Einstein's Length Contraction A problem?

[ft_c]

Hello,

I hope you can help with my question! I am a 27 year old amateur theoretical physicist about to start a bachelor's degree in physics as a mature student...
I have just been going over some bits of Einstein's theory of Special Relativity and have noticed something odd that has confused me! The theory initially states that there is no absolute frame of reference in the universe. Ok, so let's say astronauts Mary and Dave are floating next to each other in space, Dave turns on his jet pack and flies off at 500,000mph. Mary will see Dave as being thinner as he will be experiencing length contraction. So then, shouldn't Dave see Mary as being wider, as in comparison to her, he is thinner?! Instead, it could just have easily been Mary who was accelerated away from Dave, as observers they would still be equal, although in this scenario Dave would see Mary as being thinner! I am confused!

This has also brought about an interesting thought experiment. If someone were to make a million identical metal cubes and blast them off into space in a million different directions at different speeds, then by measuring which one looked biggest couldn't we potentially find a direction of motion (relative to us) that is actually an absolute point in space, i.e. the biggest cube is not actually moving in space?

Thank you for your time and help!

 

[PeterDonis]

Hi, and welcome to Physics Forums!

You're mixing up at least two distinct concepts. Separating them may help to understand what's going on.

First, let's remove the jet pack and acceleration from the scenario (we'll put them back in a bit). Suppose Dave and Mary are flying past each other at some large relative velocity; neither one has a rocket engine or any other type of thrust turned on, they are just flying past each other in free fall, both weightless. Then each one will indeed see the other as length contracted. Physically, their situation is completely symmetrical, and there is nothing that picks out either one of them as being in a special state of motion.

It may seem confusing that, if Mary sees Dave as thinner, Dave also sees Mary as thinner (instead of wider). But remember that they are flying past each other, and neither of them has any means of changing their speed (no rockets or anything else). So there is no way for them to come to rest relative to one another and actually compare their lengths while they are both in the same state of motion. What Mary sees of Dave is a "projection" of Dave into Mary's state of motion, and vice versa, and because Mary and Dave are moving relative to one another, their states of motion are "tilted" relative to one another. If you learn about spacetime diagrams, they are a good way of making this visually evident, and it becomes obvious that length contraction is just the spacetime version of a projection in geometry. The fact that Mary and Dave each see the other as thinner is then just a geometric consequence of the tilting, just as if I hold up a quarter so that it is face on to me, and you are facing at at angle to me, the quarter will look like a narrowed ellipse instead of a circle, and vice versa if you are holding another quarter face on to you (it will look like a narrowed ellipse to me); our situations are symmetrical, so our experiences are too.

Now let's put back the acceleration. Suppose Dave and Mary start out at rest relative to each other, but then Dave turns on his rocket pack and accelerates away. Now there is a real, physical difference between the two: Dave feels acceleration, and is no longer weightless, while Mary remains in free fall and is weightless. Their situations are no longer symmetric. In this particular case, Mary will still see Dave as length contracted, but Dave may be unable to "see" Mary at all while he accelerates, if he accelerates hard enough! That gets into more complications, which you may not want to get into at this stage. But the key is that, if one person is accelerating (i.e., feels an acceleration) and the other doesn't, their situations are not symmetric and the principle of relativity does not require that all of their physical experiences be symmetrical.

 

[ft_c]

Oh my god, sorry, of course...this is the whole point of 'relativity', particularly for this postulate! Please do excuse me and thank you for setting me straight :)

 

[Goodison_Lad]

This apparent paradox used to baffle me as well. A simple analogy, ignoring relativity completely, is how big they would appear to each other. From far way, Mary would see Dave as appearing quite small – he would have a small angular size.

But this, of course, doesn’t mean that Dave would see Mary as being huge – she would appear equally diminished to him. Both are right.

And each would believe (rightly) that they had not changed in size at all.

A whole bunch of odd things happen together when relativity comes into play. As well as a moving object (e.g. a spaceship) appearing to us to shrink in length, the rate of the passage of time on it appears to us to be slower than it did when it was stationary. Also, time at the front end of the spaceship appears to us to lag behind time at the rear.

But the wording is important. To the people on the ‘moving’ spaceship itself, none of these things are apparent about them and their vessel. However, they would rightly argue that it is we who had contracted in length, that our time was running slowly (compared with their ship clocks) and that our clocks, which of course appear synchronised to us, are asynchronised to them.

The strange effects of special relativity result from a strict application of the two postulates. Many special relativity paradoxes are resolved when all these effects are considered together – not just one of them. It’s very easy to remember one effect and forget another!

 

[nuclearhead]

The difference is that the one accelerating away will see the "fixed stars" contract. The average velocity of the fixed stars creates a unique frame of reference which comes from the big bang.

 

[Vandam]

Hi ftc, and welcome to Physics Forums!

You're mixing up at least two distinct concepts. Separating them may help to understand what's going on.

First, let's remove the jet pack and acceleration from the scenario (we'll put them back in a bit). Suppose Dave and Mary are flying past each other at some large relative velocity; neither one has a rocket engine or any other type of thrust turned on, they are just flying past each other in free fall, both weightless. Then each one will indeed see the other as length contracted. Physically, their situation is completely symmetrical, and there is nothing that picks out either one of them as being in a special state of motion.

It may seem confusing that, if Mary sees Dave as thinner, Dave also sees Mary as thinner (instead of wider). But remember that they are flying past each other, and neither of them has any means of changing their speed (no rockets or anything else). So there is no way for them to come to rest relative to one another and actually compare their lengths while they are both in the same state of motion. What Mary sees of Dave is a "projection" of Dave into Mary's state of motion, and vice versa, and because Mary and Dave are moving relative to one another, their states of motion are "tilted" relative to one another. If you learn about spacetime diagrams, they are a good way of making this visually evident, and it becomes obvious that length contraction is just the spacetime version of a projection in geometry. The fact that Mary and Dave each see the other as thinner is then just a geometric consequence of the tilting, just as if I hold up a quarter so that it is face on to me, and you are facing at at angle to me, the quarter will look like a narrowed ellipse instead of a circle, and vice versa if you are holding another quarter face on to you (it will look like a narrowed ellipse to me); our situations are symmetrical, so our experiences are too.

Why do you say there is a 'projection'? I thought Dave and Mary are part of different worlds of simultaneous events. Dave's and Mary's world are complete different crosssection of 4D spacetime. Dave's world is not mixed into/with Mary's world. Only certain events of past, present and future worlds of Dave are in Mary's world. And vice versa. I don't see any 'projection'.

 

[Vandam]

This apparent paradox used to baffle me as well. A simple analogy, ignoring relativity completely, is how big they would appear to each other. From far way, Mary would see Dave as appearing quite small – he would have a small angular size.

But this, of course, doesn’t mean that Dave would see Mary as being huge – she would appear equally diminished to him. Both are right.

And each would believe (rightly) that they had not changed in size at all.

A whole bunch of odd things happen together when relativity comes into play. As well as a moving object (e.g. a spaceship) appearing to us to shrink in length, the rate of the passage of time on it appears to us to be slower than it did when it was stationary. Also, time at the front end of the spaceship appears to us to lag behind time at the rear.

But the wording is important. To the people on the ‘moving’ spaceship itself, none of these things are apparent about them and their vessel. However, they would rightly argue that it is we who had contracted in length, that our time was running slowly (compared with their ship clocks) and that our clocks, which of course appear synchronised to us, are asynchronised to them.

The strange effects of special relativity result from a strict application of the two postulates. Many special relativity paradoxes are resolved when all these effects are considered together – not just one of them. It’s very easy to remember one effect and forget another!

Why do you say the moving spacehip 'appears' shorter (like an optical illusion?) and not 'is' shorter? If I stretch my arms wide open and a moving train passes (close to lightspeed ) I wil feel the steel back and steel front of the train at the same moment with respectively left and right hand. Therfore the train was really between my hands.

 

[Dale]

Why do you say there is a 'projection'? I thought Dave and Mary are part of different worlds of simultaneous events. Dave's and Mary's world are complete different crosssection of 4D spacetime. Dave's world is not mixed into/with Mary's world. Only certain events of past, present and future worlds of Dave are in Mary's world. And vice versa. I don't see any 'projection'.

Their "different crosssection of 4D spacetime" is the projection. Geometrically, the operation which does a length measurement is a dot product with a spacelike unit vector in the direction of your measuring rod. The dot product is a projection.

 

[harrylin]

[..] let's say astronauts Mary and Dave are floating next to each other in space, Dave turns on his jet pack and flies off at 500,000mph. Mary will see Dave as being thinner as he will be experiencing length contraction. So then, shouldn't Dave see Mary as being wider, as in comparison to her, he is thinner?! Instead, it could just have easily been Mary who was accelerated away from Dave, as observers they would still be equal, although in this scenario Dave would see Mary as being thinner! I am confused! [..]

That also confused me now and then, until someone reminded me that a moving length cannot be measured with just a ruler: you need to take the positions at both ends "at the same time", and he illustrated it with taking both positions using two distant, synchronized clocks. Where the other side of an object supposedly is at that same time depends on the reference system that you use. If you are not aware of that, search for "relativity of simultaneity".

If your textbook is good, it has a derivation of length contraction based on the Lorentz transformations. The moving length must be determined by comparing it to the length Δx of your ruler - "length contraction" thus refers to the ratio Δx'/Δx at either Δt=0 or at Δt'=0. Check it out and you will probably find that there the moving length Δx' is derived for Δt=0 (simultaneous in system S). Now try to work out how such lengths compare for Δt'=0 (simultaneous in system S') and make a sketch of it.

 

[Vandam]

Their "different crosssection of 4D spacetime" is the projection. Geometrically, the operation which does a length measurement is a dot product with a spacelike unit vector in the direction of your measuring rod. The dot product is a projection.

I don't get that. If in Dave's train somebody at the back of the train and somebody at the front of the train both recite the alphabet simultaneously, then from the trainstaion Mary measures the moving train between person at the back (f.ex.) saying 'L' and the person at the front of the train saying 'K'. There is no projection of the event 'person at the front saying 'L' (part of Dave's world) into Mary's world.

 

[HallsofIvy]

You are assuming that "simultaneous" makes sense here and it doesn't. Two events happening in two different places can be "simultaneous" to one observer and not to another.

 

[Vandam]

You are assuming that "simultaneous" makes sense here and it doesn't.

?

Two events happening in two different places can be "simultaneous" to one observer and not to another.

That's exactly what I said. Mary measures between two events that are simultaneous in her world of sim events, but those two events are not simultaneous in Dave's words of sim events. The shorter train in Mary's world is made of events out of different Dave worlds.

 

[Dale]

If in Dave's train somebody at the back of the train and somebody at the front of the train both recite the alphabet simultaneously, then from the trainstaion Mary measures the moving train between person at the back (f.ex.) saying 'L' and the person at the front of the train saying 'K'. There is no projection of the event 'person at the front saying 'L' (part of Dave's world) into Mary's world.

Length contraction isn't a change in the distance between events, it is a change in the distance between worldlines. That is an important distinction because the two kinds of distances are defined differently.

 

[Vandam]

Length contraction isn't a change in the distance between events, it is a change in the distance between worldlines. That is an important distinction because the two kinds of distances are defined differently.

I never state that length contraction is a change in distance between events.
Mary measures simultaneously between events that are not simultaneous in Dave's world (But Dave measures his train between the for him simultaneous events (f.ex.) 'back says 'L' and front says 'L').

Mary puts cameras along the railwaytrack. The train passes an inch before the camera's. All together the photos make one big picture of the passing train on the railwaytack
On the picture the train is shorter because the trazin is indeed shorter. Back of train says 'L' and front says 'K'. What's wrong with this? Were is the mistake?

 

[Dale]

I never state that length contraction is a change in distance between events. ... What's wrong with this? Were is the mistake?

Sorry, I misunderstood what you were saying. It is not a mistake, just a misunderstanding on my part.

 

[Vandam]

What Mary sees of Dave is a "projection" of Dave into Mary's state of motion, and vice versa, and because Mary and Dave are moving relative to one another, their states of motion are "tilted" relative to one another. If you learn about spacetime diagrams, they are a good way of making this visually evident, and it becomes obvious that length contraction is just the spacetime version of a projection in geometry. The fact that Mary and Dave each see the other as thinner is then just a geometric consequence of the tilting, just as if I hold up a quarter so that it is face on to me, and you are facing at at angle to me, the quarter will look like a narrowed ellipse instead of a circle, and vice versa if you are holding another quarter face on to you (it will look like a narrowed ellipse to me); our situations are symmetrical, so our experiences are too.

In Dave's train (orange) somebody at the back of the train and somebody at the front of the train both recite the alphabet simultaneously.

Dave measures his train between the brown and purple event; when for him person at the back and the person at the front of the train say 'L'. Notice I have added a tree.

From the trainstation Mary measures the moving train between between the purple and blue event; between person at the back saying 'L' and the person at the front of the train saying 'K'.

I do not see any projection taking place. There are no events projected from the orange line onto the green horizontal line. F.ex. the brown event 'front of train at tree' is not dropped on the green line and replacing the blue event.

Orange and green are rather different 'cuts' through the timelines. 3D cuts though 4D spacetime.

I find 'projection' rather confusing terminology. Even wrong. Why?

If I project a photo of a train (with the front of a train hitting a tree) to the wall under an oblique angle, the train will be projected shorter, but still hitting the tree. (Similar line of thought can be said about some perspective drawing). So writing for the the layman that lengthcontraction is a 'projection' will never make him understand what relativity of simultaneity is, the crux of special relativity.

 

[PeterDonis]

I do not see any projection taking place.

"Projection" may not be the best word to describe it, but it's a word that's often used. You are correct that what is projected is not events. See further comments below.

Orange and green are rather different 'cuts' through the timelines. 3D cuts though 4D spacetime.

Yes, and one way of viewing what the "cuts" do is that they "project" the object into different simultaneous spaces. But the terminology isn't important; what's important is the physics. Your diagram correctly represents the physics.

 

[Vandam]

Yes, and one way of viewing what the "cuts" do is that they "project" the object into different simultaneous spaces. But the terminology isn't important; what's important is the physics. Your diagram correctly represents the physics.

Simultaneous spaces? Let's suppose you mean by that: spaces of simultaneous events. O.K.

What 'object' do you project? There is no object to be projected!
The object 'train' in one space of sim events is a cut through the 4D train object in 4D spacetime, and the object 'train' in another space of sim events is a different cut through the 4D train object.

I do think terminology is important. If not it's very difficult to communicate...

 

[ApplePion]

The answer is very simple.

Mary will measure her width by considering two appropriate spacetime points on her body that have the same time coordinate value, and then she will measure the spatial distance between them. She will also do the same for two appropriate spacetime points on Dave's body that have the same value of the time coordinate that those first set of points had. She will measure Dave as contracted.

The reason why Dave will not get the same distance measurements as Mary is because of The Failure of Simultaneity--he will not agree that those 4 spacetime points that Mary used were spacetime points that have the same value of the time coordinate.

When you measure the length of an object you need to examine two spacetime points that are "at the same time". (If you consider that, you will see it is obvious.) It is because different observers assign different time coordinates to spacetime points that the weird phenomenon noticed by you can occur.

 

[bobc2]

What 'object' do you project? There is no object to be projected!
The object 'train' in one space of sim events is a cut through the 4D train object in 4D spacetime, and the object 'train' in another space of sim events is a different cut through the 4D train object.

I think that's the best description of the situation, and your previous space-time diagram shows exactly what you have said here.

 

[PeterDonis]

Simultaneous spaces? Let's suppose you mean by that: spaces of simultaneous events.

Yes. That's pretty standard usage.

What 'object' do you project? There is no object to be projected!

I already said that "project" may not be the best word to describe it. And I already said that your diagram correctly captures the physics. At this point you're just arguing that you don't like me calling it "projection". That's an argument about terminology, not about physics.

I do think terminology is important. If not it's very difficult to communicate...

I agree terminology is important in that everyone should try to be clear about what the terms they are using refer to. But once someone has done that (as I have), I don't see the point of objecting that they are using the "wrong" terms. We agree on the physics.