GR interpretation of a scenario

[name123]

Imagine three marbles A, B & C with each with 2 small lasers attached. All in the same rest frame, separated from each other by a light year, and forming an isosceles triangle (where AB and AC are the same length, and the centre of BC is a kilometre from A), and that the lasers of each pointed to the marbles adjacent to them in the triangle (so like pointing to the adjacent vertices). And imagine that there were two equally sized space ships, D & E each having the external shape of of opposing halves of a cylinder bisected along its axis. D and E both travel fast towards the midpoint of BC from opposite directions along a line perpendicular to BC and which transects BC at its midpoint. Imagine that, from the perspective of D and E, that in the last 30 seconds of their journey, they decelerate as fast as the laws of physics allow and come together to form a cylinder 100m in diameter with its axis orientated along BC. I'll refer to the cylinder as F. Imagine F has enough mass to (from F's perspective) pull A towards it at a rate of 0.01 mm per second from where A was when F was formed (so the rate will increase as A gets closer). Over time A will be pulled towards F such that they touch.

But what is the GR explanation using an A centred coordinate system, of the reasonably rapid close in distance between A and F. Are B and C being pulled towards A along with F, if so, how is that explained, if not, what is the explanation for the light from each still pretty much pointing at the axis of F (even when A and F touch)?

 

[A.T.]

But what is the GR explanation using an A centred coordinate system, of the reasonably rapid close in distance between A and F.

What is for you the Newtonian explanation, using an A centred coordinate system?

 

[name123]

What is for you the Newtonian explanation, using an A centred coordinate system?

That the gravitational force is proportional to mass, and that although F would have moved slightly, (perhaps) not enough to be measurably significant, the main motion was performed by A. I would assume that with Newtonian physics the coordinate system is just a mathematical device and although A moving to F could be described in an A based coordinate system, it should be understood to be to be a different description of the motion of A moving to F given by the theory. I am just guessing here, I do not know what Newton actually stated regarding this (if he stated anything regarding it at all). I had assumed the motion of each object would somehow be inversely proportional to the proportion of mass of each object to the sum of the mass of both objects involved.

 

[pervect]

Imagine three marbles A, B & C with each with 2 small lasers attached. All in the same rest frame, separated from each other by a light year, and forming an isosceles triangle (where AB and AC are the same length, and the centre of BC is a kilometre from A), and that the lasers of each pointed to the marbles adjacent to them in the triangle (so like pointing to the adjacent vertices). And imagine that there were two equally sized space ships, D & E each having the external shape of of opposing halves of a cylinder bisected along its axis. D and E both travel fast towards the midpoint of BC from opposite directions along a line perpendicular to BC and which transects BC at its midpoint. Imagine that, from the perspective of D and E, that in the last 30 seconds of their journey, they decelerate as fast as the laws of physics allow and come together to form a cylinder 100m in diameter with its axis orientated along BC. I'll refer to the cylinder as F. Imagine F has enough mass to (from F's perspective) pull A towards it at a rate of 0.01 mm per second from where A was when F was formed (so the rate will increase as A gets closer). Over time A will be pulled towards F such that they touch.

But what is the GR explanation using an A centred coordinate system, of the reasonably rapid close in distance between A and F. Are B and C being pulled towards A along with F, if so, how is that explained, if not, what is the explanation for the light from each still pretty much pointing at the axis of F (even when A and F touch)?

I don't have a complete solution to this problem, and I'm unlikely to. But I can suggest a few ideas that might be helpful. The preliminary step is to start out with linearized gravity, and some presumed flat background metric. Then we'd ask how the masses D&E perturb it. We'd also have to ask - what is causing D&E to accelerate in the indicated manner, and determine if it had a significant impact on the metric. It's not terribly clear if it would affect the answer to the problem or not.

Probably the easiest thing to do would be to assume D&E are both made to de-accelerate by some sort of photon rocket (or perhaps you could make the exhaust neutrinos, which would be almost the same thing as far as gravity goes since they are nearly massless). Then you could take a pair of solutions to Kinnersley's photon rocket solutions, and combine them. This wouldn't be possible in the full theory, because gravity is nonlinear, but I think you could get away with it in linearized theory as a linear approximation.

If that's not the mechanism you want to slow D&E down, you'd either need to specify what mechanism was doing it - or perhaps you can make an argument that the mechanism doesn't matter. At this point I couldn't say for sure, but it seems likely it will matter. If we assume that D&E are being slowed by a rocket, they must be loosing mass. If D&E are coupled by some sort of field, the field will gravitate. If D&E have significant amounts of energy compared to their rest mass due to their relativistic motion, we'd expect the interaction field that's slowing them down to have to account for where the kinetic energy of D&E is going, and it's likely that it will have a gravitational effect.

Introducing any sort of specific coordinate system would be pretty much an afterthought, after you had the metric. The metric would imply a coordinate system, but the simplest solution might or might not have the properties that you're asking. If it doesn't, you'd need to write a transformation equation to trasnform the coordinates to make them look the way you wanted, then trasnform the metric appropriately.

 

[name123]

I don't have a complete solution to this problem, and I'm unlikely to. But I can suggest a few ideas that might be helpful. The preliminary step is to start out with linearized gravity, and some presumed flat background metric. Then we'd ask how the masses D&E perturb it. We'd also have to ask - what is causing D&E to accelerate in the indicated manner, and determine if it had a significant impact on the metric. It's not terribly clear if it would affect the answer to the problem or not.

Probably the easiest thing to do would be to assume D&E are both made to de-accelerate by some sort of photon rocket (or perhaps you could make the exhaust neutrinos, which would be almost the same thing as far as gravity goes since they are nearly massless). Then you could take a pair of solutions to Kinnersley's photon rocket solutions, and combine them. This wouldn't be possible in the full theory, because gravity is nonlinear, but I think you could get away with it in linearized theory as a linear approximation.

If that's not the mechanism you want to slow D&E down, you'd either need to specify what mechanism was doing it - or perhaps you can make an argument that the mechanism doesn't matter. At this point I couldn't say for sure, but it seems likely it will matter. If we assume that D&E are being slowed by a rocket, they must be loosing mass. If D&E are coupled by some sort of field, the field will gravitate. If D&E have significant amounts of energy compared to their rest mass due to their relativistic motion, we'd expect the interaction field that's slowing them down to have to account for where the kinetic energy of D&E is going, and it's likely that it will have a gravitational effect.

Introducing any sort of specific coordinate system would be pretty much an afterthought, after you had the metric. The metric would imply a coordinate system, but the simplest solution might or might not have the properties that you're asking. If it doesn't, you'd need to write a transformation equation to trasnform the coordinates to make them look the way you wanted, then trasnform the metric appropriately.

Thanks for the response but I am not sure how significant you think the deceleration method which caused the least amount of difference would make. Would some type of timed release bungee system be out of the question? Even if D and E lost half their mass in their 30 second deceleration surely it wouldn't make that much difference if at 1km distance A and F would be gravitating towards each other at 0.01mm per second. At least I wouldn't have expected it to using Newtonian physics, and I wasn't clear that GR would offer vastly different expected results from Newtonian physics here, as perhaps we could assume that D and E were traveling at 1000km per second before deceleration, so considerably slower than the speed of light. At such speeds would you expect that a minimal impact (to the scenaro) 30 second deceleration system would be significant enough that the lasers of B and C would not still be hitting F (which has a 100m diameter) when A and F touched? If they would then there is still the question of how that is explained from a GR perspective with an A centred coordinate system.

 

[Nugatory]

If they would then there is still the question of how that is explained from a GR perspective with an A centred coordinate system.

The "explanation from a GR perspective" is the same in all coordinate systems. A, B, C, and F are following geodesic paths through spacetime, and GR will tell you what these paths are.

Before F arrives on the scene, A's path through spacetime (called a "worldline") is the same as the path of the point of intersection of the beams from B and C towards A, so A is illuminated by these laser beams. When F arrives on the scene its worldline is the same as the path of the midpoint of the BC line, so the laser beams between B and C continuously illuminate it. However, A's path through spacetime diverges from the path of the point of intersection of the beams from B and C towards A, and intersects the worldline of F; A is no longer illuminated by these laser beams and A and F will eventually collide.

Whether you describe these worldlines using coordinates in which A's position in space is fixed while B, C, and F are moving towards it or coordinates in which A is moving is irrelevant. They're the same worldlines with the same geometrical relationship to one another, and the "GR explanation" is in the geometrical relationship between them.

 

[name123]

The "explanation from a GR perspective" is the same in all coordinate systems. A, B, C, and F are following geodesic paths through spacetime, and GR will tell you what these paths are.

Before F arrives on the scene, A's path through spacetime (called a "worldline") is the same as the path of the point of intersection of the beams from B and C towards A, so A is illuminated by these laser beams. When F arrives on the scene its worldline is the same as the path of the midpoint of the BC line, so the laser beams between B and C continuously illuminate it. However, A's path through spacetime diverges from the path of the point of intersection of the beams from B and C towards A, and intersects the worldline of F; A is no longer illuminated by these laser beams and A and F will eventually collide.

Whether you describe these worldlines using coordinates in which A's position in space is fixed while B, C, and F are moving towards it or coordinates in which A is moving is irrelevant. They're the same worldlines with the same geometrical relationship to one another, and the "GR explanation" is in the geometrical relationship between them.

But unlike in Newtonian physics I thought motion was relative in GR. Such that it could be argued that A was at rest and not moving towards F, instead, F was moving towards A (from one perspective). But I did not know how could GR could argue that F was moving towards A while F was being continuously illuminated by the laser beams of B and C (which are a light year away). Since before the arrival of F, from a certain perspective, B and C were being considered at rest in the same reference frame as A, and I was assuming at rest meant involving no space-like motion through spacetime. If that was the case, and from A's perspective A underwent no space-like motion through spacetime then how is the distance between it and B and C closing to be explained. Did B and C go from no space-like motion through spacetime to space-like motion through spacetime, if so how, or did neither A, B or C undergo any space-like motion yet the distance between them reduce?

 

[A.T.]

But unlike in Newtonian physics I thought motion was relative in GR.

Uniform motion is relative in both, Galilean Relativity was around before Einstein. You should first try to understand the classical view properly, before tackling GR.

 

[name123]

Uniform motion is relative in both, Galilean Relativity was around before Einstein. You should first try to understand the classical view properly, before tackling GR.

But stating that the laws of motion is the same in all inertial frames, can be thought of as the noticing of a mathematical fact. I was not thinking that was the same as considering there to be no absolute motion and that motion itself was relative as GR did, and which was the point being discussed. I have just read that in Newtonian physics there is absolute space and that an inertial frame is a reference frame in relative uniform motion to absolute space. Which is what I had assumed, and was in line with my explanation of how the scenario would appear from a Newtonian perspective (third post down in this thread). So while I accept what you say, I do not understand its relevance to the explanation that I was asking for. As a side note, there was also thought to be absolute time in Newtonian physics

Furthermore, providing an answer to the question does not rely on my understanding, perhaps other people are reading the thread and they too would like to be told the answer.

 

[Nugatory]

I was assuming at rest meant involving no space-like motion through spacetime. If that was the case, and from A's perspective A underwent no space-like motion through spacetime then how is the distance between it and B and C closing to be explained.

The phrase "space-like motion" is altogether meaningless (in fact, it's internally inconsistent the way the phrase "the vertices of a circle" is internally inconsistent) but it sounds as if you mean "motion through space" instead.

So let's look at what it means to say that A is "moving through space". When we say that, we're really just saying that different points on A's worldline have different spatial coordinates, and whether that's true or not depends on nothing except how we choose our coordinates. If we choose what you're calling "A-centered coordinates" then A's spatial coordinates don't change while those of B, C, and F do. However, there is no physical significance to any of this; all that's going on is that you have chosen to use coordinates that happen to assign the same values to every point on A's worldline but not to every point on the other worldlines. Choose a different coordinate system and it will be A's worldline that gets different coordinates at different points. However, no matter what coordinates you use to label the various points on the various worldlines, the geometry will be as I described in post #6 above, and that description of the geometry is the GR explanation.

 

[Vitro]

But stating that the laws of motion is the same in all inertial frames, can be thought of as the noticing of a mathematical fact. I was not thinking that was the same as considering there to be no absolute motion and that motion itself was relative as GR did, and which was the point being discussed. I have just read that in Newtonian physics there is absolute space and that an inertial frame is a reference frame in relative uniform motion to absolute space.

I think you are making a distinction without a difference. Physics is as much about theory (and math) as it is about practice (experiments). A physical theory must make testable predictions which are then tested experimentally. In Galilean/Newtonian mechanics motion is relative, not just because the laws of motion (math) are the same in all inertial frames but because there is no experiment that could distinguish between a state of rest and one of uniform motion (in an absolute sense). Therefore there is no such thing as motion relative to absolute space, motion is simply an artifact of your choice of coordinates.

 

[name123]

The phrase "space-like motion" is altogether meaningless (in fact, it's internally inconsistent the way the phrase "the vertices of a circle" is internally inconsistent) but it sounds as if you mean "motion through space" instead.

So let's look at what it means to say that A is "moving through space". When we say that, we're really just saying that different points on A's worldline have different spatial coordinates, and whether that's true or not depends on nothing except how we choose our coordinates. If we choose what you're calling "A-centered coordinates" then A's spatial coordinates don't change while those of B, C, and F do. However, there is no physical significance to any of this; all that's going on is that you have chosen to use coordinates that happen to assign the same values to every point on A's worldline but not to every point on the other worldlines. Choose a different coordinate system and it will be A's worldline that gets different coordinates at different points. However, no matter what coordinates you use to label the various points on the various worldlines, the geometry will be as I described in post #6 above, and that description of the geometry is the GR explanation.

Thanks for the correction regarding the phrase "space like motion". It seems to me you are suggesting that with GR "motion through space" means nothing other than on a certain spatial coordinate system an object's spatial coordinates change. So that a lack of physical explanation of how B and C could go from pretty much at rest to A to moving towards A requires no explanation that could be regarded as a physical explanation, because the motion in GR has no physical significance, so there is no need to offer any explanation of how such motion could occur physically. Is that what you are suggesting?

If so then would it not leave room for a theory that could give a physical explanation for phenomena such as these which could assume absolute motion?

 

[name123]

I think you are making a distinction without a difference. Physics is as much about theory (and math) as it is about practice (experiments). A physical theory must make testable predictions which are then tested experimentally. In Galilean/Newtonian mechanics motion is relative, not just because the laws of motion (math) are the same in all inertial frames but because there is no experiment that could distinguish between a state of rest and one of uniform motion (in an absolute sense). Therefore there is no such thing as motion relative to absolute space, motion is simply an artifact of your choice of coordinates.

As I understand it, Newtonian physics assumed absolute space, and that in the theory there was motion relative to absolute space, and with such a theory the phenomena of B and C going from being at rest relative to A, to becoming closer to A can be given a physical explanation, such as that I gave earlier (third post down in this thread). You could have a theory which lacked a possible physical explanation for the result but why would that be favoured by physicalists over one which offered a possible physical explanation?

 

[weirdoguy]

Newtonian physics assumed absolute space,

Not absolute space, but absolute time.

 

[name123]

Not absolute space, but absolute time.

So the wiki article https://en.wikipedia.org/wiki/Galilean_invariance is wrong then? Because it states (and other articles support it):

Among the axioms from Newton's theory are:

1. There exists an absolute space, in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
2. All inertial frames share a universal time.

 

[Nugatory]

So that a lack of physical explanation of how B and C could go from pretty much at rest to A to moving towards A requires no explanation that could be regarded as a physical explanation, because the motion in GR has no physical significance, so there is no need to offer any explanation of how such motion could occur physically.

The thing that you are calling "motion", which might better be called "coordinate velocity", is just an artifact of your choice of coordinates.

However, GR does provide a complete physical explanation of why the worldlines of the various objects in your thought experiment are what they are, why the worldlines of A and F converge, why the world lines of A and the intersection of the laser beams diverge, why F remains in the path of the laser beams between A and B. The physical explanation is that all four objects are in freefall, so their worldlines are geodesics (loosely, "straight lines") through spacetime; the mass and energy of F curves spacetime in such a way that these straight lines converge and diverge as described above.

 

[Ibix]

I think you might want to think about what you mean by "an A centered frame" and "an F centered frame".

In Newtonian physics this is straightforward. You just fill space with a rigid cubic grid and a clock, and either attach the grid to A or to F. But in relativity it isn't that simple. You don't have a Euclidean background. So (assuming everything is massless except F) you could attach a rigid grid to F. But you cannot attach such a grid to A, because the curvature of spacetime is not time-independent in a coordinate system where the source of gravity is moving. The grid must flex. So whether B and C move in this coordinate system depends entirely on how the grid flexes.

 

[PAllen]

So the wiki article https://en.wikipedia.org/wiki/Galilean_invariance is wrong then? Because it states (and other articles support it):

Among the axioms from Newton's theory are:

1. There exists an absolute space, in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
2. All inertial frames share a universal time.

This was the way Newton formulated it, but he went to great lengths to show that this absolute space could never be detected. What is normally called Galilean relativity dispenses with this artifice, simply stating inertial frames are those where Newton's laws take their standard form, without reference to absolute space. This very minor modernization is what is taught nowadays as Newtonian physics.

 

[name123]

The thing that you are calling "motion", which might better be called "coordinate velocity", is just an artifact of your choice of coordinates.

However, GR does provide a complete physical explanation of why the worldlines of the various objects in your thought experiment are what they are, why the worldlines of A and F converge, why the world lines of A and the intersection of the laser beams diverge, why F remains in the path of the laser beams between A and B. The physical explanation is that all four objects are in freefall, so their worldlines are geodesics (loosely, "straight lines") through spacetime; the mass and energy of F curves spacetime in such a way that these straight lines converge and diverge as described above.

Are you suggesting that with GR a meteor could pass the Earth and fly into a star in a distant solar system, and that neither the Earth orbiting the Sun, nor the meteor traveled any distance in space?

 

[name123]

This was the way Newton formulated it, but he went to great lengths to show that this absolute space could never be detected. What is normally called Galilean relativity dispenses with this artifice, simply stating inertial frames are those where Newton's laws take their standard form, without reference to absolute space. This very minor modernization is what is taught nowadays as Newtonian physics.

So the change from the assertion that absolute space exists but could never be detected, to the assertion that there is no absolute space? How could you tell which was artifice? Out of curiosity how would you explain the scenario given in this thread using the concept of no absolute space, using an A centred coordinate system assuming A to be at rest?

 

[PAllen]

Are you suggesting that with GR a meteor could pass the Earth and fly into a star in a distant solar system, and that neither the Earth orbiting the Sun, nor the meteor traveled any distance in space?

You can certainly construct coordinate where two arbitrary non intersecting world lines both have constant position. However, there are invariant ways you could talk about their relative motion, even in such coordinates. You ask about the family of space like geodesic 4 orthogonal to one of the world lines that intersect the other. If the the length of these changes, you could say they have relative motion. However, this is still just a convention, and it would come out differently if you started from the other world line. A fundamental fact about GR is that there is NO unambiguous definition of relative motion of bodies at a distance from each other. This ambiguity follows directly from the definition of curvature.

 

[PAllen]

So the change from the assertion that absolute space exists but could never be detected, to the assertion that there is no absolute space? How could you tell which was artifice?

You can't. It is a philosophic choice outside of physics. Most physicists prefer the option without an undectable element as more elegant, but that is all.

 

[name123]

You can't. It is a philosophic choice outside of physics. Most physicists prefer the option without an undectable element as more elegant, but that is all.

Oh I wondered why you had declared the one to be artiface. Out of curiosity how would you explain the scenario given in this thread using the concept of no absolute space, using an A centred coordinate system assuming A to be at rest?

 

[name123]

You can certainly construct coordinate where two arbitrary non intersecting world lines both have constant position. However, there are invariant ways you could talk about their relative motion, even in such coordinates. You ask about the family of space like geodesic 4 orthogonal to one of the world lines that intersect the other. If the the length of these changes, you could say they have relative motion. However, this is still just a convention, and it would come out differently if you started from the other world line. A fundamental fact about GR is that there is NO unambiguous definition of relative motion of bodies at a distance from each other. This ambiguity follows directly from the definition of curvature.

So does the distance between two objects not map to any physical feature?

 

[PAllen]

So the change from the assertion that absolute space exists but could never be detected, to the assertion that there is no absolute space? How could you tell which was artifice? Out of curiosity how would you explain the scenario given in this thread using the concept of no absolute space, using an A centred coordinate system assuming A to be at rest?

I believe Nugatory has fully answered this. I have nothing to add to his posts.

 

[name123]

I believe Nugatory has fully answered this. I have nothing to add to his posts.

Nugartory answered using GR not Newtonian physics. So there was no force of gravity for example in Nugatory's answer. And I did not think there were spacetime geodesics in Newtonian physics.

 

[PAllen]

Nugartory answered using GR not Newtonian physics. So there was no force of gravity for example in Nugatory's answer.

Oh, I misunderstood your request. However, I have no interest in answering this in Newtonian physics.

 

[name123]

Oh, I misunderstood your request. However, I have no interest in answering this in Newtonian physics.

But you know how?

 

[PAllen]

So does the distance between two objects not map to any physical feature?

Correct. But a specifically described measurement of something you want to call a distance, is physical and invariant. Let me give a simple analogy on a plane. Consider two arbitrary non intersecting curves on a plane. What is the distance between them as a function of some parameter along one of them? You can't answer this without a choice how to place lines connecting them. There is no clear best way to do this. However, given a specific choice, you can compute the result. A specific measurement procedure will make such a choice. But it won't make distance between the curves unambiguous.

 

[PAllen]

But you know how?

Yes.

 

[name123]

Correct. But a specifically described measurement of something you want to call a distance, is physical and invariant. Let me give a simple analogy on a plane. Consider two arbitrary non intersecting curves on a plane. What is the distance between them as a function of some parameter along one of them? You can't answer this without a choice how to place lines connecting them. There is no clear best way to do this. However, given a specific choice, you can compute the result. A specific measurement procedure will make such a choice. But it won't make distance between the curves unambiguous.

But if distance is physical and invariant with a specifically described measurement, then if the measurement changes from object X to object Y, would the "distance" between them not be said to have changed? Could that happen without object X or object Y having been in motion?

 

[PAllen]

But if distance is physical and invariant with a specifically described measurement, then if the measurement changes from object X to object Y, would the "distance" between them not be said to have changed? Could that happen without object X or object Y having been in motion?

That could happen without either one's spatial coordinate changing. You need to learn a little differential geometry to understand this, at minimum, the notion of metric as a covariant object with different coordinate expressions.

 

[name123]

That could happen without either one's spatial coordinate changing. You need to learn a little differential geometry to understand this, at minimum, the notion of metric as a covariant object with different coordinate expressions.

Presumably you are considering a coordinate system not based on distance if distance is physical and invariant with a specifically described measurement. Is that correct? If so then could I just rephrase so as to be asking the question about a coordinate system based on distance.

 

[PAllen]

Presumably you are considering a coordinate system not based on distance if distance is physical and invariant with a specifically described measurement. Is that correct? If so then could I just rephrase so as to be asking the question about a coordinate system based on distance.

No, you would have to make the somewhat tautological statement that a coordinate system based a a specified measurement procedure that shows distance change, will then show spatial coordinate change. There may be a different measurement procedure that shows no change.

 

[name123]

No, you would have to make the somewhat tautological statement that a coordinate system based a a specified measurement procedure that shows distance change, will then show spatial coordinate change. There may be a different measurement procedure that shows no change.

I was assuming you were suggesting that different measurement procedures could give different distance readings, thus your statement: "But a specifically described measurement of something you want to call a distance, is physical and invariant. " (emphasis added).

If there was no spatial coordinate change, and the distance reading had changed, how were you thinking the specifically described measurement was invariant? Would it not be varying depending on some factor other than the difference between the spatial coordinates? Or were you considering the distance between two spatial coordinates to be something other than the difference in their values.