I The bending of starlight is twice the Newtonian prediction

[Nugatory]

Essentially the idea is that the temporal component of the bending is due to the fact that light passing near the star travels more slowly (just like sound waves traveling over water or light waves in a graded index optical fibre)

Except that this just isn't right. To see the problem we have to define exactly what "Travels more slowly" means, and that is harder than it looks. Say that we have two points A and B on the path of the light beam. We have a detector at point A and a detector at point B; the light flash passes A, a bit later it arrives at arrives at B; it seems obvious that if we divide the distance between A and B by the travel time between the two detections we'll have the speed of light... but both that distance and that time depend on our choice of coordinates.

First, what is the distance? That will be the spacetime interval between the point on A's worldline where the detection happened and that point on B's worldline which happened "at the same time", which is to say has the same time coordinate. Clearly this will depend on our simultaneity convention, we can make it come out to be pretty much any arbitrary number according to what "at the same time" means.

Second, what is the travel time? It's the difference between what A's clock read at its detection event and what B's clock reads at its detection event - but only if the two clocks are synchronized, meaning that they read the same thing at the same time, so again we require a simultaneity convention. It doesn't help to appeal to some third party, say us astronomers far from the sun and looking at our lab clock; that just means that we're choosing an arbitrary simultaneity convention between our lab clock and the two detectors.

There is a standard way out of this impasse: A and B are close enough that we can approximate the spacetime between them as locally flat and use Einstein clock synchronization and the methods of special relativity to define both the travel time and the travel distance. When we do this (and it would be somewhat perverse not to) we will always find that the distance divided by the time is $c$.

(The contributor who mentioned that in GR a geodesic maximises the proper time was confusing proper time with elapsed time. In any case the proper time of a light ray is always zero whichever way it travels.)

I fear that the confusion is on your end. What you are calling an "elapsed time" is a proper time: It is the proper time between two readings of some remote observer's clock, when the readings are made at the same time (the simultaneity convention reenters the discussion) as the two events between which you are determining the elapsed time.

 

[J O Linton]

I stand by my statements.
Tom takes a metre ruler and a clock to the bottom of a mine shaft. He sets up a device consisting of a light beam pulsing back and forth along the ruler. The device emits pulses of light from each end A and B when the light beam is reflected. He places his clock midway along the rule and calibrates it so that it reads a time interval between pulses equal to $\frac{1}{c}$ He then carries his clock to the top of the mine shaft. Due to gravitational time dilation he now discovers that the time interval between pulses is now $\gamma \frac{1}{c}$. The proper time between events A and B is, of course, zero. From Tom's point of view the elapsed time between these events is $\gamma \frac{1}{c}$ and the apparent speed with which the light travels from A to B is (from Tom's point of view) $\frac {c}{\gamma}$.

 

[PeroK]

I hope your students are spared that confusion of ideas and terminology!

 

[J O Linton]

Please. What is it that you object to in the scenario I have described?

 

[A.T.]

Firstly I don't think it is quite true to imply that the spatial effect is global which the temporal (EP) effect is local. All physical effects are local ...

But the total amount of bending after passing a large region is a cumulative effect. It's not local measurement.

... and the bending of light in any region of space near a gravitating mass is solely determined by the nature of space-time in the locality.

The bending relative to the local patch of space time is one thing. But the way how those spacetime patches are combined to form a larger region also affects the total amount of total bending, after passing that region.

 

[Nugatory]

Tom takes a metre ruler and a clock to the bottom of a mine shaft. He sets up a device consisting of a light beam pulsing back and forth along the ruler. The device emits pulses of light from each end A and B when the light beam is reflected. He places his clock midway along the rule and calibrates it so that it reads a time interval between pulses equal to $\frac{1}{c}$ He then carries his clock to the top of the mine shaft. Due to gravitational time dilation he now discovers that the time interval between pulses is now $\gamma \frac{1}{c}$.

Are we using the clock in the middle of the metre ruler to determine this? If so you are mistaken: once calibrated at the bottom of the well that clock will not need recalibration when we bring the device up to the top of the well, and it will always show the time between flashes to be $1/c$ and the speed of light to be $c$.

Or are we using Tom’s clock to determine this? If so, we have the problem described in post #31 again: we are measuring the proper time between two ticks of Tom’s clock and relying on an arbitrarily chosen simultaneity convention to calculate another quantity that you have arbitrarily chosen to treat as the “time interval between pulses”. The “speed of light” we calculate from this time interval will indeed be different at the top and the bottom of the well, but that’s just because we chose the simultaneity convention to make it come out the way. It’s an artifact of our choice of coordinates (which you haven’t stated but everything you’ve said suggests that you’ve been implicitly assuming Schwarzschild coordinates) and cannot be the cause of anything.

From Tom's point of view the elapsed time between these events is $\gamma \frac{1}{c}$ and the apparent speed with which the light travels from A to B is (from Tom's point of view) $\frac {c}{\gamma}$.

I’m not sure what this quantity $\gamma$ is in this context?

 

[J O Linton]

Yes - I made a serious error there. Sorry. I should have said that Tom calibrates his clock against his device at the top of the shaft. Then he carries the device to the bottom. When he returns to the top he will find that the pulses from the device are time dilated by a factor $\gamma = \frac{1} {\sqrt {1 - 2 \Delta \phi / c^2}}$ where $\Delta \phi$ is the gravitational potential difference between the top and the bottom of the shaft. Since there is no reason to suppose the length of the ruler to change (since it remains at all times at right angles to the assumed uniform gravitational field) Tom can legitimately conclude that the speed of light is smaller at the bottom of the shaft.

Could you explain what Schwartzchild coordinates are, please? I have implicitly been thinking of an observer comoving with the star and a long distance away. This coordinate system is rather special and cannot be described as arbitrary.

 

[Dale]

Since there is no reason to suppose the length of the ruler to change ... Tom can legitimately conclude that the speed of light is smaller at the bottom of the shaft.

The BIPM disagrees. As does Einstein.

This coordinate system is rather special and cannot be described as arbitrary.

**All** coordinate systems are arbitrary. That is indeed the whole point of the covariance of the laws of physics.

 

[PeroK]

Since there is no reason to suppose the length of the ruler to change (since it remains at all times at right angles to the assumed uniform gravitational field) Tom can legitimately conclude that the speed of light is smaller at the bottom of the shaft.

This is not the correct conclusion. In GR the local speed of light in vacuum is invariant everywhere. This is a direct consequence of moving on a null worldline.

Local measurements of the speed of light tend to $c$ as the distance over which they are measured tends to zero. I.e. the local speed of light means the instantaneous speed to which actual measurements are an approximation.

In your case, measurements of the time for the pulse to make a return journey will be slightly different at the top and bottom of the ruler. One will give a slight overestimate for the local speed of light (bottom) and one a slight underestimate (top). Note, however, this depends on the direction of the light pulse. If the light pulse is fired upward and reflects back down, then you get an overestimate; and, if the pulse is fired down, then you get an underestimate. It doesn't directly depend on where in the gravitational potential the clock is located - but on the direction of the light beam relative to the clock.

In any case, your local measurement of the speed of light will always be $c$, subject to experimental error.

There was a discussion on this recently:

https://www.physicsforums.com/threa...an-invariant-in-gr.999152/page-3#post-6454542

 

[anuttarasammyak]

Since there is no reason to suppose the length of the ruler to change (since it remains at all times at right angles to the assumed uniform gravitational field)

Prepare two 1m codes and make them straight horizontally from the poll. Higher code goes farther than the lower one.

 

[PeroK]

PS if the light beam is fired horizontally then there is no issue. The experiment at the top and bottom of a deep shaft will give the same result of $c$, independent of the gravitational potential.

 

[J O Linton]

I am sorry if I did not make myself clear and I am sorry if I forgot to mention that the ruler was horizontal (but I did say that it was at right angles to the GF). I agree that Tom at the top of the shaft and Dick at the bottom will measure the same value for the speed of light but it remains the case that the pulses of light which Tom receives from the device at the bottom will be time dilated and he can therefore legitimately conclude that from his point of view light travels slower at the bottom of the shaft then at the top. It is a trivial matter to deduce quantitatively that the refraction (i.e. bending) of a horizontal light beam is equal to g/c in perfect agreement with the EP. I am baffled as to why everyone is having so much difficulty with this argument.

 

[PeroK]

I am baffled as to why everyone is having so much difficulty with this argument.

Because it's wrong?!

 

[J O Linton]

Because it's wrong?!

Please don't shout. Instead pretend that I am an intelligent sixth former studying physics at A level and explain exactly what is wrong with the argument. Is it that you do not agree that the pulses of light received by Tom are time dilated? Or do you think that the ruler at the bottom of the shaft changes length as Tom climbs out? or do you think that Tom's conclusion is misguided?

 

[PeroK]

General Relativity is an advanced undergraduate subject.

In any case, in GR vectors are local objects, defined in a local tangent space. The speed of light only has meaning locally.

The high-school notion of a vector as a displacement or directed line segment is only valid in flat spacetime.

That's one of several mathematical hurdles to cross.

Without this focus on the proper mathematical basis, you risk the whole thing reducing to a mish-mash of ideas.

The other prerequisite is to have SR nailed down and completely understood. That's beyond A level, I would say.

 

[PeterDonis]

it remains the case that the pulses of light which Tom receives from the device at the bottom will be time dilated

Yes.

he can therefore legitimately conclude that from his point of view light travels slower at the bottom of the shaft then at the top.

The "from his point of view" is the crucial statement here. When it is unpacked, what you end up with is that Tom is calculating a coordinate speed of light. Coordinate speeds do not have physical meaning in GR. Tom cannot directly measure the speed of light at the bottom of the shaft when he is at the top. He can only calculate a coordinate speed.

This is hard for many people to grasp in GR because they are used to SR, where spacetime is flat and one can attribute a physical meaning to coordinate speeds in inertial frames. But in a curved spacetime there are no global inertial frames and this no longer works.

It is a trivial matter to deduce quantitatively that the refraction (i.e. bending) of a horizontal light beam is equal to g/c in perfect agreement with the EP.

The locally measured refraction is, yes. But that is not the same thing as a measurement of a speed at a distant location. (Nor is it the same as the global bending obtained by "adding up" all of the local bendings correctly in a curved spacetime.)

 

[Ibix]

I am baffled as to why everyone is having so much difficulty with this argument.

Apart from confusion over your experimental setup, the problem is that your interpretation of the results are coordinate dependant. It's a bit like assigning meaning to the fact that a road runs horizontally on a map. It doesn't run horizontally on every map, only the ones that happen to pick the direction it happens to run as their notion of horizontal.

Fundamentally, the problem is that to define "gravitational time dilation" you have to take the tick rate ratio between two clocks and split it into a component due to "speed through space" and a component due to "height difference". But spacetime is one thing. There is no unique way to divide it into "space" and "time" so there is no unique way to split up the tick rate ratio. In fact, in many cases, "speed through space" is a highly problematic concept.

In the specific case of a very nartow class of spacetimes called "static" spacetimes, there is an obvious way to split spacetime into space and time. Then there's an obvious way to define gravitational time dilation. Spacetime near the Earth is not one of those spacetimes - but it's not far off, so we often use the concept of gravitational time dilation as if it were a static spacetime.

But even in a static spacetime, obvious does not mean unique. So even when there's an obvious way to do it, you still have to make a conscious choice to use the obvious way. No measurement in the world will tell you that you were right to do so. Treating gravitational time dilation as a kind of refractive index is possible in static spacetimes. But it only makes sense if you choose that particular choice of coordinates and it does not work at all in more complicated spacetimes.

Here's why we'd rather you didn't teach it: people love to take models of gravitational effects and apply them to exciting phenomena like colliding black holes or the expanding universe. Neither of those spacetimes is static so the model doesn't apply. Then they come on here and ask questions based on the incorrect model and we have to waste time unpicking misunderstandings that could have been avoided by not teaching them a poor model. People often seem to find it difficult to accept that they were taught a crummy model and they either need to learn enough maths to understand why it's crummy or take our word for it that they were taught wrong.

You need to be realistic about the limitations of what you can teach without advanced maths. The most general explanation for why light paths curve differently from what a local experiment would predict is that you can't fit a chain of local labs together without distorting their right angled corners, and that angular distortion accounts for the "extra" bending. There are headaches even with that, but it's about as true an explanation as you can make without the maths.

Please either take my word for it or learn enough maths to understand why "refractive index" type explanations are problematic.

 

[PeterDonis]

Could you explain what Schwartzchild coordinates are, please?

https://en.wikipedia.org/wiki/Schwarzschild_coordinates

I have implicitly been thinking of an observer comoving with the star and a long distance away. This coordinate system is rather special and cannot be described as arbitrary.

You appear to be under the misapprehension that there is one unique coordinate system which is "at rest relative to the star". That is not the case. Schwarzschild coordinates are one such coordinate system, but not the only one. There are an infinite number of possibilities depending on how you want to define the radial coordinate. At least one other such possibility, isotropic coordinates, is commonly used in GR (and is probably the one you are intuitively picturing).

 

[J O Linton]

Once again, I thank you for your efforts at explaining things to me. You have given me a lot to think about.

I would just like to say that when it comes to teaching physics, almost nothing that we teach is strictly true. Everything is either an approximation (eg Newton's theory of gravity) or an analogy (eg the wave theory of light). But if you were to insist on teachers only teaching what is strictly true, we would not teach anything at all. Indeed, I am sure that many university teachers would positively prefer their students to come to them with as little pre-knowledge as possible (as generally happens in the States).

But the majority of my students will not study physics at university level. My job is two-fold; to give my non-scientific students a basic understanding of how the world works and why things do what they do; and secondly, to enthuse those students with the required ability and curiosity to take the subject further so that you have some undergrads to fill your lecture halls! If you have to disabuse some of them of the crazy and incorrect notions that they have picked up from their physics teachers, then I am afraid, that is life and you are going to have to live with it. You will be glad to hear that I am not often asked to explain why the bending of light round a star is twice the Newtonian prediction and I promise that if I am asked, I will bracket my replies with suitable cautionary phrases.

Thank you once again. I shall regard this thread as closed.

 

[Ibix]

I would just like to say that when it comes to teaching physics, almost nothing that we teach is strictly true.

I disagree with the "almost".

My job is two-fold; to give my non-scientific students a basic understanding of how the world works and why things do what they do; and secondly, to enthuse those students with the required ability and curiosity to take the subject further so that you have some undergrads to fill your lecture halls!

Then tell them that square labs don't fit together in a square grid, and that's why the curvatures each one measures don't add up to the total curvature. That's a much better and more general explanation than "gravity acts like a refractive index" and I don't see it as any less enthusing.

 

[vanhees71]

This is a very bad attitude for a teacher. You should not teach anything that is really wrong. Of course, you have to make simplifications and omit topics that cannot be treated with the level of mathematics your students understand and can handle. It is very well justified to teach classical Newtonian mechanics and classical electromagnetics. The latter subject is already problematic, because in principle you need vector calculus for it, but at least in electrostatic you can get pretty far by starting from Coulomb's Law and describe the electrostatic forces etc.

What's really bad is if the teaching provides even a qualitatively wrong picture, e.g., by insisting on teaching Bohr's model of the atom (cementing the idea of electrons moving on orbits around the nucleus as if it were a tiny planetary system).

Concerning general relativity it's very hard to teach right without quite advanced math (tensor calculus on non-Euclidean manifolds). You can get some of the flavor by carefully (!) referring to the equivalence principle, but as this thread shows there's always the danger to overstate this qualitative picture.

Treating the propagation of light in GR space time as if it were light going through a dielectric medium is also dangerous, and I'd not refer to this. To understand the bending of light in a gravitational field (i.e., geometrically spoken wave propagation in curved space-time) you can refer to the geodesic motion of a massless particle, but it's in fact the solution of what's called the eikonal approximation of an electromagnetic wave in this curved space-time, it describes the change of the wave vector of a locally plane em. wave. It's the description of light in terms of geometric optics.

In this and related context one should not talk about "photons" as if they were point-like particles, although that's the picture you find very often even in university-level textbooks, but that's another sin of bad physics didactics, because it provides again an even qualitatively wrong picture about electromagnetic waves/light.

 

[PeterDonis]

when it comes to teaching physics, almost nothing that we teach is strictly true

If by "strictly true" you mean "a model that is exactly correct", then yes, I agree. None of our scientific models are exactly correct.

However, the onus is still on you as a teacher to correctly describe the models themselves (as well as the experimental facts and how well the predictions of the models match the experimental facts). If you can't correctly describe the models, because you don't think your students will be able to understand a correct description (or because you're not sure you understand a correct description yourself), then IMO you should not describe the models at all; you should just describe the experimental facts, and perhaps a very simple description of an underlying model (I will give an example of that below), and point students who are interested in learning more at a reference that can help them learn a correct description of the underlying models for themselves.

My job is two-fold; to give my non-scientific students a basic understanding of how the world works and why things do what they do

There's a very simple way to do that for the topic given in the title of this thread: the bending of light is twice the Newtonian prediction because of spacetime curvature.

Most students, who as you say will not be studying physics at the university level, will probably be satisfied with that answer and you won't need to go into any more detail. The ones who will be curious for more detail are covered by my next comment below.

and secondly, to enthuse those students with the required ability and curiosity to take the subject further so that you have some undergrads to fill your lecture halls!

Students who have that ability and curiosity will be able to folllow up references themselves. They won't need to be told things in class that they then have to unlearn later.

 

[Nugatory]

Is it that you do not agree that the pulses of light received by Tom are time dilated? Or do you think that the ruler at the bottom of the shaft changes length as Tom climbs out? or do you think that Tom's conclusion is misguided?

We agree that if a source at rest (using a particular coordinate system) emits flashes of light at intervals $\Delta T_S$, a receiver at rest (using the same coordinate system) higher in that gravity well will receive them at intervals $\Delta T_R$, that these are intervals measured using the same clock and therefore are proper times, and that $\Delta T_R \gt \Delta T_S$. This is the phenomenon that is often casually referred to as gravitational time dilation; it is predicted by general relativity and confirmed by (among other tests) the Pound-Rebka experiment.

We agree that the ruler does not change length. This follows directly from the definition of the meter as the distance that light travels in $1/c$ seconds and the fact that the clock in the middle of the ruler receives a flash from one side or the other every $1/c$ seconds.

We agree that Tom can calculate the quantity $1/\Delta T_R$ and that this wil be something less than $c$. If Tom does not already know that this is an example of what is properly called a "coordinate speed", it is high time that he learn about coordinate speeds and their limitations.

Where things go off the rails is trying to assign any physical significance to this (or any other - that's one of the limitations) coordinate speed. In particular thinking that the difference between that coordinate speed and $c$ suggests that there is something different about the spacetime deeper in the well, as does your analogy with boggy and hilly ground, is just plain wrong.

I will concede that the bogs and hills analogy is consistent (at a hand-waving math-free level) with the observed deflection of starlight, but just because it works in that one case doesn't make it good pedagogy.
It would be like teaching that Apollo's chariot pulls the sun from east to west every day - a fine explanation if we care only about the motion of the sun through the sky, but one that fails for the motion of other heavenly bodies and must be unlearned before the student can make further progress.

 

[pervect]

Please. What is it that you object to in the scenario I have described?

The soldiers in the swamp are presumably minimizing travel time, rather than maximizing proper time. Other than that, it seems as good as any other popularization, though I'm not quite sure how badly it might fail if a student relies on it as their sole understanding of the issues.

A truly correct understanding of the physics would require understanding space-time as a Lorentzian geometry, and the principle of maximal aging. Which would be more ambitious, though it has the advantage of actually being correct.

It's definitely better than the bowling ball on the trampoline analogy, but that's not a high bar.

 

[PeroK]

@J O Linton

Just to add my tuppence ha'penny worth:

The book I referenced previously Gravity, an Introduction to Einstein's GR, by James Hartle is probably the theoretical minimum for GR. Even so, he doesn't cover the geometry of a spherical star and Schwarzschild coordinates until chapter 9.

I found that you need a lot of experience of mathematical physics and a certain maturity of thought before you can digest GR. The first time I looked at that book I was way beyond A-level (I found A-level physics a piece of cake even after 30 years break from maths and physics) - but I still backed off GR at the first attempt. When I came back two years later I was older and wiser, and it was much more accessible.

The other issue is that you cannot really teach GR if you are floundering in the dark. There are a great set of lectures by Professor Scott Hughes of MIT on YouTube - that's a graduate physics course and starts with a re-representation of SR in geometric terms - starting with no assumed knowledge of GR. Unless you understand GR at that level, it's difficult to teach. At the very least, you would need to have worked through Hartle's book and/or the MIT lectures to teach GR in any meaningful way - you can't just make stuff up yourself!

My final point is that even if you do soldier on and try to teach GR, you really ought to take note of what has been said in this thread.

 

[berkeman]

Instead pretend that I am an intelligent sixth former studying physics at A level and explain exactly what is wrong with the argument

I would just like to say that when it comes to teaching physics, almost nothing that we teach is strictly true.

But the majority of my students will not study physics at university level. My job is two-fold; to give my non-scientific students a basic understanding of how the world works and why things do what they do; and secondly, to enthuse those students with the required ability and curiosity to take the subject further so that you have some undergrads to fill your lecture halls!

Yeah, I Googled your username, and found your home page:

http://www.jolinton.co.uk/index.html

I am a retired physics teacher

So who are your students now? Are you referring to the folks who read your blog and previously published articles?

In any case, it does appear that your articles at your website are more "popular press" in nature, rather than scientifically rigorous.

 

[stevendaryl]

I am sorry if I did not make myself clear and I am sorry if I forgot to mention that the ruler was horizontal (but I did say that it was at right angles to the GF). I agree that Tom at the top of the shaft and Dick at the bottom will measure the same value for the speed of light but it remains the case that the pulses of light which Tom receives from the device at the bottom will be time dilated and he can therefore legitimately conclude that from his point of view light travels slower at the bottom of the shaft then at the top. It is a trivial matter to deduce quantitatively that the refraction (i.e. bending) of a horizontal light beam is equal to g/c in perfect agreement with the EP. I am baffled as to why everyone is having so much difficulty with this argument.

Well, the description of gravitational time dilation in terms of light traveling slower in a gravity well is not really correct. The equivalence principle shows this: The same dilation occurs aboard an accelerating rocket.

 

[exponent137]

Summary:: Why does GR predict starlight bending twice Newtonian?

Maybe this paper will help you at this:
A simple calculation of the deflection of light in a Schwarzschild gravitational field
https://www.researchgate.net/publication/238984034_A_simple_calculation_of_the_deflection_of_light_in_a_Schwarzschild_gravitational_field
It helped me.

But, I do not understand enough why Newtonian (or classical) and SR calculations give the same result? Newtonian version can be easily calculated if we integrate projection of gravitational acceleration on the path of a ray, we obtain such change of velocity which gives such angle 0,89 arc seconds. But SR calculation contains part $(1-\frac{2GM}{r})dt^2$, why it has the same effect?

 

[PeterDonis]

I do not understand enough why Newtonian (or classical) and SR calculations give the same result?

What "SR" calculation are you talking about?

SR calculation contains part $(1-\frac{2GM}{r})dt^2$, why it has the same effect?

Any calculation with such a factor in it is a GR calculation, not an SR calculation. The GR calculation leads to a result twice as large as the "quasi-Newtonian" calculation.

 

[exponent137]

What "SR" calculation are you talking about?Any calculation with such a factor in it is a GR calculation, not an SR calculation. The GR calculation leads to a result twice as large as the "quasi-Newtonian" calculation.

The paper which I linked gives such terminology:
obtained by the application of the equivalence principle and special relativistic considerations alone (the Newtonian value)!
For $g_{11}$ and other it gives 1, but for $g_{00}$ it gives above-mentioned part $1-2GM/r$.