultaneity.

Given the worldline of an observer, accelerated or not, in curved space-time or flat, for points sufficiently close to the observer there is a fairly natural notion of simultaneity, and of distance.

This happens because the geometry of space-time is locally Lorentzian - as is described in MTW on pg 19, if you happen to have that textbook. I'll give a short quote:

On the surface of an apple within the space of a thumbrprint, the geometry is Euclidean (Figure 1.1 - the view in the magnifying glass). In spacetime, within a limited region, the geometry is Lorentzian. On the apple the distances between point and point accord with the theorems of Euclid. In spacetime the intervals ("proper distance,", "proper time") between event and event satisfy the corresponding theorems of Lorentz-Minkowski geometry.

Given a specific metric, it's fairly easy to recover said notion of local distance. What you do is introduce a set of coordinates that make the space-time metric at that point diagonal and unity (assuming that you've set c=1). All you need to do is to find a linear transformation that diagonalizes the metric.

You can transform to new coordinates either by the usual tensor transformation laws, or by simple algebra. It's easiest if you write the old variables in terms of the new, i.e. if you have a metric in (x,t) and you want to change to (x', t') you can write:

x = ax' + bt'
t = cx' + et'

then you can just write dx = a dx'+b dt' and dt = c dx' + e dt and substitute to get the metric in terms of x' and t'.
'
Given such a swath of space-time with a locally Minkowskian metric, the coordinate differences actually represent physical distances (in the small region where space-time is flat), so you can read distances directly from the new coordinates, and you can define the natural notion of local simultaneity for said observer by setting dt' = 0.

This notion of simultaneity will make the speed of light isotropic, as should be obvious (I hope) from the Mikowskian metric, which defines the path light must take by ds^2 = 0.

In curved space-time, the notion of how to extend the notion of simultaneity beyond a small local region is not clearcut. One possibility, which however, isn't unique, is especially useful. This is to extend the definition of simultaneity by drawing geodesic curves through the locally simultaneous points as described above. This leads to "fermi normal" coordinates. Another way of saying this is that simultaneous points in time are generated by the set of space-like geodesics passing through your observer's worldline at a given point that are orthogonal to said worldline.

While this seems like (and is) a very natural choice for simultaneity, it's not the only one in common use by any means. Cosmologists, for instance, do NOT use fermi normal coordinates when they report on distances within the universe. They use surfaces of constant cosmological time, cosmological time is time elapsed in the comoving frame since the big bang, instead.

The fermi-normal defintion of simultaneity (and of distance) is useful because it's compatible with the notion of Born rigidity. You can construct a family of observers all of whom measure the distance to their neighbors as constant, which is exactly what you need for a notion of distance that's compatible with Born rigidity.

If you try this with the cosmologists notion of distance, you find that it won't work, because observers with constant coordinates don't maintain a constant distance from each other, so the conditions you need for Born rigidity aren't met by the coordinate system.

It's a bit off the topic, but https://www.physicsforums.com/showthread.php?t=435999&highlight=fermi+normal does do a series expansion for fermi-normal coordinates for observers "falling from infinity" into a black hole, which provides one answer to the question about a "constant distance" observer falling into a black hole.

On a more general note, there's some reasonable-looking discussion at the Wikipedia at http://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=392242531, about Rindler coordinates which are the flat space-time analogue of Fermi-normal coordinates which goes into some detail about distances - though it's a bit lacking in references, alas.

I'd recommend getting familiar with Rindler coordinates first before worrying too much about Fermi-normal coordinates. MTW is a good reference, if you have it, and a bit more reliable than the wiki article - not that it helps if you don't have the textbook. This would address some of your concerns about the issue of the static observers accelerating.

There are apparently exact solutions for Fermi-normal coordinates in the literature for the interior Schwarzschild space-time, unfortunately I don't have access to compare to my series expansion for the exterior region, i.e http://jmp.aip.org/resource/1/jmapaq/v51/i2/p022501_s1?isAuthorized=no .

 

[Mentz114]

Can we find the formulas for the 'raindrop'?

e.g.

- Distance to the EH (both in terms of Lorentz factoring the integrand used for a stationary observers and in Fermi coordinates (and if the 'doomsayers' there are many more, please come up with them, a few formulas more or less will not break it) ) for a given r value.
- Time it takes light to go a given physical distance away from a 'raindrop' for a given r value?

The trailing part moves obviously non inertially (a raindrop with a little rocket engine :) ) but I do not see how that matters for calculations at the head, but of course we have to watch for the acceleration at the tail to go to infinity.

Seems that is all we need, then we can start to drive this towards the EH and see what happens.

I think it's do-able. I just think the tail-end won't be able to maintain the separation after the leader goes through the EH. They will definitely lose radar contact after that time, because no light signal can be sent to outside the EH from inside.

I'll have a go at the calculation when I find a pencil and an old envlope