I Length contraction in General Relativity

[knowwhatyoudontknow]

TL;DR Summary

length contraction in General Relativity

In GR, a free falling object when viewed by a distant observer appears to be length contracted and slows down as it approaches the event horizon of a black hole. The length contraction piece, however, seems counterintuitive. I would have thought that the leading edge of the object would experience higher acceleration than the trailing edge and, therefore, the length would stretch not contract. What am I missing?

 

[PeterDonis]

In GR, a free falling object when viewed by a distant observer appears to be length contracted

In what sense? Please be specific.

and slows down as it approaches the event horizon of a black hole.

Again, in what sense? Please be specific.

What am I missing?

Respond to the above questions and then we'll see. You might find that responding to them resolves your issue all by itself.

 

[Demystifier]

Summary:: length contraction in General Relativity

I would have thought that the leading edge of the object would experience higher acceleration than the trailing edge and, therefore, the length would stretch not contract. What am I missing?

The stretching is caused by tidal forces appearing when the gravitational field is not homogeneous https://en.wikipedia.org/wiki/Spaghettification . This is an effect seen even by the observer comoving with the object. The length contraction, on the other hand, is an effect that is never seen by the comoving observer, it is only seen by another observer with respect to which the object moves. The first effect depends on the inhomogeneity of the gravitational field, the second effect depends on the relative velocity. The two effects are in a competition, but they are independent. This means that in some cases the first will prevail, while in other cases the second will prevail.

 

[PeroK]

The stretching is caused by tidal forces appearing when the gravitational field is not homogeneous https://en.wikipedia.org/wiki/Spaghettification . This is an effect seen even by the observer comoving with the object. The length contraction, on the other hand, is an effect that is never seen by the comoving observer, it is only seen by another observer with respect to which the object moves. The first effect depends on the inhomogeneity of the gravitational field, the second effect depends on the relative velocity. The two effects are in a competition, but they are independent. This means that in some cases the first will prevail, while in other cases the second will prevail.

How would you define the length of something to a distant observer in curved spacetime?

 

[Demystifier]

How would you define the length of something to a distant observer in curved spacetime?

Astronomers have no problems with that question, they see a picture in the telescope and after accounting for the fact that distant objects look smaller they compute the "true" size of the star, galaxy, or whatever. Now you will probably object that in this way they don't see Lorentz contraction, which is true, but it is true even in flat spacetime. As explained nicely in Sidney Coleman's Lectures on Relativity, Lorentz contraction cannot be seen, but can be observed. The observation consists of (i) seeing and (ii) computation of the correction for the time needed for the signal to get to the observer (page 23 in the Coleman's Lectures).

 

[PeroK]

Astronomers have no problems with that question, they see a picture in the telescope and after accounting for the fact that distant objects look smaller they compute the "true" size of the star, galaxy, or whatever. Now you will probably object that in this way they don't see Lorentz contraction, which is true, but it is true even in flat spacetime. As explained nicely in Sidney Coleman's Lectures on Relativity, Lorentz contraction cannot be seen, but can be observed. The observation consists of (i) seeing and (ii) computation of the correction for the time needed for the signal to get to the observer (page 23 in the Coleman's Lectures).

My point was that in flat spacetime there is the proper length of an object and the contracted length as measured/calculated/observed in an IRF.

But in curved spacetime the proper length may be unambiguous, and the length measured locally, but anything beyond that doesn't have any physical meaning. I would say that length contraction of a distant object only makes sense in approximately flat spacetime.

For example, if you study an object and light signals falling into a black hole in Eddington-Finkelstein coordinates, then what happens to the concept of length contraction? In my view, length contraction just isn't a thing in GR (General Relativity).

 

[Demystifier]

My point was that in flat spacetime there is the proper length of an object and the contracted length as measured/calculated/observed in an IRF.

But in curved spacetime the proper length may be unambiguous, and the length measured locally, but anything beyond that doesn't have any physical meaning. I would say that length contraction of a distant object only makes sense in approximately flat spacetime.

For example, if you study an object and light signals falling into a black hole in Eddington-Finkelstein coordinates, then what happens to the concept of length contraction? In my view, length contraction just isn't a thing in GR (General Relativity).

I agree with you that in curved spacetime the proper length is well defined only locally. But it is true even in flat spacetime if the object does not move inertially:
https://arxiv.org/abs/gr-qc/9904078

 

[PeroK]

I agree with you that in curved spacetime the proper length is well defined only locally. But it is true even in flat spacetime if the object does not move inertially:
https://arxiv.org/abs/gr-qc/9904078

If we are assuming Schwarzschild coordinates, then they have a coordinate singularity at the event horizon of a black hole, so are not a good choice for the OP's scenario. The problem with all such analyses of black holes is the overdependence on Schwarzschild coordinates as some sort of special coordinate system, such as IRF's in SR. Unless you insist on a particular coordinate system, there is no way to define length contraction.

 

[Demystifier]

If we are assuming Schwarzschild coordinates, then they have a coordinate singularity at the event horizon of a black hole, so are not a good choice for the OP's scenario. The problem with all such analyses of black holes is the overdependence on Schwarzschild coordinates as some sort of special coordinate system, such as IRF's in SR. Unless you insist on a particular coordinate system, there is no way to define length contraction.

There is no coordinate singularity if we study a regular star (instead of a black hole).

 

[Orodruin]

There is no coordinate singularity if we study a regular star (instead of a black hole).

The coordinate singularity is not what is relevant here. What is relevant is the definition of "length". In SR this is done by imposing a simultaneity convention and a following length definition within each simultaneity. The fact that the standard simultaneity convention differs between frames is what leads to length contraction in the first place and this is exacerbated in GR where the naturalness of simultaneity conventions are not necessarily as simple (although in a static spacetime there is a natural convention as spacelike surfaces orthogonal to the timelike Killing field), but this is a coordinate issue, not an issue of a far-away observer.

 

[cianfa72]

What is relevant is the definition of "length". In SR this is done by imposing a simultaneity convention and a following length definition within each simultaneity.

Yes agreed.

The fact that the standard simultaneity convention differs between frames is what leads to length contraction in the first place

standard simultaneity convention in SR (flat spacetime) basically amounts to Einstein's synchronization procedure in each inertial frame.

 

[Demystifier]

The coordinate singularity is not what is relevant here. What is relevant is the definition of "length". In SR this is done by imposing a simultaneity convention and a following length definition within each simultaneity. The fact that the standard simultaneity convention differs between frames is what leads to length contraction in the first place and this is exacerbated in GR where the naturalness of simultaneity conventions are not necessarily as simple (although in a static spacetime there is a natural convention as spacelike surfaces orthogonal to the timelike Killing field), but this is a coordinate issue, not an issue of a far-away observer.

I agree. But if a simultaneity convention is defined operationally, in terms of an experimental procedure, then the physicist who uses it does not even need to know whether the spacetime is flat or curved. My point is, if an operational procedure gives Lorentz contraction in the absence of gravity, then this operational procedure gives also (a generalized version of) Lorentz contraction in the presence of gravity.

 

[PeroK]

I agree. But if a simultaneity convention is defined operationally, in terms of an experimental procedure, then the physicist who uses it does not even need to know whether the spacetime is flat or curved. My point is, if an operational procedure gives Lorentz contraction in the absence of gravity, then this operational procedure gives also (a generalized version of) Lorentz contraction in the presence of gravity.

As an example of an operational procedure we could measure distance to an object using the round trip time of a light signal. This works for an inertial source in flat spacetime. But, for an object falling into a black hole, or objects fixed arbitrarily close to the event horizon, this gives an unlimited distance. That would give an infinite distance to the EH itself. This is clearly not what you would hope for.

 

[Demystifier]

As an example of an operational procedure we could measure distance to an object using the round trip time of a light signal. This works for an inertial source in flat spacetime. But, for an object falling into a black hole, or objects fixed arbitrarily close to the event horizon, this gives an unlimited distance. That would give an infinite distance to the EH itself. This is clearly not what you would hope for.

Fine, but this problem has nothing to do with curvature. This problem appears even in flat spacetime, for a uniformly accelerated observer. That's because such an observer "sees" the Rindler horizon.

 

[PeroK]

Fine, but this problem has nothing to do with curvature. This problem appears even in flat spacetime, for a uniformly accelerated observer. That's because such an observer "sees" the Rindler horizon.

It has everything to do with curvature, because Lorentz contraction (as far as an I level thread goes) is a coordinate effect between IRF's in flat spacetime.

The problem here is that the OP asks how the building blocks of SR apply in curved spacetime. The best answer, IMO, is that curved spacetime is a different ballgame. It's not an extension of SR.

Whereas, an answer that says "yes, you can generalise length contraction, yes you can generalise global IRF's; yes, you can generalise the Lorentz Transformation" and use these usefully to study curved spacetimes is a bad answer. However clever it might be!

You're leading the OP down a dead-end street.

 

[Demystifier]

The problem here is that the OP asks how the building blocks of SR apply in curved spacetime. The best answer, IMO, is that curved spacetime is a different ballgame. It's not an extension of SR.

I disagree that it is the best answer. I think a much better answer is possible if one first reformulates SR in purely local terms [1]. When SR is reformulated so, then GR is an extension of SR.

[1] See e.g. https://www.amazon.com/dp/3662520834/?tag=pfamazon01-20

 

[Nugatory]

because Lorentz contraction (as far as an I level thread goes) is a coordinate effect between IRF's in flat spacetime.

This discussion has some of the typical frustrations about what it means to say that something is "real" ... but I find the qualification to be unnecessarily strong. I'm reluctant to assign any special status to coordinate effects that appear when we're using Minkowski coordinates, but I also recognize that this is a matter of personal taste.

 

[PeroK]

I also recognize that this is a matter of personal taste.

I'd say it's a matter of definition and/or convention. For example, I can't find a reference to it in the index to Carroll's Spacetime and Geometry. Not even in the section on SR.

I would say if you tell the would-be student of GR that length contraction is a central concept, then you'd be misleading him or her. That's my point.

 

[knowwhatyoudontknow]

Thank you all for the spirited conversation! My 50000 foot take away is that length contraction as measured by an observer at a distance is ambiguous due the effects of curvature. However, length contraction may make sense in the context of a local comoving observer where the effects of curvature can be eliminated.

 

[PeterDonis]

My 50000 foot take away is that length contraction as measured by an observer at a distance is ambiguous due the effects of curvature.

You haven't answered the questions I asked in post #2, so "length contraction as measured by an observer at a distance" is not even a well-defined term yet in this discussion. You can't just wave your hands; you need to specify, explicitly, how the distant observer is "measuring length contraction".

 

[PeterDonis]

The length contraction, on the other hand, is an effect that is never seen by the comoving observer, it is only seen by another observer with respect to which the object moves.

I don't think this kind of "length contraction" is what the OP was asking about. (However, until the OP explicitly describes how the "length contraction" he is asking about is to be measured by the distant observer, we can't know for sure.)

 

[PeterDonis]

Astronomers have no problems with that question, they see a picture in the telescope and after accounting for the fact that distant objects look smaller they compute the "true" size of the star, galaxy, or whatever.

I don't think any of this has anything to do with what the OP is asking about.

 

[PeterDonis]

I agree with you that in curved spacetime the proper length is well defined only locally. But it is true even in flat spacetime if the object does not move inertially

This is correct, but what it means is that we need to have the OP tell us, explicitly, how the "length" he is talking about is to be measured by the distant observer. It does not mean that we should talk about some other unrelated concept, like SR "length contraction", instead.

 

[PeterDonis]

length contraction may make sense in the context of a local comoving observer where the effects of curvature can be eliminated.

A "local comoving observer" (by which I assume you mean an observer at rest relative to the object and spatially co-located with it) will not measure any "length contraction" at all.

 

[knowwhatyoudontknow]

The original intent of my question was to reconcile the idea of time dilation and length contraction near a black hole. Tidal forces stretch things in a radial direction whereas a stationary observer at a distance sees a progressive slowing of the object until it get stuck at the horizon, in a sedimentary way, and becomes something that is 2 dimensional. This is a consequence of gravitational time dilation. I think this thread is saying that length contraction doesn't make sense from the perspective of a stationary distant observer because the curvature of space makes any measurement questionable. If, on the other hand, the observer is stationary and close enough to the falling object such that curvature can be ignored, there may be a case that length contraction would be observed. My use of comoving was a mistake and I understand your comment. A stationary observer close to the accelerating object makes more sense. That said, this is potentially even more confusing to me. I am sitting close by the event horizon and I see my friend speeding by towards the BH at an ever increasing velocity in a radial direction. I see him more and more length contracted yet he should be stretched due to tidal forces. To avoid this paradox, do we just say that length contraction and gravitation are not compatible? In other words, length contraction doesn't make sense in non inertial reference frames.

I have little background in this subject so I apologize in advance if my responses lack total understanding.

 

[PeterDonis]

The original intent of my question was to reconcile the idea of time dilation and length contraction near a black hole.

And, as I have already said, before you can do that you need to describe, explicitly, how "length contraction" is measured by an observer at a distance.

For an object that is close to the hole, the observer at a distance can measure its time dilation, relative to him, by receiving light signals emitted by the object at regular intervals by the object's clock (for example, one signal per second by the object's clock) and seeing at what interval he receives the signals, by his clock. "Time dilation" means the signals are received by him at greater time intervals than they are sent (for example, they might arrive every 3 seconds by the observer's clock, which would indicate a time dilation factor of 3). This is the standard way of defining the "time dilation" of an object close to a black hole.

There is, however, no corresponding "standard" way of defining "length contraction" for such an object, as measured by an observer at a distance. That is why you need to specify explicitly how you want the distant observer to do that.

Tidal forces stretch things in a radial direction

More precisely, it causes two freely falling objects that are separated radially to increase their separation. But a single object, held together by internal forces, will not necessarily stretch in the same way; it will experience increasing internal stresses instead.

a stationary observer at a distance sees a progressive slowing of the object

More precisely, he sees light signals arriving from the object more and more redshifted, and arriving at his location larger and larger time intervals apart.

until it get stuck at the horizon, in a sedimentary way, and becomes something that is 2 dimensional.

None of this is correct.

I think this thread is saying that length contraction doesn't make sense from the perspective of a stationary distant observer because the curvature of space makes any measurement questionable.

No, what the thread is saying is that until you have defined, explicitly, how "length contraction" is to be measured, nothing useful can be said about it at all.

However, from the rest of your post, it seems like the "at a distance" question is not what you actually wanted to ask about. I'll respond to that further in a separate post.

 

[PeterDonis]

I am sitting close by the event horizon and I see my friend speeding by towards the BH at an ever increasing velocity in a radial direction. I see him more and more length contracted yet he should be stretched due to tidal forces.

This makes the issue you are asking about much clearer (and is different from the issue you appeared to be asking about in your OP).

See further comments below.

To avoid this paradox

There is no paradox. You are mixing up several different things:

(1) Suppose that you know that, if you were to sit at rest relative to the object way out in deep space, far from all gravitating bodies, you would measure its length to be $L$.

(2) If the object is then made to move past you at relativistic speed, still way out in deep space, far from all gravitating bodies, you would measure its length to be $L / \gamma$, where $\gamma = 1 / \sqrt{1 - v^2}$ and $v$ is the object's speed relative to you.

(3) Now the object is free-falling towards a black hole. What will you measure its length to be? Consider two possible cases:

(3a) You are free-falling along with the object. It's possible that tidal gravity will have caused the object to stretch--meaning, that the internal stresses inside the body, which it is subjected to because of tidal gravity, will not be sufficient to keep it from stretching. If it is stretched by tidal gravity by a factor $s$, where $s > 1$, then you will measure its length to be $s L$ if you are falling along with it (i.e., at rest relative to it).

(3b) Now suppose you are "hovering" at rest relative to the black hole, and the object is free-falling past you with the same speed $v$ as in case (2) above. Suppose also that tidal gravity is stretching the object by the same factor $s$ as in case (3a) above. Then you will measure the object's length to be $s L / \gamma$.

do we just say that length contraction and gravitation are not compatible?

No. They are perfectly compatible. See above.

In other words, length contraction doesn't make sense in non inertial reference frames.

The key underlying assumption in the "hovering" case, case (3b) above, is that you can measure the object's length in a time that is very short compared to the reciprocal of your acceleration in "natural" units. More precisely, if your acceleration is $a$, you must be able to measure the object's length in a time that is much less than $c / a$, where $c$ is the speed of light. As long as that is the case, you can just use your momentarily comoving inertial frame at the instant the object passes you to compute the object's speed $v$ and hence the length contraction you will observe.

 

[knowwhatyoudontknow]

OK, I think I have it. Maybe my language was a little sloppy at times. I meant to say looks like a 2D object from the point of view of a distant observer - not that it becomes a 2d object (actually disappears). 1, 2, 3, 3(a) and 3(b) explain things perfectly and clear up my confusion regarding a potential paradox. Finally, your point regarding how to measure length contraction is well taken. I think you are saying that if you cannot define how such a measurement could be made over large distances there is little use in going any further.

I thank you for patience.

 

[PeterDonis]

I meant to say looks like a 2D object from the point of view of a distant observer

Still not correct. Where are you getting this from?

your point regarding how to measure length contraction is well taken. I think you are saying that if you cannot define how such a measurement could be made over large distances there is little use in going any further.

That's correct. However, no matter how you define such a measurement, it will not make the infalling object "look like a 2D object" as it reaches the horizon.

 

[jartsa]

If we drop a ruler out of an accelerating rocket's window, first the ruler appears to not contract, because fast moving rulers do not appear to be contracted, for some optics related reason.

But when the ruler gets close to the Rindler-horizon, then the ruler appears to contract, as the part of ruler closer to the horizon appear to move slower than the parts of ruler further away.

If the rocket is not coordinate-accelerating, but hovering over a large black hole, the same thing is seen in that case too.

Tidal force is zero in the first case, small enough to be ignored in the second case, because the black hole is large enough, so that the gravity field is uniform enough.

 

[PeroK]

The original intent of my question was to reconcile the idea of time dilation and length contraction near a black hole.

As I pointed out above, if you study Sean Carroll's book Spacetime and Geometry in order to answer this question, then you'll be disappointed. Length contraction is not mentioned in the entire book; not even in the review of SR.

 

[Demystifier]

As I pointed out above, if you study Sean Carroll's book Spacetime and Geometry in order to answer this question, then you'll be disappointed. Length contraction is not mentioned in the entire book; not even in the review of SR.

What is really your point? Please choose one of the suggested answers that describes your opinion best.

a) Length contraction is entirely unphysical.
b) Length contraction makes sense only for flat spacetime.
c) Length contraction makes sense only when curvature is sufficiently small.
d) Length contraction makes sense only in the absence of horizons and/or coordinate singularities.

 

[PeroK]

What is really your point? Please choose one of the suggested answers that describes your opinion best.

a) Length contraction is entirely unphysical.
b) Length contraction makes sense only for flat spacetime.
c) Length contraction makes sense only when curvature is sufficiently small.
d) Length contraction makes sense only in the absence of horizons and/or coordinate singularities.

Length contraction is a useful concept in flat spacetime. Although, beyond introductory SR it becomes less useful. In particle physics, we use the energy-momentum relations and I don't recall length contraction ever being relevant to high-energy particle physics. In GR, the concept is of no practical use. There may be exceptions, but I suggest you are unikely to find discussion of length contraction in a GR text.

 

[Demystifier]

Length contraction is a useful concept in flat spacetime. Although, beyond introductory SR it becomes less useful. In particle physics, we use the energy-momentum relations and I don't recall length contraction ever being relevant to high-energy particle physics. In GR, the concept is of no practical use. There may be exceptions, but I suggest you are unikely to find discussion of length contraction in a GR text.

OK, I can agree that length contraction is not very useful in GR. But it's not totally useless. An interesting example is the submarine paradox [1]. In its standard formulation it combines length contraction and gravity, while recently I found a way to simplify it by considering a version of the paradox which does not use length contraction [2]. But just because the concept is not so much useful, it does not mean that the concept is totally wrong. I think it is perfectly legitimate to ask conceptual questions that involve length contraction in GR, even if GR is more naturally formulated without it.

[1] https://en.wikipedia.org/wiki/Supplee's_paradox
[2] https://arxiv.org/abs/2112.11162

 

[martinbn]

What is really your point? Please choose one of the suggested answers that describes your opinion best.

a) Length contraction is entirely unphysical.
b) Length contraction makes sense only for flat spacetime.
c) Length contraction makes sense only when curvature is sufficiently small.
d) Length contraction makes sense only in the absence of horizons and/or coordinate singularities.

Before asking and answering such questions, shouldn't one first give a definition of what he means by length contraction in a general space-time?

 

[Demystifier]

Before asking and answering such questions, shouldn't one first give a definition of what he means by length contraction in a general space-time?

Only if one is a mathematician.

Now seriously. No, because it seemed that the source of disagreement between PeroK and me was not in different definitions.

Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.

 

[martinbn]

Only if one is a mathematician.

Now seriously. No, because it seemed that the source of disagreement between PeroK and me was not in different definitions.

Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.

I didn't mean a strict definition vs intuitive one. I meant that one needs to say what he means when there is more than one possibility even if it is vague and intuitive definition.

 

[PeroK]

Unlike mathematicians, physicists rarely start from precise definitions. They often explain the idea intuitively and qualitatively first, while precise definitions, if needed at all, are usually given later. That's just how (most) physicists think, and it works fine for them.

Anyone teaching SR would immediately be able to give you the precise definition of (coordinate) length as the distance between simultaneous measurements of the two ends of an object (in the given reference frame - usually assumed to be inertial).

And, the proper length is defined as the length in a reference frame in which the object is at rest.

That is the starting point. Without that, you cannot make any progress and conclude that length contraction is a thing.

 

[Demystifier]

Anyone teaching SR would immediately be able to give you the precise definition of (coordinate) length ...

Sure, but here we are talking about GR, so precise definition is not so immediate.

 

[PeroK]

Sure, but here we are talking about GR, so precise definition is not so immediate.

There is none. That's the point. It's not a core concept in GR. It only appears in more obscure scenarios.

 

[PeroK]

There's a relevant discussion here:

https://physics.stackexchange.com/questions/168379/proper-length-in-gr

 

[martinbn]

Sure, but here we are talking about GR, so precise definition is not so immediate.

Can you give any definition, not necessarily precise?

 

[Demystifier]

There is none. That's the point. It's not a core concept in GR. It only appears in more obscure scenarios.

I think it's very subtle. Even in GR there is a well defined operational definition of spatial distance seen by a given observer [1]. One can then compare distances as seen by different observers in relative motion with respect to each other, and this gives a GR generalization of length contraction. Sure, it's not a core concept in GR, but it does make some sense.

[1] Landau and Lifschitz, The Classical Theory of Fields, Sec. 84.

 

[Demystifier]

Can you give any definition, not necessarily precise?

See my post above.

 

[martinbn]

I think it's very subtle. Even in GR there is a well defined operational definition of spatial distance seen by a given observer [1]. One can then compare distances as seen by different observers in relative motion with respect to each other, and this gives a GR generalization of length contraction. Sure, it's not a core concept in GR, but it does make some sense.

[1] Landau and Lifschitz, The Classical Theory of Fields, Sec. 84.

No, that is a local situation.

 

[Demystifier]

No, that is a local situation.

I'm not sure what do you mean by that. If you mean that Eq. (84.6) in LL is local, one can always integrate it $l=\int dl$ to get a global distance. The integral is taken along a geodesic with respect to the spatial metric (84.7).

 

[Demystifier]

Another, possibly deeper, way to define proper spatial size of an object in GR is to define it with respect to simultaneity in which the spatial size is maximal. It was used to define the internal black hole volume in https://arxiv.org/abs/1411.2854 .

 

[PeterDonis]

irst the ruler appears to not contract, because fast moving rulers do not appear to be contracted, for some optics related reason.

Where are you getting this from? Do you have a reference?

when the ruler gets close to the Rindler-horizon, then the ruler appears to contract, as the part of ruler closer to the horizon appear to move slower than the parts of ruler further away.

Reference?

If the rocket is not coordinate-accelerating, but hovering over a large black hole, the same thing is seen in that case too.

Reference?

 

[PeterDonis]

Before asking and answering such questions, shouldn't one first give a definition of what he means by length contraction in a general space-time?

More than that, one should first read subsequent posts to see what the OP was actually asking about. See in particular post #25 and my responses to it.

 

[knowwhatyoudontknow]

I found this: https://www.physicsforums.com/threads/gravitational-length-contraction-and-dilation.404153/. Post# 9. What bearing does this have on the current discussion?