SpaceTime Diagrams

[Alan Grayson]

Since each reference frame has its own spacetime labels, what's the justification for plotting objects moving wrt different frames on the same spacetime grid? AG

[Alan Grayson]

On Tuesday, February 4, 2025 at 9:43:18 AM UTC-7 Alan Grayson wrote:

Since each reference frame has its own spacetime labels, what's the justification for plotting objects moving wrt different frames on the same spacetime grid? AG

CORRECTION:

Since each reference frame has its own spacetime labels, what's the justification for plotting a single object moving wrt different frames on the same spacetime grid? AG 

[Alan Grayson]

I'm thinking of the Parking Paradox in SR, where the car is plotted from the pov of two frames, the car and garage frames, but the spacetime coordinates of these frames are not identical, yet the grid on which they plotted has only one set of coordinates. Something seems awry here. AG

[Jesse Mazer]

See my comment at https://groups.google.com/g/everything-list/c/gbOE5B-7a6g/m/22jbd5qZEAAJ

>Alan: Yes, except we don't have to assume the moving rod has coordinates in O2. AG

>Jesse: Do you just mean it doesn't have *fixed* coordinates in O2, or do you mean it isn't assigned coordinates at all in O2? If the latter, are you imagining it's somehow invisible to the O2 observer? If so that's not how things work in relativity, the rod is just an ordinary physical object, of course the O2 observer is going to be able to measure it as it passes by his own system of rulers and clocks, and say things like "when the clock attached to the 3-light-second mark on my ruler showed a time of 5 seconds, the back of the rod was passing right next to it (as seen in a photo taken at that location at that moment, for example), therefore the worldline of the back of the rod passes through the coordinates x=3 light seconds, t=5 seconds in my coordinate system"

In case my above comment about the O2 observer being "able to measure it as it passes by his own system of rulers and clocks", you should be clear on the idea that the coordinates of any given frame are generally defined in textbooks in terms of local readings on a system of rulers and clocks that are at rest in that frame (each clock permanently fixed to a particular ruler-marking), with the clocks having been "synchronized" in that frame using the Einstein clock synchronization convention (which has the result that O1 will consider the O2's clocks to be out of sync with one another as measured in O1's frame, and vice versa). So then if there's some event, like a firecracker going off or the back of a car passing the front of the garage, the observer just looks at a snapshot of the part of his ruler/clock system that was right next to that event when it happened. If for example the snapshot shows the firecracker going off next to the 12 light-seconds mark on my ruler and the clock of mine that's attached to that marking shows a time of 8 seconds in the snapshot, then I say the firecracker happened at coordinates x=12 light seconds, t=8 seconds in my frame. And you can imagine the ruler/clock systems of other observers are sliding smoothly past my own ruler clock/system, so that for any given event like the firecracker, each observer has a ruler-marking and clock-reading of their own that was right next to that event when it happened.

Here for example are some pages from the textbook "Spacetime Physics" by Edwin Taylor and John Wheeler which go over the concept:

 Jesse

[Alan Grayson]

Two points: I don't see what this has to do with the question on THIS thread, and I can't read your reference since it's way too small. AG

Here's my problem with the alleged solution to the Car Parking Paradox; diagreement about simultaneity means, IIUC, that the car can't fit and not fit AT THE SAME TIME. This is how Clark defined the paradox. Well, since every frame in SR has its own synchronized clocks, the concept of "at the same time" is meaningless when it is applied to two frames in SR, and the lack of simultaneity is a formal way of proving this. Now if the center of the garage has an observer situated there, and there's an observer in the car, the spacetime coordinates of the frames can be totally different in x and t when the observers are juxtaposed, yet from the pov of car observer, the car doesn't fit since it never does given the initial conditions of the paradox. OTOH,  from the pov of garage observer the car always fits. So, when the car is at the center point of garage, the two observers are juxtaposed with different coordinates. but the observers have diametrically opposite conclusions. It doesn't matter that x and t, disagree with x' and t'. So, IMO, the paradox is alive and well. AG

[Jesse Mazer]

On Tue, Feb 4, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

Two points: I don't see what this has to do with the question on THIS thread, and I can't read your reference since it's way too small. AG\

It has to do with your question "what's the justification for plotting a single object moving wrt different frames on the same spacetime grid?" The justification is that, as I said, each observer can certainly *measure* all the objects involved, it's not like different frames are parallel universes that each can only see objects at rest in that frame. They are just different ways of assigning coordinates to the same set of local physical facts about the same objects, like the firecracker exploding or the edge of one object passing next to the edge of another.

As for the text, did you try clicking on the images to expand them? Anyway the reference was just to back up what I said in the paragraph above about each observer assigning coordinates with their own ruler/clock system, if you understood that part and have no objections then there's probably no need to read the textbook images.

 

Here's my problem with the alleged solution to the Car Parking Paradox; diagreement about simultaneity means, IIUC, that the car can't fit and not fit AT THE SAME TIME.

Not if "at the same time" means both frame agreeing on a common notion of a single moment in time but disagreeing about what is happening at that moment (as you say they don't have a common notion of a single moment in time). But if John Clark did say that (I'd like to see the post to read his exact words), he might have meant something else like "there is at least one moment in the garage frame where the car is entirely inside the garage, but at no single moment in the car frame is the car wholly inside the garage", which doesn't require that they have a common definition of what events happen in a "single moment".

 

This is how Clark defined the paradox. Well, since every frame in SR has its own synchronized clocks, the concept of "at the same time" is meaningless when it is applied to two frames in SR, and the lack of simultaneity is a formal way of proving this. Now if the center of the garage has an observer situated there, and there's an observer in the car, the spacetime coordinates of the frames can be totally different in x and t when the observers are juxtaposed, yet from the pov of car observer, the car doesn't fit since it never does given the initial conditions of the paradox. OTOH,  from the pov of garage observer the car always fits. So, when the car is at the center point of garage, the two observers are juxtaposed with different coordinates. but the observers have diametrically opposite conclusions. It doesn't matter that x and t, disagree with x' and t'. So, IMO, the paradox is alive and well. AG

Does your statement "the paradox is alive and well" depend on that one phrase about fitting/not fitting "at the same time"? That isn't the usual way of formulating the paradox, you can just say they disagree about whether the car ever fits wholly inside the garage without any words like "at the same time", so if you are getting hung up on those words I'd recommend you just write them off as a confusing and non-standard way of describing the problem. As I always say, it's usually made clear explicitly or implicitly that the "paradox" is about the danger that the disagreement about fitting would lead to a disagreement about local physical facts like whether the closing garage door hits the car, and the fact that the two frames don't agree on simultaneity (or don't agree on the ordering of non-simultaneous events with a spacelike separation) is the way to show how that danger is avoided, and both frames can be in complete agreement about all local physical facts despite the disagreement about whether the car ever fits.

Jesse

 

On Tuesday, February 4, 2025 at 12:13:44 PM UTC-7 Jesse Mazer wrote:

On Tue, Feb 4, 2025 at 1:39 PM Alan Grayson <agrays...@gmail.com> wrote:

On Tuesday, February 4, 2025 at 9:43:18 AM UTC-7 Alan Grayson wrote:

Since each reference frame has its own spacetime labels, what's the justification for plotting objects moving wrt different frames on the same spacetime grid? AG

CORRECTION:

Since each reference frame has its own spacetime labels, what's the justification for plotting a single object moving wrt different frames on the same spacetime grid? AG 

See my comment at https://groups.google.com/g/everything-list/c/gbOE5B-7a6g/m/22jbd5qZEAAJ

>Alan: Yes, except we don't have to assume the moving rod has coordinates in O2. AG

>Jesse: Do you just mean it doesn't have *fixed* coordinates in O2, or do you mean it isn't assigned coordinates at all in O2? If the latter, are you imagining it's somehow invisible to the O2 observer? If so that's not how things work in relativity, the rod is just an ordinary physical object, of course the O2 observer is going to be able to measure it as it passes by his own system of rulers and clocks, and say things like "when the clock attached to the 3-light-second mark on my ruler showed a time of 5 seconds, the back of the rod was passing right next to it (as seen in a photo taken at that location at that moment, for example), therefore the worldline of the back of the rod passes through the coordinates x=3 light seconds, t=5 seconds in my coordinate system"

In case my above comment about the O2 observer being "able to measure it as it passes by his own system of rulers and clocks", you should be clear on the idea that the coordinates of any given frame are generally defined in textbooks in terms of local readings on a system of rulers and clocks that are at rest in that frame (each clock permanently fixed to a particular ruler-marking), with the clocks having been "synchronized" in that frame using the Einstein clock synchronization convention (which has the result that O1 will consider the O2's clocks to be out of sync with one another as measured in O1's frame, and vice versa). So then if there's some event, like a firecracker going off or the back of a car passing the front of the garage, the observer just looks at a snapshot of the part of his ruler/clock system that was right next to that event when it happened. If for example the snapshot shows the firecracker going off next to the 12 light-seconds mark on my ruler and the clock of mine that's attached to that marking shows a time of 8 seconds in the snapshot, then I say the firecracker happened at coordinates x=12 light seconds, t=8 seconds in my frame. And you can imagine the ruler/clock systems of other observers are sliding smoothly past my own ruler clock/system, so that for any given event like the firecracker, each observer has a ruler-marking and clock-reading of their own that was right next to that event when it happened.

Here for example are some pages from the textbook "Spacetime Physics" by Edwin Taylor and John Wheeler which go over the concept:

 Jesse

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[Alan Grayson]

On Tuesday, February 4, 2025 at 1:51:26 PM UTC-7 Jesse Mazer wrote:

On Tue, Feb 4, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

Two points: I don't see what this has to do with the question on THIS thread, and I can't read your reference since it's way too small. AG\

It has to do with your question "what's the justification for plotting a single object moving wrt different frames on the same spacetime grid?" The justification is that, as I said, each observer can certainly *measure* all the objects involved, it's not like different frames are parallel universes that each can only see objects at rest in that frame. They are just different ways of assigning coordinates to the same set of local physical facts about the same objects, like the firecracker exploding or the edge of one object passing next to the edge of another.

I don't follow your meaning. I see each frame making independent measurements when the observers are juxtaposed, and reach different conclusions about fitting and not fitting. AG  

As for the text, did you try clicking on the images to expand them?

I tried that. It didn't work. AG

 

Anyway the reference was just to back up what I said in the paragraph above about each observer assigning coordinates with their own ruler/clock system, if you understood that part and have no objections then there's probably no need to read the textbook images.

 

Here's my problem with the alleged solution to the Car Parking Paradox; diagreement about simultaneity means, IIUC, that the car can't fit and not fit AT THE SAME TIME.

Not if "at the same time" means both frame agreeing on a common notion of a single moment in time but disagreeing about what is happening at that moment (as you say they don't have a common notion of a single moment in time). But if John Clark did say that (I'd like to see the post to read his exact words), he might have meant something else like "there is at least one moment in the garage frame where the car is entirely inside the garage, but at no single moment in the car frame is the car wholly inside the garage", which doesn't require that they have a common definition of what events happen in a "single moment".

 

This is how Clark defined the paradox. Well, since every frame in SR has its own synchronized clocks, the concept of "at the same time" is meaningless when it is applied to two frames in SR, and the lack of simultaneity is a formal way of proving this. Now if the center of the garage has an observer situated there, and there's an observer in the car, the spacetime coordinates of the frames can be totally different in x and t when the observers are juxtaposed, yet from the pov of car observer, the car doesn't fit since it never does given the initial conditions of the paradox. OTOH,  from the pov of garage observer the car always fits. So, when the car is at the center point of garage, the two observers are juxtaposed with different coordinates. but the observers have diametrically opposite conclusions. It doesn't matter that x and t, disagree with x' and t'. So, IMO, the paradox is alive and well. AG

Does your statement "the paradox is alive and well" depend on that one phrase about fitting/not fitting "at the same time"?

No. I stated that when juxtaposed, x, t and x', t' need not be identical. The disagreement about simultaneity just applies to the time coordinate, and it doen't matter if they are not identical when the observers are juxtaposed, which is the only thing the simutaneity argument shows. AG

 

That isn't the usual way of formulating the paradox, you can just say they disagree about whether the car ever fits wholly inside the garage without any words like "at the same time",

I was following Clark's definition of the paradox. I'm sure I'm not misrepresenting what he meant, which was the paradox is based on a misconception that the frames share the time coordinate value. AG

 

so if you are getting hung up on those words I'd recommend you just write them off as a confusing and non-standard way of describing the problem. As I always say, it's usually made clear explicitly or implicitly that the "paradox" is about the danger that the disagreement about fitting would lead to a disagreement about local physical facts like whether the closing garage door hits the car, and the fact that the two frames don't agree on simultaneity (or don't agree on the ordering of non-simultaneous events with a spacelike separation) is the way to show how that danger is avoided, and both frames can be in complete agreement about all local physical facts despite the disagreement about whether the car ever fits.

Jesse

You can set up your clocks and rulers any way you want in both frames, and you'll find the car observer observes the car not filling and the garage observer observes it fitting, when the observers are juxtaposed, and x, t, need not be identical to x',t'. This is why I say the paradox is alive and well. Any objections? Also, FWIW, since each frame has its own distinct coordinates, it's an error to plot them using some coordinates x,t, when the frames use different coordinates. AG

[Quentin Anciaux]

Chatgpt o1:

To view this discussion visit https://groups.google.com/d/msgid/everything-list/bd23a281-47d1-48d7-91eb-534fbde6038fn%40googlegroups.com.

[Jesse Mazer]

On Tue, Feb 4, 2025 at 4:29 PM Alan Grayson <agrays...@gmail.com> wrote:

On Tuesday, February 4, 2025 at 1:51:26 PM UTC-7 Jesse Mazer wrote:

On Tue, Feb 4, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

Two points: I don't see what this has to do with the question on THIS thread, and I can't read your reference since it's way too small. AG\

It has to do with your question "what's the justification for plotting a single object moving wrt different frames on the same spacetime grid?" The justification is that, as I said, each observer can certainly *measure* all the objects involved, it's not like different frames are parallel universes that each can only see objects at rest in that frame. They are just different ways of assigning coordinates to the same set of local physical facts about the same objects, like the firecracker exploding or the edge of one object passing next to the edge of another.

I don't follow your meaning. I see each frame making independent measurements when the observers are juxtaposed, and reach different conclusions about fitting and not fitting. AG  

Sure, they disagree about fitting, but each has a grid of coordinates covering the same region of spacetime, which is used to plot the paths of both the car and the garage in that region. Is that what you meant by "plotting a single object ... on the same spacetime grid", or did you mean something different?

 

As for the text, did you try clicking on the images to expand them?

I tried that. It didn't work. AG

If you are looking at the site using a mouse or trackpad, try right-clicking on the images, and then when a menu pops up click an option like "open image in new window". If you're using a touch screen you can try just pressing down on an image with your finger until a menu like this pops up.

 

Anyway the reference was just to back up what I said in the paragraph above about each observer assigning coordinates with their own ruler/clock system, if you understood that part and have no objections then there's probably no need to read the textbook images.

 

Here's my problem with the alleged solution to the Car Parking Paradox; diagreement about simultaneity means, IIUC, that the car can't fit and not fit AT THE SAME TIME.

Not if "at the same time" means both frame agreeing on a common notion of a single moment in time but disagreeing about what is happening at that moment (as you say they don't have a common notion of a single moment in time). But if John Clark did say that (I'd like to see the post to read his exact words), he might have meant something else like "there is at least one moment in the garage frame where the car is entirely inside the garage, but at no single moment in the car frame is the car wholly inside the garage", which doesn't require that they have a common definition of what events happen in a "single moment".

 

This is how Clark defined the paradox. Well, since every frame in SR has its own synchronized clocks, the concept of "at the same time" is meaningless when it is applied to two frames in SR, and the lack of simultaneity is a formal way of proving this. Now if the center of the garage has an observer situated there, and there's an observer in the car, the spacetime coordinates of the frames can be totally different in x and t when the observers are juxtaposed, yet from the pov of car observer, the car doesn't fit since it never does given the initial conditions of the paradox. OTOH,  from the pov of garage observer the car always fits. So, when the car is at the center point of garage, the two observers are juxtaposed with different coordinates. but the observers have diametrically opposite conclusions. It doesn't matter that x and t, disagree with x' and t'. So, IMO, the paradox is alive and well. AG

Does your statement "the paradox is alive and well" depend on that one phrase about fitting/not fitting "at the same time"?

No. I stated that when juxtaposed, x, t and x', t' need not be identical. The disagreement about simultaneity just applies to the time coordinate, and it doen't matter if they are not identical when the observers are juxtaposed, which is the only thing the simutaneity argument shows. AG

By "juxtaposed" do you mean when they assign coordinates to the same event, like the event of the back of the car passing the entrance of the garage, or the event of the front of the car passing the exit of the garage? If so, I'd agree the x, t assigned to each event by one observer will in general be different from the x', t'  assigned to each event by the other observer, if that's all you're saying.

 

That isn't the usual way of formulating the paradox, you can just say they disagree about whether the car ever fits wholly inside the garage without any words like "at the same time",

I was following Clark's definition of the paradox. I'm sure I'm not misrepresenting what he meant, which was the paradox is based on a misconception that the frames share the time coordinate value. AG

 

so if you are getting hung up on those words I'd recommend you just write them off as a confusing and non-standard way of describing the problem. As I always say, it's usually made clear explicitly or implicitly that the "paradox" is about the danger that the disagreement about fitting would lead to a disagreement about local physical facts like whether the closing garage door hits the car, and the fact that the two frames don't agree on simultaneity (or don't agree on the ordering of non-simultaneous events with a spacelike separation) is the way to show how that danger is avoided, and both frames can be in complete agreement about all local physical facts despite the disagreement about whether the car ever fits.

Jesse

You can set up your clocks and rulers any way you want in both frames, and you'll find the car observer observes the car not filling and the garage observer observes it fitting, when the observers are juxtaposed, and x, t, need not be identical to x',t'.

Sure, if by "juxtaposed" you mean what I said above.

 

This is why I say the paradox is alive and well. Any objections?

I'd object to that because the mere fact that observers assign different coordinates doesn't seem like a "paradox" to me. Do you think it's a paradox that different observers assign a different velocity v and v' to the same object?

 

Also, FWIW, since each frame has its own distinct coordinates, it's an error to plot them using some coordinates x,t, when the frames use different coordinates. AG

Who has ever plotted two frames using the same coordinates? Brent gave two different diagrams, one showing how things look in the coordinates of the garage frame, and one showing how things look in the coordinates of the car frame. Both diagrams showed the same objects (the car and the garage) and events (such as the back of the car passing the entrance of the garage), but the way different events lines up with the position and time axes of each frame were different, corresponding to a given event having different x,t coordinates in one frame from its x',t' coordinates in another frame.

Jesse

 

 

On Tuesday, February 4, 2025 at 12:13:44 PM UTC-7 Jesse Mazer wrote:

On Tue, Feb 4, 2025 at 1:39 PM Alan Grayson <agrays...@gmail.com> wrote:

On Tuesday, February 4, 2025 at 9:43:18 AM UTC-7 Alan Grayson wrote:

Since each reference frame has its own spacetime labels, what's the justification for plotting objects moving wrt different frames on the same spacetime grid? AG

CORRECTION:

Since each reference frame has its own spacetime labels, what's the justification for plotting a single object moving wrt different frames on the same spacetime grid? AG 

See my comment at https://groups.google.com/g/everything-list/c/gbOE5B-7a6g/m/22jbd5qZEAAJ

>Alan: Yes, except we don't have to assume the moving rod has coordinates in O2. AG

>Jesse: Do you just mean it doesn't have *fixed* coordinates in O2, or do you mean it isn't assigned coordinates at all in O2? If the latter, are you imagining it's somehow invisible to the O2 observer? If so that's not how things work in relativity, the rod is just an ordinary physical object, of course the O2 observer is going to be able to measure it as it passes by his own system of rulers and clocks, and say things like "when the clock attached to the 3-light-second mark on my ruler showed a time of 5 seconds, the back of the rod was passing right next to it (as seen in a photo taken at that location at that moment, for example), therefore the worldline of the back of the rod passes through the coordinates x=3 light seconds, t=5 seconds in my coordinate system"

In case my above comment about the O2 observer being "able to measure it as it passes by his own system of rulers and clocks", you should be clear on the idea that the coordinates of any given frame are generally defined in textbooks in terms of local readings on a system of rulers and clocks that are at rest in that frame (each clock permanently fixed to a particular ruler-marking), with the clocks having been "synchronized" in that frame using the Einstein clock synchronization convention (which has the result that O1 will consider the O2's clocks to be out of sync with one another as measured in O1's frame, and vice versa). So then if there's some event, like a firecracker going off or the back of a car passing the front of the garage, the observer just looks at a snapshot of the part of his ruler/clock system that was right next to that event when it happened. If for example the snapshot shows the firecracker going off next to the 12 light-seconds mark on my ruler and the clock of mine that's attached to that marking shows a time of 8 seconds in the snapshot, then I say the firecracker happened at coordinates x=12 light seconds, t=8 seconds in my frame. And you can imagine the ruler/clock systems of other observers are sliding smoothly past my own ruler clock/system, so that for any given event like the firecracker, each observer has a ruler-marking and clock-reading of their own that was right next to that event when it happened.

Here for example are some pages from the textbook "Spacetime Physics" by Edwin Taylor and John Wheeler which go over the concept:

 Jesse

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[Alan Grayson]

On Tuesday, February 4, 2025 at 2:50:03 PM UTC-7 Jesse Mazer wrote:

On Tue, Feb 4, 2025 at 4:29 PM Alan Grayson <agrays...@gmail.com> wrote:

On Tuesday, February 4, 2025 at 1:51:26 PM UTC-7 Jesse Mazer wrote:

On Tue, Feb 4, 2025 at 3:09 PM Alan Grayson <agrays...@gmail.com> wrote:

Two points: I don't see what this has to do with the question on THIS thread, and I can't read your reference since it's way too small. AG\

It has to do with your question "what's the justification for plotting a single object moving wrt different frames on the same spacetime grid?" The justification is that, as I said, each observer can certainly *measure* all the objects involved, it's not like different frames are parallel universes that each can only see objects at rest in that frame. They are just different ways of assigning coordinates to the same set of local physical facts about the same objects, like the firecracker exploding or the edge of one object passing next to the edge of another.

I don't follow your meaning. I see each frame making independent measurements when the observers are juxtaposed, and reach different conclusions about fitting and not fitting. AG  

Sure, they disagree about fitting, but each has a grid of coordinates covering the same region of spacetime, which is used to plot the paths of both the car and the garage in that region. Is that what you meant by "plotting a single object ... on the same spacetime grid", or did you mean something different?

 

As for the text, did you try clicking on the images to expand them?

I tried that. It didn't work. AG

If you are looking at the site using a mouse or trackpad, try right-clicking on the images, and then when a menu pops up click an option like "open image in new window". If you're using a touch screen you can try just pressing down on an image with your finger until a menu like this pops up.

 

Anyway the reference was just to back up what I said in the paragraph above about each observer assigning coordinates with their own ruler/clock system, if you understood that part and have no objections then there's probably no need to read the textbook images.

 

Here's my problem with the alleged solution to the Car Parking Paradox; diagreement about simultaneity means, IIUC, that the car can't fit and not fit AT THE SAME TIME.

Not if "at the same time" means both frame agreeing on a common notion of a single moment in time but disagreeing about what is happening at that moment (as you say they don't have a common notion of a single moment in time). But if John Clark did say that (I'd like to see the post to read his exact words), he might have meant something else like "there is at least one moment in the garage frame where the car is entirely inside the garage, but at no single moment in the car frame is the car wholly inside the garage", which doesn't require that they have a common definition of what events happen in a "single moment".

 

This is how Clark defined the paradox. Well, since every frame in SR has its own synchronized clocks, the concept of "at the same time" is meaningless when it is applied to two frames in SR, and the lack of simultaneity is a formal way of proving this. Now if the center of the garage has an observer situated there, and there's an observer in the car, the spacetime coordinates of the frames can be totally different in x and t when the observers are juxtaposed, yet from the pov of car observer, the car doesn't fit since it never does given the initial conditions of the paradox. OTOH,  from the pov of garage observer the car always fits. So, when the car is at the center point of garage, the two observers are juxtaposed with different coordinates. but the observers have diametrically opposite conclusions. It doesn't matter that x and t, disagree with x' and t'. So, IMO, the paradox is alive and well. AG

Does your statement "the paradox is alive and well" depend on that one phrase about fitting/not fitting "at the same time"?

No. I stated that when juxtaposed, x, t and x', t' need not be identical. The disagreement about simultaneity just applies to the time coordinate, and it doen't matter if they are not identical when the observers are juxtaposed, which is the only thing the simutaneity argument shows. AG

By "juxtaposed" do you mean when they assign coordinates to the same event, like the event of the back of the car passing the entrance of the garage, or the event of the front of the car passing the exit of the garage?

I mean when juxtaposed they do any measurements necessary, to show car fitting in garage frame, but not car frame. For me this is a paradox. Presumably you disagree. AG 

 

If so, I'd agree the x, t assigned to each event by one observer will in general be different from the x', t'  assigned to each event by the other observer, if that's all you're saying.

 

That isn't the usual way of formulating the paradox, you can just say they disagree about whether the car ever fits wholly inside the garage without any words like "at the same time",

I was following Clark's definition of the paradox. I'm sure I'm not misrepresenting what he meant, which was the paradox is based on a misconception that the frames share the time coordinate value. AG

 

so if you are getting hung up on those words I'd recommend you just write them off as a confusing and non-standard way of describing the problem. As I always say, it's usually made clear explicitly or implicitly that the "paradox" is about the danger that the disagreement about fitting would lead to a disagreement about local physical facts like whether the closing garage door hits the car, and the fact that the two frames don't agree on simultaneity (or don't agree on the ordering of non-simultaneous events with a spacelike separation) is the way to show how that danger is avoided, and both frames can be in complete agreement about all local physical facts despite the disagreement about whether the car ever fits.

Jesse

You can set up your clocks and rulers any way you want in both frames, and you'll find the car observer observes the car not filling and the garage observer observes it fitting, when the observers are juxtaposed, and x, t, need not be identical to x',t'.

Sure, if by "juxtaposed" you mean what I said above.

 

This is why I say the paradox is alive and well. Any objections?

I'd object to that because the mere fact that observers assign different coordinates doesn't seem like a "paradox" to me.

That's not my claim. I am saying disagreement about simultaneity doesn't resolve the paradox because when juxtaposed, the times can be different, while the car fits in one frame and not in the other. AG

 

Do you think it's a paradox that different observers assign a different velocity v and v' to the same object?

No; I think from any frame, the object in that frame will be at rest, uncontracted, and will be in relative motion wrt the other frame. AG 

 

Also, FWIW, since each frame has its own distinct coordinates, it's an error to plot them using some coordinates x,t, when the frames use different coordinates. AG

Who has ever plotted two frames using the same coordinates? Brent gave two different diagrams, one showing how things look in the coordinates of the garage frame, and one showing how things look in the coordinates of the car frame.

In each diagram he has two objects, car and garage, as seen from one frame, and then the other, even though the objects plotted are always observed from the pov of different frames. AG

[Jesse Mazer]

So for you, the "paradox" is purely the idea that it fits in one frame but doesn't fit in another?

 

 

Do you think it's a paradox that different observers assign a different velocity v and v' to the same object?

No; I think from any frame, the object in that frame will be at rest, uncontracted, and will be in relative motion wrt the other frame. AG 

But *why* do you say it's non-paradoxical for different frames to disagree about velocity, but it is paradoxical for them to disagree about fitting? Is it just an intuitive reaction to the second that's different from your reaction to the first? To me they both seem like cases of "some statements about physical objects are frame-dependent, so different frames can disagree about them."

 

 

Also, FWIW, since each frame has its own distinct coordinates, it's an error to plot them using some coordinates x,t, when the frames use different coordinates. AG

Who has ever plotted two frames using the same coordinates? Brent gave two different diagrams, one showing how things look in the coordinates of the garage frame, and one showing how things look in the coordinates of the car frame.

In each diagram he has two objects, car and garage, as seen from one frame, and then the other, even though the objects plotted are always observed from the pov of different frames. AG

Yes, but so what? That isn't plotting trying to plot the perspective of two different FRAMES in the same graph (the 'error' you referred to above), each individual graph just plots two physical OBJECTS using a *single* frame's coordinates, say x,t. Do you have a problem with the latter? Do you think there is an error inherent in using a given frame to assign coordinates to an OBJECT that is moving relative to that frame, or that an observer in that frame would have any difficulty with making position and time measurements (using her own ruler/clock system) on objects moving relative to herself?

Jesse

 

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[Alan Grayson]

I accept the fact that different frames can make different measurements, and the situation with v and v' might be an example of that, just like measurements of E and B fields differ in different frames due to the relative motion. What I don't accept is the argument using disagreement about simultaneity resolves the paradox. I have been ridiculed for taking that position, but ISTM that showing such a disagreement, just shows what we already knew; that clocks in different frames do not necessarily agree on the time something occurs. In my model, there's no reason to expect x and t to equal x' and t', and yet with enough clocks and observers, the car will fit and not fit depending on which observer / frame is doing the measuring. So what has the disagreement about simultaneity yielded in resolving the paradox? Nothing!  Moreover, IMO, the disagreement about fitting IS the parodox. If not, what do you think it is? AG  

[Jesse Mazer]

But WHAT IS "THE PARADOX"? Is it just that it fits in one frame and doesn't in another, with no additional argument about why anyone else but you should consider this fact alone to be a "paradox"?

 

I have been ridiculed for taking that position, but ISTM that showing such a disagreement, just shows what we already knew; that clocks in different frames do not necessarily agree on the time something occurs. In my model, there's no reason to expect x and t to equal x' and t',

The x and t associated with any specific *localized* event (i.e. an event occurring at a single point in spacetime) are not the same as the x' and t' associated with that same event, for the most part (except for the event at x=0, t=0), so in that sense I agree there's no reason to expect x and t to equal x' and t' for any specific localized event. But the car fitting or not fitting is not a specific localized event, it's a statement about a multiple different localized events that are considered simultaneous in a given frame (for example if there is a moment in a frame when the back of the car is at a localized point x1 inside the garage at the same t-coordinate as the front of the car being at a different localized point x2 inside the garage, then the car is considered to fit according to that frame).

 

and yet with enough clocks and observers, the car will fit and not fit depending on which observer / frame is doing the measuring. So what has the disagreement about simultaneity yielded in resolving the paradox? Nothing!  Moreover, IMO, the disagreement about fitting IS the parodox. If not, what do you think it is? AG  

OK, what if someone said "the disagreement about speed IS the paradox", but didn't have any additional argument about WHY they thought it was paradoxical for different frames to judge speeds of objects differently? Would you say they had any rational basis for their view that there was a paradox there?

As to what *I* think the paradox is, this is something I have told you a million times including earlier on the other thread and I even repeated it earlier on this one, do you really not remember?

Jesse

 

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[Brent Meeker]

You need to describe things in a single coord system in order to model interactions between them.

Brent

On 2/4/2025 8:43 AM, Alan Grayson wrote:

Since each reference frame has its own spacetime labels, what's the justification for plotting objects moving wrt different frames on the same spacetime grid? AG --

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[Alan Grayson]

No, because I could point to relative velocity as the cause of the disagreement, ultimately on the invariance of the SoL, AG 

As to what *I* think the paradox is, this is something I have told you a million times including earlier on the other thread and I even repeated it earlier on this one, do you really not remember?

Sure; it's when there are contradictory results at some local event. In the context of the alleged paradox, I don't see that the disagreement about simultaneity proves anything, since, if proven, it just tells us what we already knew; that fitting and not fillting don't occur at the same time. OTOH, if the observers observe different results when they're juxtaposed, it's a paradox IMO for the same reason as if Clark's definition for a paradox was manifested; different results at the same time. As for your last post on the other thread, I am still working on it. It will take some time before I respond, but I will. AG

[Alan Grayson]

On Tuesday, February 4, 2025 at 9:51:30 PM UTC-7 Brent Meeker wrote:

You need to describe things in a single coord system in order to model interactions between them.

Brent

But if garage frame spacetime coordinates x, t, and car frame has spacetime coordinates x', t', and the plot represents the situation from the pov of garage frame, how can the car be plotted using x, t, when those coordinates aren't its spacetime coordinates? AG 

[Jesse Mazer]

"Relative velocity" is pretty much synonymous with different frames disagreeing in their velocity/speed measurements, so it seems circular to use the former to rebut the opinion of someone who thinks the latter is a “paradox”. It's a bit like saying "relative length is the cause of the disagreement about fitting".

 

ultimately on the invariance of the SoL, AG 

What does SoL stand for?

 

As to what *I* think the paradox is, this is something I have told you a million times including earlier on the other thread and I even repeated it earlier on this one, do you really not remember?

Sure; it's when there are contradictory results at some local event. In the context of the alleged paradox, I don't see that the disagreement about simultaneity proves anything,

Are you saying it doesn't prove anything about *your* version of the paradox, or mine? If the latter, I would say that disagreement about simultaneity (combined with disagreement about order of spacelike separated events that are not simultaneous in either frame) is crucial to understanding how they can avoid contradictory predictions about some local event. 

 

since, if proven, it just tells us what we already knew; that fitting and not fillting don't occur at the same time. OTOH, if the observers observe different results when they're juxtaposed, it's a paradox IMO for the same reason as if Clark's definition for a paradox was manifested; different results at the same time.

But they aren't seeing different results at an agreed-upon "same time" because they don't agree on which events at different locations happen at the same time.

 

As for your last post on the other thread, I am still working on it. It will take some time before I respond, but I will. AG

Sounds good--I hope you try the experiment of plugging in particular combinations of x and t coordinates (or x' and t' coordinates) into the LT equations as input to verify what I said about the resulting output.

Jesse

 

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[Alan Grayson]

Sure; it's when there are contradictory results at some local event. In the context of the alleged paradox, I don't see that the disagreement about simultaneity proves anything, since, if proven, it just tells us what we already knew; that fitting and not fillting don't occur at the same time. And we know that because there's no reason to assume x,t and x',t' are the same for both frames. OTOH, if the observers observe different results when they're juxtaposed, it's a paradox IMO for the same reason as if Clark's definition for a paradox was manifested; different results at the same time. As for your last post on the other thread, I am still working on it. It will take some time before I respond, but I will. AG

Jesse

[Alan Grayson]

Speed of Light. AG 

 

As to what *I* think the paradox is, this is something I have told you a million times including earlier on the other thread and I even repeated it earlier on this one, do you really not remember?

Sure; it's when there are contradictory results at some local event. In the context of the alleged paradox, I don't see that the disagreement about simultaneity proves anything,

Are you saying it doesn't prove anything about *your* version of the paradox, or mine?

My version. AG

 

If the latter, I would say that disagreement about simultaneity (combined with disagreement about order of spacelike separated events that are not simultaneous in either frame) is crucial to understanding how they can avoid contradictory predictions about some local event. 

That might be true, but I don't understand it. AG 

 

since, if proven, it just tells us what we already knew; that fitting and not fillting don't occur at the same time. OTOH, if the observers observe different results when they're juxtaposed, it's a paradox IMO for the same reason as if Clark's definition for a paradox was manifested; different results at the same time.

But they aren't seeing different results at an agreed-upon "same time" because they don't agree on which events at different locations happen at the same time.

I never claimed that in my model they see different results because they are simultaneous. I have been quite clear, I thought, that when the observers are juxtaposed, they do NOT generally agree that t = t'. Why should that be the case, when the frames have different coordinates? AG

[Brent Meeker]

On 2/4/2025 9:02 PM, Alan Grayson wrote:

On Tuesday, February 4, 2025 at 9:51:30 PM UTC-7 Brent Meeker wrote:

You need to describe things in a single coord system in order to model interactions between them.

Brent

But if garage frame spacetime coordinates x, t, and car frame has spacetime coordinates x', t', and the plot represents the situation from the pov of garage frame, how can the car be plotted using x, t, when those coordinates aren't its spacetime coordinates? AG

The primed and unprimed coordinates of a given point are related by the Lorentz transformation, (t,x) and (t',x') are the same point just plotted in different coordinates.

Brent

[Alan Grayson]

I think I get it. In your first plot, you're describing the situation from the pov of the garage frame, so the plot coordinates are x,t, whereas in the second plot you're describing the situation from the pov of the car frame, so technically the axes should be labeled x',t' (but you left them as x,t), AG 

[Alan Grayson]

FWIW, I came up with my model of the car observer and garage observer juxtaposed halfway within the garage because I was influenced by Clark's definition of the paradox, that the observers would observe the car fitting and not fitting AT THE SAME TIME. Then I realized that that concept makes no sense in SR because each frame has its own clocks and rulers, so if when juxtaposed, t=t', it would be the result of accident and not generally true. So Clark's definition of the paradox in one of his post, is not correct. I'm sure I'm not distorting his meaning. Further, if I claim the paradox is defined simply as car fitting and not fitting when the observers are juxtaposed, then I must also agree that v not equal v' must be paradoxical as you claimed. But since I previously claimed that v not equal v' is not paradoxical when the observers are juxtaposed, neither can I claim that car fitting and not fitting when the observers are juxtaposed defines the paradox. The consequence of this analysis is that I need a different definition of the paradox, so I will further explore the one you offered, which I do not now completely understand. Final point; I misstated what the disagreement of simultaneity means. It does not mean that t is not equal t' in my model. I forget that simultaneity is defined as two simultaneous events in one frame, hence spacelike, which are not simultaneous in a second frame when the LT is applied to the two events in the first frame. Presumably, the images of the initial pair of events under the LT have their time orders reversed, but I have never seen that done. Is it suggested in your last post on the other thread? Is there anything in this statement that you disagree with? TY, AG

[Jesse Mazer]

Any comment on the above? If someone finds it to be a "paradox" that different frames disagree about velocities, then pretty much by definition wouldn't they also find the idea of "relative velocity" to be paradoxical because this is the same idea in different words, so explaining the former in terms of the latter is circular? Similarly, since you find the idea of "the car fits in one frame but not another" to be paradoxical, presumably you find the idea of "relative length" (eg idea that different frames can disagree on which of two objects is longer) to be equally paradoxical, so explaining the fitting conflict by appealing to "relative length" would be a circular non-explanation for you. In short it seems like there's complete symmetry between how you would respond to any given attempt to say the frames' disagreement about fitting is non-paradoxical vs. how the hypothetical person I imagine above would respond to any attempt to say the frames' disagreement about velocity is non-paradoxical. I'm not seeing that you have any well-defined criteria that demand that one be seen as paradoxical while the other isn't, it just seems to be a matter of your personal intuitions.

 

ultimately on the invariance of the SoL, AG 

What does SoL stand for?

Speed of Light. AG 

How would the *agreement* on different frames about the speed of light be an "explanation" for the person who thinks it's paradoxical that they disagree about sublight speeds?

 

 

As to what *I* think the paradox is, this is something I have told you a million times including earlier on the other thread and I even repeated it earlier on this one, do you really not remember?

Sure; it's when there are contradictory results at some local event. In the context of the alleged paradox, I don't see that the disagreement about simultaneity proves anything,

Are you saying it doesn't prove anything about *your* version of the paradox, or mine?

My version. AG

 

If the latter, I would say that disagreement about simultaneity (combined with disagreement about order of spacelike separated events that are not simultaneous in either frame) is crucial to understanding how they can avoid contradictory predictions about some local event. 

That might be true, but I don't understand it. AG 

Don't understand how it answers your version of the paradox, or my version? If the latter, consider the scenario where the garage doors briefly close and open again, "at the same moment" when the car's midpoint is in the middle of the garage as defined by the garage frame's definition of simultaneity. In this case, do you understand the basic idea that the car frame says the back door closed and re-opened before the front door, and that this can explain why the car frame agrees about the local fact that neither door hits the car, despite the car being too big to fit fully inside the garage at any given moment?

 

 

since, if proven, it just tells us what we already knew; that fitting and not fillting don't occur at the same time. OTOH, if the observers observe different results when they're juxtaposed, it's a paradox IMO for the same reason as if Clark's definition for a paradox was manifested; different results at the same time.

But they aren't seeing different results at an agreed-upon "same time" because they don't agree on which events at different locations happen at the same time.

I never claimed that in my model they see different results because they are simultaneous. I have been quite clear, I thought, that when the observers are juxtaposed, they do NOT generally agree that t = t'. Why should that be the case, when the frames have different coordinates? AG

Still not clear what you mean by "juxtaposed" or what you mean by t = t' ...did your last response to Brent where you realized that his two diagrams should technically be expressed with different coordinates clear up this specific issue?

If this issue hasn't been resolved and the word "juxtaposed" is still important, since you said earlier that being juxtaposed just had to do with the sum total of measurements each observer makes to reach the conclusion "the car fit inside" or "the car didn't fit", does it actually matter if the measurements of the car observer share *any* events in common with the measurements of the garage observer, or could they be measurements on two different surfaces of simultaneity which don't intersect at any point along the body of the car or garage?

Jesse

 

 

As for your last post on the other thread, I am still working on it. It will take some time before I respond, but I will. AG

Sounds good--I hope you try the experiment of plugging in particular combinations of x and t coordinates (or x' and t' coordinates) into the LT equations as input to verify what I said about the resulting output.

Jesse

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[Alan Grayson]

I have to think about this some more. AG  

 

since, if proven, it just tells us what we already knew; that fitting and not fillting don't occur at the same time. OTOH, if the observers observe different results when they're juxtaposed, it's a paradox IMO for the same reason as if Clark's definition for a paradox was manifested; different results at the same time.

But they aren't seeing different results at an agreed-upon "same time" because they don't agree on which events at different locations happen at the same time.

I never claimed that in my model they see different results because they are simultaneous. I have been quite clear, I thought, that when the observers are juxtaposed, they do NOT generally agree that t = t'. Why should that be the case, when the frames have different coordinates? AG

Still not clear what you mean by "juxtaposed" or what you mean by t = t' ...did your last response to Brent where you realized that his two diagrams should technically be expressed with different coordinates clear up this specific issue?

Yes. AG 

If this issue hasn't been resolved and the word "juxtaposed" is still important, since you said earlier that being juxtaposed just had to do with the sum total of measurements each observer makes to reach the conclusion "the car fit inside" or "the car didn't fit", does it actually matter if the measurements of the car observer share *any* events in common with the measurements of the garage observer, or could they be measurements on two different surfaces of simultaneity which don't intersect at any point along the body of the car or garage?

Jesse

You ought to read my last long post here carefully. I agree that IF fitting and not fitting is interpreted as a paradox, then different valies of v must also be considered a paradox. My conclusion, to be consistent, must be that neither, by itself, is a paradox. What "juxtaposed" means is that the car and garage observer are at the midpoint of the garage "at the same time" (for want of any other way to describe it), although their clocks generally measure different times, t and t'.  AG