Length Contraction in SR (again)

[Alan Grayson]

In the case of a car whose rest length is greater than the length of the garage, from pov of the garage, the car will fit inside if its speed is sufficient fast due to length contraction of the car. But from the pov of the moving car, the length of garage will contract, as close to zero as one desires as its velocity approaches c, so the car will NOT fit inside the garage. Someone posted a link to an article which claimed, without proof, that this apparent contradiction can be resolved by the fact that simultaneity is frame dependent. I don't see how disagreements of simultaneity between frames solves this apparent paradox. AG

[Jesse Mazer]

On Wed, Dec 4, 2024 at 4:06 PM Alan Grayson <agrays...@gmail.com> wrote:

In the case of a car whose rest length is greater than the length of the garage, from pov of the garage, the car will fit inside if its speed is sufficient fast due to length contraction of the car. But from the pov of the moving car, the length of garage will contract, as close to zero as one desires as its velocity approaches c, so the car will NOT fit inside the garage. Someone posted a link to an article which claimed, without proof, that this apparent contradiction can be resolved by the fact that simultaneity is frame dependent. I don't see how disagreements of simultaneity between frames solves this apparent paradox. AG

Can you think of any way to define the meaning of the phrase "fit inside" other than by saying that the back end of the car is at a position inside the garage past the entrance "at the same time" as the front end of the car is at a position inside the garage but hasn't hit the back wall? (or hasn't passed through the back opening of the garage, if we imagine the garage as something like a covered bridge that's open on both ends) This way of defining it obviously depends on simultaneity, so different frames can disagree about whether there is any moment where such an event on the worldline of the back of the car is simultaneous with such an event on the worldline of the front of the car.

Jesse

[Alan Grayson]

Let's suppose that in the frame of the car, the front and back of the car are simultaneously inside the garage at some speed v.  How does this account for the fact that the length of the garage schrinks arbitarily close to zero as v approaches c, which ostensibly leads to, or tends to the opposite conclusion? AG

[Brent Meeker]

Think about what that mean operationally.  You have a photon detector at the middle of the car and mirrors at each end of the car positioned to send photons from lights at each end of the garage.  When the detector receives a photon from each direction at the same time that means the ends of your care are simultaneously at the ends of the garage IN THE CARS REFERENCE FRAME.  Now think about what it means in the garage reference frame.

Brent

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[Alan Grayson]

On Wednesday, December 4, 2024 at 10:57:16 PM UTC-7 Brent Meeker wrote:

On 12/4/2024 8:43 PM, Alan Grayson wrote:

On Wednesday, December 4, 2024 at 2:41:25 PM UTC-7 Jesse Mazer wrote:

On Wed, Dec 4, 2024 at 4:06 PM Alan Grayson <agrays...@gmail.com> wrote:

In the case of a car whose rest length is greater than the length of the garage, from pov of the garage, the car will fit inside if its speed is sufficient fast due to length contraction of the car. But from the pov of the moving car, the length of garage will contract, as close to zero as one desires as its velocity approaches c, so the car will NOT fit inside the garage. Someone posted a link to an article which claimed, without proof, that this apparent contradiction can be resolved by the fact that simultaneity is frame dependent. I don't see how disagreements of simultaneity between frames solves this apparent paradox. AG

Can you think of any way to define the meaning of the phrase "fit inside" other than by saying that the back end of the car is at a position inside the garage past the entrance "at the same time" as the front end of the car is at a position inside the garage but hasn't hit the back wall? (or hasn't passed through the back opening of the garage, if we imagine the garage as something like a covered bridge that's open on both ends) This way of defining it obviously depends on simultaneity, so different frames can disagree about whether there is any moment where such an event on the worldline of the back of the car is simultaneous with such an event on the worldline of the front of the car.

Jesse

Let's suppose that in the frame of the car, the front and back of the car are simultaneously inside the garage at some speed v.  How does this account for the fact that the length of the garage schrinks arbitarily close to zero as v approaches c, which ostensibly leads to, or tends to the opposite conclusion? AG

Think about what that mean operationally.  You have a photon detector at the middle of the car and mirrors at each end of the car positioned to send photons from lights at each end of the garage.  When the detector receives a photon from each direction at the same time that means the ends of your care are simultaneously at the ends of the garage IN THE CARS REFERENCE FRAME.  Now think about what it means in the garage reference frame.

Brent

In garage's reference frame, it might mean the car does not fit in garage, or that it can't be concluded that it does fit.  But the conclusion that it fits seems to contradict the fact that the length of the garage shrinks from the car's pov and was initially assume to be longer than the garage in the rest frame of the car. AG

[John Clark]

On Thu, Dec 5, 2024 at 1:19 AM Alan Grayson <agrays...@gmail.com> wrote:

 >But the conclusion that it fits seems to contradict the fact that the length of the garage shrinks from the car's pov

 

But from the garage man's POV the garage's length does not shrink but the car's length does.  In Special Relativity time is diluted by  the factor  γ which is equal to    1 / √(1 - v²/c²) ; and an object's length will be reduced by a factor of the inverse of γ. So Length contraction reduces the length by 1/γ, and Time Dilation increases the time interval by γ. For example, at 87% the speed of light length contracts to half its original rest length, and time dilutes by a factor of two.

The bottom line is that when two observers are in relative motion, like the garage man and the car driver are, they measure space and time differently. An event has a position and a time, and the closing of both garage doors is an event, so they will not agree if that event happened simultaneously when the entire car was in the garage or not.

John K Clark    See what's on my new list at  Extropolis

tgc

[Jesse Mazer]

Assuming the car and garage are moving inertially, whenever there's a conflict over whether the car fits in the garage in the garage frame vs. the car frame, it's always the case that it fits in the garage frame (where the car's length is contracted relative to its rest length) but not in the car frame (where the garage's length is contracted relative to its rest length), never vice versa. So if the car fits in the car frame as in your scenario, it also fits in the garage frame. Of course whether or not it fits at a given value of v depends on the numerical details of the problem, for any v < c you can always build a garage whose rest length is sufficiently large compared to the car rest length so that the car will still fit in the car's rest frame despite the length contraction of the garage.

Jesse

 

[Alan Grayson]

I don't think your proposed solution works. We're assuming the rest frame length of the car is larger than the rest frame length of the garage. So if you assume the car's velocity is sufficiently large to fit perfectly within the garage, so that its front and back end are simultaneously positioned at the open garage doors, the garage observer will not agree with that conclusion due to the failure of simultaneity of the front and rear boundaries of the car, as seen by the garage observer. IOW, unless we can establish that there's an objective reality where both observers see the same thing -- the car perfectly fitting inside the garage, or not -- we haven't resolved the apparent paradox. Moreover, since the car length decreases in the same proportion as the garage length decreases (from the pov of the garage observer and the car observer, respectively), both observers would deny, given the initial condition, that the car can perfectly fits in the garage. AG

   

[John Clark]

On Fri, Dec 6, 2024 at 12:10 AM Alan Grayson <agrays...@gmail.com> wrote:

>> from the garage man's POV the garage's length does not shrink but the car's length does.  In Special Relativity time is diluted by  the factor  γ which is equal to    1 / √(1 - v²/c²) ; and an object's length will be reduced by a factor of the inverse of γ. So Length contraction reduces the length by 1/γ, and Time Dilation increases the time interval by γ. For example, at 87% the speed of light length contracts to half its original rest length, and time dilutes by a factor of two.

The bottom line is that when two observers are in relative motion, like the garage man and the car driver are, they measure space and time differently. An event has a position and a time, and the closing of both garage doors is an event, so they will not agree if that event happened simultaneously when the entire car was in the garage or not.

 

> I don't think your proposed solution works. We're assuming the rest frame length of the car is larger than the rest frame length of the garage.

 

As Jesse Mazer points out, if the car fits in the car driver's frame of reference then it always fits in the garage man's frame of reference. However if it doesn't fit in the garage men's frame of reference then it won't fit in the driver's frame of reference either; this can happen if the car is not going fast enough, and the asymmetry between the two viewpoints occurs because when the car driver and the garage man and the car and the garage are all in the same frame of reference (a.k.a. they are not moving with respect to each other) then they both agree that the car is longer than the garage. So there is never a contradiction, there is never an occasion where one of them predicts the car will fit in the garage and the other predicts it will not.  

>  IOW, unless we can establish that there's an objective reality where both observers see the same thing 

Just because 2 observers see the same thing does not mean they are necessarily observing "objective reality" because their observation is not unique. If observer-1 and observer-2 are not moving with respect to each other then they will agree about what events are simultaneous and what events are not; however observers # 3 and 4 who are not moving with respect to each other but ARE moving with respect  to #1 and 2  will agree with each other about what events are simultaneous but they will not agree with  #1 and 2 about what events are simultaneous.

 

> since the car length decreases in the same proportion as the garage length decreases (from the pov of the garage observer and the car observer, respectively), both observers would deny, given the initial condition, that the car can perfectly fits in the garage. AG

If they both agree that the car will not fit in the garage then the car will not fit in the garage because the car is not going fast enough, and there is no contradiction. 

 John K Clark    See what's on my new list at  Extropolis

3bv

[Jesse Mazer]

On Fri, Dec 6, 2024 at 7:45 AM John Clark <johnk...@gmail.com> wrote:

On Fri, Dec 6, 2024 at 12:10 AM Alan Grayson <agrays...@gmail.com> wrote:

>> from the garage man's POV the garage's length does not shrink but the car's length does.  In Special Relativity time is diluted by  the factor  γ which is equal to    1 / √(1 - v²/c²) ; and an object's length will be reduced by a factor of the inverse of γ. So Length contraction reduces the length by 1/γ, and Time Dilation increases the time interval by γ. For example, at 87% the speed of light length contracts to half its original rest length, and time dilutes by a factor of two.

The bottom line is that when two observers are in relative motion, like the garage man and the car driver are, they measure space and time differently. An event has a position and a time, and the closing of both garage doors is an event, so they will not agree if that event happened simultaneously when the entire car was in the garage or not.

 

> I don't think your proposed solution works. We're assuming the rest frame length of the car is larger than the rest frame length of the garage.

 

As Jesse Mazer points out, if the car fits in the car driver's frame of reference then it always fits in the garage man's frame of reference. However if it doesn't fit in the garage men's frame of reference then it won't fit in the driver's frame of reference either; this can happen if the car is not going fast enough, and the asymmetry between the two viewpoints occurs because when the car driver and the garage man and the car and the garage are all in the same frame of reference (a.k.a. they are not moving with respect to each other) then they both agree that the car is longer than the garage. So there is never a contradiction, there is never an occasion where one of them predicts the car will fit in the garage and the other predicts it will not.  

You didn't really answer my question before about whether you think there is any way to define the phrase "fits in the garage" in a way that doesn't involve questions of simultaneity. If we do use a definition involving simultaneity, the natural one is to look at the two localized events A="back end of the car passes by the front door of garage" and B="front end of the car crashes into back wall of garage" (assuming the car does not brake so that everything is inertial up to the moment of the crash). In a frame where the crash B happens *after* the back end of the car entering the garage A, there will be some interval of time where the car is fully inside the garage and it hasn't yet crashed. In a frame where B happens *before* A, the car never fit in the garage because the front end crashed into the back wall before the back end had entered the garage.

When you say there is never a contradiction, do you deny we can pick values for the rest length of the car and garage and their relative velocity such if we use the Lorentz transformation, we find that B happens before A in the car rest frame (so the car doesn't fit in that frame), but A happens before B in the garage rest frame (so the car does fit in that frame)? Or do you accept that point, but think there is some other way to define the notion of "fits in the garage" that doesn't involve questions of simultaneity? If the latter you need to provide your own criterion.

Jesse

[Alan Grayson]

On Friday, December 6, 2024 at 11:13:36 AM UTC-7 Jesse Mazer wrote:

On Fri, Dec 6, 2024 at 7:45 AM John Clark <johnk...@gmail.com> wrote:

On Fri, Dec 6, 2024 at 12:10 AM Alan Grayson <agrays...@gmail.com> wrote:

>> from the garage man's POV the garage's length does not shrink but the car's length does.  In Special Relativity time is diluted by  the factor  γ which is equal to    1 / √(1 - v²/c²) ; and an object's length will be reduced by a factor of the inverse of γ. So Length contraction reduces the length by 1/γ, and Time Dilation increases the time interval by γ. For example, at 87% the speed of light length contracts to half its original rest length, and time dilutes by a factor of two.

The bottom line is that when two observers are in relative motion, like the garage man and the car driver are, they measure space and time differently. An event has a position and a time, and the closing of both garage doors is an event, so they will not agree if that event happened simultaneously when the entire car was in the garage or not.

 

> I don't think your proposed solution works. We're assuming the rest frame length of the car is larger than the rest frame length of the garage.

 

As Jesse Mazer points out, if the car fits in the car driver's frame of reference then it always fits in the garage man's frame of reference.

I think your claim is mistaken if you're using simultaneity in the car's frame. If not, then how do you define "fits in the garage"? See my comments in reply to Jesse. AG

 

However if it doesn't fit in the garage men's frame of reference then it won't fit in the driver's frame of reference either; this can happen if the car is not going fast enough, and the asymmetry between the two viewpoints occurs because when the car driver and the garage man and the car and the garage are all in the same frame of reference (a.k.a. they are not moving with respect to each other) then they both agree that the car is longer than the garage. So there is never a contradiction, there is never an occasion where one of them predicts the car will fit in the garage and the other predicts it will not.  

You didn't really answer my question before about whether you think there is any way to define the phrase "fits in the garage" in a way that doesn't involve questions of simultaneity.

Offhand, I don't know how else to structure a replywithout relying on simultaneity, but using simultaneity is useless since the garage observer will not agree with the car observer that the car fits in the garage, since he does not interpret simultaneity as the car observer does. And, in addition, the garage observer knows that the car's length decreases in the exact same proportion as the garage's length decreases, so he will deny that car fits since the relative lengths haven't changed, regardless of the car's velocity. AG

 

If we do use a definition involving simultaneity, the natural one is to look at the two localized events A="back end of the car passes by the front door of garage" and B="front end of the car crashes into back wall of garage" (assuming the car does not brake so that everything is inertial up to the moment of the crash). In a frame where the crash B happens *after* the back end of the car entering the garage A, there will be some interval of time where the car is fully inside the garage and it hasn't yet crashed. In a frame where B happens *before* A, the car never fit in the garage because the front end crashed into the back wall before the back end had entered the garage.

When you say there is never a contradiction,

 

I don't recall writing that; nor do I agree with that claim. I am saying that using simultaneity doesn't seem to solve the problem, since a solution must have both observers agree on an objective fact; whether the car fits or not. AG

[John Clark]

On Fri, Dec 6, 2024 at 1:13 PM Jesse Mazer <laser...@gmail.com> wrote:

> When you say there is never a contradiction, do you deny we can pick values for the rest length of the car and garage and their relative velocity such if we use the Lorentz transformation, we find that B happens before A in the car rest frame (so the car doesn't fit in that frame), but A happens before B in the garage rest frame (so the car does fit in that frame)? Or do you accept that point, but think there is some other way to define the notion of "fits in the garage" that doesn't involve questions of simultaneity?

Two observers, such as the car driver and the garage man, would say the car fits in the garage if and only if they see that both doors on the garage are closed and simultaneously see that both the front and the back of the car are entirely in the garage. They both agree that there is a time when the front of the car is in the garage, and they both agree there is a time when the back of the car is in the garage, and they both agreed there was a time when both doors on the garage were closed, but they may disagree if there was ever a time when those 3 events occurred simultaneously.

The faster the car goes the greater their disagreement about the length of the car and of the garage, and the greater their disagreement about the time when both doors were closed, and that resolves the logical contradiction. There is never an occasion when one observer sees the car crash into the back of the garage while the other observer does not.  

  John K Clark    See what's on my new list at  Extropolis

weq

[Jesse Mazer]

On Fri, Dec 6, 2024 at 3:23 PM Alan Grayson <agrays...@gmail.com> wrote:

On Friday, December 6, 2024 at 11:13:36 AM UTC-7 Jesse Mazer wrote:

On Fri, Dec 6, 2024 at 7:45 AM John Clark <johnk...@gmail.com> wrote:

On Fri, Dec 6, 2024 at 12:10 AM Alan Grayson <agrays...@gmail.com> wrote:

>> from the garage man's POV the garage's length does not shrink but the car's length does.  In Special Relativity time is diluted by  the factor  γ which is equal to    1 / √(1 - v²/c²) ; and an object's length will be reduced by a factor of the inverse of γ. So Length contraction reduces the length by 1/γ, and Time Dilation increases the time interval by γ. For example, at 87% the speed of light length contracts to half its original rest length, and time dilutes by a factor of two.

The bottom line is that when two observers are in relative motion, like the garage man and the car driver are, they measure space and time differently. An event has a position and a time, and the closing of both garage doors is an event, so they will not agree if that event happened simultaneously when the entire car was in the garage or not.

 

> I don't think your proposed solution works. We're assuming the rest frame length of the car is larger than the rest frame length of the garage.

 

As Jesse Mazer points out, if the car fits in the car driver's frame of reference then it always fits in the garage man's frame of reference.

I think your claim is mistaken if you're using simultaneity in the car's frame. If not, then how do you define "fits in the garage"? See my comments in reply to Jesse. AG

 

However if it doesn't fit in the garage men's frame of reference then it won't fit in the driver's frame of reference either; this can happen if the car is not going fast enough, and the asymmetry between the two viewpoints occurs because when the car driver and the garage man and the car and the garage are all in the same frame of reference (a.k.a. they are not moving with respect to each other) then they both agree that the car is longer than the garage. So there is never a contradiction, there is never an occasion where one of them predicts the car will fit in the garage and the other predicts it will not.  

You didn't really answer my question before about whether you think there is any way to define the phrase "fits in the garage" in a way that doesn't involve questions of simultaneity.

Offhand, I don't know how else to structure a replywithout relying on simultaneity, but using simultaneity is useless since the garage observer will not agree with the car observer that the car fits in the garage, since he does not interpret simultaneity as the car observer does. And, in addition, the garage observer knows that the car's length decreases in the exact same proportion as the garage's length decreases, so he will deny that car fits since the relative lengths haven't changed, regardless of the car's velocity. AG

When you say above "the garage observer knows that the car's length decreases in the exact same proportion as the garage's length decreases", presumably the latter refers to the shrinking of the garage in the car's rest frame? If so, why should the shrinking of the garage in the *car's frame* be relevant to the *garage observer's* answer to whether the car fits? In answering the question he uses his own rest frame, where there is no shrinking of the garage length. 

Below you say something about there being an "objective fact" about the matter (see my response below)--are you assuming in your answer here that the garage observer is not trying to answer the question "does the car fit in my own rest frame" but some other question like "does the car fit in objective terms"? If so, I would say this second question is simply nonsensical, like asking "do two points on a plane share the same x-coordinate in objective terms, independently of our choice of how to orient our x-y axes"--whether two points share the same x-coordinate is inherently a coordinate-dependent question that has no objective answer independent of choice of coordinate system, similarly whether the car fits is inherently a frame-dependent question that has no objective frame-independent answer.

 

 

If we do use a definition involving simultaneity, the natural one is to look at the two localized events A="back end of the car passes by the front door of garage" and B="front end of the car crashes into back wall of garage" (assuming the car does not brake so that everything is inertial up to the moment of the crash). In a frame where the crash B happens *after* the back end of the car entering the garage A, there will be some interval of time where the car is fully inside the garage and it hasn't yet crashed. In a frame where B happens *before* A, the car never fit in the garage because the front end crashed into the back wall before the back end had entered the garage.

When you say there is never a contradiction,

 

I don't recall writing that; nor do I agree with that claim. I am saying that using simultaneity doesn't seem to solve the problem, since a solution must have both observers agree on an objective fact; whether the car fits or not. AG

You said it in the last part of your older message in bold that I quoted above (right before my response beginning 'You didn't really answer my question before')--your words were "there is never a contradiction, there is never an occasion where one of them predicts the car will fit in the garage and the other predicts it will not". Did you say that by mistake, or does the wording need clarification or something?

And *why* do you think "whether the car fits or not" is an "objective fact"? Is that just a gut feeling or do you have an argument for it? I would say only local facts are really objective in relativity, like what two clocks read at the moment they pass right next to each other (one could for example compare a clock mounted on the back of the car to a clock mounted on the front of the garage and both frames would agree on what each clock reads as they pass next to each other). Since "does the car fit" cannot be stated in such localized terms, but depends on comparing the position of the front of the car and the position of the back of the car at "the same moment", I'd say any yes-or-no answer to this question is *not* an objective fact but depends on the simultaneity convention.

Jesse

[Brent Meeker]

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

Brent

[Alan Grayson]

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't? So, apparently, there's no objective answer to the question of whether car fits in garage, or not. Doesn't there have to be agreement betweem the frames to claim the apparent paradox is resolved? BTW, what velcoity did you use to get the numerical contraction values? AG

[Brent Meeker]

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

So, apparently, there's no objective answer to the question of whether car fits in garage, or not. Doesn't there have to be agreement betweem the frames to claim the apparent paradox is resolved? BTW, what velcoity did you use to get the numerical contraction values? AG

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[Alan Grayson]

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

[Alan Grayson]

On Friday, December 6, 2024 at 8:47:24 PM UTC-7 Alan Grayson wrote:

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you meant the elogated car doesn't fit in car frame, but we already knew this.  AG

[Brent Meeker]

On 12/6/2024 7:47 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you don't know the difference between a space/time diagram and a space/space diagram.

Brent

So, apparently, there's no objective answer to the question of whether car fits in garage, or not. Doesn't there have to be agreement betweem the frames to claim the apparent paradox is resolved? BTW, what velcoity did you use to get the numerical contraction values? AG

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[Brent Meeker]

On 12/6/2024 7:57 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 8:47:24 PM UTC-7 Alan Grayson wrote:

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you meant the elogated car doesn't fit in car frame, but we already knew this.  AG

In the words of Oliver Heaviside, "I've given you an argument.  I'm not obliged to give you an understanding."

Brent

So, apparently, there's no objective answer to the question of whether car fits in garage, or not. Doesn't there have to be agreement betweem the frames to claim the apparent paradox is resolved? BTW, what velcoity did you use to get the numerical contraction values? AG

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[Alan Grayson]

On Friday, December 6, 2024 at 9:24:06 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 7:57 PM, Alan Grayson wrote:

 

On Friday, December 6, 2024 at 8:47:24 PM UTC-7 Alan Grayson wrote:

 

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.

But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 

But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you meant the elogated car doesn't fit in car frame, but we already knew this.  AG

In the words of Oliver Heaviside, "I've given you an argument.  I'm not obliged to give you an understanding."

Brent

Heavyside isn't well known and now we know why. You can do better, much better, but don't want to. And that's where this story presumably ends. AG 

[Alan Grayson]

On Friday, December 6, 2024 at 9:32:53 PM UTC-7 Alan Grayson wrote:

On Friday, December 6, 2024 at 9:24:06 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 7:57 PM, Alan Grayson wrote:

 

On Friday, December 6, 2024 at 8:47:24 PM UTC-7 Alan Grayson wrote:

 

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.

But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 

But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you meant the elogated car doesn't fit in car frame, but we already knew this.  AG

In the words of Oliver Heaviside, "I've given you an argument.  I'm not obliged to give you an understanding."

Brent

Heavyside isn't well known and now we know why. You can do better, much better, but don't want to. And that's where this story presumably ends. AG

Ultimately, what you've presented relies on handwaving since your argument is incomplete. Maybe someone else here can complete it. AG 

[Alan Grayson]

On Friday, December 6, 2024 at 9:22:33 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 7:47 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you don't know the difference between a space/time diagram and a space/space diagram.

Brent

 

I can read English well and what you've presented is a space/TIME diagram, with TIME on the vertical axis, and where the car perfectly fits from the pov of the garage, and then you have the enlongated car which doesn't fit, supposedly based on simultaneity as measured in the car's frame, which, as I previously wrote, we already know doesn't fit. I have a high IQ but absolutely no clue what you have proven, if anything. AG

[Alan Grayson]

On Saturday, December 7, 2024 at 12:53:00 AM UTC-7 Alan Grayson wrote:

On Friday, December 6, 2024 at 9:22:33 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 7:47 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 8:00:56 PM UTC-7 Brent Meeker wrote:

On 12/6/2024 6:09 PM, Alan Grayson wrote:

On Friday, December 6, 2024 at 5:30:59 PM UTC-7 Brent Meeker wrote:

I thought I had put this to bed long ago.  Here's how is looks in the cars reference frame.  The garage ,which is 10' long, is moving fast toward the car.  It's length is Lorentz contracted to only 6', so the car doesn't fit.  No surprise since the car is 12' long.



But now from the garage's reference frame.  The car is contracted to only 8' and so fits nicely, both ends of the car are inside the garage at the same time (where simultaneity is defined in the garage reference frame). 


But doesn't this contradict the car's observation that the garage was way to short?  No, because what the car measure to be simultaneous is shown as the slanted car.  His front bumper was well beyond the end of the garage when his rear bumper had just entered.  These two diagrams are just the Lorentz transform of one another.

 Please elaborate on this point. TY, AG

Brent

So, from the car's frame, the car won't fit in garage due to contraction of garage's length, but from garage frame the car fits perfectly due to contraction of car's length, and then it doesn't?

How did you get "and then it doesn't out of what I wrote."  Study the diagram.

Brent

First the car doesn't fit from car's frame, then it does fit from garage frame, then the elogated car doesn't fit from garage frame. AG

Maybe you don't know the difference between a space/time diagram and a space/space diagram.

Brent

 

I can read English well and what you've presented is a space/TIME diagram, with TIME on the vertical axis, and where the car perfectly fits from the pov of the garage, and then you have the enlongated car which doesn't fit, supposedly based on simultaneity as measured in the car's frame, which, as I previously wrote, we already know doesn't fit. I have a high IQ but absolutely no clue what you have proven, if anything. AG

Concerning your Heavyside remark, I understand your impulse to put me down, but the fact is your "proof" fails what it means to prove anything. That's why I regard it as handwaving. When attempting a proof, you are minimally required to clearly state exactly what you're trying to prove, and then do it if you can. What exactly do you think you're trying to prove? What exactly is the apparent paradox you're trying to resolve, and how have your diagrams done that? Failing to meet these requirements, you can't expect to be understood. AG 

[Alan Grayson]

On Wednesday, December 4, 2024 at 2:41:25 PM UTC-7 Jesse Mazer wrote:

Are you claiming that the apparent paradox can be resolved by accepting the fact that the car and garage frames have different conclusions about whether the car fits in the garage? AG 

[Quentin Anciaux]

I've already explained it to you last month, there is no contradiction, just non agreement on simultaneity... from the cars pov, the doors aren't closed simultaneously, that's it.

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[Alan Grayson]

On Saturday, December 7, 2024 at 4:16:38 AM UTC-7 Quentin Anciaux wrote:

I've already explained it to you last month, there is no contradiction, just non agreement on simultaneity... from the cars pov, the doors aren't closed simultaneously, that's it.

I think you have it backwards. From the garage frame, simultaneity in car frame is invalid. But this doesn't address the issue of whether the car fits in the garage or not. AG 

[Alan Grayson]

Let me restate the problem where the car length is assumed to be longer than the garage length in the rest frame. So, the car can never fit in the garage when moving, since the garage length, which is initially smaller than the car's length, contracts. So there's no possibility of a perfect fit within the garage where simultaneity would apply. However, the car can fit in the garage, from the garage frame, since the car's length contracts. So, we have a situation where the car fits in the garage, but only from the garage frame. This seems paradoxical insofar as there's no objective reality of whether the car fits in the garage or not. AG 

[Alan Grayson]

On Saturday, December 7, 2024 at 4:16:38 AM UTC-7 Quentin Anciaux wrote:

I've already explained it to you last month, there is no contradiction, just non agreement on simultaneity... from the cars pov, the doors aren't closed simultaneously, that's it.

I disagree with your conclusion. GIven the initial conditions of the scenario, the car cannot fit in the garage from the car's frame. And from the frame of the garage, the car can easily fit if its velocity is suffiently close to c. So, although I know that disagreements about simultaneity is the alleged solution to the apparent paradox, I don't see it as relevant. What we have are two frames which reach opposiite conclusions about whether the car can fit in the garage. This seems paradoxical to me, but perhaps I am mistaken. AG 

[John Clark]

On Fri, Dec 6, 2024 at 11:32 PM Alan Grayson <agrays...@gmail.com> wrote:

>> In the words of Oliver Heaviside, "I've given you an argument.  I'm not obliged to give you an understanding."
Brent

> Heavyside isn't well known

Perhaps not on Twitter, but Oliver Heaviside is certainly well known and respected among professional physicists!  

[John Clark]

On Sat, Dec 7, 2024 at 1:43 PM Alan Grayson <agrays...@gmail.com> wrote:

  > What we have are two frames which reach opposiite conclusions about whether the car can fit in the garage.

If that is true then Special Relativity is nonsense, and so is General Relativity because it is built on that foundation. But physicists tell me neither idea is nonsense and I find the argument they present to defend their view to be very VERY persuasive. 

  > This seems paradoxical to me, but perhaps I am mistaken. 

Either Einstein was wrong in 1905 and all professional physicists for the last 120 years have been wrong in accepting Einstein's argument, or you Alan Grayson are correct and have found a flaw that nobody in the last 120 years had noticed before. Which do you honestly think is more probable?  

  John K Clark    See what's on my new list at  Extropolis

pnt

[Alan Grayson]

On Sunday, December 8, 2024 at 5:13:04 AM UTC-7 John Clark wrote:

On Sat, Dec 7, 2024 at 1:43 PM Alan Grayson <agrays...@gmail.com> wrote:

  > What we have are two frames which reach opposiite conclusions about whether the car can fit in the garage.

If that is true then Special Relativity is nonsense, and so is General Relativity because it is built on that foundation. But physicists tell me neither idea is nonsense and I find the argument they present to defend their view to be very VERY persuasive. 

I do, as well. AG 

  > This seems paradoxical to me, but perhaps I am mistaken. 

Either Einstein was wrong in 1905 and all professional physicists for the last 120 years have been wrong in accepting Einstein's argument, or you Alan Grayson are correct and have found a flaw that nobody in the last 120 years had noticed before. Which do you honestly think is more probable?  

Much more likely I am mistaken. HOWEVER, if, say, the garage is 10 meters in length, the velocity of the car can be assumed high enough to contract the car, from the pov of the garage, to, say, .000001 meter (or less), since its length approaches zero as v --> c, do you doubt the car can fit fully inside the garage? AG 

 

  John K Clark    pnt

[Jesse Mazer]

On Sat, Dec 7, 2024 at 6:49 AM Alan Grayson <agrays...@gmail.com> wrote:

On Saturday, December 7, 2024 at 3:51:30 AM UTC-7 Alan Grayson wrote:

On Wednesday, December 4, 2024 at 2:41:25 PM UTC-7 Jesse Mazer wrote:

On Wed, Dec 4, 2024 at 4:06 PM Alan Grayson <agrays...@gmail.com> wrote:

In the case of a car whose rest length is greater than the length of the garage, from pov of the garage, the car will fit inside if its speed is sufficient fast due to length contraction of the car. But from the pov of the moving car, the length of garage will contract, as close to zero as one desires as its velocity approaches c, so the car will NOT fit inside the garage. Someone posted a link to an article which claimed, without proof, that this apparent contradiction can be resolved by the fact that simultaneity is frame dependent. I don't see how disagreements of simultaneity between frames solves this apparent paradox. AG

Can you think of any way to define the meaning of the phrase "fit inside" other than by saying that the back end of the car is at a position inside the garage past the entrance "at the same time" as the front end of the car is at a position inside the garage but hasn't hit the back wall? (or hasn't passed through the back opening of the garage, if we imagine the garage as something like a covered bridge that's open on both ends) This way of defining it obviously depends on simultaneity, so different frames can disagree about whether there is any moment where such an event on the worldline of the back of the car is simultaneous with such an event on the worldline of the front of the car.

Jesse

Are you claiming that the apparent paradox can be resolved by accepting the fact that the car and garage frames have different conclusions about whether the car fits in the garage? AG 

Yes, there are many facts in physics that are not truly "objective" in the sense of being independent of the choice of reference frame, like "which of these two objects has a higher velocity" (even in classical physics there is the notion of 'Galilean relativity' where the laws of physics work the same in different classical inertial frames related by the Galilei transformation--Isaac Newton postulated the idea of absolute space and therefore absolute velocity, but this was an untestable philosophical idea even in classical physics). Any facts about whether the car fits in the garage are similarly non-objective from a physical point of view, depending irreducibly on the choice of frame.

 

Let me restate the problem where the car length is assumed to be longer than the garage length in the rest frame. So, the car can never fit in the garage when moving, since the garage length, which is initially smaller than the car's length, contracts. So there's no possibility of a perfect fit within the garage where simultaneity would apply. However, the car can fit in the garage, from the garage frame, since the car's length contracts. So, we have a situation where the car fits in the garage, but only from the garage frame. This seems paradoxical insofar as there's no objective reality of whether the car fits in the garage or not. AG 

Right, as I said above there are many questions in physics which have no unique "objective reality" in that sense, so there's nothing especially unusual about the "does the car fit in the garage" question. The word "paradox" can refer either to a genuine logical paradox or just to something that runs counter to everyday "common sense" intuitions but involves no genuine contradiction, see definition 2a at https://www.merriam-webster.com/dictionary/paradox -- the car/garage paradox fits this latter definition.

Also note that in relativity, the idea of the car's position and physical state at a given instant in time can be thought of as a 3-dimensional cross-section of its 4-dimensional "world-tube", so it's possible to come up with analogous "paradoxes" in a purely geometric scenario that doesn't involve relativity or time. For example suppose we have two cylinders in 3D space, arranged in an X-shaped intersecting pattern so there is some overlap in their interior regions. And suppose we have a Cartesian coordinate system with x-y-z coordinate axes, and we define a "level plane" to be a 2D surface with constant z-coordinate. A given level plane will then slice through the two cylinders and contain oval or circular cross-sections of each one, so we can ask the question "is there any level plane where the cross section of cylinder #1 lies wholly inside the cross section of cylinder #2"? In this case the answer depends on the orientation of our x-y-z coordinate axes and the angle that the level planes slice through the cylinders--for some choice of coordinates the answer might be yes, for others it might be no, there is no coordinate-independent answer to the question.

Jesse

[Alan Grayson]

On Sunday, December 8, 2024 at 4:57:12 AM UTC-7 John Clark wrote:

On Fri, Dec 6, 2024 at 11:32 PM Alan Grayson <agrays...@gmail.com> wrote:

>> In the words of Oliver Heaviside, "I've given you an argument.  I'm not obliged to give you an understanding."
Brent

> Heavyside isn't well known

Perhaps not on Twitter, but Oliver Heaviside is certainly well known and respected among professional physicists!  

So presumably is Deutsch and those of us with common sense we know he's deeply delusional. AG 

[Alan Grayson]

On Sunday, December 8, 2024 at 5:13:04 AM UTC-7 John Clark wrote:

On Sat, Dec 7, 2024 at 1:43 PM Alan Grayson <agrays...@gmail.com> wrote:

  > What we have are two frames which reach opposiite conclusions about whether the car can fit in the garage.

If that is true then Special Relativity is nonsense, and so is General Relativity because it is built on that foundation. But physicists tell me neither idea is nonsense and I find the argument they present to defend their view to be very VERY persuasive. 

But that's what we have! -- two frames which reach opposite conclusions about whether the car can fit in the garage! Introducing simultaneity obfusates this simple fact! The car frame indicates the car cannot fit due to the initial condition, and the garage frame indicates it does fit. Why is this so hard to understand? If the initial condition says the car cannot fit because its length is too large, why cannot the garage observer assert it can fit because via length contraction, the car's length can be measured arbitrarily short? AG

[John Clark]

On Sun, Dec 8, 2024 at 7:39 PM Alan Grayson <agrays...@gmail.com> wrote:

>  two frames which reach opposite conclusions about whether the car can fit in the garage!

Yes but disagreeing about how long something is is the inevitable consequence of disagreeing about simultaneity. And after the thought experiment is over and the car slows down so it's in the same frame of reference as the garage there is no evidence of any paradox having occurred, there would be one if one observed the car making a big hole in the back door and the other did not, but nobody observed such a thing.  

 

> Why is this so hard to understand?

Because it's very hard to understand something if it's not true.  I think I'll repeat what I said before:

 

"Either Einstein was wrong in 1905 and all professional physicists for the last 120 years have been wrong in accepting Einstein's argument, or you Alan Grayson are correct and have found a flaw that nobody in the last 120 years had noticed before. Which do you honestly think is more probable?"

  
  John K Clark    See what's on my new list at  Extropolis

hrz

[Alan Grayson]

Simultaneity is an unnecessary red herring to cover up something obvious; that the two frames do NOT agree if the car fits in the garage. AG 

[Alan Grayson]

On Sunday, December 8, 2024 at 7:03:13 PM UTC-7 John Clark wrote:

On Sun, Dec 8, 2024 at 7:39 PM Alan Grayson <agrays...@gmail.com> wrote:

>  two frames which reach opposite conclusions about whether the car can fit in the garage!

Yes but disagreeing about how long something is is the inevitable consequence of disagreeing about simultaneity. And after the thought experiment is over and the car slows down so it's in the same frame of reference as the garage there is no evidence of any paradox having occurred, there would be one if one observed the car making a big hole in the back door and the other did not, but nobody observed such a thing.  

This is nonsense. Only if the car slows down to be at rest, will the car and garage be in the same frame, where there is no length contraction. This is just the initial condition where there is no paradox, none alleged, and no paradox to be observed. Obviously, the paradox exists only when the car speeds up, to a sufficient speed so it can fit exactly in the garage, from the pov of the garage frame. But why stop here? If the car continues to speed up, we have the well-defined paradox which you refuse to admit since the issue of simultaneity ceases to give ambiguous results. AG

 

> Why is this so hard to understand?

Because it's very hard to understand something if it's not true.  I think I'll repeat what I said before:

 

"Either Einstein was wrong in 1905 and all professional physicists for the last 120 years have been wrong in accepting Einstein's argument, or you Alan Grayson are correct and have found a flaw that nobody in the last 120 years had noticed before. Which do you honestly think is more probable?"

I don't how this problem is resolved within the context of relativity. All I can say is that the frames differ on what's observed; they come to opposite conclusions. No way of denying that, without introducing an artificial, unnecessary issue of simultaneity. AG 

[John Clark]

On Mon, Dec 9, 2024 at 3:55 AM Alan Grayson <agrays...@gmail.com> wrote:

> This is nonsense. [...] Obviously, the paradox exists only when the car speeds up, to a sufficient speed so it can fit exactly in the garage.which you refuse to admit.

I admit that in different frames of reference clocks running at different speeds and meter-sticks having different lengths is odd, but you need more than strangeness to have a paradox. My high school physics textbook explained, and I'm sure many others did too, that Einstein discovered that these two effects cancel out in such a way that no logical contradiction is ever produced, and that's what's required to have a paradox. I'd suggest that you read such a textbook but I know you never will, and even if you did you'd just say the textbook was wrong as were all physicists since 1905.   

  John K Clark    See what's on my new list at  Extropolis

tq4
hr

[Alan Grayson]

Since the contracted length of the car, from the pov of the garage, can be arbitraily close to zero, as v --> c, there's no way that the car cannot fit in the garage, from the pov of the garage. But from the pov of the car, assumed to have length greater than the garage as an initial condition, the car can never be contained within the garage, as the garage length shrinks. These are the facts. Why don't you focus on the facts, instead of on other BS, such as what I will or won't read? These results are paradoxical if and only if it is assumed there is some objective reality, such as the objective reality that the car can OR cannot fit in the garage. On the issue of time dilation, please be specific about how time dilation somehow cancels out space contraction. AG 

hr

[Alan Grayson]

On Monday, December 9, 2024 at 5:31:17 AM UTC-7 John Clark wrote:

On Mon, Dec 9, 2024 at 3:55 AM Alan Grayson <agrays...@gmail.com> wrote:

> This is nonsense. [...] Obviously, the paradox exists only when the car speeds up, to a sufficient speed so it can fit exactly in the garage.which you refuse to admit.

I admit that in different frames of reference clocks running at different speeds and meter-sticks having different lengths is odd,

Not odd if you understand where those results come from. They come from the LT, which is, I believe, the only frame transformation which preserves the frame invariance of the SoL. If we assume the SoL is frame invariant, then length contraction and time dilation are necessarily implied. AG

[John Clark]

On Mon, Dec 9, 2024 at 8:05 AM Alan Grayson <agrays...@gmail.com> wrote:

> Since the contracted length of the car, from the pov of the garage, can be arbitraily close to zero, as v --> c, there's no way that the car cannot fit in the garage, from the pov of the garage. But from the pov of the car, assumed to have length greater than the garage as an initial condition, the car can never be contained within the garage, as the garage length shrinks.

For heaven sake! Nobody is denying that two observers in two different frames of reference can and will observe different things and thus disagree if the car was ever entirely in the garage or not; just as they disagree about how long a meter stick is and how fast a clock ticks. But that's not a logical paradox, that's just strange. And objective reality does exist in relativity because some things DO remain constant in ANY frame of reference, such as the speed of light and the distance through spacetime of ANY two events. 

An event, such as the closing of both the front and back doors of the garage, is a specific point in space and time, the contraction of length and the stretching of time are not independent properties; they always change in such a way that the garage man in the car driver agree that the distance through spacetime between the front of the car entering the front of the garage in the back of the car exiting the back of the garage is exactly the same. But because they disagreed about length and time (but not when both are considered together) they will sometimes disagree if there was ever a time when both doors were closed AND the front and the back of the car were both in the garage. 

  John K Clark    See what's on my new list at  Extropolis

1o$

[Alan Grayson]

1

If that's your present position, why, when I first stated that the frames differed on whether the car will fit in the garage, did you claim the alleged flaw I described, would, if true, undermine 120 years of professional thinking about relativity? That is, why did you think I described a logical inconsistency or paradox, when now you've made a 180 degree turn on its implication? AG

[Alan Grayson]

I don't see the relevance in this situation of the fact that the spacetime distance is independent of the path between two fixed points. Moreover, there is a fixed rate of dilation and contraction between two frames for some relative velocity. aAG 

[John Clark]

On Mon, Dec 9, 2024 at 9:28 AM Alan Grayson <agrays...@gmail.com> wrote:

On Monday, December 9, 2024 at 7:17:07 AM UTC-7 John Clark wrote:

For heaven sake! Nobody is denying that two observers in two different frames of reference can and will observe different things and thus disagree if the car was ever entirely in the garage or not; just as they disagree about how long a meter stick is and how fast a clock ticks. But that's not a logical paradox, that's just strange. And objective reality does exist in relativity because some things DO remain constant in ANY frame of reference, such as the speed of light and the distance through spacetime of ANY two events. 

An event, such as the closing of both the front and back doors of the garage, is a specific point in space and time, the contraction of length and the stretching of time are not independent properties; they always change in such a way that the garage man in the car driver agree that the distance through spacetime between the front of the car entering the front of the garage in the back of the car exiting the back of the garage is exactly the same. But because they disagreed about length and time (but not when both are considered together) they will sometimes disagree if there was ever a time when both doors were closed AND the front and the back of the car were both in the garage. 

  John K Clark 

1

> If that's your present position,

It has always been my position that there is nothing paradoxical about two observers disagreeing if a car had fit in the garage for an instant or not, and it has always been my position that there will NEVER be an occasion where one observer sees the car make a car shaped hole in the back door of the garage but the other observer sees no such damage because THAT would have been a paradox, it would've been a logical contradiction which is what you need to produce a paradox.   

 

> why, when I first stated that the frames differed on whether the car will fit in the garage, did you claim the alleged flaw I described, would, if true, undermine 120 years of professional thinking about relativity?

Because you kept making a big deal about it and because you kept calling it a "paradox" which it is not. It's just odd.  

> now you've made a 180 degree turn on its implication

Actually I've made a 360 degree turn on its implication. 

If γ = 1 / √(1 - v²/c²) then Length Contraction reduces the length by 1/γ, and Time Dilation increases the time interval by γ, so they disagree about length and they disagree about time but they agree when space and time are both considered together in spacetime.

  John K Clark    See what's on my new list at  Extropolis

tst 

[Alan Grayson]

Nothing odd about dilation and contraction when you know its cause. But what is odd is the fact that each frame sees the result differently -- that the car fits in one frame, but not in the other -- and you see nothing odd about that, that there's no objective reality despite the symmetry. AG 

tst 

[Alan Grayson]

What's odd for me is how the consenus opinion goes out of its way to use difference of simultaneity to dispose of the different conclusions of the frames, when simply increasing the speed of the car shows otherwise. AG 

tst 

[Quentin Anciaux]

You discovered that in relativity there is no absolute frame of reference and so no objective simultaneity which in the end the "paradox" reduce to it, there is no objective truth about simultaneity of two spacetime events.

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[Quentin Anciaux]

Front door, back door, front of the car, rear of the car are four spacetime locations, so there can't be any objective truth about the simultaneity of both doors being closed and the front and rear of the car being fully inside the garage.

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[Alan Grayson]

No. I knew that 50+ years ago. But I thought the LT gave an objective fact about how events transform between frames. Now I doubt that, and what the LT gives. AG

[Alan Grayson]

So if the length of car is contracted close to zero, one cannot be sure the car fits in garage? This cannot be correct. AG

[Alan Grayson]

If you latest claim were true, GR would be useless in making predictions. AG 

[Brent Meeker]

On 12/9/2024 5:05 AM, Alan Grayson wrote:

Since the contracted length of the car, from the pov of the garage, can be arbitraily close to zero, as v --> c, there's no way that the car cannot fit in the garage, from the pov of the garage. But from the pov of the car, assumed to have length greater than the garage as an initial condition, the car can never be contained within the garage, as the garage length shrinks. These are the facts. Why don't you focus on the facts, instead of on other BS, such as what I will or won't read? These results are paradoxical if and only if it is assumed there is some objective reality, such as the objective reality that the car can OR cannot fit in the garage.

Turns out that's not "objective reality".  It's viewpoint dependent.  Surprising, but well understood.

On the issue of time dilation, please be specific about how time dilation somehow cancels out space contraction. AG

Time dilation and space contraction don't "cancel out".  What they do is keep the speed of light the same in all frames of reference.

Brent

[Brent Meeker]

>
> Nothing odd about dilation and contraction when you know its cause.
> But what is odd is the fact that each frame sees the result
> differently -- that the car fits in one frame, but not in the other --
> and you see nothing odd about that, that there's no objective reality
> despite the symmetry. AG

The facts are events in spacetime.  There's an event F at which the
front of the car is even with the exit of the garage and there's an
event R at which the rear of the car is even with the entrance to the
garage.  If R is before F we say the car fitted in the garage. If R is
after F we say the car did not fit.  But if F and  R are spacelike, then
there is no fact of the matter about their time order.  The time order
will depend on the state of motion.

Brent

[Alan Grayson]

I never made the "cancel out" claim and I don't believe it. That was the claim of our local genius who can see multiple non-existent worlds. AG 

[Alan Grayson]

Since the car length can be assumed as arbitrarily small from the

pov of the garage, why worry about fitting the car in garage perfectly,

and then appealing to difference in spontaneity to prove no direct

contradiction between the frames? It seems like a foolish effort to 

avoid a contradiction, when one clearly exists. AG 

[Brent Meeker]

On 12/9/2024 3:24 PM, Alan Grayson wrote:

On Monday, December 9, 2024 at 2:01:28 PM UTC-7 Brent Meeker wrote:

Since the car's length can be assumed to be arbitrarily small from the 

pov of the garage, why worry about fitting the car in garage perfectly,

and then appealing to difference in spontaneity to prove no direct

contradiction between the frames? It seems like a foolish effort to 

avoid a contradition, when one clearly exists. AG

What's the contradiction? 

And it's "simultaneity" not "spontaneity".

Brent

[Alan Grayson]

ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not. If one avoids the issue of simultaneity, by not requiring the car to perfectly fit in the garage, we get opposite conclusions from the frames. AG

[Jesse Mazer]

On Mon, Dec 9, 2024 at 8:28 PM Alan Grayson <agrays...@gmail.com> wrote:

On Monday, December 9, 2024 at 4:54:34 PM UTC-7 Brent Meeker wrote:

On 12/9/2024 3:24 PM, Alan Grayson wrote:

On Monday, December 9, 2024 at 2:01:28 PM UTC-7 Brent Meeker wrote:

>
> Nothing odd about dilation and contraction when you know its cause.
> But what is odd is the fact that each frame sees the result
> differently -- that the car fits in one frame, but not in the other --
> and you see nothing odd about that, that there's no objective reality
> despite the symmetry. AG

The facts are events in spacetime.  There's an event F at which the
front of the car is even with the exit of the garage and there's an
event R at which the rear of the car is even with the entrance to the
garage.  If R is before F we say the car fitted in the garage. If R is
after F we say the car did not fit.  But if F and  R are spacelike, then
there is no fact of the matter about their time order.  The time order
will depend on the state of motion.

Brent

Since the car's length can be assumed to be arbitrarily small from the 

pov of the garage, why worry about fitting the car in garage perfectly,

and then appealing to difference in spontaneity to prove no direct

contradiction between the frames? It seems like a foolish effort to 

avoid a contradition, when one clearly exists. AG

What's the contradiction? 

ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not.

Can you define what "fits in the garage" would mean in objective terms, i.e. a definition that does not depend on simultaneity or choice of frame? If you can't even define that phrase in any objective terms, why do you believe there is any objective reality here? Is it just some kind of strong intuitive hunch that there "should" be some kind of objective reality attached to such a notion?

If one avoids the issue of simultaneity, by not requiring the car to perfectly fit in the garage, we get opposite conclusions from the frames. AG

The paradox does not depend on the assumption that the car is the same length as the garage in either the car frame or the garage frame (or that they have the same rest length), if that's what you mean by "perfectly fit in the garage". Even if the car's rest length is much shorter than the garage's rest length, so it fits easily in the garage frame, you could always pick a sufficiently large relative velocity such that the car would be longer than the garage in the car's rest frame, and thus it would not fit in that frame, so the paradox remains. Not sure what "opposite conclusions from the frames" could mean if you don't have a specific way to define "fits in the garage" in a way that doesn't depend on picking some frame or another.

Jesse

[Alan Grayson]

"Fits" means the car's length is equal to garage's length, or less. If car exactly fits, this is ambiguous from the pov of garage frame due to lack of simultaneity, and this is the consensus solution to an alleged paradox. But what happens if the car's velocity is increased, so car fits with room to spare? This is the case I have been posting about. AG 

If one avoids the issue of simultaneity, by not requiring the car to perfectly fit in the garage, we get opposite conclusions from the frames. AG

The paradox does not depend on the assumption that the car is the same length as the garage in either the car frame or the garage frame (or that they have the same rest length),

I don't make this assumption. AG

 

if that's what you mean by "perfectly fit in the garage". Even if the car's rest length is much shorter than the garage's rest length, so it fits easily in the garage frame,

No, this isn't the initial assumed car length. Its length is assumed larger than the garage, and the question is whether it can fit due to its motion which causes length contraction. AG

 

you could always pick a sufficiently large relative velocity such that the car would be longer than the garage in the car's rest frame,

No, the car's length decreases in the garage frame only due to its motion. In the car's frame, the car's length doesn't change. AG

 

and thus it would not fit in that frame, so the paradox remains. Not sure what "opposite conclusions from the frames" could mean if you don't have a specific way to define "fits in the garage" in a way that doesn't depend on picking some frame or another.

Jesse

As I understand the problem, in the initial rest frame the car is assumed to be larger than the garage. Then the question is whether it can fit when the car is in motion due to length contraction. In the car's frame, the garage length decreases, so there is no possibility of the car fitting. OTOH, from the pov of the garage frame, the car's length shrinks, so there is some velocity where it fits perfectly. If the velocity continues to increase, the car fits with room to spare. So, I have shown that the frames differ in concluding whether the car fits, or not, and the question is whether this is a paradox. If you conclude it is not, then you deny there's an objective reality such that the car fits, or doesn't fit. And "fits" just means the car's contracted length is EQUAL TO or LESS than the garage's length. AG

[Brent Meeker]

If you define "fit in" it requires using the concept of simultaneity. 

Brent

[Alan Grayson]

I have been considering this possibility. If true, does it mean a car fitting in a garage can never be compared to how it appears in the garage frame, and that there's no objective reality? AG

[Alan Grayson]

I forget; what problem is ostensibly solved in the car's perfect fit scenario in the garage, by the fact that simultaneous events for the front and rear end of the car, are not simultaneous in the frame of the garage. TY, AG 

[Alan Grayson]

The contradiction is precisely this; assuming the initial rest state is that the length of the car is larger than the length of the garage, we get the car never fitting in the garage from the pov of the car, and the car fitting in the garage from the pov of the garage. The car can't fit and not fit in the garage. The former result is easy to see, since the car's motion shrinks the garage's length, so the car, initially longer than the garage, can never fit inside the garage. The latter result follows from the fact that from the pov of the garage, the car's length shrinks, and for a sufficient velocity, it will shrink enough to fit in the garage. Further, the issue of simultaneity is a non-issue, since measurements of the front and back end of the car occur in the car's frame, and since the car never fits in the garage, such measurements can never be made when the car perfectly fits in the garage, or even loosely, since this condition never occurs. In summary, I think I've done for relativity, what Bertrand Russell did for Cantor's set theory; proving the existence of a contradiction. AG

Brent

[Jesse Mazer]

Do you define the "length" of each in the standard relativistic frame-dependent way?

 

If car exactly fits, this is ambiguous from the pov of garage frame due to lack of simultaneity, and this is the consensus solution to an alleged paradox. But what happens if the car's velocity is increased, so car fits with room to spare? This is the case I have been posting about. AG 

If one avoids the issue of simultaneity, by not requiring the car to perfectly fit in the garage, we get opposite conclusions from the frames. AG

The paradox does not depend on the assumption that the car is the same length as the garage in either the car frame or the garage frame (or that they have the same rest length),

I don't make this assumption. AG

Then what did you mean by "not requiring the car to perfectly fit in the garage"? What does perfect vs. imperfect fit mean here, if "perfect" does not refer to them being exactly the same length?

 

if that's what you mean by "perfectly fit in the garage". Even if the car's rest length is much shorter than the garage's rest length, so it fits easily in the garage frame,

No, this isn't the initial assumed car length. Its length is assumed larger than the garage, and the question is whether it can fit due to its motion which causes length contraction. AG

The paradox can refer to any scenario where the car fits in the garage frame but not in the car frame, this can happen regardless of whether the car's rest length is larger or smaller than the garage's rest length.

 

 

you could always pick a sufficiently large relative velocity such that the car would be longer than the garage in the car's rest frame,

No, the car's length decreases in the garage frame only due to its motion. In the car's frame, the car's length doesn't change. AG

I didn't say the car's length changed in the car's frame, I just said that if you pick a sufficiently high relative velocity, then in the car's rest frame the car will be longer than the garage (in this case due to the garage's length being shortened in that frame).

 

and thus it would not fit in that frame, so the paradox remains. Not sure what "opposite conclusions from the frames" could mean if you don't have a specific way to define "fits in the garage" in a way that doesn't depend on picking some frame or another.

Jesse

As I understand the problem, in the initial rest frame the car is assumed to be larger than the garage. Then the question is whether it can fit when the car is in motion due to length contraction. In the car's frame, the garage length decreases, so there is no possibility of the car fitting. OTOH, from the pov of the garage frame, the car's length shrinks, so there is some velocity where it fits perfectly. If the velocity continues to increase, the car fits with room to spare. So, I have shown that the frames differ in concluding whether the car fits, or not, and the question is whether this is a paradox. If you conclude it is not, then you deny there's an objective reality such that the car fits, or doesn't fit. And "fits" just means the car's contracted length is EQUAL TO or LESS than the garage's length. AG

Yes, I've said before that there's no objective frame-independent reality about whether the car fits, that's part of the standard answer to this paradox. Do you accept that this is a valid way of resolving it? You seemed to be objecting in your earlier comment when you said "ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not", but maybe I misunderstood?

Jesse

[Jesse Mazer]

On Tue, Dec 10, 2024 at 4:33 AM Alan Grayson <agrays...@gmail.com> wrote:

On Monday, December 9, 2024 at 4:54:34 PM UTC-7 Brent Meeker wrote:

On 12/9/2024 3:24 PM, Alan Grayson wrote:

On Monday, December 9, 2024 at 2:01:28 PM UTC-7 Brent Meeker wrote:

>
> Nothing odd about dilation and contraction when you know its cause.
> But what is odd is the fact that each frame sees the result
> differently -- that the car fits in one frame, but not in the other --
> and you see nothing odd about that, that there's no objective reality
> despite the symmetry. AG

The facts are events in spacetime.  There's an event F at which the
front of the car is even with the exit of the garage and there's an
event R at which the rear of the car is even with the entrance to the
garage.  If R is before F we say the car fitted in the garage. If R is
after F we say the car did not fit.  But if F and  R are spacelike, then
there is no fact of the matter about their time order.  The time order
will depend on the state of motion.

Brent

Since the car's length can be assumed to be arbitrarily small from the 

pov of the garage, why worry about fitting the car in garage perfectly,

and then appealing to difference in spontaneity to prove no direct

contradiction between the frames? It seems like a foolish effort to 

avoid a contradition, when one clearly exists. AG

What's the contradiction? 

The contradiction is precisely this; assuming the initial rest state is that the length of the car is larger than the length of the garage, we get the car never fitting in the garage from the pov of the car, and the car fitting in the garage from the pov of the garage. The car can't fit and not fit in the garage.

It would only be a contradiction if "the car fits" and "the car doesn't fit" were meant in exactly the same sense, but they aren't, they are referring to different coordinate systems. If I have some geometric shapes on a plane, and I then overlay one set of x-y coordinate axes to get the conclusion "the circle has a greater x-coordinate than the triangle", but then I overlay a *different* set of x-y coordinates and get the conclusion "the circle has a smaller x-coordinate than the triangle", the two statements are referring to different coordinate systems which define the x and y coordinates of both shapes differently, so there is no true contradiction--do you agree? It's exactly the same with "the car fits" and "the car doesn't fit", they are made from the POV of different spacetime coordinate systems so they are not using "fit" in exactly the same sense, and there is no true contradiction.

Further, the issue of simultaneity is a non-issue, since measurements of the front and back end of the car occur in the car's frame, and since the car never fits in the garage, such measurements can never be made when the car perfectly fits in the garage, or even loosely, since this condition never occurs.

Yes, but different measurements of simultaneity can be made in the coordinates of the garage's frame, and it's in these coordinates that the car *does* fit, so clearly the differing definitions of simultaneity *are* relevant to understanding why the two frames have different answers.

Jesse

[Alan Grayson]

If we assume as an initial condition that the car's length is smaller than the garage's length, then for some velocity and greater, the car will not fit in the garage from the car's frame, so we're in the same situation as when we assumed as an initial condition that the car's length is greater than the garage's length.  The contradition persists and differences in simultaneity does not come to the rescue, since it's measured the car's frame and couldn't be used to show the car fits, exactly, or loosely, since it doesn't fit. AG

 

you could always pick a sufficiently large relative velocity such that the car would be longer than the garage in the car's rest frame,

No, the car's length decreases in the garage frame only due to its motion. In the car's frame, the car's length doesn't change. AG

I didn't say the car's length changed in the car's frame, I just said that if you pick a sufficiently high relative velocity, then in the car's rest frame the car will be longer than the garage (in this case due to the garage's length being shortened in that frame).

I don't follow. If the garage's length is shorten, the length of the car remains unchanged. AG 

 

and thus it would not fit in that frame, so the paradox remains. Not sure what "opposite conclusions from the frames" could mean if you don't have a specific way to define "fits in the garage" in a way that doesn't depend on picking some frame or another.

Jesse

As I understand the problem, in the initial rest frame the car is assumed to be larger than the garage. Then the question is whether it can fit when the car is in motion due to length contraction. In the car's frame, the garage length decreases, so there is no possibility of the car fitting. OTOH, from the pov of the garage frame, the car's length shrinks, so there is some velocity where it fits perfectly. If the velocity continues to increase, the car fits with room to spare. So, I have shown that the frames differ in concluding whether the car fits, or not, and the question is whether this is a paradox. If you conclude it is not, then you deny there's an objective reality such that the car fits, or doesn't fit. And "fits" just means the car's contracted length is EQUAL TO or LESS than the garage's length. AG

Yes, I've said before that there's no objective frame-independent reality about whether the car fits, that's part of the standard answer to this paradox. Do you accept that this is a valid way of resolving it? You seemed to be objecting in your earlier comment when you said "ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not", but maybe I misunderstood?

Jesse

My conclusion is that there's a contradiction in evidence, since the situation of the car fitting and not fitting makes no sense. AG 

[Quentin Anciaux]

And the answer only lies in disagreement of simultaneity of both doors being closed and the car fully inside the garage, and that is dependent on the frame of reference,  there aren't any contradictions.

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[Alan Grayson]

I don't agree that coordinate systems can resolve the contradiction. IIUC, measurements of simultaneity are made in car's frame, not in garage's frame. From that it's concluded that from garage frame, it's indeterminate whether the car really fits. But I'm confused on this point. How can the garage frame know about those measurements in car frame, and know they differ? AG 

[Alan Grayson]

But if the initial rest condition is that the car's length is greater than the garage's length, the car can never be within the garage, exactly or with space to spare. AG 

[John Clark]

On Tue, Dec 10, 2024 at 5:47 AM Quentin Anciaux <allc...@gmail.com> wrote:

>> My conclusion is that there's a contradiction in evidence, since the situation of the car fitting and not fitting makes no sense. AG 

>>And the answer only lies in disagreement of simultaneity of both doors being closed and the car fully inside the garage, and that is dependent on the frame of reference,  there aren't any contradictions.

Exactly! In Special Relativity two observers in different frames of reference can disagree about what order a series of events should be in, provided that the events are "spacelike separated", that is to say if even the speed of light is not fast enough to send a signal between them. One observer could say events should be ordered XYZ and another insist the order should be XZY. The classic example is two lightning strikes hitting the front and back of a very fast moving train, an observer on the train might see them as simultaneous but an observer on the ground see one hit happening before the other.  

And in the example that Alan has trouble with, one observer might say the back of the car entered the garage before the back door of the garage was opened, and another observer in a different frame of reference would say that happened after the back door of the garage was opened. So one would say the car was able to fit into the garage and the other would say it could not.

 

 John K Clark    See what's on my new list at  Extropolis

057

[Alan Grayson]

Apparently you haven't read my posts or don't understand them. If the initial conditions are set so the car's length is longer than the garage, and then the car is set in motion, the car is NEVER inside the garage since the garage is contracting! In this situation, there would be no measurements that you allege for which simultaneity would apply. There is NO entering of the garage for the car! AG 

 

[Alan Grayson]

Further, you bring up spacelike separated events to allegedly make your points. But as far as I can tell, no events in this situation are spacelike separated. AG 

 

[Quentin Anciaux]

So the front door, the back door, the front of the car, the rear of the car are all at the same spacelike location?  I don't call that doors or cars, I call that a point, and there are no paradoxes 

 

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[Alan Grayson]

On Tuesday, December 10, 2024 at 11:36:19 AM UTC-7 Quentin Anciaux wrote:

Le mar. 10 déc. 2024, 19:31, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:07:25 AM UTC-7 Alan Grayson wrote:

On Tuesday, December 10, 2024 at 6:29:25 AM UTC-7 John Clark wrote:

On Tue, Dec 10, 2024 at 5:47 AM Quentin Anciaux <allc...@gmail.com> wrote:

>> My conclusion is that there's a contradiction in evidence, since the situation of the car fitting and not fitting makes no sense. AG 

>>And the answer only lies in disagreement of simultaneity of both doors being closed and the car fully inside the garage, and that is dependent on the frame of reference,  there aren't any contradictions.

Exactly! In Special Relativity two observers in different frames of reference can disagree about what order a series of events should be in, provided that the events are "spacelike separated", that is to say if even the speed of light is not fast enough to send a signal between them. One observer could say events should be ordered XYZ and another insist the order should be XZY. The classic example is two lightning strikes hitting the front and back of a very fast moving train, an observer on the train might see them as simultaneous but an observer on the ground see one hit happening before the other.  

And in the example that Alan has trouble with, one observer might say the back of the car entered the garage before the back door of the garage was opened, and another observer in a different frame of reference would say that happened after the back door of the garage was opened. So one would say the car was able to fit into the garage and the other would say it could not.

 

 John K Clark  

Apparently you haven't read my posts or don't understand them. If the initial conditions are set so the car's length is longer than the garage, and then the car is set in motion, the car is NEVER inside the garage since the garage is contracting! In this situation, there would be no measurements that you allege for which simultaneity would apply. There is NO entering of the garage for the car! AG 

Further, you bring up spacelike separated events to allegedly make your points. But as far as I can tell, no events in this situation are spacelike separated. AG 

So the front door, the back door, the front of the car, the rear of the car are all at the same spacelike location?  I don't call that doors or cars, I call that a point, and there are no paradoxes 

You don't get it. The car is never inside the garage, given the initial conditions. AG 

[Quentin Anciaux]

Someone doesn't get it that's true, if somehow you could stop the I'm the genius versus the dumb world attitude, that'll surely help you, you have a big ego problem. 

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[Alan Grayson]

On Tuesday, December 10, 2024 at 11:52:10 AM UTC-7 Quentin Anciaux wrote:

Le mar. 10 déc. 2024, 19:42, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:36:19 AM UTC-7 Quentin Anciaux wrote:

Le mar. 10 déc. 2024, 19:31, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:07:25 AM UTC-7 Alan Grayson wrote:

On Tuesday, December 10, 2024 at 6:29:25 AM UTC-7 John Clark wrote:

On Tue, Dec 10, 2024 at 5:47 AM Quentin Anciaux <allc...@gmail.com> wrote:

>> My conclusion is that there's a contradiction in evidence, since the situation of the car fitting and not fitting makes no sense. AG 

>>And the answer only lies in disagreement of simultaneity of both doors being closed and the car fully inside the garage, and that is dependent on the frame of reference,  there aren't any contradictions.

Exactly! In Special Relativity two observers in different frames of reference can disagree about what order a series of events should be in, provided that the events are "spacelike separated", that is to say if even the speed of light is not fast enough to send a signal between them. One observer could say events should be ordered XYZ and another insist the order should be XZY. The classic example is two lightning strikes hitting the front and back of a very fast moving train, an observer on the train might see them as simultaneous but an observer on the ground see one hit happening before the other.  

And in the example that Alan has trouble with, one observer might say the back of the car entered the garage before the back door of the garage was opened, and another observer in a different frame of reference would say that happened after the back door of the garage was opened. So one would say the car was able to fit into the garage and the other would say it could not.

 

 John K Clark  

Apparently you haven't read my posts or don't understand them. If the initial conditions are set so the car's length is longer than the garage, and then the car is set in motion, the car is NEVER inside the garage since the garage is contracting! In this situation, there would be no measurements that you allege for which simultaneity would apply. There is NO entering of the garage for the car! AG 

Further, you bring up spacelike separated events to allegedly make your points. But as far as I can tell, no events in this situation are spacelike separated. AG 

So the front door, the back door, the front of the car, the rear of the car are all at the same spacelike location?  I don't call that doors or cars, I call that a point, and there are no paradoxes 

You don't get it. The car is never inside the garage, given the initial conditions. AG 

Someone doesn't get it that's true, if somehow you could stop the I'm the genius versus the dumb world attitude, that'll surely help you, you have a big ego problem. 

Why can't you see the obvious? The car can never fit in the garage, given the initial conditions. AG 

[Jesse Mazer]

Difference in simultaneity does come to the rescue by showing that it's a difference in coordinate-based descriptions of the situation, and that there is no objective coordinate-independent way to define the question of whether the car fits.

 

 

you could always pick a sufficiently large relative velocity such that the car would be longer than the garage in the car's rest frame,

No, the car's length decreases in the garage frame only due to its motion. In the car's frame, the car's length doesn't change. AG

I didn't say the car's length changed in the car's frame, I just said that if you pick a sufficiently high relative velocity, then in the car's rest frame the car will be longer than the garage (in this case due to the garage's length being shortened in that frame).

I don't follow. If the garage's length is shorten, the length of the car remains unchanged. AG 

"the car will be longer than the garage" is just a relative comparison, it doesn't imply anything about the length of the car changing. If the garage's length shrinks to a length shorter than the car in the car's frame, then that means the car is longer than the garage in that frame.

 

 

and thus it would not fit in that frame, so the paradox remains. Not sure what "opposite conclusions from the frames" could mean if you don't have a specific way to define "fits in the garage" in a way that doesn't depend on picking some frame or another.

Jesse

As I understand the problem, in the initial rest frame the car is assumed to be larger than the garage. Then the question is whether it can fit when the car is in motion due to length contraction. In the car's frame, the garage length decreases, so there is no possibility of the car fitting. OTOH, from the pov of the garage frame, the car's length shrinks, so there is some velocity where it fits perfectly. If the velocity continues to increase, the car fits with room to spare. So, I have shown that the frames differ in concluding whether the car fits, or not, and the question is whether this is a paradox. If you conclude it is not, then you deny there's an objective reality such that the car fits, or doesn't fit. And "fits" just means the car's contracted length is EQUAL TO or LESS than the garage's length. AG

Yes, I've said before that there's no objective frame-independent reality about whether the car fits, that's part of the standard answer to this paradox. Do you accept that this is a valid way of resolving it? You seemed to be objecting in your earlier comment when you said "ISTM that the car can, or cannot fit in garage given the initial condition that in the rest frame, the car is longer than the garage; in other words there is an objective reality, but the frames differ on whether the car fits or not", but maybe I misunderstood?

Jesse

My conclusion is that there's a contradiction in evidence, since the situation of the car fitting and not fitting makes no sense. AG 

Why does it make no sense? Did you read my earlier analogy of the purely geometric scenario with two cylinders in 3D space that are at an angle relative to each other and cross each other in something like an X-shape, so there is some region where their interiors overlap? If you define a Cartesian x-y-z coordinate system and call the z-dimension the "vertical" one so that a "horizontal cross-section" of each cylinder would be where they intersect with a 2D plane of fixed z-coordinate, then the question "is there any horizontal cross-section of cylinder #1 that fits wholly inside the horizontal cross-section of cylinder #2" can depend on your choice of how the x-y-z axes are oriented, i.e. it's a coordinate-dependent question. Do you follow this (if not I can draw a diagram), and if so do you agree that "is there any horizontal cross-section of cylinder #1 that fits wholly inside the horizontal cross-section of cylinder #2" is a question with no objective coordinate-independent answer? If this makes sense to you, why is the question about the car and the garage any different? Instead of 2D cross-sections of 3D cylinders, this is basically just a question about 3D cross-sections of 4D world-tubes, and the answer similarly depends on the angle of the cross-section.

Jesse

 

[Jesse Mazer]

Where did you get that idea? Every inertial frame in relativity has its own definition of simultaneity, based on the assumption that light travels at the same speed in both directions in that frame. And the only reason a physicist would say the car "fits" in the garage frame is that according to the garage frame's definition of simultaneity, if you measure the position of the front and back end of both car and garage simultaneously, you will find at some moments the back end of the car is past the front door of the garage while the front end of the car has not yet reached the back wall (or back door) of the garage.

 

From that it's concluded that from garage frame, it's indeterminate whether the car really fits. But I'm confused on this point. How can the garage frame know about those measurements in car frame, and know they differ? AG

If you know the coordinates of events in one frame you can simply use the Lorentz transformation to know what they are in a different frame, it's just that the coordinates in the other frame won't correspond to measurements on your own rulers and clocks at rest relative to you.

Jesse

 

[Brent Meeker]

On 12/10/2024 1:33 AM, Alan Grayson wrote:

On Monday, December 9, 2024 at 4:54:34 PM UTC-7 Brent Meeker wrote:

On 12/9/2024 3:24 PM, Alan Grayson wrote:

On Monday, December 9, 2024 at 2:01:28 PM UTC-7 Brent Meeker wrote:

>
> Nothing odd about dilation and contraction when you know its cause.
> But what is odd is the fact that each frame sees the result
> differently -- that the car fits in one frame, but not in the other --
> and you see nothing odd about that, that there's no objective reality
> despite the symmetry. AG

The facts are events in spacetime.  There's an event F at which the
front of the car is even with the exit of the garage and there's an
event R at which the rear of the car is even with the entrance to the
garage.  If R is before F we say the car fitted in the garage. If R is
after F we say the car did not fit.  But if F and  R are spacelike, then
there is no fact of the matter about their time order.  The time order
will depend on the state of motion.

Brent

Since the car's length can be assumed to be arbitrarily small from the 

pov of the garage, why worry about fitting the car in garage perfectly,

and then appealing to difference in spontaneity to prove no direct

contradiction between the frames? It seems like a foolish effort to 

avoid a contradition, when one clearly exists. AG

What's the contradiction? 

The contradiction is precisely this; assuming the initial rest state is that the length of the car is larger than the length of the garage, we get the car never fitting in the garage from the pov of the car, and the car fitting in the garage from the pov of the garage. The car can't fit and not fit in the garage.

You think that because you have not carefully defined "fit", which does require reference to simultaneity.

The former result is easy to see, since the car's motion shrinks the garage's length, so the car, initially longer than the garage, can never fit inside the garage.

Within the cars reference frame.

The latter result follows from the fact that from the pov of the garage, the car's length shrinks, and for a sufficient velocity, it will shrink enough to fit in the garage. Further, the issue of simultaneity is a non-issue,

No it is the essential issue.  The car (or the garage) don't actually undergo some physical shrinkage.  If they did they wouldn't keep their dimensions in their own frame.  So it is a question of measurement and simultaneity.

Brent

since measurements of the front and back end of the car occur in the car's frame, and since the car never fits in the garage, such measurements can never be made when the car perfectly fits in the garage, or even loosely, since this condition never occurs. In summary, I think I've done for relativity, what Bertrand Russell did for Cantor's set theory; proving the existence of a contradiction. AG

Brent

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[Alan Grayson]

I defined "fit" to mean the car's length in any frame is less than the garage's length.  AG

The former result is easy to see, since the car's motion shrinks the garage's length, so the car, initially longer than the garage, can never fit inside the garage.

Within the cars reference frame.

Yes. AG 

The latter result follows from the fact that from the pov of the garage, the car's length shrinks, and for a sufficient velocity, it will shrink enough to fit in the garage. Further, the issue of simultaneity is a non-issue,

No it is the essential issue.  The car (or the garage) don't actually undergo some physical shrinkage.

Yes. It's all about appearances, or so it seems. And yet, physicists claim the LT gives the actual measurements in one frame, using the measurements in another frame. AG

 

  If they did they wouldn't keep their dimensions in their own frame.  So it is a question of measurement and simultaneity.

Why then do physicists agree that the distance to Andromeda will be immensely shortened if a traveler's velocity is close to c? Never a mention of simultaneiry in this case. AG 

[Alan Grayson]

I don't see how your complicated geometry has any relevance. In the situation at hand, we essentially have a one-dimensional coordinate system consisting of a straight line, the path along the garage. AG 

[Jesse Mazer]

Different frames can define their coordinates in terms of local measurements on a grid of rulers and synchronized clocks like the one in the image at http://www.upscale.utoronto.ca/GeneralInterest/Harrison/SpecRel/SpecRel.html#Exploring -- for example if a collision between two space rocks happens right next to the X=15 meters mark on my ruler, and the clock at that mark reads T=10 seconds when the collision happens right next to it, all observers agree that those events coincide in space and time. But a different frame will have its own parallel set of rulers and clocks, so that the same collision might happen next to say the X'=20 meters on your ruler, with your clock at that location reading T'=5 seconds, and all observers will all also agree that these events coincide. Each set of rulers measures the rulers of other frames to be shrunk, and the clocks of other frames to be running slow and out-of-sync with one another, I gave some illustrations showing how the ruler markings and clock readings of two frames line up with one another at https://physics.stackexchange.com/a/155016/59406

 

 

  If they did they wouldn't keep their dimensions in their own frame.  So it is a question of measurement and simultaneity.

Why then do physicists agree that the distance to Andromeda will be immensely shortened if a traveler's velocity is close to c? Never a mention of simultaneiry in this case. AG 

All length/distance claims do involve simultaneity since one is talking about the distance between two events or ends of an object when both are measured at the same moment in a given frame. Imagine a ruler long enough to stretch from our galaxy to Andromeda which was moving at a high fraction of c relative to the two galaxies, with clocks mounted at regular intervals along it (so at rest relative to the ruler, moving close to c relative to the galaxies) and synchronized in the ruler's frame using the Einstein synchronization convention. Then if you looked for a pair of clocks such that one was passing through our galaxy when it showed a given time T, and the other was passing through Andromeda when it showed the same time T, and looked at the number of light-year markings on the ruler between their two positions, you would find this was greatly shortened relative to the distance to Andromeda as measured by a similar ruler/clock system at rest relative to ourselves.

Jesse

[Jesse Mazer]

Since length is always *simultaneous* distance you need at least two dimensions, one space and one time, and the different frames are measuring length along different 1D cross-sections of the world-strips of the car and garage (using world-strip to refer to the region enclosed within the wordlines of the front and back of the car, and the front and back of the garage). And since simultaneity is relative, the 1D cross-sections used by different frames are tilted at an angle relative to one another. Similarly, if you like you can replace the 3D cylinders in my example with 2D rectangles (or strips) in my example, and ask whether there is any "horizontal" 1D cross-section where the cross-section of rectangle #1 fits within the cross-section of rectangle #2, with the answer depending on the orientation of the x-axis which determines what "horizontal" means. 

Whether we use rectangle or cylinders, do you agree or disagree that in my geometric example the question "is there a horizontal cross section where the cross-section of cylinder/rectangle #1 fits within cylinder/rectangle #2" is coordinate-dependent, that the answer may be yes for one choice of orientation of x-y axes but no for a different orientation? Do you think there is something wrong with defining the question about the car fitting in the garage in similar terms, as a question about spatial cross-sections of a space-time diagram with world-strips for the car and garage? (this is just a way of visualizing the notion that all questions about spatial arrangements of objects depend on momentary snapshots which depend on the choice of simultaneity convention).

Jesse

[Brent Meeker]

On 12/10/2024 10:07 AM, Alan Grayson wrote:
> Apparently you haven't read my posts or don't understand them. If the
> initial conditions are set so the car's length is longer than the
> garage, and then the car is set in motion, the car is NEVER inside the
> garage since the garage is contracting!

But in the garage's system the car is contracting!!

Brent

[Alan Grayson]

On Tuesday, December 10, 2024 at 4:55:57 PM UTC-7 Brent Meeker wrote:

On 12/10/2024 10:07 AM, Alan Grayson wrote:
> Apparently you haven't read my posts or don't understand them. If the
> initial conditions are set so the car's length is longer than the
> garage, and then the car is set in motion, the car is NEVER inside the
> garage since the garage is contracting!

 

But in the garage's system the car is contracting!!

Brent

Correct. So garage and car are both contracting by the same gamma function, 

and the car still won't fit in the garage. Why then is the "solution" started with

the car perfectly fitting in the garage? And if the velocity is high enough, from

the pov of the garage, the car will eventually fit, thus yielding the contradiction

of the car not fitting in the car's frame, but fitting in the garage's frame. AG

[Brent Meeker]

Wrong!

The former result is easy to see, since the car's motion shrinks the garage's length, so the car, initially longer than the garage, can never fit inside the garage.

Within the cars reference frame.

Yes. AG 

The latter result follows from the fact that from the pov of the garage, the car's length shrinks, and for a sufficient velocity, it will shrink enough to fit in the garage. Further, the issue of simultaneity is a non-issue,

No it is the essential issue.  The car (or the garage) don't actually undergo some physical shrinkage.

Yes. It's all about appearances, or so it seems.

No, it's all about measurements and simultaneity.

And yet, physicists claim the LT gives the actual measurements in one frame, using the measurements in another frame. AG

 

  If they did they wouldn't keep their dimensions in their own frame.  So it is a question of measurement and simultaneity.

Why then do physicists agree that the distance to Andromeda will be immensely shortened if a traveler's velocity is close to c? Never a mention of simultaneiry in this case. AG

Because they're not concerned with two events, just with the duration of the trip.  In this case you must consider two events: One when the front of the car is adjacent to the exit of the garage and the other when the rear of the car is adjacent to the entrance of the garage.  If these two events are spacelike relative to one another then there is a reference frame in which they are simultaneous.

Brent

Brent

since measurements of the front and back end of the car occur in the car's frame, and since the car never fits in the garage, such measurements can never be made when the car perfectly fits in the garage, or even loosely, since this condition never occurs. In summary, I think I've done for relativity, what Bertrand Russell did for Cantor's set theory; proving the existence of a contradiction. AG

Brent

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[Alan Grayson]

How would you define fit? By allleging the car perfectly fits within the garage, even though it is longer than garage? AG 

The former result is easy to see, since the car's motion shrinks the garage's length, so the car, initially longer than the garage, can never fit inside the garage.

Within the cars reference frame.

Yes. AG 

The latter result follows from the fact that from the pov of the garage, the car's length shrinks, and for a sufficient velocity, it will shrink enough to fit in the garage. Further, the issue of simultaneity is a non-issue,

No it is the essential issue.  The car (or the garage) don't actually undergo some physical shrinkage.

Yes. It's all about appearances, or so it seems.

No, it's all about measurements and simultaneity.

But when the measurements give different results depending on which frame they are made from, the LT de facto tells us how things appear from such frames, so about appearances. AG

And yet, physicists claim the LT gives the actual measurements in one frame, using the measurements in another frame. AG

If they did they wouldn't keep their dimensions in their own frame.  So it is a question of measurement and simultaneity.

Why then do physicists agree that the distance to Andromeda will be immensely shortened if a traveler's velocity is close to c? Never a mention of simultaneiry in this case. AG

Because they're not concerned with two events, just with the duration of the trip.  In this case you must consider two events: One when the front of the car is adjacent to the exit of the garage and the other when the rear of the car is adjacent to the entrance of the garage.  If these two events are spacelike relative to one another then there is a reference frame in which they are simultaneous.

Brent

In the case of the car perfectly fitting in the garage, are the ends of the car spacelike separated? I don't think so, so why bring it up? AG 

[Alan Grayson]

The above comment is wrong. I was confusing spacelike separated from events which are not causally connected. AG 

[Alan Grayson]

The primary flaw in the attempt to model the problem is to assume the car fits perfectly in the garage (meaning the front and back ends of car are juxtaposed to back and front end of the garage) in contradicton to the initial conditions (rest length of car is assumed greater than the rest length of the garage), so that when the car is moving, and the same gamma factor applied to yield the transformed lengths observed by each frame, each of these rest lengths changes in the same proportion, making the assumed initial perfect fit while moving, impossible. AG    

[Brent Meeker]

Do I not only have provide a diagram I also have to explain it in detail just to end this silly thread??

First note by comparing the two diagrams that the car is longer than the garage, 12' vs 10'.  So the car doesn't fit at small relative speed.  What does "fit" mean?  It means that the event of the front of the car coinciding with the right-hand end of the garage is after or at the same time as the rear of the car coinciding with the left-had end of the garage.  In both diagrams the car is moving to the right at 0.8c so \gamma=sqrt{1-0.8^2}=0.6.  Consequently, in the car's reference frame, the garage is contracted to 6' length and when the rear of the car is just entering the garage, the front is simultaneously, in the car's reference frame, already 6' beyond the right-hand end of the garage.



Then in the garage's reference frame the car's length is contracted to 0.6*12'=7.2' so at the moment the front of the car coincides with the right end of the garage, the rear of the car will simultaneously, in the garage reference system, be 2.8' inside the garage as shown below.

Note that in the above diagram I have marked two simultaneous events with small \delta's.  The diagram below is just the Lorentz transform of the one above.  The two simultaneous \delta's are also in the diagram below.  You can confirm they are the same events by referring to the time blips along the world lines, which are also just the Lorentz transforms of those above.  But clearly the events marking the simultaneous locations of the rear and front of the car above are NOT simultaneous in the garage  frame below.  Conversely, the front and rear simultaneous locations of the car below are not simultaneous in the above diagram, as the reader is invited to confirm by plotting them.   Simultaneity is frame dependent.



Incidentally, when I was in graduate school this was still know as the "Tank Trap Paradox".  The idea was that if one dug a tank trap shorter than the enemy tank, then the tank would just bridge the hole, UNLESS the tank were going very fast in which its contracted length would allow it to fall into the trap.  This was being explained to me by Jurgen Ehlers, whom you may correctly infer from his name was a German professor recently hired at Univ Texas.  I said, "What is it with you Germans, illustrating things with tank traps and cats in boxes with poison gas?"  Jurgen who was too young to have fought in the war didn't realize I was pulling his leg and he was struck speechless.

Brent

[Alan Grayson]

On Tuesday, December 10, 2024 at 11:15:16 PM UTC-7 Brent Meeker wrote:

Do I not only have provide a diagram I also have to explain it in detail just to end this silly thread??

Yes you do. Providing plots without the numerical values in the LT, is useless. I can't tell if you're drawing plots to satisfy your biases, or if the numbers support the case you're making. Lesson learned; always do a real proof, which means supplying the arguments, or STFU. AG 

[Quentin Anciaux]

Le mer. 11 déc. 2024, 07:40, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:15:16 PM UTC-7 Brent Meeker wrote:

Do I not only have provide a diagram I also have to explain it in detail just to end this silly thread??

Yes you do. Providing plots without the numerical values in the LT, is useless. I can't tell if you're drawing plots to satisfy your biases, or if the numbers support the case you're making. Lesson learned; always do a real proof, which means supplying the arguments, or STFU. AG 

Wake up, your ego will kill your brain.

First note by comparing the two diagrams that the car is longer than the garage, 12' vs 10'.  So the car doesn't fit at small relative speed.  What does "fit" mean?  It means that the event of the front of the car coinciding with the right-hand end of the garage is after or at the same time as the rear of the car coinciding with the left-had end of the garage.  In both diagrams the car is moving to the right at 0.8c so \gamma=sqrt{1-0.8^2}=0.6.  Consequently, in the car's reference frame, the garage is contracted to 6' length and when the rear of the car is just entering the garage, the front is simultaneously, in the car's reference frame, already 6' beyond the right-hand end of the garage.



Then in the garage's reference frame the car's length is contracted to 0.6*12'=7.2' so at the moment the front of the car coincides with the right end of the garage, the rear of the car will simultaneously, in the garage reference system, be 2.8' inside the garage as shown below.

Note that in the above diagram I have marked two simultaneous events with small \delta's.  The diagram below is just the Lorentz transform of the one above.  The two simultaneous \delta's are also in the diagram below.  You can confirm they are the same events by referring to the time blips along the world lines, which are also just the Lorentz transforms of those above.  But clearly the events marking the simultaneous locations of the rear and front of the car above are NOT simultaneous in the garage  frame below.  Conversely, the front and rear simultaneous locations of the car below are not simultaneous in the above diagram, as the reader is invited to confirm by plotting them.   Simultaneity is frame dependent.



Incidentally, when I was in graduate school this was still know as the "Tank Trap Paradox".  The idea was that if one dug a tank trap shorter than the enemy tank, then the tank would just bridge the hole, UNLESS the tank were going very fast in which its contracted length would allow it to fall into the trap.  This was being explained to me by Jurgen Ehlers, whom you may correctly infer from his name was a German professor recently hired at Univ Texas.  I said, "What is it with you Germans, illustrating things with tank traps and cats in boxes with poison gas?"  Jurgen who was too young to have fought in the war didn't realize I was pulling his leg and he was struck speechless.

Brent

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[Jesse Mazer]

In the garage rest frame where the car fits, the gamma factor is not applied to the garage, only to the car. Likewise in the car rest frame where the car doesn't fit, the gamma factor is not applied to the car, only to the garage. Again you seem to want to allude to some frame-independent notion of what it means for the car to "fit" without ever giving a clear definition.

Jesse

[Alan Grayson]

On Tuesday, December 10, 2024 at 11:51:46 PM UTC-7 Quentin Anciaux wrote:

Le mer. 11 déc. 2024, 07:40, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:15:16 PM UTC-7 Brent Meeker wrote:

Do I not only have provide a diagram I also have to explain it in detail just to end this silly thread??

Yes you do. Providing plots without the numerical values in the LT, is useless. I can't tell if you're drawing plots to satisfy your biases, or if the numbers support the case you're making. Lesson learned; always do a real proof, which means supplying the arguments, or STFU. AG 

Wake up, your ego will kill your brain.

I don't suppose you could explain the resolution clearly. Relying on slogans, which you have done throughout without knowing it, gets us nowhere. I am studying the numbers, and hoping this resolves the issue. AG 

[Quentin Anciaux]

Le mer. 11 déc. 2024, 07:59, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:51:46 PM UTC-7 Quentin Anciaux wrote:

Le mer. 11 déc. 2024, 07:40, Alan Grayson <agrays...@gmail.com> a écrit :

On Tuesday, December 10, 2024 at 11:15:16 PM UTC-7 Brent Meeker wrote:

Do I not only have provide a diagram I also have to explain it in detail just to end this silly thread??

Yes you do. Providing plots without the numerical values in the LT, is useless. I can't tell if you're drawing plots to satisfy your biases, or if the numbers support the case you're making. Lesson learned; always do a real proof, which means supplying the arguments, or STFU. AG 

Wake up, your ego will kill your brain.

I don't suppose you could explain the resolution clearly. Relying on slogans, which you have done throughout without knowing it, gets us nowhere. I am studying the numbers, and hoping this resolves the issue. AG 

To resolve the issue is simple, stop believing you're the most brilliant being in the universe and all others are dumb shit... resolve your childish ego problem, then you'll figure it out hopefully. 

First note by comparing the two diagrams that the car is longer than the garage, 12' vs 10'.  So the car doesn't fit at small relative speed.  What does "fit" mean?  It means that the event of the front of the car coinciding with the right-hand end of the garage is after or at the same time as the rear of the car coinciding with the left-had end of the garage.  In both diagrams the car is moving to the right at 0.8c so \gamma=sqrt{1-0.8^2}=0.6.  Consequently, in the car's reference frame, the garage is contracted to 6' length and when the rear of the car is just entering the garage, the front is simultaneously, in the car's reference frame, already 6' beyond the right-hand end of the garage.



Then in the garage's reference frame the car's length is contracted to 0.6*12'=7.2' so at the moment the front of the car coincides with the right end of the garage, the rear of the car will simultaneously, in the garage reference system, be 2.8' inside the garage as shown below.

Note that in the above diagram I have marked two simultaneous events with small \delta's.  The diagram below is just the Lorentz transform of the one above.  The two simultaneous \delta's are also in the diagram below.  You can confirm they are the same events by referring to the time blips along the world lines, which are also just the Lorentz transforms of those above.  But clearly the events marking the simultaneous locations of the rear and front of the car above are NOT simultaneous in the garage  frame below.  Conversely, the front and rear simultaneous locations of the car below are not simultaneous in the above diagram, as the reader is invited to confirm by plotting them.   Simultaneity is frame dependent.



Incidentally, when I was in graduate school this was still know as the "Tank Trap Paradox".  The idea was that if one dug a tank trap shorter than the enemy tank, then the tank would just bridge the hole, UNLESS the tank were going very fast in which its contracted length would allow it to fall into the trap.  This was being explained to me by Jurgen Ehlers, whom you may correctly infer from his name was a German professor recently hired at Univ Texas.  I said, "What is it with you Germans, illustrating things with tank traps and cats in boxes with poison gas?"  Jurgen who was too young to have fought in the war didn't realize I was pulling his leg and he was struck speechless.

Brent

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[Alan Grayson]

I didn't apply the gamma factor in the rest frame, since we need relative motion to apply it. But when motion exists, I can use the gamma factor to determine the contraction of the motion being observed from either frame, to the other. I thought I gave a clear definition of fit, namely, the car length being LESS than the garage. AG