... paradox help needed ...

[zetetic56]

Two V shaped bodies are moving towards one another.   One is moving up and the other is moving down.  The two V’s are in relative constant linear motion.

When two bodies are in relative constant linear motion, the one body could be at rest while the other body is in motion, or the other body could be at rest while the one body is in motion.

The moving body length contracts.  The moving body length contracts in its direction of motion.  In the one inertial frame of reference the other moving V length contracts and becomes obtuse while the one resting V remains a right angle.  And in the other inertial frame of reference the one moving V length contracts and becomes obtuse while the other resting V remains a right angle.

The V’s are surrounded by a fluid.  It is a nearly incompressible fluid.

(As the V’s move the fluid is displaced around them.  And as the V’s move they collide with the surrounding fluid.  And so, for the V’s to remain in constant relative motion with one another there must be another force to offset the deceleration of the V’s due to their collisions with the fluid.  (Perhaps there are a bunch of little rocket packs on and along each of the V’s and calibrated just right to keep the two V’s in constant linear motion with each other as they move towards one another while colliding with the surrounding fluid.)  The fluid is in its own, third, inertial frame of reference.)

In the one inertial frame of reference the first parts of the V’s to collide are the ends of the arms.  The collision between the one V body and the other V body is an inelastic collision.  When the bodies of the two V's come into contact with one another they stick/fuse together.

(It should be noted that the two V’s start out in two different inertial frames of reference but after the collision the V’s are no longer inertial.)

 

And so, after the two ends of the V’s collide and fuse together, this means some of the fluid ends up trapped between the two V’s.  There is no way for the fluid to get out from between the two V bodies.

(Assume that there is something blocking the fluid from moving “out of the page” or “into the page”.)

(Before the two V’s collide, as the openings between their ends get smaller, it gets harder for the fluid to get out from between the two V’s and so even before the two V’s collide they will necessarily start to accelerate and decelerate.  No amount of force from the rocket packs will be enough to keep them inertial.  (This should be noted.  But I don’t think it's important to the overall point/idea here.))

In the other inertial frame of reference the first parts of the V’s to collide are the angled midpoints of the two V’s.  The collision between the other V body and the one V body is an inelastic collision.  When the bodies of the two V's come into contract with one another they stick/fuse together.

(Again, after the collision, or perhaps just prior to the collision, the two V’s will necessarily start to accelerate and decelerate and will no longer be in inertial frames of reference.  (And again, noteworthy but perhaps/probably not pivotal.))

And so, in the other inertial frame of reference, after the two angled midpoints of the V’s collide and fuse together, this means the fluid between the two V’s does not end up trapped between the two V’s.  There is a way for the fluid to get out from between the two V bodies.

And after the two V’s collide and fuse together either at the end points in the one inertial frame of reference or at their angled midpoints in the other inertial frame of reference, the same particles of fluid trapped between the two V’s in the one inertial frame of reference are the same particles of fluid not trapped and are possibly pushed out from between the two V’s in the other inertial frame of reference.  If the fluid is pushed out from between the two V’s in the other inertial frame of reference then this is a paradox.

How can the fluid between the two V's in the other inertial frame of reference not get pushed out but instead somehow get trapped between the two V's?

How can the same particles of fluid and how can the same amount of fluid get trapped between the two V's in the other inertial frame of reference as is trapped between the two V's in the one inertial frame of reference?

Thank you for any assistance.  : )

---

And a coupla things to note:

1. No body is a perfectly rigid body.  But we can assume the bodies of the V’s are much much more rigid (as rigid as theoretically possible) than the body of fluid.

2. When the V’s disturb the fluid, the fluid is then no longer in an inertial frame of reference.  Swirling around fluid is not inertial, and displaced fluid is not inertial.  (This does not seem to affect the overall logic here, but perhaps it should be noted.)

3. Collisions at “relativistic speeds” (speeds at which the effects of Relativity become noticeable) can end up in explosions and if this happens then where the fluid between the two V’s ends up could then be unknowable.  We could say the two V’s here are moving slow relative to one another while the length of their arms are light-years long, and so then explosions are unlikely while the effects of Relativity with these “relativistic distances” are noticeable.  Or, we can just stipulate in this thought experiment “no explosions”.  “Explosions” are not a necessary element here.

 

 

[studiot]

24 minutes ago, zetetic56 said:

How can the fluid between the two V's in the other inertial frame of reference not get pushed out but instead somehow get trapped between the two V's?

How can the same particles of fluid and how can the same amount of fluid get trapped between the two V's in the other inertial frame of reference as is trapped between the two V's in the one inertial frame of reference?

Thank you for any assistance.  : )

 

There is no paradox, just a failure of analysis.

 

What exactly does 'Relativity (SR) say about length contraction ?

How far have you studied it ? This scenario is a more complicated version of the PI-Meson first direct experimental proof of SR.

Your analysis above directly violates the second part of the statement of SR, Notably "In the direction of motion"

[Bufofrog]

30 minutes ago, zetetic56 said:

In the other inertial frame of reference the first parts of the V’s to collide are the angled midpoints of the two V’s.

No, you don't seem to understand reference frames.  Each frame would see the other one as length contracted. 

[Genady]

1 hour ago, zetetic56 said:

And after the two V’s collide and fuse together either at the end points in the one inertial frame of reference or at their angled midpoints in the other inertial frame of reference, the same particles of fluid trapped between the two V’s in the one inertial frame of reference are the same particles of fluid not trapped and are possibly pushed out from between the two V’s in the other inertial frame of reference.

To see what happens to the fluid you need to analyze the order of events in its reference frame. The result will be different depending on how the V's move relative to the fluid / how the fluid moves relative to each V.

[zetetic56]

1 hour ago, studiot said:

Your analysis above directly violates the second part of the statement of SR, Notably "In the direction of motion"

Thank you for the reply.

I have studies STR to some extent.  But I am not expert.

However, it is my understanding of STR that "length contraction occurs in the direction of motion" is a very basic concept in STR.

If a car is driving down the road at a constant speed and I am standing on the sidewalk, then the length of the car (from the front bumper to the back bumper) will become shorter for me in my sidewalk frame of reference.  However, the height of the car (from the bottom of the wheels to the roof of the car) will not length contract in any way and will be the same height for me on the sidewalk as for the person in the car and in the car's inertial frame of reference.

And so, based on this understanding of STR, if there are two V shaped bodies moving towards one another (one moving up and the other moving down) then it is my understanding that these V shaped bodies will also length contract in their directions of motion (up or down) and so we end up with the moving V bodies both being obtuse in each other's inertial frame of reference.

 

1 hour ago, Bufofrog said:

No, you don't seem to understand reference frames.  Each frame would see the other one as length contracted. 

I apologize.

I was sloppy with my language.

I was referring to the green-moving-up-V as the "one V" and I was referring to the purple-moving-down-V as the "other V".

And I use these two terms ("one" and "other") for the green V ("one") and for the purple V ("other") consistently.

However, as you point out, we use the term "other" to mean "the other one".  And so, using the language this way, in each V's (green or purple) inertial frame of reference, it is the "other moving V" (purple or green) that is length contracted.

And so when I say in my initial post here that in the "other V's" inertial frame of reference (meaning the purple-resting-V) the "one V" (meaning the green-moving-V) I am being confusing and sloppy with my language.

Again, apologies.  : )

31 minutes ago, Genady said:

To see what happens to the fluid you need to analyze the order of events in its reference frame. The result will be different depending on how the V's move relative to the fluid / how the fluid moves relative to each V.

It seems to me (and I may be wrong here) that the two V's will moving towards one another, in both inertial frames of reference, and then collide.

And in the green-at-rest V's inertial frame of reference (where the purple V is moving), the first parts of the V's that will collide (given that the green V is a right angle in its own inertial frame of reference and the moving-purple-V length contracts and becomes obtuse) will be the end points of the V's.  And in the purple-at-rest V's inertial frame of reference (where the green V is moving), the first parts of the V's that will collide (given that the purple V is a right angle in its own inertial frame of reference and the moving-green-V length contracts and becomes obtuse) will be the angled midpoints of the V's.

What happens to the fluid in each inertial frame of reference after the two V's first collide in the two different inertial frames of reference is what I then can't figure out (without ending up in a paradox).

Did I misunderstand you?

Have I misunderstood any of you?

Please let me know. : )

And, thank you all for considering my idea/problem and responding. : ) !

 

 

[Genady]

4 minutes ago, zetetic56 said:

It seems to me (and I may be wrong here) that the two V's will moving towards one another, in both inertial frames of reference, and then collide.

And in the green-at-rest V's inertial frame of reference (where the purple V is moving), the first parts of the V's that will collide (given that the green V is a right angle in its own inertial frame of reference and the moving-purple-V length contracts and becomes obtuse) will be the end points of the V's.  And in the purple-at-rest V's inertial frame of reference (where the green V is moving), the first parts of the V's that will collide (given that the purple V is a right angle in its own inertial frame of reference and the moving-green-V length contracts and becomes obtuse) will be the angled midpoints of the V's.

What happens to the fluid in each inertial frame of reference after the two V's first collide in the two different inertial frames of reference is what I then can't figure out (without ending up in a paradox).

My response was, and still is unless I am wrong, that the answer depends on how the fluid moves. If it moves together with the green V, i.e., rests relative to the green V, it will be trapped. If it moves together with the purple V, i.e., rests relative to the purple V, it will escape.

[zetetic56]

11 minutes ago, Genady said:

My response was, and still is unless I am wrong, that the answer depends on how the fluid moves. If it moves together with the green V, i.e., rests relative to the green V, it will be trapped. If it moves together with the purple V, i.e., rests relative to the purple V, it will escape.

Ah, yes, we are in total agreement.

I am sorry.

I thought I was clear in my original post, but here is another example of me needing to be clearer.

3 hours ago, zetetic56 said:

The fluid is in its own, third, inertial frame of reference.)

 

In a tiny tiny little footnote in my original post I buried some important information in the set up.

Apologies.

Both V's are moving relative to the fluid.

Each V is, one, moving relative to the other V, and, each V is, two, moving relative to the fluid.

The fluid is in its own inertial frame of reference.

And so, in each V's inertial frame of reference the other V will be moving towards the resting V and the fluid will be moving towards the resting V.

(In each V's inertial frame of reference the other moving V will be moving faster and in each V's inertial frame of reference the fluid will be moving slower (relative to each V at rest).  The moving V length contracts and the moving fluid length contracts (in their direction of motion).  The velocity of each moving V will be the same in both inertial frames of reference that the velocity of the fluid will be the same in both inertial frames of reference.)

The fact that the fluid moves (and so length contracts in its direction of motion) does not seem affect the analysis of "how is the fluid trapped in the purple inertial frame of reference in the same way that the fluid is trapped in the green inertial frame of reference?".

(But, again, the reason why I'm here to to find out if there are parts of this that I am missing, and so I could very well be wrong.)

Thank you and I hope this makes things clear (or at least clearer).  (?)

[Genady]

15 minutes ago, zetetic56 said:

Ah, yes, we are in total agreement.

I am sorry.

I thought I was clear in my original post, but here is another example of me needing to be clearer.

 

In a tiny tiny little footnote in my original post I buried some important information in the set up.

Apologies.

Both V's are moving relative to the fluid.

Each V is, one, moving relative to the other V, and, each V is, two, moving relative to the fluid.

The fluid is in its own inertial frame of reference.

And so, in each V's inertial frame of reference the other V will be moving towards the resting V and the fluid will be moving towards the resting V.

(In each V's inertial frame of reference the other moving V will be moving faster and in each V's inertial frame of reference the fluid will be moving slower (relative to each V at rest).  The moving V length contracts and the moving fluid length contracts (in their direction of motion).  The velocity of each moving V will be the same in both inertial frames of reference that the velocity of the fluid will be the same in both inertial frames of reference.)

The fact that the fluid moves (and so length contracts in its direction of motion) does not seem affect the analysis of "how is the fluid trapped in the purple inertial frame of reference in the same way that the fluid is trapped in the green inertial frame of reference?".

(But, again, the reason why I'm here to to find out if there are parts of this that I am missing, and so I could very well be wrong.)

Thank you and I hope this makes things clear (or at least clearer).  (?)

If the two V's move toward each other with the same speed relative to the fluid, then in the fluid's reference frame they have the same flattened shape and they nicely go into each other.

The fluid being at rest relative to one V or to another, or neither, are different physical scenarios, and the different outcomes don't constitute a paradox.

The whole point of the analysis is that you need to consider not only the length contraction effect, but relativity of simultaneity. Compare to the ladder/barn paradox with one V being a "ladder" and the other being a "barn". Ladder paradox - Wikipedia

Edited April 25, 2023 by Genady

[studiot]

44 minutes ago, zetetic56 said:

Thank you for the reply.

I have studies STR to some extent.  But I am not expert.

However, it is my understanding of STR that "length contraction occurs in the direction of motion" is a very basic concept in STR.

If a car is driving down the road at a constant speed and I am standing on the sidewalk, then the length of the car (from the front bumper to the back bumper) will become shorter for me in my sidewalk frame of reference.  However, the height of the car (from the bottom of the wheels to the roof of the car) will not length contract in any way and will be the same height for me on the sidewalk as for the person in the car and in the car's inertial frame of reference.

And so, based on this understanding of STR, if there are two V shaped bodies moving towards one another (one moving up and the other moving down) then it is my understanding that these V shaped bodies will also length contract in their directions of motion (up or down) and so we end up with the moving V bodies both being obtuse in each other's inertial frame of reference.

That's good you know this point. But you have yet to understand it.

Draw a vector triangle for the contractions, giving due consideration to the shape of your Vs.

Parallel to the direction of motion you have a very short distance, just the thickness of the lines in your drawing..

So this component of the total length change will be very small indeed, as a fraction of an already small quantity.

At right angles to the direction of motion, ie across the wings of the V, you will have exactly zero contraction.

This is the other component of the triangle whose hypotenuse is the length of wings of the V as drawn or observed.

Put these two components of contraction together and the result will be an almost imperceptible  flattening of the V, until at light speed it appears completely flat.

 

Conventionally we apply the length contraction to significant distance such as the distance travelled, not the size of the travelling objects, which are often considered as point particles, as in the PI-Meson example I gave and you did not reply to.

 

I have set aside for the moment matters of this fluid until the basics relativistic mechanics of the travelling objects is resolved.

 

I am sorry that I am no ,longer able to post maths or pictures on this website.

[md65536]

6 hours ago, Genady said:

The whole point of the analysis is that you need to consider not only the length contraction effect, but relativity of simultaneity.

I concur. This sort of thing has cropped up before, and it has always been due to relativity of simultaneity (usually).

 

Forget about the fluid for now. Consider the frame where both Vs are traveling toward each other at the same speed. They're length-contracted to an identical shape, and should therefore collide at all points along the V simultaneously. Therefore neither is there an enclosed volume to trap fluid nor extra space for the fluid to escape.

The same events happen in all frames, ie. all points along the Vs must collide. The main difference is that the events aren't all simultaneous in other frames. Yes, in one frame the point will contact first, and in another the ends collide first.

When you add the fluid back, in all frames the outcome is the same: either the fluid is moved early enough so it can escape in all frames, or it doesn't have enough time to move in any frame.

You can maintain a paradox by stipulating some impossible requirement. Eg. have the 2 Vs come to mutual rest instantly in multiple frames. Eg. accelerate a perfectly rigid body (including a volume of incompressible liquid).

Edited April 26, 2023 by md65536

[zetetic56]

Thank you for all of the feedback and replies.

I will go and think more about this and about your replies.

: )

Thank you and take care : ) !!!